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Reporting Research Results in APA Style | Tips & Examples

Published on December 21, 2020 by Pritha Bhandari . Revised on January 17, 2024.

The results section of a quantitative research paper is where you summarize your data and report the findings of any relevant statistical analyses.

The APA manual provides rigorous guidelines for what to report in quantitative research papers in the fields of psychology, education, and other social sciences.

Use these standards to answer your research questions and report your data analyses in a complete and transparent way.

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Table of contents

What goes in your results section, introduce your data, summarize your data, report statistical results, presenting numbers effectively, what doesn’t belong in your results section, frequently asked questions about results in apa.

In APA style, the results section includes preliminary information about the participants and data, descriptive and inferential statistics, and the results of any exploratory analyses.

Include these in your results section:

• Participant flow and recruitment period. Report the number of participants at every stage of the study, as well as the dates when recruitment took place.
• Missing data . Identify the proportion of data that wasn’t included in your final analysis and state the reasons.
• Any adverse events. Make sure to report any unexpected events or side effects (for clinical studies).
• Descriptive statistics . Summarize the primary and secondary outcomes of the study.
• Inferential statistics , including confidence intervals and effect sizes. Address the primary and secondary research questions by reporting the detailed results of your main analyses.
• Results of subgroup or exploratory analyses, if applicable. Place detailed results in supplementary materials.

Write up the results in the past tense because you’re describing the outcomes of a completed research study.

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Before diving into your research findings, first describe the flow of participants at every stage of your study and whether any data were excluded from the final analysis.

Participant flow and recruitment period

It’s necessary to report any attrition, which is the decline in participants at every sequential stage of a study. That’s because an uneven number of participants across groups sometimes threatens internal validity and makes it difficult to compare groups. Be sure to also state all reasons for attrition.

If your study has multiple stages (e.g., pre-test, intervention, and post-test) and groups (e.g., experimental and control groups), a flow chart is the best way to report the number of participants in each group per stage and reasons for attrition.

Also report the dates for when you recruited participants or performed follow-up sessions.

Missing data

Another key issue is the completeness of your dataset. It’s necessary to report both the amount and reasons for data that was missing or excluded.

Data can become unusable due to equipment malfunctions, improper storage, unexpected events, participant ineligibility, and so on. For each case, state the reason why the data were unusable.

Some data points may be removed from the final analysis because they are outliers—but you must be able to justify how you decided what to exclude.

If you applied any techniques for overcoming or compensating for lost data, report those as well.

Adverse events

For clinical studies, report all events with serious consequences or any side effects that occured.

Descriptive statistics summarize your data for the reader. Present descriptive statistics for each primary, secondary, and subgroup analysis.

Don’t provide formulas or citations for commonly used statistics (e.g., standard deviation) – but do provide them for new or rare equations.

Descriptive statistics

The exact descriptive statistics that you report depends on the types of data in your study. Categorical variables can be reported using proportions, while quantitative data can be reported using means and standard deviations . For a large set of numbers, a table is the most effective presentation format.

Include sample sizes (overall and for each group) as well as appropriate measures of central tendency and variability for the outcomes in your results section. For every point estimate , add a clearly labelled measure of variability as well.

Be sure to note how you combined data to come up with variables of interest. For every variable of interest, explain how you operationalized it.

According to APA journal standards, it’s necessary to report all relevant hypothesis tests performed, estimates of effect sizes, and confidence intervals.

When reporting statistical results, you should first address primary research questions before moving onto secondary research questions and any exploratory or subgroup analyses.

Present the results of tests in the order that you performed them—report the outcomes of main tests before post-hoc tests, for example. Don’t leave out any relevant results, even if they don’t support your hypothesis.

Inferential statistics

For each statistical test performed, first restate the hypothesis , then state whether your hypothesis was supported and provide the outcomes that led you to that conclusion.

Report the following for each hypothesis test:

• the test statistic value,
• the degrees of freedom ,
• the exact p- value (unless it is less than 0.001),
• the magnitude and direction of the effect.

When reporting complex data analyses, such as factor analysis or multivariate analysis, present the models estimated in detail, and state the statistical software used. Make sure to report any violations of statistical assumptions or problems with estimation.

Effect sizes and confidence intervals

For each hypothesis test performed, you should present confidence intervals and estimates of effect sizes .

Confidence intervals are useful for showing the variability around point estimates. They should be included whenever you report population parameter estimates.

Effect sizes indicate how impactful the outcomes of a study are. But since they are estimates, it’s recommended that you also provide confidence intervals of effect sizes.

Subgroup or exploratory analyses

Briefly report the results of any other planned or exploratory analyses you performed. These may include subgroup analyses as well.

Subgroup analyses come with a high chance of false positive results, because performing a large number of comparison or correlation tests increases the chances of finding significant results.

If you find significant results in these analyses, make sure to appropriately report them as exploratory (rather than confirmatory) results to avoid overstating their importance.

While these analyses can be reported in less detail in the main text, you can provide the full analyses in supplementary materials.

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To effectively present numbers, use a mix of text, tables , and figures where appropriate:

• To present three or fewer numbers, try a sentence ,
• To present between 4 and 20 numbers, try a table ,
• To present more than 20 numbers, try a figure .

Since these are general guidelines, use your own judgment and feedback from others for effective presentation of numbers.

Tables and figures should be numbered and have titles, along with relevant notes. Make sure to present data only once throughout the paper and refer to any tables and figures in the text.

Formatting statistics and numbers

It’s important to follow capitalization , italicization, and abbreviation rules when referring to statistics in your paper. There are specific format guidelines for reporting statistics in APA , as well as general rules about writing numbers .

If you are unsure of how to present specific symbols, look up the detailed APA guidelines or other papers in your field.

It’s important to provide a complete picture of your data analyses and outcomes in a concise way. For that reason, raw data and any interpretations of your results are not included in the results section.

It’s rarely appropriate to include raw data in your results section. Instead, you should always save the raw data securely and make them available and accessible to any other researchers who request them.

Making scientific research available to others is a key part of academic integrity and open science.

Interpretation or discussion of results

This belongs in your discussion section. Your results section is where you objectively report all relevant findings and leave them open for interpretation by readers.

While you should state whether the findings of statistical tests lend support to your hypotheses, refrain from forming conclusions to your research questions in the results section.

Explanation of how statistics tests work

For the sake of concise writing, you can safely assume that readers of your paper have professional knowledge of how statistical inferences work.

In an APA results section , you should generally report the following:

• Participant flow and recruitment period.
• Missing data and any adverse events.
• Descriptive statistics about your samples.
• Inferential statistics , including confidence intervals and effect sizes.
• Results of any subgroup or exploratory analyses, if applicable.

According to the APA guidelines, you should report enough detail on inferential statistics so that your readers understand your analyses.

• the test statistic value
• the degrees of freedom
• the exact p value (unless it is less than 0.001)
• the magnitude and direction of the effect

You should also present confidence intervals and estimates of effect sizes where relevant.

In APA style, statistics can be presented in the main text or as tables or figures . To decide how to present numbers, you can follow APA guidelines:

• To present three or fewer numbers, try a sentence,
• To present between 4 and 20 numbers, try a table,
• To present more than 20 numbers, try a figure.

Results are usually written in the past tense , because they are describing the outcome of completed actions.

The results chapter or section simply and objectively reports what you found, without speculating on why you found these results. The discussion interprets the meaning of the results, puts them in context, and explains why they matter.

In qualitative research , results and discussion are sometimes combined. But in quantitative research , it’s considered important to separate the objective results from your interpretation of them.

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How to write the results section of a research paper

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Table of Contents

At its core, a research paper aims to fill a gap in the research on a given topic. As a result, the results section of the paper, which describes the key findings of the study, is often considered the core of the paper. This is the section that gets the most attention from reviewers, peers, students, and any news organization reporting on your findings. Writing a clear, concise, and logical results section is, therefore, one of the most important parts of preparing your manuscript.

Difference between results and discussion

Before delving into how to write the results section, it is important to first understand the difference between the results and discussion sections. The results section needs to detail the findings of the study. The aim of this section is not to draw connections between the different findings or to compare it to previous findings in literature—that is the purview of the discussion section. Unlike the discussion section, which can touch upon the hypothetical, the results section needs to focus on the purely factual. In some cases, it may even be preferable to club these two sections together into a single section. For example, while writing  a review article, it can be worthwhile to club these two sections together, as the main results in this case are the conclusions that can be drawn from the literature.

Structure of the results section

Although the main purpose of the results section in a research paper is to report the findings, it is necessary to present an introduction and repeat the research question. This establishes a connection to the previous section of the paper and creates a smooth flow of information.

Next, the results section needs to communicate the findings of your research in a systematic manner. The section needs to be organized such that the primary research question is addressed first, then the secondary research questions. If the research addresses multiple questions, the results section must individually connect with each of the questions. This ensures clarity and minimizes confusion while reading.

Consider representing your results visually. For example, graphs, tables, and other figures can help illustrate the findings of your paper, especially if there is a large amount of data in the results.

Remember, an appealing results section can help peer reviewers better understand the merits of your research, thereby increasing your chances of publication.

Practical guidance for writing an effective results section for a research paper

• Always use simple and clear language. Avoid the use of uncertain or out-of-focus expressions.
• The findings of the study must be expressed in an objective and unbiased manner. While it is acceptable to correlate certain findings in the discussion section, it is best to avoid overinterpreting the results.
• If the research addresses more than one hypothesis, use sub-sections to describe the results. This prevents confusion and promotes understanding.
• Ensure that negative results are included in this section, even if they do not support the research hypothesis.
• Wherever possible, use illustrations like tables, figures, charts, or other visual representations to showcase the results of your research paper. Mention these illustrations in the text, but do not repeat the information that they convey.
• For statistical data, it is adequate to highlight the tests and explain their results. The initial or raw data should not be mentioned in the results section of a research paper.

The results section of a research paper is usually the most impactful section because it draws the greatest attention. Regardless of the subject of your research paper, a well-written results section is capable of generating interest in your research.

For detailed information and assistance on writing the results of a research paper, refer to Elsevier Author Services.

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How to Write the Results/Findings Section in Research

What is the research paper Results section and what does it do?

The Results section of a scientific research paper represents the core findings of a study derived from the methods applied to gather and analyze information. It presents these findings in a logical sequence without bias or interpretation from the author, setting up the reader for later interpretation and evaluation in the Discussion section. A major purpose of the Results section is to break down the data into sentences that show its significance to the research question(s).

The Results section appears third in the section sequence in most scientific papers. It follows the presentation of the Methods and Materials and is presented before the Discussion section —although the Results and Discussion are presented together in many journals. This section answers the basic question “What did you find in your research?”

What is included in the Results section?

The Results section should include the findings of your study and ONLY the findings of your study. The findings include:

• Data presented in tables, charts, graphs, and other figures (may be placed into the text or on separate pages at the end of the manuscript)
• A contextual analysis of this data explaining its meaning in sentence form
• All data that corresponds to the central research question(s)
• All secondary findings (secondary outcomes, subgroup analyses, etc.)

If the scope of the study is broad, or if you studied a variety of variables, or if the methodology used yields a wide range of different results, the author should present only those results that are most relevant to the research question stated in the Introduction section .

As a general rule, any information that does not present the direct findings or outcome of the study should be left out of this section. Unless the journal requests that authors combine the Results and Discussion sections, explanations and interpretations should be omitted from the Results.

How are the results organized?

The best way to organize your Results section is “logically.” One logical and clear method of organizing research results is to provide them alongside the research questions—within each research question, present the type of data that addresses that research question.

Let’s look at an example. Your research question is based on a survey among patients who were treated at a hospital and received postoperative care. Let’s say your first research question is:

“What do hospital patients over age 55 think about postoperative care?”

This can actually be represented as a heading within your Results section, though it might be presented as a statement rather than a question:

Attitudes towards postoperative care in patients over the age of 55

Now present the results that address this specific research question first. In this case, perhaps a table illustrating data from a survey. Likert items can be included in this example. Tables can also present standard deviations, probabilities, correlation matrices, etc.

Following this, present a content analysis, in words, of one end of the spectrum of the survey or data table. In our example case, start with the POSITIVE survey responses regarding postoperative care, using descriptive phrases. For example:

“Sixty-five percent of patients over 55 responded positively to the question “ Are you satisfied with your hospital’s postoperative care ?” (Fig. 2)

Include other results such as subcategory analyses. The amount of textual description used will depend on how much interpretation of tables and figures is necessary and how many examples the reader needs in order to understand the significance of your research findings.

Next, present a content analysis of another part of the spectrum of the same research question, perhaps the NEGATIVE or NEUTRAL responses to the survey. For instance:

  “As Figure 1 shows, 15 out of 60 patients in Group A responded negatively to Question 2.”

After you have assessed the data in one figure and explained it sufficiently, move on to your next research question. For example:

  “How does patient satisfaction correspond to in-hospital improvements made to postoperative care?”

This kind of data may be presented through a figure or set of figures (for instance, a paired T-test table).

Explain the data you present, here in a table, with a concise content analysis:

“The p-value for the comparison between the before and after groups of patients was .03% (Fig. 2), indicating that the greater the dissatisfaction among patients, the more frequent the improvements that were made to postoperative care.”

Let’s examine another example of a Results section from a study on plant tolerance to heavy metal stress . In the Introduction section, the aims of the study are presented as “determining the physiological and morphological responses of Allium cepa L. towards increased cadmium toxicity” and “evaluating its potential to accumulate the metal and its associated environmental consequences.” The Results section presents data showing how these aims are achieved in tables alongside a content analysis, beginning with an overview of the findings:

“Cadmium caused inhibition of root and leave elongation, with increasing effects at higher exposure doses (Fig. 1a-c).”

The figure containing this data is cited in parentheses. Note that this author has combined three graphs into one single figure. Separating the data into separate graphs focusing on specific aspects makes it easier for the reader to assess the findings, and consolidating this information into one figure saves space and makes it easy to locate the most relevant results.

Following this overall summary, the relevant data in the tables is broken down into greater detail in text form in the Results section.

• “Results on the bio-accumulation of cadmium were found to be the highest (17.5 mg kgG1) in the bulb, when the concentration of cadmium in the solution was 1×10G2 M and lowest (0.11 mg kgG1) in the leaves when the concentration was 1×10G3 M.”

Captioning and Referencing Tables and Figures

Tables and figures are central components of your Results section and you need to carefully think about the most effective way to use graphs and tables to present your findings . Therefore, it is crucial to know how to write strong figure captions and to refer to them within the text of the Results section.

The most important advice one can give here as well as throughout the paper is to check the requirements and standards of the journal to which you are submitting your work. Every journal has its own design and layout standards, which you can find in the author instructions on the target journal’s website. Perusing a journal’s published articles will also give you an idea of the proper number, size, and complexity of your figures.

Regardless of which format you use, the figures should be placed in the order they are referenced in the Results section and be as clear and easy to understand as possible. If there are multiple variables being considered (within one or more research questions), it can be a good idea to split these up into separate figures. Subsequently, these can be referenced and analyzed under separate headings and paragraphs in the text.

To create a caption, consider the research question being asked and change it into a phrase. For instance, if one question is “Which color did participants choose?”, the caption might be “Color choice by participant group.” Or in our last research paper example, where the question was “What is the concentration of cadmium in different parts of the onion after 14 days?” the caption reads:

 “Fig. 1(a-c): Mean concentration of Cd determined in (a) bulbs, (b) leaves, and (c) roots of onions after a 14-day period.”

Steps for Composing the Results Section

Because each study is unique, there is no one-size-fits-all approach when it comes to designing a strategy for structuring and writing the section of a research paper where findings are presented. The content and layout of this section will be determined by the specific area of research, the design of the study and its particular methodologies, and the guidelines of the target journal and its editors. However, the following steps can be used to compose the results of most scientific research studies and are essential for researchers who are new to preparing a manuscript for publication or who need a reminder of how to construct the Results section.

Step 1 : Consult the guidelines or instructions that the target journal or publisher provides authors and read research papers it has published, especially those with similar topics, methods, or results to your study.

• The guidelines will generally outline specific requirements for the results or findings section, and the published articles will provide sound examples of successful approaches.
• Note length limitations on restrictions on content. For instance, while many journals require the Results and Discussion sections to be separate, others do not—qualitative research papers often include results and interpretations in the same section (“Results and Discussion”).
• Reading the aims and scope in the journal’s “ guide for authors ” section and understanding the interests of its readers will be invaluable in preparing to write the Results section.

Step 2 : Consider your research results in relation to the journal’s requirements and catalogue your results.

• Focus on experimental results and other findings that are especially relevant to your research questions and objectives and include them even if they are unexpected or do not support your ideas and hypotheses.
• Catalogue your findings—use subheadings to streamline and clarify your report. This will help you avoid excessive and peripheral details as you write and also help your reader understand and remember your findings. Create appendices that might interest specialists but prove too long or distracting for other readers.
• Decide how you will structure of your results. You might match the order of the research questions and hypotheses to your results, or you could arrange them according to the order presented in the Methods section. A chronological order or even a hierarchy of importance or meaningful grouping of main themes or categories might prove effective. Consider your audience, evidence, and most importantly, the objectives of your research when choosing a structure for presenting your findings.

Step 3 : Design figures and tables to present and illustrate your data.

• Tables and figures should be numbered according to the order in which they are mentioned in the main text of the paper.
• Information in figures should be relatively self-explanatory (with the aid of captions), and their design should include all definitions and other information necessary for readers to understand the findings without reading all of the text.
• Use tables and figures as a focal point to tell a clear and informative story about your research and avoid repeating information. But remember that while figures clarify and enhance the text, they cannot replace it.

Step 4 : Draft your Results section using the findings and figures you have organized.

• The goal is to communicate this complex information as clearly and precisely as possible; precise and compact phrases and sentences are most effective.
• In the opening paragraph of this section, restate your research questions or aims to focus the reader’s attention to what the results are trying to show. It is also a good idea to summarize key findings at the end of this section to create a logical transition to the interpretation and discussion that follows.
• Try to write in the past tense and the active voice to relay the findings since the research has already been done and the agent is usually clear. This will ensure that your explanations are also clear and logical.
• Make sure that any specialized terminology or abbreviation you have used here has been defined and clarified in the  Introduction section .

Step 5 : Review your draft; edit and revise until it reports results exactly as you would like to have them reported to your readers.

• Double-check the accuracy and consistency of all the data, as well as all of the visual elements included.
• Read your draft aloud to catch language errors (grammar, spelling, and mechanics), awkward phrases, and missing transitions.
• Ensure that your results are presented in the best order to focus on objectives and prepare readers for interpretations, valuations, and recommendations in the Discussion section . Look back over the paper’s Introduction and background while anticipating the Discussion and Conclusion sections to ensure that the presentation of your results is consistent and effective.
• Consider seeking additional guidance on your paper. Find additional readers to look over your Results section and see if it can be improved in any way. Peers, professors, or qualified experts can provide valuable insights.

One excellent option is to use a professional English proofreading and editing service  such as Wordvice, including our paper editing service . With hundreds of qualified editors from dozens of scientific fields, Wordvice has helped thousands of authors revise their manuscripts and get accepted into their target journals. Read more about the  proofreading and editing process  before proceeding with getting academic editing services and manuscript editing services for your manuscript.

As the representation of your study’s data output, the Results section presents the core information in your research paper. By writing with clarity and conciseness and by highlighting and explaining the crucial findings of their study, authors increase the impact and effectiveness of their research manuscripts.

For more articles and videos on writing your research manuscript, visit Wordvice’s Resources page.

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The results section is where you report the findings of your study based upon the methodology [or methodologies] you applied to gather information. The results section should state the findings of the research arranged in a logical sequence without bias or interpretation. A section describing results should be particularly detailed if your paper includes data generated from your own research.

Annesley, Thomas M. "Show Your Cards: The Results Section and the Poker Game." Clinical Chemistry 56 (July 2010): 1066-1070.

Importance of a Good Results Section

When formulating the results section, it's important to remember that the results of a study do not prove anything . Findings can only confirm or reject the hypothesis underpinning your study. However, the act of articulating the results helps you to understand the problem from within, to break it into pieces, and to view the research problem from various perspectives.

The page length of this section is set by the amount and types of data to be reported . Be concise. Use non-textual elements appropriately, such as figures and tables, to present findings more effectively. In deciding what data to describe in your results section, you must clearly distinguish information that would normally be included in a research paper from any raw data or other content that could be included as an appendix. In general, raw data that has not been summarized should not be included in the main text of your paper unless requested to do so by your professor.

Avoid providing data that is not critical to answering the research question . The background information you described in the introduction section should provide the reader with any additional context or explanation needed to understand the results. A good strategy is to always re-read the background section of your paper after you have written up your results to ensure that the reader has enough context to understand the results [and, later, how you interpreted the results in the discussion section of your paper that follows].

Bavdekar, Sandeep B. and Sneha Chandak. "Results: Unraveling the Findings." Journal of the Association of Physicians of India 63 (September 2015): 44-46; Brett, Paul. "A Genre Analysis of the Results Section of Sociology Articles." English for Specific Speakers 13 (1994): 47-59; Go to English for Specific Purposes on ScienceDirect;Burton, Neil et al. Doing Your Education Research Project . Los Angeles, CA: SAGE, 2008; Results. The Structure, Format, Content, and Style of a Journal-Style Scientific Paper. Department of Biology. Bates College; Kretchmer, Paul. Twelve Steps to Writing an Effective Results Section. San Francisco Edit; "Reporting Findings." In Making Sense of Social Research Malcolm Williams, editor. (London;: SAGE Publications, 2003) pp. 188-207.

Structure and Writing Style

I.  Organization and Approach

For most research papers in the social and behavioral sciences, there are two possible ways of organizing the results . Both approaches are appropriate in how you report your findings, but use only one approach.

• Present a synopsis of the results followed by an explanation of key findings . This approach can be used to highlight important findings. For example, you may have noticed an unusual correlation between two variables during the analysis of your findings. It is appropriate to highlight this finding in the results section. However, speculating as to why this correlation exists and offering a hypothesis about what may be happening belongs in the discussion section of your paper.
• Present a result and then explain it, before presenting the next result then explaining it, and so on, then end with an overall synopsis . This is the preferred approach if you have multiple results of equal significance. It is more common in longer papers because it helps the reader to better understand each finding. In this model, it is helpful to provide a brief conclusion that ties each of the findings together and provides a narrative bridge to the discussion section of the your paper.

NOTE:   Just as the literature review should be arranged under conceptual categories rather than systematically describing each source, you should also organize your findings under key themes related to addressing the research problem. This can be done under either format noted above [i.e., a thorough explanation of the key results or a sequential, thematic description and explanation of each finding].

II.  Content

In general, the content of your results section should include the following:

• Introductory context for understanding the results by restating the research problem underpinning your study . This is useful in re-orientating the reader's focus back to the research problem after having read a review of the literature and your explanation of the methods used for gathering and analyzing information.
• Inclusion of non-textual elements, such as, figures, charts, photos, maps, tables, etc. to further illustrate key findings, if appropriate . Rather than relying entirely on descriptive text, consider how your findings can be presented visually. This is a helpful way of condensing a lot of data into one place that can then be referred to in the text. Consider referring to appendices if there is a lot of non-textual elements.
• A systematic description of your results, highlighting for the reader observations that are most relevant to the topic under investigation . Not all results that emerge from the methodology used to gather information may be related to answering the " So What? " question. Do not confuse observations with interpretations; observations in this context refers to highlighting important findings you discovered through a process of reviewing prior literature and gathering data.
• The page length of your results section is guided by the amount and types of data to be reported . However, focus on findings that are important and related to addressing the research problem. It is not uncommon to have unanticipated results that are not relevant to answering the research question. This is not to say that you don't acknowledge tangential findings and, in fact, can be referred to as areas for further research in the conclusion of your paper. However, spending time in the results section describing tangential findings clutters your overall results section and distracts the reader.
• A short paragraph that concludes the results section by synthesizing the key findings of the study . Highlight the most important findings you want readers to remember as they transition into the discussion section. This is particularly important if, for example, there are many results to report, the findings are complicated or unanticipated, or they are impactful or actionable in some way [i.e., able to be pursued in a feasible way applied to practice].

NOTE:   Always use the past tense when referring to your study's findings. Reference to findings should always be described as having already happened because the method used to gather the information has been completed.

III.  Problems to Avoid

When writing the results section, avoid doing the following :

• Discussing or interpreting your results . Save this for the discussion section of your paper, although where appropriate, you should compare or contrast specific results to those found in other studies [e.g., "Similar to the work of Smith [1990], one of the findings of this study is the strong correlation between motivation and academic achievement...."].
• Reporting background information or attempting to explain your findings. This should have been done in your introduction section, but don't panic! Often the results of a study point to the need for additional background information or to explain the topic further, so don't think you did something wrong. Writing up research is rarely a linear process. Always revise your introduction as needed.
• Ignoring negative results . A negative result generally refers to a finding that does not support the underlying assumptions of your study. Do not ignore them. Document these findings and then state in your discussion section why you believe a negative result emerged from your study. Note that negative results, and how you handle them, can give you an opportunity to write a more engaging discussion section, therefore, don't be hesitant to highlight them.
• Including raw data or intermediate calculations . Ask your professor if you need to include any raw data generated by your study, such as transcripts from interviews or data files. If raw data is to be included, place it in an appendix or set of appendices that are referred to in the text.
• Be as factual and concise as possible in reporting your findings . Do not use phrases that are vague or non-specific, such as, "appeared to be greater than other variables..." or "demonstrates promising trends that...." Subjective modifiers should be explained in the discussion section of the paper [i.e., why did one variable appear greater? Or, how does the finding demonstrate a promising trend?].
• Presenting the same data or repeating the same information more than once . If you want to highlight a particular finding, it is appropriate to do so in the results section. However, you should emphasize its significance in relation to addressing the research problem in the discussion section. Do not repeat it in your results section because you can do that in the conclusion of your paper.
• Confusing figures with tables . Be sure to properly label any non-textual elements in your paper. Don't call a chart an illustration or a figure a table. If you are not sure, go here .

Annesley, Thomas M. "Show Your Cards: The Results Section and the Poker Game." Clinical Chemistry 56 (July 2010): 1066-1070; Bavdekar, Sandeep B. and Sneha Chandak. "Results: Unraveling the Findings." Journal of the Association of Physicians of India 63 (September 2015): 44-46; Burton, Neil et al. Doing Your Education Research Project . Los Angeles, CA: SAGE, 2008;  Caprette, David R. Writing Research Papers. Experimental Biosciences Resources. Rice University; Hancock, Dawson R. and Bob Algozzine. Doing Case Study Research: A Practical Guide for Beginning Researchers . 2nd ed. New York: Teachers College Press, 2011; Introduction to Nursing Research: Reporting Research Findings. Nursing Research: Open Access Nursing Research and Review Articles. (January 4, 2012); Kretchmer, Paul. Twelve Steps to Writing an Effective Results Section. San Francisco Edit ; Ng, K. H. and W. C. Peh. "Writing the Results." Singapore Medical Journal 49 (2008): 967-968; Reporting Research Findings. Wilder Research, in partnership with the Minnesota Department of Human Services. (February 2009); Results. The Structure, Format, Content, and Style of a Journal-Style Scientific Paper. Department of Biology. Bates College; Schafer, Mickey S. Writing the Results. Thesis Writing in the Sciences. Course Syllabus. University of Florida.

Writing Tip

Why Don't I Just Combine the Results Section with the Discussion Section?

It's not unusual to find articles in scholarly social science journals where the author(s) have combined a description of the findings with a discussion about their significance and implications. You could do this. However, if you are inexperienced writing research papers, consider creating two distinct sections for each section in your paper as a way to better organize your thoughts and, by extension, your paper. Think of the results section as the place where you report what your study found; think of the discussion section as the place where you interpret the information and answer the "So What?" question. As you become more skilled writing research papers, you can consider melding the results of your study with a discussion of its implications.

Driscoll, Dana Lynn and Aleksandra Kasztalska. Writing the Experimental Report: Methods, Results, and Discussion. The Writing Lab and The OWL. Purdue University.

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How to Write an APA Results Section

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Emily is a board-certified science editor who has worked with top digital publishing brands like Voices for Biodiversity, Study.com, GoodTherapy, Vox, and Verywell.

Verywell / Nusha Ashjaee 

What to Include in an APA Results Section

• Justify Claims
• Summarize Results

Report All Relevant Results

• Report Statistical Findings

Include Tables and Figures

What not to include in an apa results section.

Psychology papers generally follow a specific structure. One important section of a paper is known as the results section. An APA results section of a psychology paper summarizes the data that was collected and the statistical analyses that were performed. The goal of this section is to report the results of your study or experiment without any type of subjective interpretation.

At a Glance

The results section is a vital part of an APA paper that summarizes a study's findings and statistical analysis. This section often includes descriptive text, tables, and figures to help summarize the findings.

The focus is purely on summarizing and presenting the findings and should not include any interpretation, since you'll cover that in the subsequent discussion section.

This article covers how to write an APA results section, including what to include and what to avoid.

The results section is the third section of a psychology paper. It will appear after the introduction and methods sections and before the discussion section.

The results section should include:

• A summary of the research findings.
• Information about participant flow, recruitment , retention, and attrition. If some participants started the study and later left or failed to complete the study, then this should be described. 
• Information about any reasons why some data might have been excluded from the study. 
• Statistical information including samples sizes and statistical tests that were used. It should report standard deviations, p-values, and other measures of interest.

Results Should Justify Your Claims

Report data in order to sufficiently justify your conclusions. Since you'll be talking about your own interpretation of the results in the discussion section, you need to be sure that the information reported in the results section justifies your claims.

When you start writing your discussion section, you can then look back on your results to ensure that all the data you need are there to fully support your conclusions. Be sure not to make claims in your discussion section that are not supported by the findings described in your results section.

Summarize Your Results

Remember, you are summarizing the results of your psychological study, not reporting them in full detail. The results section should be a relatively brief overview of your findings, not a complete presentation of every single number and calculation.

If you choose, you can create a supplemental online archive where other researchers can access the raw data if they choose.

How long should a results section be?

The length of your results section will vary depending on the nature of your paper and the complexity of your research. In most cases, this will be the shortest section of your paper.

Just as the results section of your psychology paper should sufficiently justify your claims, it should also provide an accurate look at what you found in your study. Be sure to mention all relevant information.

Don't omit findings simply because they failed to support your predictions.

Your hypothesis may have expected more statistically significant results or your study didn't support your hypothesis , but that doesn't mean that the conclusions you reach are not useful. Provide data about what you found in your results section, then save your interpretation for what the results might mean in the discussion section.

While your study might not have supported your original predictions, your finding can provide important inspiration for future explorations into a topic.

How is the results section different from the discussion section?

The results section provides the results of your study or experiment . The goal of the section is to report what happened and the statistical analyses you performed. The discussion section is where you will examine what these results mean and whether they support or fail to support your hypothesis.

Report Your Statistical Findings

Always assume that your readers have a solid understanding of statistical concepts. There's no need to explain what a t-test is or how a one-way ANOVA works. Your responsibility is to report the results of your study, not to teach your readers how to analyze or interpret statistics.

Include Effect Sizes

The Publication Manual of the American Psychological Association recommends including effect sizes in your results section so that readers can appreciate the importance of your study's findings.

Your results section should include both text and illustrations. Presenting data in this way makes it easier for readers to quickly look at your results.

Structure your results section around tables or figures that summarize the results of your statistical analysis. In many cases, the easiest way to accomplish this is to first create your tables and figures and then organize them in a logical way. Next, write the summary text to support your illustrative materials.

Only include tables and figures if you are going to talk about them in the body text of your results section.

In addition to knowing what you should include in the results section of your psychology paper, it's also important to be aware of things that you should avoid putting in this section:

Cause-and-Effect Conclusions

Don't draw cause-effect conclusions. Avoid making any claims suggesting that your result "proves" that something is true. 

Interpretations

Present the data without editorializing it. Save your comments and interpretations for the discussion section of your paper. 

Statistics Without Context

Don't include statistics without narration. The results section should not be a numbers dump. Instead, you should sequentially narrate what these numbers mean.

Don't include the raw data in the results section. The results section should be a concise presentation of the results. If there is raw data that would be useful, include it in the appendix .

Don't only rely on descriptive text. Use tables and figures to present these findings when appropriate. This makes the results section easier to read and can convey a great deal of information quickly.

Repeated Data

Don't present the same data twice in your illustrative materials. If you have already presented some data in a table, don't present it again in a figure. If you have presented data in a figure, don't present it again in a table.

All of Your Findings

Don't feel like you have to include everything. If data is irrelevant to the research question, don't include it in the results section.

But Don't Skip Relevant Data

Don't leave out results because they don't support your claims. Even if your data does not support your hypothesis, including it in your findings is essential if it's relevant.

More Tips for Writing a Results Section

If you are struggling, there are a few things to remember that might help:

• Use the past tense . The results section should be written in the past tense.
• Be concise and objective . You will have the opportunity to give your own interpretations of the results in the discussion section.
• Use APA format . As you are writing your results section, keep a style guide on hand. The Publication Manual of the American Psychological Association is the official source for APA style .
• Visit your library . Read some journal articles that are on your topic. Pay attention to how the authors present the results of their research.
• Get a second opinion . If possible, take your paper to your school's writing lab for additional assistance.

What This Means For You

Remember, the results section of your paper is all about providing the data from your study. This section is often the shortest part of your paper, and in most cases, the most clinical.

Be sure not to include any subjective interpretation of the results. Simply relay the data in the most objective and straightforward way possible. You can then provide your own analysis of what these results mean in the discussion section of your paper.

Bavdekar SB, Chandak S. Results: Unraveling the findings . J Assoc Physicians India . 2015 Sep;63(9):44-6. PMID:27608866.

Snyder N, Foltz C, Lendner M, Vaccaro AR. How to write an effective results section .  Clin Spine Surg . 2019;32(7):295-296. doi:10.1097/BSD.0000000000000845

American Psychological Association.  Publication Manual of the American Psychological Association  (7th ed.). Washington DC: The American Psychological Association; 2019.

Purdue Online Writing Lab. APA sample paper: Experimental psychology .

Berkeley University. Reviewing test results .

Tuncel A, Atan A. How to clearly articulate results and construct tables and figures in a scientific paper ? Turk J Urol . 2013;39(Suppl 1):16-19. doi:10.5152/tud.2013.048

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

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Writing a scientific paper.

• Writing a lab report
• INTRODUCTION

Writing a "good" results section

Figures and Captions in Lab Reports

"Results Checklist" from: How to Write a Good Scientific Paper. Chris A. Mack. SPIE. 2018.

Additional tips for results sections.

• LITERATURE CITED
• Bibliography of guides to scientific writing and presenting
• Peer Review
• Presentations
• Lab Report Writing Guides on the Web

This is the core of the paper. Don't start the results sections with methods you left out of the Materials and Methods section. You need to give an overall description of the experiments and present the data you found.

• Factual statements supported by evidence. Short and sweet without excess words
• Present representative data rather than endlessly repetitive data
• Discuss variables only if they had an effect (positive or negative)
• Use meaningful statistics
• Avoid redundancy. If it is in the tables or captions you may not need to repeat it

A short article by Dr. Brett Couch and Dr. Deena Wassenberg, Biology Program, University of Minnesota

• Present the results of the paper, in logical order, using tables and graphs as necessary.
• Explain the results and show how they help to answer the research questions posed in the Introduction. Evidence does not explain itself; the results must be presented and then explained. 
• Avoid: presenting results that are never discussed;  presenting results in chronological order rather than logical order; ignoring results that do not support the conclusions; 
• Number tables and figures separately beginning with 1 (i.e. Table 1, Table 2, Figure 1, etc.).
• Do not attempt to evaluate the results in this section. Report only what you found; hold all discussion of the significance of the results for the Discussion section.
• It is not necessary to describe every step of your statistical analyses. Scientists understand all about null hypotheses, rejection rules, and so forth and do not need to be reminded of them. Just say something like, "Honeybees did not use the flowers in proportion to their availability (X2 = 7.9, p<0.05, d.f.= 4, chi-square test)." Likewise, cite tables and figures without describing in detail how the data were manipulated. Explanations of this sort should appear in a legend or caption written on the same page as the figure or table.
• You must refer in the text to each figure or table you include in your paper.
• Tables generally should report summary-level data, such as means ± standard deviations, rather than all your raw data.  A long list of all your individual observations will mean much less than a few concise, easy-to-read tables or figures that bring out the main findings of your study.  
• Only use a figure (graph) when the data lend themselves to a good visual representation.  Avoid using figures that show too many variables or trends at once, because they can be hard to understand.

From:  https://writingcenter.gmu.edu/guides/imrad-results-discussion

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How to Write an Effective Results Section

Affiliation.

• 1 Rothman Orthopaedics Institute, Philadelphia, PA.
• PMID: 31145152
• DOI: 10.1097/BSD.0000000000000845

Developing a well-written research paper is an important step in completing a scientific study. This paper is where the principle investigator and co-authors report the purpose, methods, findings, and conclusions of the study. A key element of writing a research paper is to clearly and objectively report the study's findings in the Results section. The Results section is where the authors inform the readers about the findings from the statistical analysis of the data collected to operationalize the study hypothesis, optimally adding novel information to the collective knowledge on the subject matter. By utilizing clear, concise, and well-organized writing techniques and visual aids in the reporting of the data, the author is able to construct a case for the research question at hand even without interpreting the data.

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Results Section Of A Research Paper: How To Write It Properly

The results section of a research paper refers to the part that represents the study’s core findings from the methods that the researcher used to collect and analyze data. This section presents the results logically without interpretation or bias from the author.

Thus, this part of a research paper sets up the read for evaluation and analysis of the findings in the discussion section. Essentially, this section breaks down the information into several sentences, showing its importance to the research question. Writing results section in a research paper entails summarizing the gathered data and the performed statistical analysis. That way, the author presents or reports the results without subjective interpretation.

What Is The Results Section Of A Research Paper?

In its simplest definition, a research paper results section is where the researcher reports the findings of a study based on the applied methodology for gathering information. It’s the part where the author states the research findings in a logical sequence without interpreting them. If the research paper has data from actual research, this section should feature a detailed description of the results.

When writing a dissertation, a thesis, or any other academic paper, the result section should come third in sections’ sequence. It should follow the Methods and Materials presentation and the Discussion section comes after it. But most scientific papers present the Results and Discussion sections together. However, the results section answers the question, “What did your research uncover?”

Ideally, this section allows you to report findings in research paper, creating the basis for sufficiently justified conclusions. After writing the study findings in the results section, you interpret them in the subsequent discussion part. Therefore, your results section should report information that will justify your claims. That way, you can look back on the results section when writing the discussion part to ensure that your report supports your conclusions.

What Goes in the Results Section of a Research Paper?

This section should present results in research paper. The findings part of a research paper can differ in structure depending on the study, discipline, and journal. Nevertheless, the results section presents a description of the experiment while presenting the research results. When writing this part of your research paper, you can use graphs and tables if necessary.

However, state the findings without interpreting them. For instance, you can find a correlation between variables when analyzing data. In that case, your results section can explain this correlation without speculating about the causes of this correlation.

Here’s what to include in the results section of research paper:

A brief introductory of the context, repeating the research questions to help the readers understand the results A report about information collection, participants, and recruitment: for instance, you can include a demographic summary with the participants’ characteristics A systematic findings’ description, with a logical presentation highlighting relevant and crucial results A contextual data analysis explaining the meaning in sentences Information corresponding to the primary research questions Secondary findings like subgroup analysis and secondary outcomes Visual elements like charts, figures, tables, and maps, illustrating and summarizing the findings

Ensure that your results section cites and numbers visual elements in an orderly manner. Every table or figure should stand alone without text. That means visual elements should have adequate non-textual content to enable the audiences to understand their meanings.

If your study has a broad scope, several variables, or used methodologies that yielded different results, state the most relevant results only based on the research question you presented in your Introduction section.

The general rule is to leave out any data that doesn’t present your study’s direct outcome or findings. Unless the professor, advisor, university faulty, or your target journal requests you to combine the Results and Discussion sections, omit the interpretations and explanations of the results in this section.

How Long Should A Results Section Be?

The findings section of a research paper ranges between two and three pages, with tables, text, and figures. In most cases, universities and journals insist that this section shouldn’t exceed 1,000 words over four to nine paragraphs, usually with no references.

But a good findings section occupies 5% of the entire paper. For instance, this section should have 500 words if a dissertation has 10,000 words. If the educator didn’t specify the number of words to include in this chapter, use the data you collect to determine its length. Nevertheless, be as concise as possible by featuring only relevant results that answer your research question.

How To Write Results Section Of Research Paper

Perhaps, you have completed researching and writing the preceding sections, and you’re now wondering how to write results. By the time you’re composing this section, you already have findings or answers to your research questions. However, you don’t even know how to start a results section. And your search for guidelines landed you on this page.

Well, every research project is different and unique. That’s why researchers use different strategies when writing this section of their research papers. The scientific or academic discipline, specialization field, target journal, and the author are factors influencing how you write this section. Nevertheless, there’s a general way of writing this section, although it might differ slightly between disciplines. Here’s how to write results section in a research paper.

Check the instructions or guidelines. Check their instructions or guidelines first, whether you’re writing the research paper as part of your coursework or for an academic journal. These guidelines outline the requirements for presenting results in research papers. Also, check the published articles to know how to approach this section. When reviewing the procedures, check content restrictions and length. Essentially, learn everything you can about this section from the instructions or guidelines before you start writing. Reflect on your research findings. With instructions and guidelines in mind, reflect on your research findings to determine how to present them in your research paper. Decide on the best way to show the results so that they can answer the research question. Also, strive to clarify and streamline your report, especially with a complex and lengthy results section. You can use subheadings to avoid peripheral and excessive details. Additionally, consider breaking down the content to make it easy for the readers to understand or remember. Your hypothesis, research question, or methodologies might influence the structure of the findings sections. Nevertheless, a hierarchy of importance, chronological order, or meaningful grouping of categories or themes can be an effective way of presenting your findings. Design your visual presentations. Visual presentations improve the textual report of the research findings. Therefore, decide on the figures and styles to use in your tables, graphs, photos, and maps. However, check the instructions and guidelines of your faculty or journal to determine the visual aids you can use. Also, check what the guidelines say about their formats and design elements. Ideally, number the figures and tables according to their mention in the text. Additionally, your figures and tables should be self-explanatory. Write your findings section. Writing the results section of a research paper entails communicating the information you gathered from your study. Ideally, be as objective and factual as possible. If you gathered complex information, try to simplify and present it accurately, precisely, and clearly. Therefore, use well-structured sentences instead of complex expressions and phrases. Also, use an active voice and past tense since you’ve already done the research. Additionally, use correct spelling, grammar, and punctuation. Take your time to present the findings in the best way possible to focus your readers on your study objectives while preparing them for the coming speculations, interpretations, and recommendations. Edit Your Findings Section. Once you’ve written the results part of your paper, please go through it to ensure that you’ve presented your study findings in the best way possible. Make sure that the content of this section is factual, accurate, and without errors. You’ve taken a considerable amount of time to compose the results scientific paper audiences will find interesting to read. Therefore, take a moment to go through the draft and eliminate all errors.

Practical Tips on How to Write a Results Section of a Research Paper

The results part of a research paper aims to present the key findings objectively in a logical and orderly sequence using text and illustrative materials. A common mistake that many authors make is confusing the information in the discussion and the results sections. To avoid this, focus on presenting your research findings without interpreting them or speculating about them.

The following tips on how to write a results section should make this task easier for you:

Summarize your study results: Instead of reporting the findings in full detail, summarize them. That way, you can develop an overview of the results. Present relevant findings only: Don’t report everything you found during your research. Instead, present pertinent information only. That means taking time to analyze your results to know what your audiences want to know. Report statistical findings: When writing this section, assume that the audiences understand statistical concepts. Therefore, don’t try to explain the nitty-gritty in this section. Remember that your work is to report your study’s findings in this section. Be objective and concise: You can interpret the findings in the discussion sections. Therefore, focus on presenting the results objectively and concisely in this section. Use the suitable format: Use the correct style to present the findings depending on your study field.

Get Professional Help with the Research Section

Maybe you’re pursuing your graduate or undergraduate studies but cannot write the results part of your paper. Perhaps, you’re done researching and analyzing information, but this section proves too tricky for you to write. Well, you’re not alone because many students across the world struggle to present their research findings.

Luckily, our highly educated, talented, and experienced writers are always ready to assist such learners. If you are stuck with the results part of your paper, our professionals can help you . We offer high-quality, custom writing help online. We’re a reliable team of experts with a sterling reputation for providing comprehensive assistance to college, high school, and university learners. We deliver highly informative academic papers after conducting extensive and in-depth research. Contact us saying something like, “please do my thesis” to get quality help with your paper!

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How to Write the Results Section of a Research Paper

This article is part of an ongoing series on academic writing help of scholarly articles. Previous parts explored how to write an introduction for a research paper , literature review outline and format , and how to write a research methodology .

Academics and researchers publish their scholarly articles to show the results they have obtained using gathered or collected data. Research papers present the process of testing hypotheses or models and how their findings help shape or advance a particular research topic. Thus, the ‘Results’ section is essential in expressing the significance of an academic article.

The findings of your research should be included in a separate section of your academic article, as it is the only section that contains data and results.

Aspects to Consider in Writing the Results Section of a Research Paper

A good place to start for your results section, it’s to restate the aim and objective of your research paper , so that your readers can refocus on the core of your academic article. So far in your research paper, your readers covered the introduction , literature review , research methodology and now it’s the time and place to bring their attention back to the purpose. A short paragraph is sufficient to restate your paper’s purpose.

Then, it’s key to consider that this is main section of your research paper where you present and explain the data you have collected or gathered and the findings of your data analysis and interpretation .

The academic writing should be clear, impartial, and objective . Each result, which confirms or refutes your assumptions, should be noted in an unbiased manner to increase the credibility of your study.

The results section gives you the opportunity to:

• summarize the collected data in the form of descriptive statistics and
• report on the findings from relevant and appropriate inferential statistical analyses and interpretation that are aimed at answering your academic article’s research questions or supporting your hypotheses, and show your research significance.

For an organized Research Results section, it’s best to use sub-sections. These sub-sections or divisions can be based on:

• Your research questions, hypotheses or models , or
• The statistical tests you have conducted.

How to Clearly Report Your Research Findings

If you have used statistical analyses in your academic article, and found answers to your research questions, report those facts in relation to your question.

A clear, coherent presentation of your research paper’s results should exhibit logical explanations without bias.

Confirming or Rejecting Hypotheses in Your Research Results

While defining the section of your research’s outcomes area, it’s important to keep in mind that the research results do not prove or demonstrate anything.

Your research findings can only affirm/ confirm or reject the hypotheses and assumptions elaborated upon in your academic article. In any case, your results:

• help with the understanding of a research problem from within,
• assist in dividing the research problem into different parts and concepts,
• add to the exploration of an issue from various vantage points.

Summarizing Key Findings in Your Results Section

In a coherent results presentation, you should:

• offer summarizing notes of your outcomes and
• save the explanations of your key discoveries for your Discussion section.

For example, in your empirical analysis you notice an uncommon correlation between two variables. In the Results section, it is okay to bring up this outcome, however, posing new hypotheses for this uncommon result should be presented in the Discussion section.

Using Tables and Figures to Highlight Research Results

Any valuable academic article should focus on using tables, figures and/or graphs to:

• provide accurate views about the research findings,
• summarize the analysis,
• help with the interpretation of these outcomes, and
• offer better understanding of the overall study.

Instead of using only descriptive text for your scholarly article, consider other visual ways and representations that improve the academic writing of your research paper.

Figures, tables and graphs are useful methods for gathering a great deal of information into one place that can then be mentioned in the content of your article. If any research question or hypothesis is confirmed by your data and analysis, you can point to a table or figure that illustrates your finding.

When you present tables or figures in your results section, make sure to describe at least some of the data included in these visual representations so that readers can clearly understand how the table works and what interpretations can be concluded from them.

You can also use appendices if you have many other helpful figures or tables that cannot be fully included in the text of your academic article.

By using a helpful combination of text, figures, and tables, you, as Authors and Academics, can use this section to effectively share your studies’ findings with the scientific community.

Presenting Research Findings and Statistical Significance

A systematic description of your research results and a correct data analysis and interpretation are related to statistical significance, as they help avoid speculations or misinterpretations by readers of your academic article.

In a valuable research paper:

• data must be directly and clearly presented,
• statistical tests need to be used, and
• the figures obtained and included in the study have to be explained.

Tests of statistical significance should always be presented with your results to show that your research findings objectively confirm or disprove your hypotheses. You need to report the research results with enough details so that readers can see which statistical analyses were conducted and validated to justify or disprove your hypotheses. It is important to mention relevant research findings, including those that were are statistical insignificant, not validated within your model’s framework, and are at odds with your initial assumptions.

Even if not all of your research results are confirmed, you should not ignore them. These negative results that do not support a particular hypothesis should be noted in the results section, and then explained in the Discussion section.

Writing a Research Results section that do not address the negative results, invalidates the research paper and does not reflect appropriate academic writing.

Research Results Comparison with Similar Academic Articles

The largest part of interpreting and discussing your research findings should be reserved for the Discussion / Conclusion section.

However, there are instances when it is appropriate to compare or contrast your results with findings from previous and similar studies. For example:

• Similar to Author [Year], one of the findings of this study is the strong relationship between…
• While Author [Year] found an indirect relationship between, our study highlighted ….

Key Aspects for Your Research Results Section

For a good structure and organization of your research, keep in mind these aspects:

• Start your research results section by restating the purpose of your research, so that your readers can re-focus on core of your academic article
• Include helpful and quality tables, figures, graphs that can synthesize your research
• Make sure you include details about your data analysis and interpretation, as well as statistical significance tests
• Report the statistical insignificant research findings for your academic article’s credibility
• Use the past tense when describing to your research results
• Do not use vague terms and be as concise as possible when you are reporting your research findings
• Conclude your section with a short paragraph that summarizes your study’s key outcomes.

Which aspects do you focus on when writing your research results section?

This blog series focuses on useful academic writing tips. Next, we examine the Discussion and Conclusion section . Find our more on writing high-quality research papers

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Related policies and links, responsibilities of the publisher in the relationship with journal editors, general duties of publisher.

Organizing Academic Research Papers: 7. The Results

• Purpose of Guide
• Design Flaws to Avoid
• Glossary of Research Terms
• Narrowing a Topic Idea
• Broadening a Topic Idea
• Extending the Timeliness of a Topic Idea
• Academic Writing Style
• Choosing a Title
• Making an Outline
• Paragraph Development
• Executive Summary
• Background Information
• The Research Problem/Question
• Theoretical Framework
• Citation Tracking
• Content Alert Services
• Evaluating Sources
• Primary Sources
• Secondary Sources
• Tertiary Sources
• What Is Scholarly vs. Popular?
• Qualitative Methods
• Quantitative Methods
• Using Non-Textual Elements
• Limitations of the Study
• Common Grammar Mistakes
• Avoiding Plagiarism
• Footnotes or Endnotes?
• Further Readings
• Annotated Bibliography
• Dealing with Nervousness
• Using Visual Aids
• Grading Someone Else's Paper
• How to Manage Group Projects
• Multiple Book Review Essay
• Reviewing Collected Essays
• About Informed Consent
• Writing Field Notes
• Writing a Policy Memo
• Writing a Research Proposal
• Acknowledgements

The results section of the research paper is where you report the findings of your study based upon the information gathered as a result of the methodology [or methodologies] you applied. The results section should simply state the findings, without bias or interpretation, and arranged in a logical sequence. The results section should always be written in the past tense. A section describing results [a.k.a., "findings"] is particularly necessary if your paper includes data generated from your own research.

Importance of a Good Results Section

When formulating the results section, it's important to remember that the results of a study do not prove anything . Research results can only confirm or reject the research problem underpinning your study. However, the act of articulating the results helps you to understand the problem from within, to break it into pieces, and to view the research problem from various perspectives.

The page length of this section is set by the amount and types of data to be reported . Be concise, using non-textual elements, such as figures and tables, if appropriate, to present results more effectively. In deciding what data to describe in your results section, you must clearly distinguish material that would normally be included in a research paper from any raw data or other material that could be included as an appendix. In general, raw data should not be included in the main text of your paper unless requested to do so by your professor.

Avoid providing data that is not critical to answering the research question . The background information you described in the introduction section should provide the reader with any additional context or explanation needed to understand the results. A good rule is to always re-read the background section of your paper after you have written up your results to ensure that the reader has enough context to understand the results [and, later, how you interpreted the results in the discussion section of your paper].

Bates College; Burton, Neil et al. Doing Your Education Research Project . Los Angeles, CA: SAGE, 2008; Results . The Structure, Format, Content, and Style of a Journal-Style Scientific Paper. Department of Biology. Bates College.

Structure and Writing Style

I. Structure and Approach

For most research paper formats, there are two ways of presenting and organizing the results .

• Present the results followed by a short explanation of the findings . For example, you may have noticed an unusual correlation between two variables during the analysis of your findings. It is correct to point this out in the results section. However, speculating as to why this correlation exists, and offering a hypothesis about what may be happening, belongs in the discussion section of your paper.
• Present a section and then discuss it, before presenting the next section then discussing it, and so on . This is more common in longer papers because it helps the reader to better understand each finding. In this model, it can be helpful to provide a brief conclusion in the results section that ties each of the findings together and links to the discussion.

NOTE: The discussion section should generally follow the same format chosen in presenting and organizing the results.

II.  Content

In general, the content of your results section should include the following elements:

• An introductory context for understanding the results by restating the research problem that underpins the purpose of your study.
• A summary of your key findings arranged in a logical sequence that generally follows your methodology section.
• Inclusion of non-textual elements, such as, figures, charts, photos, maps, tables, etc. to further illustrate the findings, if appropriate.
• In the text, a systematic description of your results, highlighting for the reader observations that are most relevant to the topic under investigation [remember that not all results that emerge from the methodology that you used to gather the data may be relevant].
• Use of the past tense when refering to your results.
• The page length of your results section is guided by the amount and types of data to be reported. However, focus only on findings that are important and related to addressing the research problem.

Using Non-textual Elements

• Either place figures, tables, charts, etc. within the text of the result, or include them in the back of the report--do one or the other but never do both.
• In the text, refer to each non-textual element in numbered order [e.g.,  Table 1, Table 2; Chart 1, Chart 2; Map 1, Map 2].
• If you place non-textual elements at the end of the report, make sure they are clearly distinguished from any attached appendix materials, such as raw data.
• Regardless of placement, each non-textual element must be numbered consecutively and complete with caption [caption goes under the figure, table, chart, etc.]
• Each non-textual element must be titled, numbered consecutively, and complete with a heading [title with description goes above the figure, table, chart, etc.].
• In proofreading your results section, be sure that each non-textual element is sufficiently complete so that it could stand on its own, separate from the text.

III. Problems to Avoid

When writing the results section, avoid doing the following :

• Discussing or interpreting your results . Save all this for the next section of your paper, although where appropriate, you should compare or contrast specific results to those found in other studies [e.g., "Similar to Smith [1990], one of the findings of this study is the strong correlation between motivation and academic achievement...."].
• Reporting background information or attempting to explain your findings ; this should have been done in your Introduction section, but don't panic! Often the results of a study point to the need to provide additional background information or to explain the topic further, so don't think you did something wrong. Revise your introduction as needed.
• Ignoring negative results . If some of your results fail to support your hypothesis, do not ignore them. Document them, then state in your discussion section why you believe a negative result emerged from your study. Note that negative results, and how you handle them, often provides you with the opportunity to write a more engaging discussion section, therefore, don't be afraid to highlight them.
• Including raw data or intermediate calculations . Ask your professor if you need to include any raw data generated by your study, such as transcripts from interviews or data files. If raw data is to be included, place it in an appendix or set of appendices that are referred to in the text.
• Be as factual and concise as possible in reporting your findings . Do not use phrases that are vague or non-specific, such as, "appeared to be greater or lesser than..." or "demonstrates promising trends that...."
• Presenting the same data or repeating the same information more than once . If you feel the need to highlight something, you will have a chance to do that in the discussion section.
• Confusing figures with tables . Be sure to properly label any non-textual elements in your paper. If you are not sure, look up the term in a dictionary.

Burton, Neil et al. Doing Your Education Research Project . Los Angeles, CA: SAGE, 2008;  Caprette, David R. Writing Research Papers . Experimental Biosciences Resources. Rice University; Hancock, Dawson R. and Bob Algozzine. Doing Case Study Research: A Practical Guide for Beginning Researchers . 2nd ed. New York: Teachers College Press, 2011; Introduction to Nursing Research: Reporting Research Findings. Nursing Research: Open Access Nursing Research and Review Articles. (January 4, 2012); Reporting Research Findings. Wilder Research, in partnership with the Minnesota Department of Human Services. (February 2009); Results . The Structure, Format, Content, and Style of a Journal-Style Scientific Paper. Department of Biology. Bates College; Schafer, Mickey S. Writing the Results . Thesis Writing in the Sciences. Course Syllabus. University of Florida.

Writing Tip

Why Don't I Just Combine the Results Section with the Discussion Section?

It's not unusual to find articles in social science journals where the author(s) have combined a description of the findings from the study with a discussion about their implications. You could do this. However, if you are inexperienced writing research papers, consider creating two sections for each element in your paper as a way to better organize your thoughts and, by extension, your  paper. Think of the results section as the place where you report what your study found; think of the discussion section as the place where you interpret your data and answer the "so what?" question. As you become more skilled writing research papers, you may want to meld the results of your study with a discussion of its implications.

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How to clearly articulate results and construct tables and figures in a scientific paper?

The writing of the results section of a scientific paper is very important for the readers for clearly understanding of the study. This review summarizes the rules for writing the results section of a scientific paper and describes the use of tables and figures.

Introduction

Medical articles consist of review articles, case reports, and letters to the editor which are prepared with the intention of publishing in journals related to the medical discipline of the author. For an academician to be able to progress in carreer, and make his/her activities known in the academic environment, require preparation of the protocol of his/her academic research article, and acquiring sufficient information, and experience related to the composition of this article. In this review article, the information related to the writing of the ‘Results’ section, and use of tables, and figures will be presented to the attention of the readers.

Writing the ‘Results’ section

The ‘Results’ section is perhaps the most important part of a research article. In fact the authors will share the results of their research/study with their readers. Renown British biologist Thomas Henry Huxley (1825–1895) indicated his feelings as “The great tragedy of science: the slaying of a beautiful hypothesis by an ugly fact.” which emphasizes the importance of accurately, and impressively written results.

In essence results provide a response for the question” What is found in the research performed?”. Therefore, it is the most vital part of the article. As a priority, while drafting the ‘Results’ section of a manuscript one should not firstly write down methods in the ‘Material and Method’ section. The first sentence should give information about the number of patients who met the inclusion criteria, and thus enrolled in the study. [ 1 ] Besides information about the number of patients excluded from the study, and the reasons for exclusion is very important in that they will enlighten the readers, and reviewers who critically evaluate the manuscript, and also reflect the seriousness of the study. On the other hand, the results obtained should be recorded in chronological order, and without any comments. [ 2 ] In this section use of simple present tense is more appropriate. The findings should be expressed in brief, lucid, and explicable words. The writing style should not be boring for the reader. During writing process of a research article, a generally ill-conceived point is that positive, and significant findings are more important, attractive, and valuable, while negative, and insignificant findings are worthless, and less attractive. A scientific research is not performed to confirm a hypothesis, rather to test it. Not only positive, and significant results are worth writing, on the other hand negative or statistically insignificant result which support fallacy of a widely accepted opinion might be valuable. Therefore, all findings obtained during research should be inclıuded in the ‘Results’ section. [ 1 ]

While writing the ‘Results’ section, the sequence of results, tabulated data, and information which will be illustrated as figures should be definitively indicated. In indicating insignificant changes, do not use expressions as “decreased” or “increased”, these words should be reserved for significant changes. If results related to more than one parameter would be reported, it is appropriate to write the results under the subheading of its related parameter so as to facilitate reading, and comprehension of information. [ 2 ] Only data, and information concerning the study in question should be included in the ‘Results’ section. Results not mentioned in this section should not be included in the ‘Discussion’ and ‘Summary’ sections. Since the results obtained by the authors are cited in the ‘Results’ section, any reference should not be indicated in this section. [ 3 ]

In the ‘Results’ section, numerical expressions should be written in technically appropriate terms. The number of digits (1, 2 or 3 digits) to be written after a comma (in Turkish) or a point (in especially American English) should be determined The number of digits written after the punctuation marks should not be changed all throughout the text. Data should be expressed as mean/median ± standard deviation. Data as age, and scale scores should be indicated together with ranges of values. Absolute numerical value corresponding to a percentage must be also indicated. P values calculated in statistical analysis should be expressed in their absolute values. While writing p values of statistically significant data, instead of p<0.05 the actual level of significance should be recorded. If p value is smaller than 0.001, then it can be written as p <0.01. [ 2 ] While writing the ‘Results’ section, significant data which should be recalled by the readers must be indicated in the main text. It will be appropriate to indicate other demographic numerical details in tables or figures.

As an example elucidating the abovementioned topics a research paper written by the authors of this review article, and published in the Turkish Journal of Urology in the year 2007 (Türk Üroloji Dergisi 2007;33:18–23) is presented below:

“A total of 9 (56.2%) female, and 7 (43.8%) male patients with were included in this study. Mean age of all the patients was 44.3±13.8 (17–65) years, and mean dimensions of the adrenal mass was 4.5±3.4 (1–14) cm. Mean ages of the male, and female patients were 44.1 (30–65), and 42.4 (17–64) years, while mean diameters of adrenal masses were 3.2 (1–5), and 4.5 (1–14) cm (p age =0.963, p mass size =0.206). Surgical procedures were realized using transperitoneal approach through Chevron incision in 1 (6.2%), and retroperitoneal approach using flank incision with removal of the 11. rib in 15 (93.7%) patients. Right (n=6; 37.5%), and left (n=2; 12.5%) adrenalectomies were performed. Two (12.5%) patients underwent bilateral adrenalectomy in the same session because of clinical Cushing’s syndrome persisted despite transsphenoidal hipophysectomy. Mean operative time, and length of the hospital stay were 135 (65–190) min, and 3 (2–6) days, respectively. While resecting 11. rib during retroperitoneal adrenalectomy performed in 1 patient, pleura was perforated for nearly 1.5 cm. The perforated region was drained, and closed intraoperatively with 4/0 polyglyctan sutures. The patient did not develop postoperative pneumothorax. In none of the patients postoperative complications as pneumothorax, bleeding, prolonged drainage were seen. Results of histopathological analysis of the specimens retrieved at the end of the operation were summarized in Table 1 .” Table 1. Histopathological examination results of the patients Histopathological diagnosis Men n (%) Women n (%) Total n (%) Adrenal cortical adenoma 5 (31.3) 6 (37.6) 11 (68.8) Pheochromocytoma 1 (6.2) 1 (6.2) 2 (12.6) Ganglioneuroma 1 (6.2) - 1 (6.2) Myelolipoma - 1 (6.2) 1 (6.2) Adrenal carcinoma - 1 (6.2) 1 (6.2) Total 7 (43.7) 9 (56.2) 16 (100) Open in a separate window

Use of tables, and figures

To prevent the audience from getting bored while reading a scientific article, some of the data should be expressed in a visual format in graphics, and figures rather than crowded numerical values in the text. Peer-reviewers frequently look at tables, and figures. High quality tables, and figures increase the chance of acceptance of the manuscript for publication.

Number of tables in the manuscript should not exceed the number recommended by the editorial board of the journal. Data in the main text, and tables should not be repeated many times. Tables should be comprehensible, and a reader should be able to express an opinion about the results just at looking at the tables without reading the main text. Data included in tables should comply with those mentioned in the main text, and percentages in rows, and columns should be summed up accurately. Unit of each variable should be absolutely defined. Sampling size of each group should be absolutely indicated. Values should be expressed as values±standard error, range or 95% confidence interval. Tables should include precise p values, and level of significance as assessed with statistical analysis should be indicated in footnotes. [ 2 ] Use of abbreviations in tables should be avoided, if abbreviations are required they should be defined explicitly in the footnotes or legends of the tables. As a general rule, rows should be arranged as double-spaced Besides do not use pattern coloring for cells of rows, and columns. Values included in tables should be correctly approximated. [ 1 , 2 ]

As an example elucidating the abovementioned topics a research paper written by the authors of this review article, and published in the Turkish Journal of Urology in the year 2007 (Türk Üroloji Dergisi 2007;33:18–23).is shown in Table 1 .

Most of the readers priorly prefer to look at figures, and graphs rather than reading lots of pages. Selection of appropriate types of graphs for demonstration of data is a critical decision which requires artist’s meticulousness. As is the case with tables, graphs, and figures should also disploay information not provided in the text. Bar, line, and pie graphs, scatter plots, and histograms are some examples of graphs. In graphs, independent variables should be represented on the horizontal, and dependent variables on the vertical axis. Number of subjects in every subgroup should be indicated The labels on each axis should be easily understandable. [ 2 ] The label of the Y axis should be written vertically from bottom to top. The fundamental point in writing explanatory notes for graphs, and figures is to help the readers understand the contents of them without referring to the main text. Meanings of abbreviations, and acronyms used in the graphs, and figures should be provided in explanatory notes. In the explanatory notes striking data should be emphasized. Statistical tests used, levels of significance, sampling size, stains used for analyses, and magnification rate should be written in order to facilitate comprehension of the study procedures. [ 1 , 2 ]

Flow diagram can be utilized in the ‘Results’ section. This diagram facilitates comprehension of the results obtained at certain steps of monitorization during the research process. Flow diagram can be used either in the ‘Results’ or ‘Material and Method’ section. [ 2 , 3 ]

Histopathological analyses, surgical technique or radiological images which are considered to be more useful for the comprehension of the text by the readers can be visually displayed. Important findings should be marked on photos, and their definitions should be provided clearly in the explanatory legends. [ 1 ]

As an example elucidating the abovementioned issues, graphics, and flow diagram in the ‘Results’ section of a research paper written by the authors of this review article, and published in the World Journal of Urology in the year 2010 (World J Urol 2010;28:17–22.) are shown in Figures 1 , and ​ and2 2 .

a The mean SHIM scores of the groups before and after treatment. SHIM sexual health inventory for male. b The mean IPSS scores of the groups before and after treatment. IPSS international prostate symptom score

Flowchart showing patients’ progress during the study. SHIM sexual health inventory for male, IIEF international index of erectile function, IPSS international prostate symptom score, QoL quality of life, Q max maximum urinary flow rate. PRV post voiding residual urine volume

In conclusion, in line with the motto of the famous German physicist Albert Einstein (1879–1955). ‘If you are out to describe the truth, leave elegance to the tailor .’ results obtained in a scientific research article should be expressed accurately, and with a masterstroke of a tailor in compliance with certain rules which will ensure acceptability of the scientific manuscript by the editorial board of the journal, and also facilitate its intelligibility by the readers.

Research Voyage

Research Tips and Infromation

How to Write Results Section of Your Research Paper

Introduction

How to summarize the data preprocessing steps in the results section, how to summarize the research findings in the results section, common phrasal verbs used in results section, what are common mistakes observed in the results section, how long should a results section be of a research paper, should the results of a research paper be given in the introduction or in another section.

• What is the difference between the "discussion" and the "results" section of a research paper?

Does the summary be part of the result section in the research article?

Why do some scientific papers not include a ‘methods and results’ section, how do you introduce a results section, why do researchers need to avoid making speculations in the results section of a research paper.

The result section is the third major part of the research paper and it’s probably the most important part because it contains actual outcomes about your experiment. The other sections contain a plan, hope and interpretations but the result section is the actual truth of your study.

In the result section, one should aim to narrate his/her finding without trying to interpret or evaluate them. Basically, the result section explains any issues you faced during your data collection, the main results of the experiment and any other interesting trends in the data.

With the results, we want to convey our data in the most accessible way, so we usually use visual elements like graphs and tables to make it easier to understand. The facts, figures, and findings are to be presented in a logical manner leading to the hypothesis and following the sequence of the method section. Mention must be made for the negative results as it would substantiate the discussion section later on. Interpretation of the meaning of the results section is done in the discussion section .

How Results Section is Structured?

When structuring the results section, it is important that your information is presented in a logical order. 

Now, when it comes to the organization of the result section, as a generic rule

• Always start with textual content, not a Table or Figure
• Make sure you show the Tables and Figures after they are mentioned in the text
• Explain any missing data or problems you had while collecting the data.

The results section gives you the opportunity to:

• Summarize the  Data Preprocessing Steps

2. Report on the Findings 

3. Summarize the Research Findings

At the beginning of the result section, you can discuss how you have collected, transformed and analyzed your data. This step is usually known as data preprocessing.

The data collection step may involve collecting data from various hardware, software or internet sources.

If your research requires data cleaning, then explain the steps and procedures used for data cleaning. Here, the researchers can describe how they transformed data to facilitate analysis (e.g. converting data from one format to another format). If there was missing data, explain how you have substituted missing values and with what techniques you have substituted your data.

You can mention what software or statistical procedures you have used to analyze and interpret the data.  Demonstrate with the help of charts or tables the cleansed data ready to be used for getting results.   In a few research papers, you may find these steps appearing at the end of the method section. 

How to Present your Research Findings in Research Section?

Second, present your findings in a structured way (such as thematically or chronologically), bringing the readers’ attention to any important, interesting, or significant findings.

Be sure to include a combination of text and visuals. Data illustrations should not be used to substitute or replace text, but to enhance the narrative of your findings.  

Resultant data are to be presented either through text, figures, graphs or tables or in a combination of all of the best suited for leading to the hypothesis. Care should be taken to prevent any duplication of the text, figures, graphs, and tables. If any result is presented in figures or graphs, it need not be explained through text. Similarly, any data presented through the graph should not be repeated in the table.

Each table and graph should be clearly labelled and titled. Each different finding should be made in a separate sub-section under the proper sub-heading following the sequence adopted in Method Section.

If you are not comfortable with data analysis then you can take professional services for research data analysis .

Figures 

 Identify and list the figures which are relevant to your results. For example, if you are working on the problem statement of ” Identifying the pathological issues with pomegranate fruits”, then you can add the figures of pomegranate fruits with good quality and bad quality along with their stage of infection. If you are working on pomegranate cultivar-related issues, put the figures of pomegranate fruits belonging to different cultivars. 

The key takeaway here is not to add any figures which may not directly contribute to results. These diagrams may include generic block diagrams, and images conveying generic information like farm fields, plantations etc.

While putting the figures, as much as possible use grayscale images as many users take the photocopies in black and white mode. In certain scenarios you are 

 In the case of figures, the captions should come below, called Fig. 1, Fig. 2 and so on. 

You can visit my article on The Power of Images in Research Papers: How They Enhance the Quality of Your Paper? . This article will help you how images or figures enhances the possibility of selection of your paper to top quality journals and conferences.

Tables are good for showing the exact values or showing much different information in one place. Graphs are good for showing overall trends and are much easier to understand quickly. It also depends on your data.

Tables are labelled at the top as Table 1,  Table 2 and so on.  Every table must have a caption. It’s good if one can put independent variable conditions on the left side vertically, and the things you have measured horizontally so one can easily compare the measurements across the categories. But you need to decide for each table you make, what is easiest to understand, and what fits on the paper.

Visit article on Best Practices for Designing and Formatting Tables in Research Papers for further details on proper representation of tables at proper places.

You can use various types of graphs in your results like a line graph, bar graph, scatter plot, a line graph with colours, a box with whiskers plot and a histogram.

In general, continuous variables like temperature, growth, age, and time can be better displayed in a line graph on a scatter plot or maybe on histograms.

If you have comparative data that you would like to represent through a chart then a bar chart would be the best option. This type of chart is one of the more familiar options as it is easy to interpret.

These charts are useful for displaying data that is classified into nominal or ordinal categories. In any case, you need to decide which is the best option for each particular example you have,  but never put a graph and a table with the same data in your paper.

In the case of graphs, the captions should come below, called Fig. 1, Fig. 2 and so on. 

A limited number of professional tools provide you the chance to add some life to your graphs, charts, and figures and present your data in a way that will astound your audience as much as your astounding results.

My article on Maximizing the Impact of Your Research Paper with Graphs and Charts will help you in drawing eye catching and informative graphs and charts for your research paper.

The results section should include a closing paragraph that clearly summarizes the key findings of the study. This paves the way for the discussion section of the research paper, wherein the results are interpreted and put in conversation with existing literature.

Any unusual correlation observed between variables should be noted in the result section. But any speculation about the reason for such an unusual correlation should be avoided. Such speculations are the domains of the discussion section.

Comparisons between samples or controls are to be clearly defined by specifically mentioning the common quality and the degree of difference between the comparable samples or controls. Results should always be presented in the past tense.

Common academic phrases that can be used in the results section of a paper or research article. I have included a table with examples to illustrate how these phrases might be used:

Let’s look at some of the common mistakes which can be observed in the result section.

• One should not include raw data which are not directly related to your objectives. Readers will not be able to interpret your intentions and may unnecessarily collect unwanted data while replicating your experiments.
• Do not just tell the readers to look at the Table and Figure and figure it out by themselves, e.g “The results are shown in the following Tables and Graphs”.
• Do not give too much explanation about Figures and Tables.

“An Optimized Fuzzy Based Short Term Object Motion Prediction for Real-Life Robot Navigation Environment”  ( Paper Link )

Object motions with different motion patterns are generated by a simulator in different directions to generate the initial rule base. The rules generated are clustered based on the direction of the motion pattern into the directional space clusters. Table 1 shows the number of rules that remained in each directional space after removing inconsistencies and redundancies.

Our predictor algorithm is tested for a real-life benchmark dataset (EC Funded CAVIAR project/IST 2001 37540) to check for relative error. The data set consists of different human motion patterns observed at INRIA Lab at Grenoble, France and Shop Centre. These motion patterns consist of frames captured at 25 frames/second. A typical scenario of the INRIA Lab and the Shop Centre is shown in the Figure below.

                                                      Fig.1: A typical scenario of the INRIA Lab and the Shop Centre

For each test case, the average response time is calculated to find its suitability for a real-life environment. The prediction algorithm is tested by processing the frame data of moving human patterns stored in the database at intervals of 50 frames (02 Seconds).

The navigation environment is presented in the form of a Prediction graph where the x-axis represents the Range parameter and the y-axis represents the Angle parameter. The predicted Angle and Range values are compared with actual values obtained from the real-life environment.

The performance of the predictor is tested when more than one object is sensed by the sensor. The tests are carried out assuming at most 6-8 objects can be visible and can affect the decisions to be made regarding robot traversal.

The results section is an essential component of any research paper, as it provides readers with a detailed understanding of the study’s findings. In this blog post, we discussed three important steps for writing a results section: summarizing the data preprocessing steps, reporting on the findings, and summarizing the research findings.

Firstly, summarizing the data preprocessing steps is crucial in the results section, as it provides readers with an understanding of how the raw data was processed and transformed. This step includes data cleaning, data transformation, and data reduction techniques. By summarizing the data preprocessing steps, readers can understand how the data was prepared for analysis, which is critical for interpreting the study’s findings accurately.

Secondly, reporting on the findings is an important step in the results section. It involves presenting the study’s results in a clear and concise manner, using tables, graphs, and statistical analyses where necessary. This step should be focused on answering the research question or hypothesis and should present the findings in a way that is easily understood by the reader. Reporting on the findings can also include providing detailed interpretations of the results, as well as any potential limitations of the study.

Finally, summarizing the research findings is crucial in the results section, as it provides readers with a concise summary of the study’s main results and conclusions. This step should be written in a clear and straightforward manner, highlighting the most important findings and explaining their significance. Additionally, it should relate the study’s findings to the research question or hypothesis and provide a conclusion that is well-supported by the results.

Overall, the results section of a research paper is a critical component that requires careful attention to detail. By following the guidelines discussed in this blog post, researchers can present their findings in a clear and concise manner, helping readers to understand the research process and the resulting conclusions.

Frequently Asked Questions

An IMRaD paper format suggests around 35% of the text should be dedicated to the results and discussion section. For a research paper of length 10 pages, the results and discussion section should occupy 3-4 pages.

The results of a research paper should be given in a separate section. However, the highlights of the results can be discussed in the introduction section.

What is the difference between the “discussion” and the “results” section of a research paper?

The results section only depicts the results obtained by implementing the methodology used. The results will be in the form of figures, tables, charts or graphs. The discussion section elaborates the analysis of the results obtained in the results section.

The summary can be part of the results section of a research paper. However, the results obtained can be summarized in the form of a table in results section of a research paper.

Survey papers and papers which are focussed on theoretical proofs do not involve separate methods and results sections.

The results section is introduced by the data collection steps and the setting up of equipment in different scenarios for obtaining the results.

Making speculations in the results section may lead to wrong interpretations by the researcher who is planning to replicate the methodology used for obtaining the results. This may further lead to wrong comparative analysis.

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How to Write the Results Section of A Research Paper

How to Evaluate Research

Writing a research paper can be a daunting experience. Whether it is your first college paper, the very first lab report you have done in high school or something you are hoping to have published, it can feel like a lot to master. Clearly, when you are doing an experiment, you want to list the results of the experiment or the research.

How do you structure the results of an experiment, and how do you communicate the results of your research? By writing clearly and answering any questions you raised earlier in your paper, you can be sure that your results section will be easily comprehensible and will bring your paper to a strong conclusion.

What Is a Research Paper?

Defined broadly, a research paper is any sort of written account of work that you undertook in order to learn more about a specific topic or set of conditions. Whether you read books in a library about ancient Egyptian hieroglyphics or wanted to conduct an experiment to see whether the sun can melt pennies, the written account of this exploration can be termed a "research paper."

In some situations, a research paper is also called a "lab report." This is usually the case when the investigation in question pertains to an experiment that was conducted in lab-like conditions. Either way, it essentially functions as a research paper.

Most research papers begin as a school assignment. While people in the academic or scientific fields and individuals who are interested in pursuing topics independently may take on research papers as a work assignment, most people's first research paper takes place while in school. Whatever you are researching, begin to think about it as an investigation. That can help you to set the paper up for a results section that will reveal what you have learned.

How to Imagine Research as an Investigation

What questions do you have about your topic? How can you get curious about the subject if it's something you at first think is boring? The key to writing the results of the research is having a question to investigate in the first place. If you need to do a research paper on something having to do with Italy, get curious about Italy. What about Italy is interesting to you? If you can choose the topic yourself, try to find something about the topic that engages you or makes you think and ask questions.

If the topic of your paper has something to do with science, and you really dislike science, this is an opportunity to get curious. Do you need to do an experiment demonstrating that sponges soak up water? Why might that be interesting or important? These are great places to begin.

Getting curious about the topic you are researching is critical. Getting curious can help you connect to the research and can make the experience more unique and interesting than it would ever have been otherwise. It can also make your research paper stand out as being distinctly yours. A lot of people can write a lazy summary of something, but only you can bring your individuality to the proceedings and use it as a lens to guide your research.

How Do I Structure a Research Paper?

The structure of a research paper or a lab report on an experiment is critical. Because research is scientific, you want to be as meticulous as possible so that all the necessary information is conveyed. You will want to begin your paper by explaining why the topic you are pursuing is worth researching. Explain why it interests you.

Explain what you hope to gain from conducting this research or this experiment. Tell the reader what your hypothesis is and explain why you've come to believe this to be true. Next, lay out your strategy or methodology. What are you going to do in this paper?

How are you planning to discover whether or not your hypothesis is correct? Explain your plan for figuring out whether your hypothesis is correct or explain the way that you decided to research the topic. Offer a list of resources that you consulted. Make it clear why you chose to go about the research in the way that you did.

What Sections Does a Research Paper Have?

Typically, a research paper has five major parts:

• Introduction
• Review of literature

The introduction is the section of the research paper where you introduce the question you are looking to investigate and explain why you are doing so. If there are statistics or quotes or other writing you have found that lends itself to supporting your investigation, you can introduce it here. For example, if you are writing about whether or not the Loch Ness monster is real, you can share quotes or statistics about the number of times that people have said they've seen it.

The next section of the paper, the review of literature, should be a synthesis of the research that you've done thus far that has informed your hypothesis. Gather and summarize the information that has led you to this point and make it clear that going into your research, you were aware of this literature, and you used it to develop your methods. In the methods section, you will begin to detail the way that you went about conducting your experiment or conducting your research.

What Is the Results and Discussion Section?

The results and discussion pieces are the two most critical parts of the research paper. This tells us in factual terms exactly what you discovered. The results section is not the place for analysis. The results section is not the place for narrative discussion or emotion either. The results section is only for the results of the research.

The results section provides the facts about what you discovered in the course of your research or experiment. The discussion section is where you can get analytical or reflective about exactly what you have discovered. This is the place where you can tell us what the results mean. Does it mean that your hypothesis was correct, or does it mean that you need to do further research or experiments before you can come to a definitive conclusion about this issue?

How Do You Write the Results of the Research?

In the results section of your paper, you need to list what you have discovered. If your experiment confirmed your hypothesis, save the discussion about that for the discussion section. The results section should simply be hard facts written in the passive voice.

Many students get confused between the active and passive voices when writing a research paper. Unlike the rest of the paper, the results section should be written in the passive voice in order to draw attention to the action and not to the person performing the action.

Once you have clearly defined what your experiment or research has yielded, you can move on to the discussion section.

How Do You Write the Discussion Section?

The discussion section is where you can analyze and make inferences about your research or your experiment. Tell the reader what it means to you now that your hypothesis was confirmed or proved to be incorrect. Moreover, what does it mean for the future of this research?

If your hypothesis was proven to be correct, can that be brought to bear on any other research or hypotheses? If your experiment was wholly inconclusive, can you say why that was? What went wrong? Is it something that could be corrected?

In What Tone Should You Write a Research Paper?

Many people who read research papers, including teachers, editors and professors, hate the passive voice. They consider the passive voice to be an example of poor writing. Many colleges have writing centers where they can help students to improve the quality of their writing, and one of the tasks they face most often is getting students out of the passive voice.

A sentence written in the active voice shows the subject acting on a direct object. "David mailed the package" is an example of a sentence in the active voice. On the contrary, a sentence written in the passive voice shows the object being acted on by a verb. An example is: "The package was mailed by David."

While technically the passive voice is not grammatically incorrect – and in some cases, given literary license, it is necessary – the passive voice is considered an example of less-than-ideal writing. Active and passive voice can change the quality of a piece of writing, particularly academic writing. If you find that you have written any sentences in your research paper in the passive voice anywhere other than in the results section, it is a good rule of thumb to go back and do a passive-to-active conversion.

What Part of My Research Paper Should Be in the Passive Voice?

The passive voice is not gramatically incorrect. It is used correctly when the intention of the sentence is to draw attention to the action and not the person performing the action. This is why when you write the results section of the research paper, you will want to employ the passive voice.

The passive voice tells us that the results of the experiment or the research are more important in this instance than the way that the research was carried out. Said another way, the results section is not about you. It is not about the way that you performed the research or the way that you set up the experiment. It is purely and simply about the results.

What Are Some Active and Passive Voice Rules?

There are some tips to make sure that you are writing in the active voice. However, keep in mind that in some cases, such as in the results section, you will need to use the passive voice. After all, if you talk about something that happened in the past or that happened to someone, you will need to use words like "was" and "had."

Sometimes, it can be effective to make something passive. For example, the phrase "the city of Rome was attacked by invaders" shows that the subject of the sentence is Rome and that is the thing that is being acted on, even if it is in a passive sense. "Invaders attacked the city of Rome" turns the focus to the invaders. A reader may well expect the following sentence to be about the invaders.

Pay close attention to the subject in the sentence. Is the subject the one carrying out the activities described in the verb? If not, go back and fix it.

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• Towson University: Active/Passive Voice
• College of Western Idaho: What Are the Differences Between Active and Passive Voice?
• University of Wisconsin-Madison Writing Center: Use the Active Voice
• A Research Guide For Students: How to Write a Research Paper
• American Psychological Association
• Try starting this section by creating an outline of each hypothesis or research question followed by each statistical test you will use for it.
• Where relevant, state the effect size of the particular statistical test.
• Only include statistical tests that are relevant to your particular hypotheses or research questions. Excessive irrelevant statistical tests detract from the big picture and make it difficult for the reader to follow.
• Do not include specific calculations used to determine the statistic.

Ashley Friedman is a freelance writer with experience writing about education for a variety of organizations and educational institutions as well as online media sites. She has written for Pearson Education, The University of Miami, The New York City Teaching Fellows, New Visions for Public Schools, and a number of independent secondary schools. She lives in Los Angeles.

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Writing a Results Section

The next stage of any research paper: writing the results section, announcing your findings to the world.

This article is a part of the guide:

• Outline Examples
• Example of a Paper
• Write a Hypothesis
• Introduction

Browse Full Outline

• 1 Write a Research Paper
• 2 Writing a Paper
• 3.1 Write an Outline
• 3.2 Outline Examples
• 4.1 Thesis Statement
• 4.2 Write a Hypothesis
• 5.2 Abstract
• 5.3 Introduction
• 5.4 Methods
• 5.5 Results
• 5.6 Discussion
• 5.7 Conclusion
• 5.8 Bibliography
• 6.1 Table of Contents
• 6.2 Acknowledgements
• 6.3 Appendix
• 7.1 In Text Citations
• 7.2 Footnotes
• 7.3.1 Floating Blocks
• 7.4 Example of a Paper
• 7.5 Example of a Paper 2
• 7.6.1 Citations
• 7.7.1 Writing Style
• 7.7.2 Citations
• 8.1.1 Sham Peer Review
• 8.1.2 Advantages
• 8.1.3 Disadvantages
• 8.2 Publication Bias
• 8.3.1 Journal Rejection
• 9.1 Article Writing
• 9.2 Ideas for Topics

In theory, this is the easiest part to write, because it is a straightforward commentary of exactly what you observed and found. In reality, it can be a little tricky, because it is very easy to include too much information and bury the important findings.

Too Much Information?

The results section is not for interpreting the results in any way; that belongs strictly in the discussion section. You should aim to narrate your findings without trying to interpret or evaluate them, other than to provide a link to the discussion section.

For example, you may have noticed an unusual correlation between two variables during the analysis of your results. It is correct to point this out in the results section.

Speculating why this correlation is happening, and postulating about what may be happening, belongs in the discussion section .

It is very easy to put too much information into the results section and obscure your findings underneath reams of irrelevance.

If you make a table of your findings, you do not need to insert a graph highlighting the same data. If you have a table of results, refer to it in the text, but do not repeat the figures - duplicate information will be penalized.

One common way of getting around this is to be less specific in the text. For example, if the result in table one shows 23.9%, you could write….

Table One shows that almost a quarter of…..

Tips for Writing a Results Section

Perhaps the best way to use the results section is to show the most relevant information in the graphs, figures and tables.

The text, conversely, is used to direct the reader to those, also clarifying any unclear points. The text should also act as a link to the discussion section, highlighting any correlations and findings and leaving plenty of open questions.

For most research paper formats , there are two ways of presenting and organizing the results. The first method is to present the results and add a short discussion explaining them at the end, before leading into the discussion proper.

This is very common where the research paper is straightforward, and provides continuity. The other way is to present a section and then discuss it, before presenting the next section with a short discussion. This is common in longer papers, and your discussion part of the paper will generally follow the same structure.

Be sure to include negative results - writing a results section without them not only invalidate the paper, but it is extremely bad science. The negative results, and how you handle them, often gives you the makings of a great discussion section, so do not be afraid to highlight them.

Using an Appendix to Streamline Writing the Results Section

If you condense your raw data down, there is no need to include the initial findings in the results, because this will simply confuse the reader.

If you are in doubt about how much to include, you can always insert your raw data into the appendix section, allowing others to follow your calculations from the start. This is especially useful if you have used many statistical manipulations, so that people can check your calculations and ensure that you have not made any mistakes.

In the age of spreadsheets, where the computer program prepares all of the calculations for you, this is becoming less common, although you should specify the program that you used and the version. On that note, it is unnecessary show your working - assume that the reader understands what a Chi Squared test, or a Students t-test is, and can perform it themselves.

Once you have a streamlined and informative results section, you can move onto the discussion section, where you begin to elaborate your findings.

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Martyn Shuttleworth (Mar 2, 2009). Writing a Results Section. Retrieved Jun 30, 2024 from Explorable.com: https://explorable.com/writing-a-results-section

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Rubber-based agroforestry systems associated with food crops: a solution for sustainable rubber and food production.

1. Introduction

2. materials and methods, 3.1. structure of the library, 3.2. evolution of the number of research studies related to rubber-based agroforestry, 3.3. number of journal articles on rubber per country, 3.4. analysis of intercrop types in rubber agroforestry systems, 3.5. analysis of intercrop products and level of usage in rubber-based agroforestry systems, 3.6. analysis of the disciplines studied in the journal articles, 4. discussion, 4.1. evolution of research on rubber agroforestry, 4.2. breeding food crops for agroforestry systems.

4.3. Crop Management for Food Crops in Agroforestry

4.4. tentative recommendation for rass with food crops, 5. conclusions, supplementary materials, author contributions, data availability statement, acknowledgments, conflicts of interest.

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Cahyo, A.N.; Dong, Y.; Taryono; Nugraha, Y.; Junaidi; Sahuri; Penot, E.; Hairmansis, A.; Purwestri, Y.A.; Akbar, A.; et al. Rubber-Based Agroforestry Systems Associated with Food Crops: A Solution for Sustainable Rubber and Food Production? Agriculture 2024 , 14 , 1038. https://doi.org/10.3390/agriculture14071038

Cahyo AN, Dong Y, Taryono, Nugraha Y, Junaidi, Sahuri, Penot E, Hairmansis A, Purwestri YA, Akbar A, et al. Rubber-Based Agroforestry Systems Associated with Food Crops: A Solution for Sustainable Rubber and Food Production? Agriculture . 2024; 14(7):1038. https://doi.org/10.3390/agriculture14071038

Cahyo, Andi Nur, Ying Dong, Taryono, Yudhistira Nugraha, Junaidi, Sahuri, Eric Penot, Aris Hairmansis, Yekti Asih Purwestri, Andrea Akbar, and et al. 2024. "Rubber-Based Agroforestry Systems Associated with Food Crops: A Solution for Sustainable Rubber and Food Production?" Agriculture 14, no. 7: 1038. https://doi.org/10.3390/agriculture14071038

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Tariff Tracker: Tracking the Economic Impact of the Trump-Biden Tariffs

Key finding.

• The Trump administration imposed nearly $80 billion worth of new taxes on Americans by levying tariffs on thousands of products valued at approximately $380 billion in 2018 and 2019, amounting to one of the largest tax A tax is a mandatory payment or charge collected by local, state, and national governments from individuals or businesses to cover the costs of general government services, goods, and activities. increases in decades.
• The Biden administration has kept most of the Trump administration tariffs in place, and in May 2024, announced tariff Tariffs are taxes imposed by one country on goods or services imported from another country. Tariffs are trade barriers that raise prices and reduce available quantities of goods and services for U.S. businesses and consumers. hikes on an additional $18 billion of Chinese goods, including semiconductors and electric vehicles, for an additional tax increase of $3.6 billion.
• We estimate the Trump-Biden tariffs will reduce long-run GDP by 0.2 percent, the capital stock by 0.1 percent, and employment by 142,000 full-time equivalent jobs.
• Altogether, the trade war policies currently in place add up to $79 billion in tariffs based on trade levels at the time of tariff implementation and excluding behavioral and dynamic effects.
• Before accounting for behavioral effects, the $79 billion in higher tariffs amounts to an average annual tax increase on US households of $625. Based on actual revenue collections data, trade war tariffs have directly increased tax collections by $200 to $300 annually per US household, on average. Both estimates understate the cost to US households because they do not factor in the lost output, lower incomes, and loss in consumer choice the tariffs have caused.
• Candidate Trump has proposed significant tariff hikes as part of his presidential campaign; we estimate that if imposed, his proposed tariff increases would hike taxes by another $524 billion annually and shrink GDP by at least 0.8 percent, the capital stock by 0.7 percent, and employment by 684,000 full-time equivalent jobs. Our estimates do not capture the effects of retaliation, nor the additional harms that would stem from starting a global trade war.
• Academic and governmental studies find the Trump-Biden tariffs have raised prices and reduced output and employment, producing a net negative impact on the US economy.

The Trade War Timeline

The Trump administration imposed several rounds of tariffs on steel, aluminum, washing machines, solar panels, and goods from China, affecting more than $380 billion worth of trade at the time of implementation and amounting to a tax increase of nearly $80 billion. The Biden administration has maintained most tariffs, except for the suspension of certain tariffs on imports from the European Union , the replacement of tariffs with tariff-rate quotas (TRQs) on steel and aluminum from the European Union and United Kingdom and imports of steel from Japan , and the expiration of the tariffs on washing machines after a two-year extension. In May 2024, the Biden administration announced additional tariffs on $18 billion of Chinese goods for a tax increase of $3.6 billion.

Altogether, the trade war policies currently in place add up to $79 billion in tariffs based on trade levels at the time of tariff implementation. Note the total revenue generated will be less than our static estimate because tariffs reduce the volume of imports and are subject to evasion and avoidance (which directly lowers tariff revenues) and they reduce real income (which lowers other tax revenues).

Section 232, Steel and Aluminum

In March 2018 , President Trump announced the administration would impose a 25 percent tariff on imported steel and a 10 percent tariff on imported aluminum. The value of imported steel totaled $29.4 billion and the value of imported aluminum totaled $17.6 billion in 2018. Based on 2018 levels, the steel tariffs would have amounted to $9 billion and the aluminum tariffs to $1.8 billion. Several countries, however, have been excluded from the tariffs.

In early 2018 , the US reached agreements to permanently exclude Australia from steel and aluminum tariffs, use quotas for steel imports from Brazil and South Korea , and use quotas for steel and aluminum imports from Argentina.

In May 2019 , President Trump announced that the US was lifting tariffs on steel and aluminum from Canada and Mexico .

In 2020 , President Trump expanded the scope of steel and aluminum tariffs to cover certain derivative products, totaling approximately $0.8 billion based on 2018 import levels.

In August 2020 , President Trump announced that the US was reimposing tariffs on aluminum imports from Canada . The US imported approximately $2.5 billion worth of non-alloyed unwrought aluminum, resulting in a $0.25 billion tax increase. About a month later, the US eliminated the 10 percent tariff on Canadian aluminum that had just been reimposed.

In 2021 and 2022 , the Biden administration reached deals to replace certain steel and aluminum tariffs with tariff rate quota systems, whereby certain levels of imports will not face tariffs, but imports above the thresholds will. TRQs for the European Union took effect on January 1, 2022; TRQs for Japan took effect on April 1, 2022; and TRQs for the UK took effect on June 1, 2022. Though the agreements on steel and aluminum tariffs will reduce the cost of tariffs paid by some US businesses, a quota system similarly leads to higher prices, and further, retaining tariffs at the margin continues the negative economic impact of the previous tariff policy.

Tariffs on steel, aluminum, and derivative goods currently account for $2.7 billion of the $79 billion in tariffs , based on initial import values. Current retaliation against Section 232 steel and aluminum tariffs targets more than $6 billion worth of American products for an estimated total tax of approximately $1.6 billion.

Section 301, Chinese Products

Under the Trump administration, the United States Trade Representative began an investigation of China in August 2017, which culminated in a March 2018 report that found China was conducting unfair trade practices.

In March 2018, President Trump announced tariffs on up to $60 billion of imports from China. The administration soon published a list of about $50 billion worth of Chinese products to be subject to a new 25 percent tariff. The first tariffs began July 6, 2018, on $34 billion worth of Chinese imports, while tariffs on the remaining $16 billion went into effect August 23, 2018. These tariffs amount to a $12.5 billion tax increase.

In September 2018, the Trump administration imposed another round of Section 301 tariffs—10 percent on $200 billion worth of goods from China, amounting to a $20 billion tax increase.

In May 2019, the 10 percent tariffs increased to 25 percent, amounting to a $30 billion increase. That increase had been scheduled to take effect beginning in January 2019, but was delayed .

In August 2019, the Trump administration announced plans to impose a 10 percent tariff on approximately $300 billion worth of additional Chinese goods beginning on September 1, 2019, but soon followed with an announcement of schedule changes and certain exemptions.

In September 2019, the Trump administration imposed “List 4a,” a 10 percent tariff on $112 billion of imports, an $11 billion tax increase. They announced plans for tariffs on the remaining $160 billion to take effect on December 15, 2019.

In August 2019, the Trump administration decided that 4a tariffs would be 15 percent rather than the previously announced 10 percent, a $5.6 billion tax increase.

In December 2019 , the administration reached a “Phase One” trade deal with China and agreed to postpone indefinitely the stage 4b tariffs of 15 percent on approximately $160 billion worth of goods that were scheduled to take effect December 15 and to reduce the stage 4a tariffs from 15 percent to 7.5 percent in January 2020, reducing tariff revenues by $8.4 billion.

In May 2024, the Biden administration published its required statutory review of the Section 301 tariffs, deciding to retain them and impose higher rates on $18 billion worth of goods. The new tariff rates range from 25 to 100 percent on semiconductors, steel and aluminum products, electric vehicles, batteries and battery parts, natural graphite and other critical materials, medical goods, magnets, cranes, and solar cells. Some of the tariff increases go into effect immediately, while others are scheduled for 2025 or 2026. Based on 2023 import values, the increases will add $3.6 billion in new taxes.

Section 301 tariffs on China currently account for $77 billion of the $79 billion in tariffs , based on initial import values. China has responded to the United States’ Section 301 tariffs with several rounds of tariffs on more than $106 billion worth of US goods , for an estimated tax of nearly $11.6 billion.

WTO Dispute, European Union

In October 2019 , the United States won a nearly 15-year-long World Trade Organization (WTO) dispute against the European Union. The WTO ruling authorized the United States to impose tariffs of up to 100 percent on $7.5 billion worth of EU goods. Beginning October 18, 2019, tariffs of 10 percent were to be applied on aircraft and 25 percent on agricultural and other products.

In summer 2021, the Biden administration reached an agreement to suspend the tariffs on the European Union for five years .

Section 201, Solar Panels and Washing Machines

In January 2018 , the Trump administration announced it would begin imposing tariffs on washing machine imports for three years and solar cell and module imports for four years as the result of a Section 201 investigation.

In 2021 , the Trump administration extended the washing machine tariffs for two years through February 2023, and they have now expired .

In 2022 , the Biden administration extended the solar panel tariffs for four years , though later provided temporary two-year exemptions for imports from four Southeast Asian nations beginning in 2022 , which account for a significant share of solar panel imports.

In 2024, the Biden administration removed separate exemptions for bifacial solar panels from the Section 201 tariffs. Additionally, the temporary two-year exemptions expired and the Biden administration is further investigating solar panel imports from the four Southeast Asian nations for additional tariffs.

We estimate the solar cell and module tariffs amounted to a $0.2 billion tax increase based on 2018 import values and quantities, while the washing machine tariffs amounted to a $0.4 billion tax increase based on 2018 import values and quantities.

We exclude the tariffs from our tariff totals given the broad exemptions and small magnitudes.

Tariff Revenue Collections under the Trump-Biden Tariffs

As of March 2024, the trade war tariffs have generated more than $233 billion of higher taxes collected for the US government from US consumers. Of that total, $89 billion, or about 38 percent, was collected during the Trump administration, while the remaining $144 billion, or about 62 percent, has been collected during the Biden administration.

Before accounting for behavioral effects, the $79 billion in higher tariffs amount to an average annual tax increase on US households of $625. Based on actual revenue collections data, trade war tariffs have directly increased tax collections by $200 to $300 annually per US household, on average. The actual cost to households is higher than both the $600 estimate before behavioral effects and the $200 to $300 after, because neither accounts for lower incomes as tariffs shrink output, nor the loss in consumer choice as people switch to alternatives that do not face tariffs.

Economic Effects of Imposed and Retaliatory Tariffs

Using the Tax Foundation’s General Equilibrium Model, we estimate the Trump-Biden Section 301 and Section 232 tariffs will reduce long-run GDP by 0.2 percent, the capital stock by 0.1 percent, and hours worked by 142,000 full-time equivalent jobs. The reason tariffs have no impact on pre-tax wages in our estimates is that, in the long run, the capital stock shrinks in proportion to the reduction in hours worked, so that the capital-to-labor ratio, and thus the level of wages, remains unchanged. Removing the tariffs would boost GDP and employment, as Tax Foundation estimates have shown for the Section 232 steel and aluminum tariffs.

We estimate the retaliatory tariffs stemming from Section 232 and Section 301 actions total approximately $13.2 billion in tariff revenues. Retaliatory tariffs are imposed by foreign governments on their country’s importers. While they are not direct taxes on US exports, they raise the after-tax price of US goods in foreign jurisdictions, making them less competitively priced in foreign markets. We estimate the retaliatory tariffs will reduce US GDP and the capital stock by less than 0.05 percent and reduce full-time employment by 27,000 full-time equivalent jobs. Unlike the tariffs imposed by the United States, which raise federal revenue, tariffs imposed by foreign jurisdictions raise no revenue for the US but result in lower US output.

Trade Volumes since Tariffs Were Imposed

Since the tariffs were imposed, imports of affected goods have fallen, even before the onset of the COVID-19 pandemic. Some of the biggest drops are the result of decreased trade with China, as affected imports decreased significantly after the tariffs and still remain below their pre-trade war levels. Even though trade with China fell after the imposition of tariffs, it did not fundamentally alter the overall balance of trade, as the reduction in trade with China was diverted to increased trade with other countries .

Economic Effects of Proposed Tariffs

Tariffs have become a flashpoint in the 2024 presidential campaign as candidate Trump has proposed a new 10 percent universal tariff on all imports and a 60 percent tariff on all imports from China, as well as potentially higher tariffs on EVs from China or across the board.

In 2023, goods imports totaled $3.1 trillion and imports from China totaled $421.4 billion. With no behavioral effects, the universal tariff would raise taxes by $311 billion, while separately lifting the average tariff rate on Chinese goods to 60 percent would raise about $213 billion. Actual revenue raised would be significantly lower because of avoidance and evasion, falling imports, and lower incomes resulting in lower payroll and income tax revenues.

We estimate the proposed tariffs would reduce long-run GDP by 0.8 percent, the capital stock by 0.7 percent, and hours worked by 684,000 full-time equivalent jobs. The reason tariffs have no impact on pre-tax wages in our estimates is that, in the long run, the capital stock shrinks in proportion to the reduction in hours worked, so that the capital-to-labor ratio, and thus the level of wages, remains unchanged.

Tariffs Raise Prices and Reduce Economic Growth

Economists generally agree free trade increases the level of economic output and income, while conversely, trade barriers reduce economic output and income. Historical evidence shows tariffs raise prices and reduce available quantities of goods and services for US businesses and consumers, which results in lower income, reduced employment, and lower economic output.

Tariffs could reduce US output through a few channels. One possibility is a tariff may be passed on to producers and consumers in the form of higher prices. Tariffs can raise the cost of parts and materials, which would raise the price of goods using those inputs and reduce private sector output. This would result in lower incomes for both owners of capital and workers. Similarly, higher consumer prices due to tariffs would reduce the after-tax value of both labor and capital income. Because higher prices would reduce the return to labor and capital, they would incentivize Americans to work and invest less, leading to lower output.

Alternatively, the US dollar may appreciate in response to tariffs, offsetting the potential price increase for US consumers. The more valuable dollar, however, would make it more difficult for exporters to sell their goods on the global market, resulting in lower revenues for exporters. This would also result in lower US output and incomes for both workers and owners of capital, reducing incentives for work and investment and leading to a smaller economy.

Many economists have evaluated the consequences of the trade war tariffs on the American economy, with results suggesting the tariffs have raised prices and lowered economic output and employment since the start of the trade war in 2018.

• A February 2018 analysis by economists Kadee Russ and Lydia Cox found that steel‐​consuming jobs outnumber steel‐​producing jobs 80 to 1, indicating greater job losses from steel tariffs than job gains.
• A March 2018 Chicago Booth survey of 43 economic experts revealed that 0 percent thought a US tariff on steel and aluminum would improve Americans’ welfare.
• An August 2018 analysis from economists at the Federal Reserve Bank of New York warned the Trump administration’s intent to use tariffs to narrow the trade deficit would reduce imports and US exports, resulting in little to no change in the trade deficit.
• A March 2019 National Bureau of Economic Research study conducted by Pablo D. Fajgelbaum and others found that the trade war tariffs did not lower the before-duties import prices of Chinese goods, resulting in US importers taking on the entire burden of import duties in the form of higher after-duty prices.
• An April 2019 University of Chicago study conducted by Aaron Flaaen, Ali Hortacsu, and Felix Tintelnot found that after the Trump administration imposed tariffs on washing machines, washer prices increased by $86 per unit and dryer prices increased by $92 per unit, due to package deals, ultimately resulting in an aggregate increase in consumer costs of over $1.5 billion.
• An April 2019 research publication from the International Monetary Fund used a range of general equilibrium models to estimate the effects of a 25 percent increase in tariffs on all trade between China and the US, and each model estimated that the higher tariffs would bring both countries significant economic losses.
• An October 2019 study by Alberto Cavallo and coauthors found tariffs on importsfrom China were almost fully passed through to US import prices but only partially to retail consumers, implying some businesses absorbed the higher tariffs, reducing retail margins, instead of passing them on to retail consumers.
• In December 2019, Federal Reserve economists Aaron Flaaen and Justin Pierce found a net decrease in manufacturing employment due to the tariffs, suggesting that the benefit of increased production in protected industries was outweighed by the consequences of rising input costs and retaliatory tariffs.
• A February 2020 paper from economists Kyle Handley, Fariha Kamal, and Ryan Monarch estimated the 2018–2019 import tariffswere equivalent to a 2 percent tariff on all US exports.
• A December 2021 review of the data and methods used to estimate the trade war effects through 2021, by Pablo Fajgelbaum and Amit Khandelwal, concluded that “US consumers of imported goods have borne the brunt of the tariffs through higher prices, and that the trade war has lowered aggregate real income in both the US and China, although not by large magnitudes relative to GDP.”
• A January 2022 study from the US Department of Agriculture estimated the direct export losses from the retaliatory tariffs totaled $27 billion from 2018 through the end of 2019.
• A May 2023 United States International Trade Commission report from Peter Herman and others found evidence for near complete pass-through of the steel, aluminum, and Chinese tariffs to US prices. It also found an estimated $2.8 billion production increase in industries protected by the steel and aluminum tariffs was met with a $3.4 billion production decrease in downstream industries affected by higher input prices.
• A January 2024 International Monetary Fund paper found that unexpected tariff shocks tend to reduce imports more than exports, leading to slight decreases in the trade deficit at the expense of persistent gross domestic product losses—for example, the study estimates reversing the 2018–2019 tariffs would increase US output by 4 percent over three years.
• A January 2024 study by David Autor and others concludes that the 2018–2019 tariffs failed to provide economic help to the heartland: import tariffs had “neither a sizable nor significant effect on US employment in regions with newly‐​protected sectors” and foreign retaliation “by contrast had clear negative employment impacts, particularly in agriculture.”

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Timeline of Activity

• The update adds import data through 2023, new data on tariff collections, and updated model results for imposed, retaliatory, and proposed tariffs. The modeling updates reflect President Biden’s tariff increases and former President Trump’s tariff proposals. Nicolo Pastrone assisted with the research for this update.
• The update adds a new column to the “Imports Affected by U.S. Tariffs” table, reflecting import data for calendar year 2022, data updates for prior years, and tariff-rate quotas that took effect in 2022 for certain steel and aluminum imports.
• Tariffs on washing machines expired in February 2023 after an initial three-year period and a two-year extension. The Biden administration provided a two-year suspension of solar panel tariffs for four Southeast Asian nations beginning in 2022. The update adjusts the revenue and economic results for imposed tariffs.
• The Biden administration has reached deals to replace steel and aluminum tariffs with tariff rate quotas for the European Union and United Kingdom and steel tariffs with tariff-rate quotas for Japan. The deals also eliminate tariffs on derivative goods from the same jurisdictions and will bring an end to related retaliatory tariffs. The update adjusts revenue and economic estimates for imposed and retaliatory tariffs and adds a new table illustrating how import levels of affected goods have changed since 2017.
• Under President Biden, the U.S. will suspend tariffs on aircrafts and other goods from the E.U. under a five-year pause in the ongoing Boeing-Airbus dispute. We have reorganized the layout of the tracker.
• U.S. to eliminate tariffs on $2.5 billion worth of Canadian aluminum that had been imposed on August 16, 2020, to avoid Canadian retaliatory tariffs.
• U.S. to reimpose tariffs on $2.5 billion worth of Canadian aluminum on August 16, 2020, and Canada to impose retaliatory tariffs.
• U.S. reduces tariffs on $120 billion of Chinese goods by half to 7.5% and China reduces tariffs on approximately $75 billion of US goods in half to 2.5% and 5%.
• U.S. postpones indefinitely the scheduled tariff of 15% on $160 billion worth of goods from China and announces plans to decrease the 15% tariff on $120 billion worth of goods from China to 7.5% (date unknown, will be included in the model when the decrease takes effect). China took corresponding measures and canceled their schedule tariff increase.
• U.S. concludes Section 301 investigation into France's Digital Services Tax, threatens tariffs on $2.4 billion French products. Our analysis now includes tariffs on solar panels and washing machines.
• U.S. imposes 10% and 25% tariffs on $7.5 billion European Union goods under WTO ruling.
• U.S. postpones scheduled tariff hike from 25% to 30% on $250 billion worth of goods from China.
• U.S. announces 10% and 25% tariffs on $7.5 billion European Union goods under WTO ruling, with the authority to raise the tariffs to 100%.
• U.S. delays tariff increase from 25% to 30% on $250 billion worth of Chinese goods from Oct. 1 until Oct. 15.
• U.S. announces the 25% tariff on $250 billion of Chinese goods would increase to 30 percent, effective Oct. 1, after a comment period.
• China announces additional tariffs on $75 billion of U.S. imports, from 5-10%, and will resume tariffs on U.S. cars and car parts suspended earlier in 2019. Tariffs to begin Sept. 1 and end Dec. 15. U.S. announces 10% tariff on $300 billion of Chinese goods to increase to 15%, some beginning Sept. 1, others on Dec. 15.
• U.S. announces 10% tariff on $300 billion of Chinese goods would be delayed from Sept. 1 until Dec. 15.
• U.S. announces 10% tariff on $300 billion Chinese goods, to be levied on Sept. 1, lowered from the previously announced 25% on $325 billion.
• U.S. confirms announced July 5 plans to impose tariffs on all Chinese imports, roughly $500 billion of goods, modeled as a 10% tariff.
• U.S. again threatens additional tariffs on Chinese imports if China further retaliates, increasing threats from levies on $200 billion and another $200 billion to $200 billion and $300 billion.
• U.S. “indefinitely suspended” previously announced tariffs against Mexican products, set to begin at a 5% rate in June and gradually rise to 25%.
• U.S. threatens 5% tariff beginning June 10 on $346.5 billion of imports from Mexico until illegal immigration across the southern border stops. It would rise to 10% on July 1; 15% on Aug. 1; 20% on Sept. 1; and 25% on Oct. 1.
• U.S. announces it will lift steel and aluminum tariffs on Canada and Mexico, and those nations will lift their retaliatory tariffs.
• U.S. announces it will raise tariffs on $200 billion of imports from China from 10% to 25%, with threats to impose an additional 25% on $325 billion of goods.
• Tax Foundation separated our automobile tariff estimate to show auto imports from Canada, and made slight estimate adjustments to correct for rounding.
• U.S. doubles the tariffs on steel and aluminum imports from Turkey, which responds by doubling its tariffs on 22 U.S. products.
• U.S. threatens a 10% tariff on $200 billion of Chinese goods if China retaliates for the previous 10% tariff, and that would extend to an additional $200 billion of goods. This would amount to a $40 billion tax increase.
• U.S. considers increasing the proposed 10% tariff to 25% on $200 billion of Chinese imports. That would be a $30 billion tax increase.
• U.S. reaffirms plans to impose tariffs on all Chinese imports (roughly $500 billion).
• Russia will begin placing tariffs on U.S. goods, worth about $87.6 million. (Slight adjustments were made to our estimates to correct for rounding.)
• U.S. announces readiness to target an additional $200 billion in Chinese imports, and an additional $300 billion after that—an increase of $100 billion from previous threats.
• Turkey will begin placing tariffs on U.S. goods, worth about $266.5 million.

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• Published: 05 January 2024

Enhancing geometric representations for molecules with equivariant vector-scalar interactive message passing

• Yusong Wang 1 , 2   na1 ,
• Tong Wang   ORCID: orcid.org/0000-0002-9483-0050 1   na1 ,
• Shaoning Li 1   na1 ,
• Xinheng He 1 , 3 , 4 ,
• Mingyu Li 1 , 5 ,
• Zun Wang   ORCID: orcid.org/0000-0002-8763-8327 1 ,
• Nanning Zheng 2 ,
• Bin Shao   ORCID: orcid.org/0000-0002-9790-5687 1 &
• Tie-Yan Liu   ORCID: orcid.org/0000-0002-0476-8020 1  

Nature Communications volume  15 , Article number:  313 ( 2024 ) Cite this article

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• Chemical biology
• Computational biology and bioinformatics
• Computational models
• Molecular modelling
• Protein structure predictions

Geometric deep learning has been revolutionizing the molecular modeling field. Despite the state-of-the-art neural network models are approaching ab initio accuracy for molecular property prediction, their applications, such as drug discovery and molecular dynamics (MD) simulation, have been hindered by insufficient utilization of geometric information and high computational costs. Here we propose an equivariant geometry-enhanced graph neural network called ViSNet, which elegantly extracts geometric features and efficiently models molecular structures with low computational costs. Our proposed ViSNet outperforms state-of-the-art approaches on multiple MD benchmarks, including MD17, revised MD17 and MD22, and achieves excellent chemical property prediction on QM9 and Molecule3D datasets. Furthermore, through a series of simulations and case studies, ViSNet can efficiently explore the conformational space and provide reasonable interpretability to map geometric representations to molecular structures.

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Introduction.

Molecular modeling plays a crucial role in modern scientific and engineering fields, aiding in the understanding of chemical reactions, facilitating new drug development, and driving scientific and technological advancements 1 , 2 , 3 , 4 . One commonly used method in molecular modeling is density functional theory (DFT). DFT enables accurate calculations of energy, forces, and other chemical properties of molecules 5 , 6 . However, due to the large computational requirements, DFT calculations often demand significant computational resources and time, particularly for large molecular systems or high-precision calculations. Machine learning (ML) offers an alternative solution by learning from reference data with ab initio accuracy and high computational efficiency 7 , 8 . Gradient-domain machine learning (GDML) 9 constructs accurate molecular force fields using conservation of energy and limited samples from ab initio molecular dynamics trajectories, enabling cost-effective simulations while maintaining accuracy. Symmetric GDML (sGDML) 10 further improves force field construction by incorporating physical symmetries, achieving CCSD(T)-level accuracy for flexible molecules. An exact iterative approach (Global sGDML) 11 extends sGDML to global force fields for molecules with several hundred atoms, maintaining correlations of atomic degree and accurately describing complex molecules and materials. In recent years, deep learning (DL) has demonstrated its powerful ability to learn from raw data without any hand-crafted features in many fields and thus attracted more and more attention. However, the inherent drawback of deep learning, which requires large amounts of data, has become a bottleneck for its application to more scenarios 12 . To alleviate the dependency on data for DL potentials, recent works have incorporated the inductive bias of symmetry into neural network design, known as geometric deep learning (GDL). Symmetry describes the conservation of physical laws, i.e., the unchanged physical properties with any transformations such as translations or rotations. It allows GDL to be extended to limited data scenarios without any data augmentation.

Equivariant graph neural network (EGNN) is one of the representative approaches in GDL, which has extensive capability to model molecular geometry 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 . A popular kind of EGNN conducts equivariance from directional information and involves geometric features to predict molecular properties. GemNet 20 extends the invariant DimeNet/DimeNet++ 16 , 17 with dihedral information. They explicitly extract geometric information in the Euclidean space with first-order geometric tensor, i.e., setting l max  = 1. PaiNN 18 and equivariant transformer 19 further adopt vector embedding and scalarize the angular representation implicitly via the inner product of the vector embedding itself. They reduce the complexity of explicit geometry extraction by taking the angular information into consideration. Another mainstream approach to achieving equivariance is through group representation theory, which can achieve higher accuracy but comes with large computational costs. NequIP, Allegro, and MACE 12 , 22 , 23 achieve state-of-the-art performance on several molecular dynamics simulation datasets leveraging high-order geometric tensors. On the one hand, algorithms based on group representation theory have strong mathematical foundations and are able to fully utilize geometric information using high-order geometric tensors. On the other hand, these algorithms often require computationally expensive operations such as the Clebsch–Gordan product (CG-product) 24 , making them possibly suitable for periodic systems with elaborate model design but impractical for large molecular systems such as chemical and biological molecules without periodic boundary conditions.

In this study, we propose ViSNet (short for “Vector-Scalar interactive graph neural Network"), which alleviates the dilemma between computational costs and sufficient utilization of geometric information. By incorporating an elaborate runtime geometry calculation (RGC) strategy, ViSNet implicitly extracts various geometric features, i.e., angles, dihedral torsion angles, and improper angles in accordance with the force field of classical MD with linear time complexity, thus significantly accelerating model training and inference while reducing the memory consumption. To extend the vector representation, we introduce spherical harmonics and simplify the computationally expensive Clebsch–Gordan product with the inner product. Furthermore, we present a well-designed vector–scalar interactive equivariant message passing (ViS-MP) mechanism, which fully utilizes the geometric features by interacting vector hidden representations with scalar ones. When comprehensively evaluated on some benchmark datasets, ViSNet outperforms all state-of-the-art algorithms on all molecules in MD17, revised MD17 and MD22 datasets and shows superior performance on QM9, Molecule3D dataset indicating the powerful capability of molecular geometric representation. ViSNet also has won the PCQM4Mv2 track in the OGB-LCS@NeurIPS2022 competition ( https://ogb.stanford.edu/neurips2022/results/ ). We then performed molecular dynamics simulations for each molecule on MD17 driven by ViSNet trained only with limited data (950 samples). The highly consistent interatomic distance distributions and the explored potential energy surfaces between ViSNet and quantum simulation illustrate that ViSNet is genuinely data-efficient and can perform simulations with high fidelity. To further explore the usefulness of ViSNet to real-world applications, we used an in-house dataset that consists of about 10,000 different conformations of the 166-atom mini-protein Chignolin derived from replica exchange molecular dynamics and calculated at the DFT level. When evaluated on the dataset, ViSNet also achieved significantly better performance than empirical force fields, and the simulations performed by ViSNet exhibited very close force calculation to DFT. In addition, ViSNet exhibits reasonable interpretability to map geometric representation to molecular structures. The contributions of ViSNet can be summarized as follows:

Proposing an RGC module that utilizes high-order geometric tensors to implicitly extract various geometric features, including angles, dihedral torsion angles, and improper angles, with linear time complexity.

Introducing ViS-MP mechanism to enable efficient interaction between vector hidden representations and scalar ones and fully exploit the geometric information.

Achieving state-of-the-art performance in six benchmarks for predicting energy, forces, HOMO-LUMO gap, and other quantum properties of molecules.

Performing molecular dynamics simulations driven by ViSNet on both small molecules and 166-atom Chignolin with high fidelity.

Demonstrating reasonable model interpretability between geometric features and molecular structures.

Overview of ViSNet

ViSNet is a versatile EGNN that predicts potential energy, atomic forces as well as various quantum chemical properties by taking atomic coordinates and numbers as inputs. As shown in Fig.  1 a, the model is composed of an embedding block and multiple stacked ViSNet blocks, followed by an output block. The atomic number and coordinates are fed into the embedding block followed by ViSNet blocks to extract and encode geometric representations. The geometric representations are then used to predict molecular properties through the output block. It is worth noting that ViSNet is an energy-conserving potential, i.e., the predicted atomic forces are derived from the negative gradients of the potential energy with respect to the coordinates 9 , 10 .

a Model sketch of ViSNet. ViSNet embeds the 3D structures of molecules and extracts the geometric information through a series of ViSNet blocks and outputs the molecule properties such as energy, forces, and HOMO–LUMO gap through an output block. b Flowchart of one ViSNet Block. One ViSNet block consists of two modules: (i) Scalar2Vec , responsible for attaching scalar embeddings to vectors.; (ii) Vec2Scalar , renovates scalar embeddings built on RGC strategy. The inputs of Scalar2Vec are the node embedding h i , edge embedding f i j , direction unit \({\overrightarrow{v}}_{i}\) and the relative positions between two atoms. The edge-fusion graph attention module (serves as \({\phi }_{{\rm {m}}}^{{\rm {s}}}\) ) takes as input h i and the output of the dense layer following f i j , and outputs scalar messages. Before aggregation, each scalar message is transformed through a dense layer, and then fused with the unit of the relative position \({\overrightarrow{u}}_{ij}\) and its own direction unit \({\overrightarrow{v}}_{j}\) . We further compute the vector messages and aggregate them all among the neighborhood. Through a gated residual connection, the final residual \({{\Delta }}{\overrightarrow{v}}_{i}\) is produced. In Vec2Scalar module, by Hadamard production of aggregated scalar messages and the output of RGC-Angle calculation and adding a gated residual connection, the final Δ h i is figured out. Likewise, combining the projected f i j and the output of RGC-Dihedral calculation, the final Δ f i j is determined.

The success of classical force fields shows that geometric features such as interatomic distances, angles, dihedral torsion angles, and improper angles in Fig.  2 are essential to determine the total potential energy of molecules. The explicit extraction of invariant geometric representations in previous studies often suffers from a large amount of time or memory consumption during model training and inference. Given an atom, the calculation of angular information scales \({{{{{{{\mathcal{O}}}}}}}}({{{{{{{{\mathcal{N}}}}}}}}}^{2})\) with the number of neighboring atoms, while the computational complexity is even \({{{{{{{\mathcal{O}}}}}}}}({{{{{{{{\mathcal{N}}}}}}}}}^{3})\) for dihedrals 20 . To alleviate this problem, inspired by Sch¨utt et al. 18 , we propose runtime geometry calculation (RGC), which uses an equivariant vector representation (termed as direction unit) for each node to preserve its geometric information. RGC directly calculates the geometric information from the direction unit which only sums the vectors from the target node to its neighbors once. Therefore, the computational complexity can be reduced to \({{{{{{{\mathcal{O}}}}}}}}({{{{{{{\mathcal{N}}}}}}}})\) . Notably, beyond employing angular information that has been used in PaiNN 18 and ET 19 , ViSNet further considers the dihedral torsion and improper angle calculation with higher geometric tensors.

The bonded terms consist of bond length, bond angle, dihedral torsion, and improper angle. The RGC module depicts all bonded terms of classical MD as model operations in linear time complexity. Yellow arrow \({\overrightarrow{v}}_{i}\) denotes the direction unit in Eq. ( 1 ).

Considering the sub-structure of a toy molecule with four atoms shown in Fig.  2 , the angular information of the target node i could be conducted from the vector \({\overrightarrow{r}}_{ij}\) as follows:

where \({\overrightarrow{r}}_{ij}\) is the vector from node i to its neighboring node j , \({\overrightarrow{u}}_{ij}\) is the unit vector of \({\overrightarrow{r}}_{ij}\) . Here, we define the direction unit \({\overrightarrow{v}}_{i}\) as the sum of all unit vectors from node i to its all neighboring nodes j , where node i is the intersection of all unit vectors. As shown in Eq. ( 2 ), we calculate the inner product of the direction unit \({\overrightarrow{v}}_{i}\) which represents the sum of the inner products of unit vectors from node i to all its neighboring nodes. Combining with Eq. ( 1 ), the inner product of direction \({\overrightarrow{v}}_{i}\) finally stands for the sum of cosine values of all angles formed by node i and any two of its neighboring nodes.

Similar to runtime angle calculation, we also calculate the vector rejection 25 of the direction unit \({\overrightarrow{v}}_{i}\) of node i and \({\overrightarrow{v}}_{j}\) of node j on the vector \({\overrightarrow{u}}_{ij}\) and \({\overrightarrow{u}}_{ji}\) , respectively.

where \({{{{{{{{\rm{Rej}}}}}}}}}_{\overrightarrow{b}}(\overrightarrow{a})\) represents the vector component of \(\overrightarrow{a}\) perpendicular to \(\overrightarrow{b}\) , termed as the vector rejection. \({\overrightarrow{u}}_{ij}\) and \({\overrightarrow{v}}_{i}\) are defined in Eq. ( 1 ). \({\overrightarrow{w}}_{ij}\) represents the sum of the vector rejection \({{{{{{{{\rm{Rej}}}}}}}}}_{{\overrightarrow{u}}_{ij}}({\overrightarrow{u}}_{im})\) and \({\overrightarrow{w}}_{ji}\) represents the sum of the vector rejection \({{{{{{{{\rm{Rej}}}}}}}}}_{{\overrightarrow{u}}_{ji}}({\overrightarrow{u}}_{jn})\) . The inner product between \({\overrightarrow{w}}_{ij}\) and \({\overrightarrow{w}}_{ji}\) is then calculated to conduct dihedral torsion angle information of the intersecting edge e i j as follows:

The improper angle is derived from a pyramid structure forming by 4 nodes. As the last toy molecule shown in Fig.  2 , node i is the vertex of the pyramid, and the improper torsion angle is formed by two adjacent planes with an intersecting edge e i j . We can also calculate the improper angle by vector rejection:

In the same way, the inner product between \({\overrightarrow{t}}_{ij}\) and \({\overrightarrow{t}}_{ji}\) indicates the summation of improper angle information formed by e i j :

Multiple works have shown the effectiveness of high-order geometric tensors for molecular modeling 12 , 22 , 26 , 27 . However, the computational overheads of these approaches are generally expansive due to the CG-product, impeding their further application for large systems. In this work, we convert the vectors to high-order representation with spherical harmonics but discard CG-product with the inner product following the idea of RGC. We find that the extended high-order geometric tensors can still represent the above angular information in the form of Legendre polynomials according to the addition theorem:

where the P l is the Legendre polynomial of degree l , Y l , m denotes the spherical harmonics function and \({Y}_{l,m}^{*}\) denotes its complex conjugation. We sum the product of different order l to obtain the scalar angular representation, which is the same operation as the inner product. It is worth noting that such an extension does not increase the model size and keeps the model architecture unchanged. We also provide proof about the rotational invariance of the RGC strategy in the section “Proofs of the rotational invariance of RGC ”.

In order to make full use of geometric information and enhance the interaction between scalars and vectors, we designed an effective vector–scalar interactive message-passing mechanism with respect to the intersecting nodes and edges for angles and dihedrals, respectively. It is important to note that previous studies 18 , 19 primarily focused on updating node features, whereas our approach updates both node and edge features during message passing, leading to a more comprehensive geometric representation. The key operations in ViS-MP are given as follows:

where h i denotes the scalar embedding of node i , f i j stands for the edge feature between node i and node j . \({\overrightarrow{v}}_{i}\) represents the embedding of the direction unit mentioned in RGC. The superscript of variables indicates the index of the block that the variables belong to. We omit the improper angle here for brevity. A comprehensive version is depicted in Supplementary. ViS-MP extends the conventional message passing, aggregation, and update processes with vector–scalar interactions. Eqs. ( 8 ) and ( 9 ) depict our message-passing and aggregation processes. To be concrete, scalar messages m i j incorporating scalar embedding h j , h i , and f i j are passed and then aggregated to node i through a message function \({\phi }_{m}^{s}\) (Eq. ( 8 )). Similar operations are applied for vector messages \({\overrightarrow{m}}_{i}^{l}\) of node i that incorporates scalar message m i j , vector \({\overrightarrow{r}}_{ij}\) and vector embedding \({\overrightarrow{v}}_{j}\) (Eq. ( 9 )). Equations ( 10 ) and ( 11 ) demonstrate the update processes. h i is updated by the aggregated scalar message output m i while the inner product of \({\overrightarrow{v}}_{i}\) is updated through an update function \({\phi }_{un}^{s}\) . Then \({\overrightarrow{f}}_{ij}\) is updated by the inner product of the rejection of the vector embedding \({\overrightarrow{v}}_{i}\) and \({\overrightarrow{v}}_{j}\) through an update function \({\phi }_{ue}^{s}\) . Finally, the vector embedding \({\overrightarrow{v}}_{i}\) is updated by both scalar and vector messages through an update function \({\phi }_{un}^{v}\) . Notably, the vectors update function, i.e., ϕ v require to be equivariant. The detailed message and update functions can be found in the Methods section. A proof about the equivariance of ViS-MP can be found in Supplementary Methods.

In summary, the geometric features are extracted by inner products in the RGC strategy and the scalar and vector embeddings are cyclically updating each other in ViS-MP so as to learn a comprehensive geometric representation from molecular structures.

Accurate quantum chemical property predictions

We evaluated ViSNet on several prevailing benchmark datasets including MD17 9 , 10 , 28 , revised MD17 29 , MD22 30 , QM9 31 , Molecule3D 32 , and OGB-LSC PCQM4Mv2 33 for energy, force, and other molecular property prediction. MD17 consists of the MD trajectories of seven small organic molecules; the number of conformations in each molecule dataset ranges from 133,700 to 993,237. The dataset rMD17 is a reproduced version of MD17 with higher accuracy. MD22 is a recently proposed MD trajectories dataset that presents challenges with respect to larger system sizes (42–370 atoms). Large molecules such as proteins, lipids, carbohydrates, nucleic acids, and supramolecules are included in MD22. QM9 consists of 12 kinds of quantum chemical properties of 133,385 small organic molecules with up to 9 heavy atoms. Molecule3D is a recently proposed dataset including 3,899,647 molecules collected from PubChemQC with their ground-state structures and corresponding properties calculated by DFT. We focus on the prediction of the HOMO–LUMO gap following ComENet 34 . OGB-LSC PCQM4Mv2 is a quantum chemistry dataset originally curated under the PubChemQC including a DFT-calculated HOMO–LUMO gap of 3,746,619 molecules. The 3D conformations are provided for 3,378,606 training molecules but not for the validation and test sets. The training details of ViSNet on each benchmark are described in the “Methods” section.

We compared ViSNet with the state-of-the-art algorithms, including DimeNet 16 , PaiNN 18 , SpookyNet 21 , ET 19 , GemNet 20 , UNiTE 35 , NequIP 12 , SO3KRATES 36 , Allegro 22 , MACE 23 and so on. As shown in Table  1 (MD17), Table  2 (rMD17), and Table  3 (MD22), it is remarkable that ViSNet outperformed the compared algorithms for both small (MD17 and rMD17) and large molecules (MD22) with the lowest mean absolute errors (MAE) of predicted energy and forces. On the one hand, compared with PaiNN, ET, and GemNet, ViSNet incorporated more geometric information and made full use of geometric information in ViS-MP, which contributes to the performance gains. On the other hand, compared with NequIP, Allegro, SO3KRATES, MACE, etc., ViSNet testified the effect of introducing spherical harmonics in the RGC module.

As shown in Table  4 , ViSNet also achieved superior performance for chemical property predictions on QM9. It outperformed the compared algorithms for 9 of 12 chemical properties and achieved comparable results on the remaining properties. Elaborated evaluations on Molecule3D confirmed the high prediction accuracy of ViSNet as shown in Table  5 . ViSNet achieved 33.6% and 6.51% improvements than the second-best for random split and scaffold split, respectively. Furthermore, ViSNet exhibited good portability to other multimodality methods, e.g., Transformer-M 37 and outperformed other approaches on OGB-LSC PCQM4Mv2 (see Supplementary Fig.  S1) . ViSNet also achieved the winners of PCQM4Mv2 track in the OGB-LCS@NeurIPS2022 competition when testing on unseen molecules 38 ( https://ogb.stanford.edu/neurips2022/results/ ).

To evaluate the computational efficiency of our ViSNet, following 23 , we compare the time latency of ViSNet with prevailing models in Supplementary Fig.  S2 . The latency is defined as the time it takes to compute forces on a structure (i.e., the gradient calculation for a set of input coordinates through the whole deep neural network). As shown in Supplementary Fig.  S2 , ViSNet ( L  = 2) saved 42.8% time latency compared with MACE ( L  = 2). Notably, despite the use of CG-product, Allegro had a significant speed improvement compared to NequIP and BOTNet. However, ViSNet still saved 6.1%, 4.1%, and 61% time latency compared to Allegro with L  = 1, 2, and 3, respectively.

Efficient molecular dynamics simulations

To evaluate ViSNet as the potential for MD simulations, we incorporated ViSNet that trained only with 950 samples on MD17 into the ASE simulation framework 39 to perform MD simulations for all seven kinds of organic molecules. All simulations are run with a time step τ  = 0.5 fs under the Berendsen thermostat with the other settings the same as those of the MD17 dataset. As shown in Fig.  3 , we analyzed the interatomic distance distributions derived from both AIMD simulations with ViSNet as the potential and ab initio molecular dynamics simulations at the DFT level for all seven molecules, respectively. As shown in Fig.  3 a, the interatomic distance distribution h ( r ) is defined as the ensemble average of atomic density at a radius r 9 . Figure  3 b–h illustrates the distributions derived from ViSNet are very close to those generated by DFT. We also compared the potential energy surfaces sampled by ViSNet and DFT for these molecules, respectively (Supplementary Fig.  S3 ). The consistent potential energy surfaces suggest that ViSNet can recover the conformational space from the simulation trajectories. Moreover, compared to DFT, numerous groundbreaking machine learning force fields (MLFFs), including sGDML 10 , ANI 40 , DPMD 41 , and PhysNet 42 have proven their exceptional speeds in MD simulations. Similar to such algorithms, ViSNet also exhibited significant computational cost reduction compared to DFT as shown in Supplementary Fig.  S4 and Table  S2 .

a An illustration about the atomic density at a radius r with the arbitrary atom as the center. The interatomic distance distribution is defined as the ensemble average of atomic density. b – h The interatomic distance distributions comparison between simulations by ViSNet and DFT for all seven organic molecules in MD17. The curve of ViSNet is shown using a solid blue line, while the dashed orange line is used for the DFT curve. The structures of the corresponding molecules are shown in the upper right corner. Source data are provided as a Source Data file.

To further examine the molecular properties derived from simulations driven by ViSNet, we performed 500 ps MD simulations at a constant energy ensemble (NVE) for ethanol in the MD17 dataset with a time step of τ  = 0.5 fs and 200 ps Ac-Ala3-NHMe in the MD22 dataset with a time step of τ  = 1 fs. The simulations were driven by ViSNet, sGDML, and DFT, respectively. For ethanol, we analyzed its vibrational spectra and the probability distribution of dihedral angles. For Ac-Ala3-NHMe, we investigated its vibrational spectra and potential energy surface (PES) via the Ramachandran plot. To analyze the Ramachandran plot of different simulations, the free energy value was estimated using the potential of mean force (PMF). ϕ and ψ were set as two reaction coordinates ( x , y ). All three ϕ and ψ dihedrals in Ac-Ala3-NHMe were calculated and plotted. The relative free energy value was calculated and referred to with the minimum value. To generate the landscape, 40 bins were used in both the x and y directions. Supplementary Fig.  S5 a and b demonstrate that both ViSNet and sGDML generate similar vibrational spectra, with slight differences in peak intensities compared to DFT. The probability distribution of hydroxyl angles in ethanol (Supplementary Fig.  S5 c) reveals three minima: gauche ± ( M g ± ) and trans ( M t ). Furthermore, even though ViSNet showed better performance than sGDML for various conformations in the MD22 dataset, starting from the same structure of the alanine tetrapeptide, the performance difference may not have a notable impact on the sampling efficiency for such small molecules, and thus may also lead to similar dynamics on the Ramachandran plots as shown in the Supplementary Fig.  S5 d–f. These results demonstrate that with only a few training samples, ViSNet can act with the potential to perform high-fidelity molecular dynamics simulations with much less computational cost and higher accuracy.

Applications for real-world full-atom proteins

To examine the usefulness of ViSNet in real-world applications, we made evaluations on the 166-atom mini-protein Chignolin (Fig.  4 a). Based on a Chignolin dataset consisting of about 10,000 conformations that sampled by replica exchange MD 43 and calculated at DFT level by Gaussian 16 44 in our another study 45 , 46 , we split it as training, validation, and test sets by the ratio of 8:1:1. We trained ViSNet as well as other prevailing MLFFs including ET 19 , PaiNN 18 , GemNet-OC 47 , MACE 23 , NequIP 12 and Allegro 22 and compared them with molecular mechanics (MM) 48 . The DFT results were used as the ground truth. Figure  4 b shows the free energy landscape of Chignolin and is depicted by d D3−G7 (the distance between carbonyl oxygen on the D3 backbone and nitrogen on the G7 backbone) and d E5−T8 (the distance between carbonyl oxygen on the E5 backbone and nitrogen on T8 backbone). The concentrated energy basin on the left shows the folded state and the scattered energy basin on the right shows the unfolded state. We randomly selected six structures from different regions of the potential energy surface for visualization. Among them, four structures were predicted by the model with smaller errors than the MAE while the other two with larger errors. Interestingly, all models consistently performed poorly on the structures with high potential energies (low probability of sampling) and performed well on the other structures. This implies that the sampling of conformations with high potential energies could be enhanced to ensure the generalization ability of the models.

a The visualization of Chignolin structure. The backbone is colored grey while the side chains of each residue in Chignolin are highlighted with a ball and stick. b The energy landscape of Chignolin sampled by REMD. The x -axis of the landscape is the distance between carbonyl oxygen on the D3 backbone and nitrogen on the G7 backbone, while the y -axis is the distance between carbonyl oxygen on the E5 backbone and nitrogen on the T8 backbone. Six structures were then selected for visualization. Each structure is shown as a cartoon and residues are depicted in sticks. The histograms show the absolute error between the energy difference predicted by MLFFs including ViSNet, ET, PaiNN, GemNet-OC, NequIP, Allegro, and MACE or calculated by MM, and the ground truth calculated by DFT on the corresponding structure. c The average root mean square deviation (RMSD) of the Chignolin trajectories simulated by ViSNet was calculated from 10 different trajectories. The shaded areas indicate the standard deviation range. d The MAE of each component of atomic forces during the simulations driven by ViSNet. The ground truth energies and forces were calculated using Gaussian 16. The shaded areas indicate the standard deviation range. Source data are provided as a Source Data file.

Supplementary Fig.  S6 shows the correlations between the energies predicted by MLFFs or MM and the ground truth values calculated by DFT for all conformations in the test set. ViSNet achieved a lower MAE and a higher R 2 score. From the violin plot of the absolute errors shown in Supplementary Fig.  S7 , ViSNet, PaiNN and ET exhibited smaller errors than other MLFFs while MM got a much wider range of prediction errors. Similar results can be seen in the force correlations in each component shown in Supplementary Fig.  S8 . Detailed settings about DFT and MM calculations are shown in Supplementary Materials. Furthermore, we also made a comprehensive comparison by taking model performance, training time consumption, and model size into consideration. ViSNet and other state-of-the-art algorithms such as PaiNN, ET, GemNet-OC, MACE, NequIP, and Allegro were analyzed on the Chignolin dataset and shown in Fig.  5 . Although ViSNet is marginally slower than ET and PaiNN, it introduces more geometric information, significantly enhancing its performance. When compared to GemNet, which also incorporates dihedral angles, ViSNet’s computational cost is significantly more affordable. Similarly, ViSNet proves to be computationally efficient when compared to models employing the CG-product method, such as MACE, Allegro, and NequIP.

PaiNN and ET are faster and smaller as ViSNet further incorporates dihedral calculation. ViSNet outperforms GemNet-OC due to its Runtime Geometry Calculation, reducing the explicit extraction of dihedral complexity from \({{{{{{{\mathcal{O}}}}}}}}({{{{{{{{\mathcal{N}}}}}}}}}^{3})\) to \({{{{{{{\mathcal{O}}}}}}}}({{{{{{{\mathcal{N}}}}}}}})\) . Additionally, ViSNet is also faster and smaller than MACE, Allegro, and NequIP for streamlining the CG-product. ViSNet achieves the best performance for its elaborate design, i.e., runtime geometric calculation and vector–scalar interactive message passing. Source data are provided as a Source Data file.

In addition, we performed MD simulations for Chignolin driven by ViSNet. 10 conformations were randomly selected as initial structures, and 100 ps simulations were run for each. As shown in Fig.  4 c, the RMSD for 10 simulation trajectories is shown against the simulation time. In Fig.  4 d, we displayed the MAE values of each component of the atomic forces between ViSNet and those calculated by Gaussian 16 44 at the DFT level. The simulation trajectory driven by ViSNet exhibited a small force difference for each component to quantum mechanics, which implies that ViSNet has no bias towards any force component, and thus consolidates the accuracy and potential usefulness for real-world applications.

Interpretability of ViSNet on molecular structures

Prior works have shown the effectiveness of incorporating geometric features, such as angles 16 , 20 . The primary method of geometry extraction utilized by ViSNet is the distinct inner product in its runtime geometry calculation. To this end, we illustrate a reasonable model interpretability of ViSNet by mapping the angle representations derived from the inner product of direction units in the model to the atoms in the molecular structure. We aim to bridge the gap between geometric representation in ViSNet and molecular structures. We visualized the embeddings after the inner product of direction units \(\langle {\overrightarrow{v}}_{i},{\overrightarrow{v}}_{i}\rangle\) extracted from 50 aspirin samples on the validation set. The high-dimensional embeddings were reduced to 2-dimensional space using T-SNE 49 and then clustered using DBSCAN 50 without the prior of number of clusters.

Supplementary Fig.  S9 exhibits the clustering results of nodes’ embeddings after the inner product of their corresponding direction units. We further map the clustered nodes to the atoms of aspirin chemical structure. Interestingly, the embeddings for these nodes could be distinctly gathered into several clusters shown in different colors. For example, although carbon atom C 11 and carbon atom C 12 possess different positions and connect with different atoms, their inner product \(\langle {\overrightarrow{v}}_{i},{\overrightarrow{v}}_{i}\rangle\) are clustered into the same class for holding similar substructures ({ C 11 − O 2 O 3 C 6 } and { C 12 − O 1 O 4 C 13 }). To summarize, ViSNet can discriminate different molecular substructures in the embedding space.

Ablation study

To further explore where the performance gains of ViSNet come from, we conducted a comprehensive ablation study. Specifically, we excluded the runtime angle calculation (w/o A), runtime dihedral calculation (w/o D), and both of them (w/o A&D) in ViSNet, in order to evaluate the usefulness of each part. ViSNet-improper denotes the additional improper angles and ViSNet l =1 uses the first-order spherical harmonics.

We designed some model variants with different message-passing mechanisms based on ViS-MP for scalar and vector interaction. ViSNet-N directly aggregates the dihedral information to intersecting nodes, and ViSNet-T leverages another form of dihedral calculation. The details of these model variants are elaborated in Supplementary. The results of the ablation study are shown in Supplementary Table  S3 and Supplementary Fig.  S10 . Based on the results, we can see that both kinds of directional geometric information are useful and the dihedral information contributes a little bit more to the final performance. The significant performance drop from ViSNet-N and ViSNet-T further validates the effectiveness of the ViS-MP mechanism. ViSNet-improper achieves similar performance to ViSNet for small molecules, but the contribution of improper angles is more obvious for large molecules (see Table  3 ). Furthermore, ViSNet using higher-order spherical harmonics achieves better performance.

We propose ViSNet, a geometric deep learning potential for molecular dynamics simulation. The group representation theory-based methods and the directional information-based methods are two mainstream classes of geometric deep learning potentials to enforce SE(3) equivariance 20 . ViSNet takes advantage of both sides in designing the RGC strategy and ViS-MP mechanism. On the one hand, the RGC strategy explicitly extracts and exploits the directional geometric information with computationally lightweight operations, making the model training and inference fast. On the other hand, ViS-MP employs a series of effective and efficient vector-scalar interactive operations, leading to the full use of geometric information. Furthermore, according to the many-body expansion theory 51 , 52 , 53 , the potential energy of the whole system equals the potential of each single atom plus the energy corrections from two-bodies to many-bodies. Most of the previous studies model the truncated energy correction terms hierarchically with k -hop information via stacking k message passing blocks. Different from these approaches, ViSNet encodes the angle, dihedral torsion, and improper information in a single block, which empowers the model to have a much more powerful representation ability. In addition, ViSNet’s universality or completeness is not validated by the geometric Weisfeiler–Leman (GWL) test 54 due to the inner product operation, which is computationally efficient but fails to distinguish certain atom reflection structures with the same angular information. To pass counterexamples or the GWL test, incorporating the CG-product with higher-order spherical harmonics is necessary in future studies.

Besides predicting energy, force, and chemical properties with high accuracy, performing molecular dynamics simulations with ab initio accuracy at the cost of the empirical force field is a grand challenge. ViSNet proves its usefulness in real-world ab initio molecular dynamics simulations with less computational costs and the ability of scaling to large molecules such as proteins. Extending ViSNet to support larger and more complex molecular systems will be our future research direction.

Equivariance

In the context of machine learning for atomic systems, equivariance is a pervasive concept. Specifically, the atomic vectors such as dipoles or forces must rotate in a manner consistent with the conformation coordinates. In molecular dynamics, such equivariance can be ensured by computing gradients based on a predicted conservative scalar energy. Formally, a function \({{{{{{{\mathcal{F}}}}}}}}:{{{{{{{\mathcal{X}}}}}}}}\to {{{{{{{\mathcal{Y}}}}}}}}\) is equivariant should guarantee:

where \({\rho }_{{{{{{{{\mathcal{X}}}}}}}}}(g)\) and \({\rho }_{{{{{{{{\mathcal{X}}}}}}}}}(g)\) are group representations in input and output spaces. The integration of equivariance into model parameterization has been shown to be effective, as seen in the implementation of shift-equivariance in CNNs, which is critical for enhancing the generalization capacity.

Proofs of the rotational invariance of RGC

Assume that the molecule rotates in 3D space, i.e.,

where, R   ∈   S O (3) is an arbitrary rotation matrix that satisfies:

The angular information after rotation is calculated as follows:

As shown in Eq. ( 18 ), the angle information does not change after rotation. The dihedral angular and improper information is also rotationally invariant since:

As Eq. ( 18 ) proved, the inner product has rotational invariance. Then, Eq. ( 19 ) can be further simplified as

The dihedral or improper angular information after rotation is calculated as:

As a result, Eqs. ( 18 ) and ( 21 ) have proved the rotational invariance of our proposed runtime geometry calculation (RGC).

We also provide proof of the equivariance of our ViS-MP in Supplementary Methods.

Detailed operations and modules in ViSNet

ViSNet predicts the molecular properties (e.g., energy \(\hat{E}\) , forces \(\overrightarrow{F}\in {{\mathbb{R}}}^{N\times 3}\) , dipole moment μ ) from the current states of atoms, including the atomic positions \(X\in {{\mathbb{R}}}^{N\times 3}\) and atomic numbers \(Z\in {{\mathbb{N}}}^{N}\) . The architecture of the proposed ViSNet is shown in Fig.  1 . The overall design of ViSNet follows the vector–scalar interactive message passing as illustrated from Eqs. ( 8 )–( 11 ). First, an embedding block encodes the atom numbers and edge distances into the embedding space. Then, a series of ViSNet blocks update the node-wise scalar and vector representations based on their interactions. A residual connection is placed between two ViSNet blocks. Finally, stacked corresponding gated equivariant blocks proposed by 18 are attached to the output block for specific molecular property prediction.

The embedding block

ViSNet expands the direct node and edge embedding with their neighbors. It first embeds atomic chemical symbol z i , and calculates the edge representation whose distances within the cutoff through radial basis functions (RBF). Then the initial embedding of the atom i , its 1-hop neighbors j and the directly connected edge e i j within cutoff are fused together as the initial node embedding \({h}_{i}^{0}\) and edge embedding \({f}_{ij}^{0}\) . In summary, the embedding block is given by:

\({{{{{{{\mathcal{N}}}}}}}}(i)\) denotes the set of 1-hop neighboring nodes of node i , and j is one of its neighbors. The embedding process is elaborated in Supplementary. The initial vector embedding \({\overrightarrow{v}}_{i}\) is set to \(\overrightarrow{0}\) . The vector embeddings \(\overrightarrow{v}\) are projected into the embedding space by following 18 ; \(\overrightarrow{v}\in {{\mathbb{R}}}^{N\times 3\times F}\) and F is the size of hidden dimension. The advantage of such projection is to assign a unique high-dimensional representation for each embedding to discriminate from each other. Further discussions on its effectiveness and interpretability are given in the Results section.

The Scalar2Vec module

In the Scalar2Vec module, the vector embedding \(\overrightarrow{v}\) is updated by both the scalar messages derived from node and edge scalar embeddings (Eq. ( 8 )) and the vector messages with inherent geometric information (Eq. ( 9 )). The message of each atom is calculated through an Edge-Fusion Graph Attention module, which fuses the node and edge embeddings and computes the attention scores. The fusion of the node and edge embeddings could be the concatenation operation, Hadamard product, or adding a learnable bias 55 . We leverage the Hadamard product and the vanilla multi-head attention mechanism borrowed from Transformer 56 for edge-node fusion.

Following 19 , we pass the fused representations through a nonlinear activation function as shown in Eq. ( 23 ). The value ( V ) in the attention mechanism is also fused by edge features before being multiplied by attention scores weighted by a cosine cutoff as shown in Eq. ( 24 ),

where l   ∈  {0, 1, 2,  ⋯   ,  L } is the index of the block, σ denotes the activation function (SiLU in this paper), W is the learnable weight matrix,  ⊙  represents the Hadamard product, ϕ (  ⋅  ) denotes the cosine cutoff and Dense(  ⋅  ) refers to one learnable weight matrix with an activation function. For brevity, we omit the learnable bias for linear transformation on scalar embedding in equations, and there is no bias for vector embedding to ensure the equivariance.

Then, the computed \({m}_{ij}^{l}\) is used to produce the geometric messages \({\overrightarrow{m}}_{ij}^{l}\) for vectors:

And the vector embedding \({\overrightarrow{v}}^{l}\) is updated by:

The Vec2Scalar module

In the Vec2Scalar module, the node embedding \({h}_{i}^{l}\) and edge embedding \({f}_{ij}^{l}\) are updated by the geometric information extracted by the RGC strategy, i.e., angles (Eq. ( 10 )) and dihedrals (Eq. ( 11 )), respectively. The residual node embedding \({{\Delta }}{h}_{i}^{l+1}\) , is calculated by a Hadamard product between the runtime angle information and the aggregated scalar messages with a gated residual connection:

To compute the residual edge embedding \({{\Delta }}{f}_{ij}^{l+1}\) , we perform the Hadamard product of the runtime dihedral information with the transformed edge embedding:

After the residual hidden representations are calculated, we add them to the original input of block l and feed them to the next block.

A comprehensive version that includes improper angles is depicted in Supplementary Methods.

The output block

Following PaiNN 18 , we update the scalar embedding and vector embedding of nodes with multiple gated equivariant blocks:

where [  ⋅  ,  ⋅  ] is the tensor concatenation operation. The final scalar embedding \({h}_{i}^{L}\in {{\mathbb{R}}}^{N\times 1}\) and vector embedding \({\overrightarrow{v}}_{i}^{L}\in {{\mathbb{R}}}^{N\times 3\times 1}\) are used to predict various molecular properties.

On QM9, the molecular dipole is calculated as follows:

where \({\overrightarrow{r}}_{c}\) denotes the center of mass. Similarly, for the prediction of electronic spatial extent 〈 R 2 〉, we use the following equation:

For the remaining 10 properties y , we simply aggregate the final scalar embedding of nodes as follows:

For models trained on the molecular dynamics datasets including MD17, revised MD17, and Chignolin, the total potential energy is obtained as the sum of the final scalar embedding of the nodes. As an energy-conserving potential, the forces are then calculated using the negative gradients of the predicted total potential energy with respect to the atomic coordinates:

Statistics and reproducibility

For the QM9 dataset, we randomly split it into 110,000 samples as the train set, 10,000 samples as the validation set, and the rest as the test set by following the previous studies 18 , 19 . For the Molecule3D and OGB-LSC PCQM4Mv2 datasets, the splitting has been provided in their paper 32 , 33 .

To evaluate the effectiveness of ViSNet in simulation data, ViSNet was trained on MD17 and rMD17 with a limited data setting, which consists of only 950 uniformly sampled conformations for model training and 50 conformations for validation for each molecule. For the MD22 dataset, we use the same number of molecules as in ref. 30 for training and validation, and the rest as the test set.

Furthermore, the whole Chignolin dataset was randomly split into 80%, 10%, and 10% as the training, validation, and test datasets. Six representative conformations were picked from the test set for illustration.

Experimental settings

For the QM9 dataset, we adopted a batch size of 32 and a learning rate of 1e−4 for all the properties. For the Molecule3D dataset, we adopted a larger batch size of 512 and a learning rate of 2e−4. For the OGB-LSC PCQM4Mv2 dataset, we trained our model in a mixed 2D/3D mode with a batch size of 256 and a learning rate of 2e−4. The mean squared error (MSE) loss was used for model training. For the molecular dynamic dataset including MD17, rMD17, MD22, and Chignolin, we leveraged a combined MSE loss for energy and force prediction. The weight of energy loss was set to 0.05. The weight of force loss was set to 0.95. The batch size was chosen from 2, 4, 8 due to the GPU memory and the learning rate was chosen from 1e−4 to 4e−4 for different molecules. The cutoff was set to 5 for small molecules in QM9, MD17, rMD17, and Molecule3D, and changed to 4 for Chignolin in order to reduce the number of edges in the molecular graphs. For the MD22 dataset, the cutoff of relatively small molecules was set to 5, that of bigger molecules was set to 4. Cutoff was not used in the OGB-LSC PCQM4Mv2 dataset. We used the learning rate decay if the validation loss stopped decreasing. The patience was set to 5 epochs for Molecule3D, 15 epochs for QM9, and 30 epochs for MD17, rMD17, MD22, and Chignolin. The learning rate decay factor was set to 0.8 for these models. Training is stopped if a maximum number of epochs is reached, or the validation loss does not improve for a maximum number of early stopping patience. The ViSNet model trained on the molecular dynamic datasets and Molecule3D had 9 hidden layers and the embedding dimension was set to 256. We used a larger model for the QM9 dataset, i.e., the embedding dimension changed to 512. For the OGB-LSC PCQM4Mv2 dataset, we use the 12-layer and 768-dimension Transformer-M 37 as the backbone. More details about the hyperparameters of ViSNet can be found in Supplementary Table  S4 . Experiments were conducted on NVIDIA 32G-V100 GPUs.

Reporting summary

Further information on research design is available in the  Nature Portfolio Reporting Summary linked to this article.

Data availability

All relevant data supporting the key findings of this study are available within the article and its Supplementary Information files. MD17 dataset [ http://www.quantum-machine.org/gdml/data/npz ], MD22 dataset [ http://www.quantum-machine.org/gdml/data/npz ], rMD17 dataset [ https://archive.materialscloud.org/record/file?filename=rmd17.tar.bz2&record_id=466 ], QM9 dataset [ https://deepchemdata.s3-us-west-1.amazonaws.com/datasets/molnet_publish/qm9.zip ], Molecule3D dataset [ https://github.com/divelab/MoleculeX/tree/molx/Molecule3D ], OGB-LSC PCQM4Mv2 dataset [ https://ogb.stanford.edu/docs/lsc/pcqm4mv2 ] and Chignolin dataset [ https://github.com/microsoft/AI2BMD/tree/ViSNet/chignolin_data ].  Source data are provided with this paper.

Code availability

Most experiments were run with Python with version 3.9.15, Pytorch with version 1.11.0, Pytorch Geometric with version 2.1.0, and Pytorch Lightning with version 1.8.0. The code used to reproduce our results is available at https://github.com/microsoft/AI2BMD/tree/ViSNet 57 . Matplotlib and Seaborn were used for plotting figures.

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Acknowledgements

We would like to express our sincere gratitude to S. Chmiela, H.E. Sauceda, K.R. Müller, and A. Tkatchenko, for their invaluable assistance in performing the simulations and analyzing the vibrational spectra. Their extensive expertise and knowledge greatly contributed to the completion of the supplementary experiments, making our manuscript more solid.

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These authors contributed equally: Yusong Wang, Tong Wang, Shaoning Li.

Authors and Affiliations

Microsoft Research AI4Science, 100080, Beijing, China

Yusong Wang, Tong Wang, Shaoning Li, Xinheng He, Mingyu Li, Zun Wang, Bin Shao & Tie-Yan Liu

National Key Laboratory of Human–Machine Hybrid Augmented Intelligence, National Engineering Research Center for Visual Information and Applications, and Institute of Artificial Intelligence and Robotics, Xi’an Jiaotong University, 710049, Xi’an, China

Yusong Wang & Nanning Zheng

The CAS Key Laboratory of Receptor Research and State Key Laboratory of Drug Research, Shanghai Institute of Materia Medica, Chinese Academy of Sciences, 201203, Shanghai, China

University of Chinese Academy of Sciences, 100049, Beijing, China

Medicinal Chemistry and Bioinformatics Center, School of Medicine, Shanghai Jiaotong University, Shanghai, 200025, China

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Contributions

T.W. led, conceived, and designed the study. T.W. is the lead contact. Y.W., S.L., X.H., and M.L. conducted the work when they were visiting Microsoft Research. S.L., Y.W., and T.W. carried out algorithm design. Y.W., S.L., X.H., and T.W. carried out experiments, evaluations, analysis, and visualization. Y.W. and S.L. wrote the original manuscript. T.W., X.H., M.L., Z.W., and B.S. revised the manuscript. N.Z. and T.-Y.L. contributed to the writing. All authors reviewed the final manuscript.

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Correspondence to Tong Wang or Bin Shao .

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T.W., B.S., and T.-Y.L. have been filing a patent on ViSNet model. The remaining authors declare no competing interests.

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Wang, Y., Wang, T., Li, S. et al. Enhancing geometric representations for molecules with equivariant vector-scalar interactive message passing. Nat Commun 15 , 313 (2024). https://doi.org/10.1038/s41467-023-43720-2

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CHARMed collaboration creates a potent therapy candidate for fatal prion diseases

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Drug development is typically slow: The pipeline from basic research discoveries that provide the basis for a new drug to clinical trials and then production of a widely available medicine can take decades. But decades can feel impossibly far off to someone who currently has a fatal disease. Broad Institute of MIT and Harvard Senior Group Leader Sonia Vallabh is acutely aware of that race against time, because the topic of her research is a neurodegenerative and ultimately fatal disease — fatal familial insomnia, a type of prion disease — that she will almost certainly develop as she ages. 

Vallabh and her husband, Eric Minikel, switched careers and became researchers after they learned that Vallabh carries a disease-causing version of the prion protein gene and that there is no effective therapy for fatal prion diseases. The two now run a lab at the Broad Institute, where they are working to develop drugs that can prevent and treat these diseases, and their deadline for success is not based on grant cycles or academic expectations but on the ticking time bomb in Vallabh’s genetic code.

That is why Vallabh was excited to discover, when she entered into a collaboration with Whitehead Institute for Biomedical Research member Jonathan Weissman, that Weissman’s group likes to work at full throttle. In less than two years, Weissman, Vallabh, and their collaborators have developed a set of molecular tools called CHARMs that can turn off disease-causing genes such as the prion protein gene — as well as, potentially, genes coding for many other proteins implicated in neurodegenerative and other diseases — and they are refining those tools to be good candidates for use in human patients. Although the tools still have many hurdles to pass before the researchers will know if they work as therapeutics, the team is encouraged by the speed with which they have developed the technology thus far.

“The spirit of the collaboration since the beginning has been that there was no waiting on formality,” Vallabh says. “As soon as we realized our mutual excitement to do this, everything was off to the races.”

Co-corresponding authors Weissman and Vallabh and co-first authors Edwin Neumann, a graduate student in Weissman’s lab, and Tessa Bertozzi, a postdoc in Weissman’s lab, describe CHARM — which stands for Coupled Histone tail for Autoinhibition Release of Methyltransferase — in a paper published today in the journal Science .

“With the Whitehead and Broad Institutes right next door to each other, I don’t think there’s any better place than this for a group of motivated people to move quickly and flexibly in the pursuit of academic science and medical technology,” says Weissman, who is also a professor of biology at MIT and a Howard Hughes Medical Institute Investigator. “CHARMs are an elegant solution to the problem of silencing disease genes, and they have the potential to have an important position in the future of genetic medicines.”

To treat a genetic disease, target the gene

Prion disease, which leads to swift neurodegeneration and death, is caused by the presence of misshapen versions of the prion protein. These cause a cascade effect in the brain: the faulty prion proteins deform other proteins, and together these proteins not only stop functioning properly but also form toxic aggregates that kill neurons. The most famous type of prion disease, known colloquially as mad cow disease, is infectious, but other forms of prion disease can occur spontaneously or be caused by faulty prion protein genes.

Most conventional drugs work by targeting a protein. CHARMs, however, work further upstream, turning off the gene that codes for the faulty protein so that the protein never gets made in the first place. CHARMs do this by epigenetic editing, in which a chemical tag gets added to DNA in order to turn off or silence a target gene. Unlike gene editing, epigenetic editing does not modify the underlying DNA — the gene itself remains intact. However, like gene editing, epigenetic editing is stable, meaning that a gene switched off by CHARM should remain off. This would mean patients would only have to take CHARM once, as opposed to protein-targeting medications that must be taken regularly as the cells’ protein levels replenish.

Research in animals suggests that the prion protein isn’t necessary in a healthy adult, and that in cases of disease, removing the protein improves or even eliminates disease symptoms. In a person who hasn’t yet developed symptoms, removing the protein should prevent disease altogether. In other words, epigenetic editing could be an effective approach for treating genetic diseases such as inherited prion diseases. The challenge is creating a new type of therapy.

Fortunately, the team had a good template for CHARM: a research tool called CRISPRoff that Weissman’s group previously developed for silencing genes. CRISPRoff uses building blocks from CRISPR gene editing technology, including the guide protein Cas9 that directs the tool to the target gene. CRISPRoff silences the targeted gene by adding methyl groups, chemical tags that prevent the gene from being transcribed, or read into RNA, and so from being expressed as protein. When the researchers tested CRISPRoff’s ability to silence the prion protein gene, they found that it was effective and stable.

Several of its properties, though, prevented CRISPRoff from being a good candidate for a therapy. The researchers’ goal was to create a tool based on CRISPRoff that was just as potent but also safe for use in humans, small enough to deliver to the brain, and designed to minimize the risk of silencing the wrong genes or causing side effects.

From research tool to drug candidate

Led by Neumann and Bertozzi, the researchers began engineering and applying their new epigenome editor. The first problem that they had to tackle was size, because the editor needs to be small enough to be packaged and delivered to specific cells in the body. Delivering genes into the human brain is challenging; many clinical trials have used adeno-associated viruses (AAVs) as gene-delivery vehicles, but these are small and can only contain a small amount of genetic code. CRISPRoff is way too big; the code for Cas9 alone takes up most of the available space.

The Weissman lab researchers decided to replace Cas9 with a much smaller zinc finger protein (ZFP). Like Cas9, ZFPs can serve as guide proteins to direct the tool to a target site in DNA. ZFPs are also common in human cells, meaning they are less likely to trigger an immune response against themselves than the bacterial Cas9.

Next, the researchers had to design the part of the tool that would silence the prion protein gene. At first, they used part of a methyltransferase, a molecule that adds methyl groups to DNA, called DNMT3A. However, in the particular configuration needed for the tool, the molecule was toxic to the cell. The researchers focused on a different solution: Instead of delivering outside DNMT3A as part of the therapy, the tool is able to recruit the cell’s own DNMT3A to the prion protein gene. This freed up precious space inside of the AAV vector and prevented toxicity.

The researchers also needed to activate DNMT3A. In the cell, DNMT3A is usually inactive until it interacts with certain partner molecules. This default inactivity prevents accidental methylation of genes that need to remain turned on. Neumann came up with an ingenious way around this by combining sections of DNMT3A’s partner molecules and connecting these to ZFPs that bring them to the prion protein gene. When the cell’s DNMT3A comes across this combination of parts, it activates, silencing the gene.

“From the perspectives of both toxicity and size, it made sense to recruit the machinery that the cell already has; it was a much simpler, more elegant solution,” Neumann says. “Cells are already using methyltransferases all of the time, and we’re essentially just tricking them into turning off a gene that they would normally leave turned on.”

Testing in mice showed that ZFP-guided CHARMs could eliminate more than 80 percent of the prion protein in the brain, while previous research has shown that as little as 21 percent elimination can improve symptoms.

Once the researchers knew that they had a potent gene silencer, they turned to the problem of off-target effects. The genetic code for a CHARM that gets delivered to a cell will keep producing copies of the CHARM indefinitely. However, after the prion protein gene is switched off, there is no benefit to this, only more time for side effects to develop, so they tweaked the tool so that after it turns off the prion protein gene, it then turns itself off.

Meanwhile, a complementary project from Broad Institute scientist and collaborator Benjamin Deverman’s lab, focused on brain-wide gene delivery and published in Science on May 17, has brought the CHARM technology one step closer to being ready for clinical trials. Although naturally occurring types of AAV have been used for gene therapy in humans before, they do not enter the adult brain efficiently, making it impossible to treat a whole-brain disease like prion disease. Tackling the delivery problem, Deverman’s group has designed an AAV vector that can get into the brain more efficiently by leveraging a pathway that naturally shuttles iron into the brain. Engineered vectors like this one make a therapy like CHARM one step closer to reality.

Thanks to these creative solutions, the researchers now have a highly effective epigenetic editor that is small enough to deliver to the brain, and that appears in cell culture and animal testing to have low toxicity and limited off-target effects.

“It’s been a privilege to be part of this; it’s pretty rare to go from basic research to therapeutic application in such a short amount of time,” Bertozzi says. “I think the key was forming a collaboration that took advantage of the Weissman lab’s tool-building experience, the Vallabh and Minikel lab’s deep knowledge of the disease, and the Deverman lab’s expertise in gene delivery.”

Looking ahead

With the major elements of the CHARM technology solved, the team is now fine-tuning their tool to make it more effective, safer, and easier to produce at scale, as will be necessary for clinical trials. They have already made the tool modular, so that its various pieces can be swapped out and future CHARMs won’t have to be programmed from scratch. CHARMs are also currently being tested as therapeutics in mice. 

The path from basic research to clinical trials is a long and winding one, and the researchers know that CHARMs still have a way to go before they might become a viable medical option for people with prion diseases, including Vallabh, or other diseases with similar genetic components. However, with a strong therapy design and promising laboratory results in hand, the researchers have good reason to be hopeful. They continue to work at full throttle, intent on developing their technology so that it can save patients’ lives not someday, but as soon as possible.

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• http://orcid.org/0000-0002-2780-7897 Morganne Wilbourne 1 ,
• http://orcid.org/0000-0002-9347-8702 Frances Hand 2 ,
• Sophie McAllister 3 ,
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Many first-time mothers (primiparous) within UK National Health Service (NHS) settings require an obstetric intervention to deliver their babies safely. While the antepartum period allows time for conversations about consent for planned interventions, such as elective caesarean section, current practice is that, in emergencies, consent is addressed in the moments before the intervention takes place. This paper explores whether there are limitations on the validity of consent offered in time-pressured and emotionally charged circumstances, specifically concerning emergency obstetric interventions. Using the legal framework of the Mental Capacity Act, Montgomery v. Lanarkshire Health Board (2015) and McCulloch v Forth Valley Health Board (2023), we argue that while women have the capacity to consent during labour, their autonomy is best supported by providing more information about instrumental delivery (ID) during the antepartum period. This conclusion is supported by some national guidelines, including those developed by the Royal College of Obstetricians and Gynaecologists, but not all. Further, we examine the extent to which these principles are upheld in modern-day practice. Data suggest there is relatively little antepartum information provision regarding ID within NHS settings, and that primiparous women do not report a thorough understanding of ID before labour. Based on these results, and bearing in mind the pressures under which NHS obstetric services currently operate, we recommend further research into patient and clinician perceptions of the consent process for ID. Pending these results, we discuss possible modes of information delivery in future practice.

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Contributors FH and MW have contributed equally to this project and should be considered to have joint first authorship. They are entitled to reference their own name first on curricula vitae. They have planned the project and drafted the original manuscript. SA, LP-L and MB provided revisions to the original manuscript. LP-L provided data from the Oxfordshire Maternity and Neonatal Voices Partnership. MB and SM provided supervision. MB and SM provided clinical insight and ensured accuracy with respect to current NHS practice in obstetrics. MB is the guarantor and accepts full responsibility for the finished work and the conduct of the study, had access to the data and controlled the decision to publish.

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