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Introductory Applied Statistics pp 39–55 Cite as

Statistics and Data Analysis in an ANOVA Model

• Bruce Blaine 2  
• First Online: 06 May 2023

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In Chap. 1 we reviewed a set of fundamental statistical concepts and tools and used them to summarize the properties of a numeric variable. In Chap. 2 we learned that data analysis and interpretation are closely tied to design elements of the study that produced the data, including the statistical model inherent in the research question, how the study sample was created, and whether sample participants were randomly allocated to treatment or comparison groups in the study. Building on the foundation of Chaps. 1 and 2 , starting in this chapter and continuing through Chap. 6 , we cover statistics and data analytic methods for describing relationships between variables.

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Note that we call statistics that describe an X - Y relationship effect size statistics, but that doesn’t imply that there’s a causal effect between the predictor and outcome. Remember, evidence for a causal relationship between x and y is determined by the study design (see Chap. 2 ).

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Blaine, B. (2023). Statistics and Data Analysis in an ANOVA Model. In: Introductory Applied Statistics. Springer, Cham. https://doi.org/10.1007/978-3-031-27741-2_3

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Analysis of variance (ANOVA) comparing means of more than two groups

Hae-young kim.

Department of Dental Laboratory Science and Engineering, College of Health Science & Department of Public Health Science, Graduate School & BK21+ Program in Public Health Sciences, Korea University, Seoul, Korea.

Mean values obtained from different groups with different conditions are frequently compared in clinical studies. For example, two mean bond strengths between tooth surface and resin cement may be compared using the parametric Student's t test when independent groups are subjected to the comparison under the assumptions of normal distribution and equal variances (or standard deviation). In a condition of unequal variances we may apply the Welch's t test as an adaptation of the t test. As the nature and specific shape of distributions are predetermined by the assumption, the t test compares only the locations of the distribution represented by means, which is simple and intuitive. The t statistic is the ratio of mean difference and standard errors of the mean difference.

Even when more than two groups are compared, some researchers erroneously apply the t test by implementing multiple t tests on multiple pairs of means. It is inappropriate because the repetition of the multiple tests may repeatedly add multiple chances of error, which may result in a larger α error level than the pre-set α level. When we try to compare means of three groups, A, B, and C, using the t test, we need to implement 3 pairwise tests, i.e., A vs B, A vs C, and B vs C. Similarly if comparisons are repeated k times in an experiment and the α level 0.05 was set for each comparison, an unacceptably increased total error rate of 1-(0.95) k may be expected for the total comparison procedure in the experiment. For a comparison of more than two group means the one-way analysis of variance (ANOVA) is the appropriate method instead of the t test. As the ANOVA is based on the same assumption with the t test, the interest of ANOVA is on the locations of the distributions represented by means too. Then why is the method comparing several means the 'analysis of variance', rather than 'analysis of means' themselves? It is because that the relative location of the several group means can be more conveniently identified by variance among the group means than comparing many group means directly when number of means are large.

The ANOVA method assesses the relative size of variance among group means (between group variance) compared to the average variance within groups (within group variance). Figure 1 shows two comparative cases which have similar 'between group variances' (the same distance among three group means) but have different 'within group variances'. When the between group variances are the same, mean differences among groups seem more distinct in the distributions with smaller within group variances (a) compared to those with larger within group variances (b). Therefore the ratio of between group variance to within group variance is of the main interest in the ANOVA.

Distributions with the same between group variance. (a) smaller variance within groups; (b) larger variance within groups.

Table 1 displays an artificial data of bond strength according to different resin types and Table 2 shows the result of the one-way ANOVA. The 'SSB' represents the sum of squares between groups which is the variation of group means from the total grand mean, and the mean of squares between groups (MSB) is subsequently obtained by dividing SSB with degrees of freedom. The 'SSW' represents sum of squares within groups which is the sum of squared deviations from the group means and individual observations because the equal variances in all the groups were already assumed. The mean of square within groups (MSW) is subsequently obtained by dividing SSW with degrees of freedom, in the same way. The ratio of MSB and MSW determines the degree of how relatively greater the difference is between group means (between group variance) compared to within group variance. If the ratio is greater than expected by chance we may think not all the group means are the same which means that at least one mean is substantially different. As the result is interpreted about the whole set of groups, it is called as a global or overall test. The ratio of MSB and MSW is known to follow the F distribution. Therefore, to get a statistical conclusion we may compare the F value calculated from the observed data with the critical value at an α error level of 0.05 in the F table.

Measurements of bonding strength according to three different types of resin (artificial data)

One-way ANOVA table

Larger F value implies that means of the groups are greatly different from each other compared to the variation of the individual observations in each groups. Larger F value than the critical value supports that the differences between group means are larger than what would be expected by chance. In this example the critical F value is 3.23 in the F table when the degrees of freedom of numerator and denominator are 2 and 42 respectively at the α error level 0.05. As the observed F value 8.4 is larger than the critical value, the result in Table 2 may be interpreted as statistically significant difference among the means of the groups at the α error level 0.05. The result suggests to rejection of the null hypothesis that all the group means are the same, and coincidently supports that at least one group mean differs from other group means.

If any significant difference is detected by the 'overall F test' above, we need to examine what specific pair of group means shows difference and what pairs do not. While many different kinds of post-hoc multiple comparison procedures have been proposed, the choice needs to be made according to the specific research question. One basic method is implementing multiple pairwise t tests using the common variance as MSW and appropriately adjusting α error level to get the optimal α error level for the whole experiment. For example, the Bonferroni correction is a simple method that adjusts comparisonwise type α error level as the usual experiment-wise α error level divided by the number of comparisons, e.g., 0.05/k. However, caution is needed because in some situations the Bonferroni correction may be substantially conservative that actual experiment-wise α error level applied may be lower than 0.05. Tukey's HSD, Schaffe method, and Duncan multiple range test are more frequently preferred methods for the multiple comparison procedures. Table 3 displays the analysis results by both the ANOVA and multiple comparison procedure. We usually need to report the p -value of overall F test and the result of the post-hoc multiple comparison. Table 3 shows that 'C' resin has the highest bond strength and 'A' resin shows the lowest.

Comparative mean bond strength according to different types of resin (display of ANOVA results)

* Different superscripts mean statistically different.

The comparison of more than two group means by ANOVA using the SPSS statistical package (SPSS Inc., Chicago, Il) according to the following procedures:

The one-way ANOVA test explained

Affiliation.

• 1 University of Limerick, Limerick, Republic of Ireland.
• PMID: 37317616
• DOI: 10.7748/nr.2023.e1885

Background: Quantitative methods and statistical analysis are essential tools in nursing research, as they support researchers testing phenomena, illustrate their findings clearly and accurately, and provide explanation or generalisation of the phenomenon being investigated. The most popular inferential statistics test is the one-way analysis of variance (ANOVA), as it is the test designated for comparing the means of a study's target groups to identify if they are statistically different to the others. However, the nursing literature has identified that statistical tests are not being used correctly and findings are being reported incorrectly.

Aim: To present and explain the one-way ANOVA.

Discussion: The article presents the purpose of inferential statistics and explains one-way ANOVA. It uses relevant examples to examine the steps needed to successfully apply the one-way ANOVA. The authors also provide recommendations for other statistical tests and measurements in parallel to one-way ANOVA.

Conclusion: Nurses need to develop their understanding and knowledge of statistical methods, to engage in research and evidence-based practice.

Implications for practice: This article enhances the understanding and application of one-way ANOVAs by nursing students, novice researchers, nurses and those engaged in academic studies. Nurses, nursing students and nurse researchers need to familiarise themselves with statistical terminology and develop their understanding of statistical concepts, to support evidence-based, quality, safe care.

Keywords: data analysis; quantitative research; research; study design.

©2023 RCN Publishing Company Ltd. All rights reserved. Not to be copied, transmitted or recorded in any way, in whole or part, without prior permission of the publishers.

• Analysis of Variance
• Correlation of Data
• Nursing Research*
• Research Design
• Students, Nursing*

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One-way ANOVA | When and How to Use It (With Examples)

Published on March 6, 2020 by Rebecca Bevans . Revised on June 22, 2023.

ANOVA , which stands for Analysis of Variance, is a statistical test used to analyze the difference between the means of more than two groups.

A one-way ANOVA uses one independent variable , while a two-way ANOVA uses two independent variables.

Table of contents

When to use a one-way anova, how does an anova test work, assumptions of anova, performing a one-way anova, interpreting the results, post-hoc testing, reporting the results of anova, other interesting articles, frequently asked questions about one-way anova.

Use a one-way ANOVA when you have collected data about one categorical independent variable and one quantitative dependent variable . The independent variable should have at least three levels (i.e. at least three different groups or categories).

ANOVA tells you if the dependent variable changes according to the level of the independent variable. For example:

• Your independent variable is social media use , and you assign groups to low , medium , and high levels of social media use to find out if there is a difference in hours of sleep per night .
• Your independent variable is brand of soda , and you collect data on Coke , Pepsi , Sprite , and Fanta to find out if there is a difference in the price per 100ml .
• You independent variable is type of fertilizer , and you treat crop fields with mixtures 1 , 2 and 3 to find out if there is a difference in crop yield .

The null hypothesis ( H 0 ) of ANOVA is that there is no difference among group means. The alternative hypothesis ( H a ) is that at least one group differs significantly from the overall mean of the dependent variable.

If you only want to compare two groups, use a t test instead.

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ANOVA determines whether the groups created by the levels of the independent variable are statistically different by calculating whether the means of the treatment levels are different from the overall mean of the dependent variable.

If any of the group means is significantly different from the overall mean, then the null hypothesis is rejected.

ANOVA uses the F test for statistical significance . This allows for comparison of multiple means at once, because the error is calculated for the whole set of comparisons rather than for each individual two-way comparison (which would happen with a t test).

The F test compares the variance in each group mean from the overall group variance. If the variance within groups is smaller than the variance between groups , the F test will find a higher F value, and therefore a higher likelihood that the difference observed is real and not due to chance.

The assumptions of the ANOVA test are the same as the general assumptions for any parametric test:

• Independence of observations : the data were collected using statistically valid sampling methods , and there are no hidden relationships among observations. If your data fail to meet this assumption because you have a confounding variable that you need to control for statistically, use an ANOVA with blocking variables.
• Normally-distributed response variable : The values of the dependent variable follow a normal distribution .
• Homogeneity of variance : The variation within each group being compared is similar for every group. If the variances are different among the groups, then ANOVA probably isn’t the right fit for the data.

While you can perform an ANOVA by hand , it is difficult to do so with more than a few observations. We will perform our analysis in the R statistical program because it is free, powerful, and widely available. For a full walkthrough of this ANOVA example, see our guide to performing ANOVA in R .

The sample dataset from our imaginary crop yield experiment contains data about:

• fertilizer type (type 1, 2, or 3)
• planting density (1 = low density, 2 = high density)
• planting location in the field (blocks 1, 2, 3, or 4)
• final crop yield (in bushels per acre).

This gives us enough information to run various different ANOVA tests and see which model is the best fit for the data.

For the one-way ANOVA, we will only analyze the effect of fertilizer type on crop yield.

Sample dataset for ANOVA

After loading the dataset into our R environment, we can use the command aov() to run an ANOVA. In this example we will model the differences in the mean of the response variable , crop yield, as a function of type of fertilizer.

To view the summary of a statistical model in R, use the summary() function.

The summary of an ANOVA test (in R) looks like this:

The ANOVA output provides an estimate of how much variation in the dependent variable that can be explained by the independent variable.

• The first column lists the independent variable along with the model residuals (aka the model error).
• The Df column displays the degrees of freedom for the independent variable (calculated by taking the number of levels within the variable and subtracting 1), and the degrees of freedom for the residuals (calculated by taking the total number of observations minus 1, then subtracting the number of levels in each of the independent variables).
• The Sum Sq column displays the sum of squares (a.k.a. the total variation) between the group means and the overall mean explained by that variable. The sum of squares for the fertilizer variable is 6.07, while the sum of squares of the residuals is 35.89.
• The Mean Sq column is the mean of the sum of squares, which is calculated by dividing the sum of squares by the degrees of freedom.
• The F value column is the test statistic from the F test: the mean square of each independent variable divided by the mean square of the residuals. The larger the F value, the more likely it is that the variation associated with the independent variable is real and not due to chance.
• The Pr(>F) column is the p value of the F statistic. This shows how likely it is that the F value calculated from the test would have occurred if the null hypothesis of no difference among group means were true.

Because the p value of the independent variable, fertilizer, is statistically significant ( p < 0.05), it is likely that fertilizer type does have a significant effect on average crop yield.

ANOVA will tell you if there are differences among the levels of the independent variable, but not which differences are significant. To find how the treatment levels differ from one another, perform a TukeyHSD (Tukey’s Honestly-Significant Difference) post-hoc test.

The Tukey test runs pairwise comparisons among each of the groups, and uses a conservative error estimate to find the groups which are statistically different from one another.

The output of the TukeyHSD looks like this:

First, the table reports the model being tested (‘Fit’). Next it lists the pairwise differences among groups for the independent variable.

Under the ‘$fertilizer’ section, we see the mean difference between each fertilizer treatment (‘diff’), the lower and upper bounds of the 95% confidence interval (‘lwr’ and ‘upr’), and the p value , adjusted for multiple pairwise comparisons.

The pairwise comparisons show that fertilizer type 3 has a significantly higher mean yield than both fertilizer 2 and fertilizer 1, but the difference between the mean yields of fertilizers 2 and 1 is not statistically significant.

When reporting the results of an ANOVA, include a brief description of the variables you tested, the  F value, degrees of freedom, and p values for each independent variable, and explain what the results mean.

If you want to provide more detailed information about the differences found in your test, you can also include a graph of the ANOVA results , with grouping letters above each level of the independent variable to show which groups are statistically different from one another:

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Chi square test of independence
• Statistical power
• Descriptive statistics
• Degrees of freedom
• Pearson correlation
• Null hypothesis

Methodology

• Double-blind study
• Case-control study
• Research ethics
• Data collection
• Hypothesis testing
• Structured interviews

Research bias

• Hawthorne effect
• Unconscious bias
• Recall bias
• Halo effect
• Self-serving bias
• Information bias

The only difference between one-way and two-way ANOVA is the number of independent variables . A one-way ANOVA has one independent variable, while a two-way ANOVA has two.

• One-way ANOVA : Testing the relationship between shoe brand (Nike, Adidas, Saucony, Hoka) and race finish times in a marathon.
• Two-way ANOVA : Testing the relationship between shoe brand (Nike, Adidas, Saucony, Hoka), runner age group (junior, senior, master’s), and race finishing times in a marathon.

All ANOVAs are designed to test for differences among three or more groups. If you are only testing for a difference between two groups, use a t-test instead.

A factorial ANOVA is any ANOVA that uses more than one categorical independent variable . A two-way ANOVA is a type of factorial ANOVA.

Some examples of factorial ANOVAs include:

• Testing the combined effects of vaccination (vaccinated or not vaccinated) and health status (healthy or pre-existing condition) on the rate of flu infection in a population.
• Testing the effects of marital status (married, single, divorced, widowed), job status (employed, self-employed, unemployed, retired), and family history (no family history, some family history) on the incidence of depression in a population.
• Testing the effects of feed type (type A, B, or C) and barn crowding (not crowded, somewhat crowded, very crowded) on the final weight of chickens in a commercial farming operation.

In ANOVA, the null hypothesis is that there is no difference among group means. If any group differs significantly from the overall group mean, then the ANOVA will report a statistically significant result.

Significant differences among group means are calculated using the F statistic, which is the ratio of the mean sum of squares (the variance explained by the independent variable) to the mean square error (the variance left over).

If the F statistic is higher than the critical value (the value of F that corresponds with your alpha value, usually 0.05), then the difference among groups is deemed statistically significant.

Quantitative variables are any variables where the data represent amounts (e.g. height, weight, or age).

Categorical variables are any variables where the data represent groups. This includes rankings (e.g. finishing places in a race), classifications (e.g. brands of cereal), and binary outcomes (e.g. coin flips).

You need to know what type of variables you are working with to choose the right statistical test for your data and interpret your results .

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Reporting and Interpreting One-Way Analysis of Variance (ANOVA) Using a Data-Driven Example: A Practical Guide for Social Science Researchers

• Simon NTUMI University of Education, Winneba, West Africa, Ghana

One-way ( between-groups) analysis of variance (ANOVA) is a statistical tool or procedure used to analyse variation in a response variable (continuous random variable) measured under conditions defined by discrete factors (classification variables, often with nominal levels). The tool is used to detect a difference in means of 3 or more independent groups. It compares the means of the samples or groups in order to make inferences about the population means. It can be construed as an extension of the independent t-test. Given the omnibus nature of ANOVA, it appears that most researchers in social sciences and its related fields have difficulties in reporting and interpreting ANOVA results in their studies. This paper provides detailed processes and steps on how researchers can practically analyse and interpret ANOVA in their research works. The paper expounded that in applying ANOVA in analysis, a researcher must first formulate the null and in other cases alternative hypothesis. After the data have been gathered and cleaned, the researcher must test statistical assumptions to see if the data meet those assumptions. After this, the researcher must then do the necessary statistical computations and calculate the F-ratio (ANOVA result) using a software. To this end, the researcher then compares the critical value of the F-ratio with the table value or simply look at the p -value against the established alpha. If the calculated critical value is greater than the table value, the null hypothesis will be rejected and the alternative hypothesis is upheld.

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Home » ANOVA (Analysis of variance) – Formulas, Types, and Examples

ANOVA (Analysis of variance) – Formulas, Types, and Examples

Table of Contents

Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA) is a statistical method used to test differences between two or more means. It is similar to the t-test, but the t-test is generally used for comparing two means, while ANOVA is used when you have more than two means to compare.

ANOVA is based on comparing the variance (or variation) between the data samples to the variation within each particular sample. If the between-group variance is high and the within-group variance is low, this provides evidence that the means of the groups are significantly different.

ANOVA Terminology

When discussing ANOVA, there are several key terms to understand:

• Factor : This is another term for the independent variable in your analysis. In a one-way ANOVA, there is one factor, while in a two-way ANOVA, there are two factors.
• Levels : These are the different groups or categories within a factor. For example, if the factor is ‘diet’ the levels might be ‘low fat’, ‘medium fat’, and ‘high fat’.
• Response Variable : This is the dependent variable or the outcome that you are measuring.
• Within-group Variance : This is the variance or spread of scores within each level of your factor.
• Between-group Variance : This is the variance or spread of scores between the different levels of your factor.
• Grand Mean : This is the overall mean when you consider all the data together, regardless of the factor level.
• Treatment Sums of Squares (SS) : This represents the between-group variability. It is the sum of the squared differences between the group means and the grand mean.
• Error Sums of Squares (SS) : This represents the within-group variability. It’s the sum of the squared differences between each observation and its group mean.
• Total Sums of Squares (SS) : This is the sum of the Treatment SS and the Error SS. It represents the total variability in the data.
• Degrees of Freedom (df) : The degrees of freedom are the number of values that have the freedom to vary when computing a statistic. For example, if you have ‘n’ observations in one group, then the degrees of freedom for that group is ‘n-1’.
• Mean Square (MS) : Mean Square is the average squared deviation and is calculated by dividing the sum of squares by the corresponding degrees of freedom.
• F-Ratio : This is the test statistic for ANOVAs, and it’s the ratio of the between-group variance to the within-group variance. If the between-group variance is significantly larger than the within-group variance, the F-ratio will be large and likely significant.
• Null Hypothesis (H0) : This is the hypothesis that there is no difference between the group means.
• Alternative Hypothesis (H1) : This is the hypothesis that there is a difference between at least two of the group means.
• p-value : This is the probability of obtaining a test statistic as extreme as the one that was actually observed, assuming that the null hypothesis is true. If the p-value is less than the significance level (usually 0.05), then the null hypothesis is rejected in favor of the alternative hypothesis.
• Post-hoc tests : These are follow-up tests conducted after an ANOVA when the null hypothesis is rejected, to determine which specific groups’ means (levels) are different from each other. Examples include Tukey’s HSD, Scheffe, Bonferroni, among others.

Types of ANOVA

Types of ANOVA are as follows:

One-way (or one-factor) ANOVA

This is the simplest type of ANOVA, which involves one independent variable . For example, comparing the effect of different types of diet (vegetarian, pescatarian, omnivore) on cholesterol level.

Two-way (or two-factor) ANOVA

This involves two independent variables. This allows for testing the effect of each independent variable on the dependent variable , as well as testing if there’s an interaction effect between the independent variables on the dependent variable.

Repeated Measures ANOVA

This is used when the same subjects are measured multiple times under different conditions, or at different points in time. This type of ANOVA is often used in longitudinal studies.

Mixed Design ANOVA

This combines features of both between-subjects (independent groups) and within-subjects (repeated measures) designs. In this model, one factor is a between-subjects variable and the other is a within-subjects variable.

Multivariate Analysis of Variance (MANOVA)

This is used when there are two or more dependent variables. It tests whether changes in the independent variable(s) correspond to changes in the dependent variables.

Analysis of Covariance (ANCOVA)

This combines ANOVA and regression. ANCOVA tests whether certain factors have an effect on the outcome variable after removing the variance for which quantitative covariates (interval variables) account. This allows the comparison of one variable outcome between groups, while statistically controlling for the effect of other continuous variables that are not of primary interest.

Nested ANOVA

This model is used when the groups can be clustered into categories. For example, if you were comparing students’ performance from different classrooms and different schools, “classroom” could be nested within “school.”

ANOVA Formulas

ANOVA Formulas are as follows:

Sum of Squares Total (SST)

This represents the total variability in the data. It is the sum of the squared differences between each observation and the overall mean.

• yi represents each individual data point
• y_mean represents the grand mean (mean of all observations)

Sum of Squares Within (SSW)

This represents the variability within each group or factor level. It is the sum of the squared differences between each observation and its group mean.

• yij represents each individual data point within a group
• y_meani represents the mean of the ith group

Sum of Squares Between (SSB)

This represents the variability between the groups. It is the sum of the squared differences between the group means and the grand mean, multiplied by the number of observations in each group.

• ni represents the number of observations in each group
• y_mean represents the grand mean

Degrees of Freedom

The degrees of freedom are the number of values that have the freedom to vary when calculating a statistic.

For within groups (dfW):

For between groups (dfB):

For total (dfT):

• N represents the total number of observations
• k represents the number of groups

Mean Squares

Mean squares are the sum of squares divided by the respective degrees of freedom.

Mean Squares Between (MSB):

Mean Squares Within (MSW):

F-Statistic

The F-statistic is used to test whether the variability between the groups is significantly greater than the variability within the groups.

If the F-statistic is significantly higher than what would be expected by chance, we reject the null hypothesis that all group means are equal.

Examples of ANOVA

Examples 1:

Suppose a psychologist wants to test the effect of three different types of exercise (yoga, aerobic exercise, and weight training) on stress reduction. The dependent variable is the stress level, which can be measured using a stress rating scale.

Here are hypothetical stress ratings for a group of participants after they followed each of the exercise regimes for a period:

• Yoga: [3, 2, 2, 1, 2, 2, 3, 2, 1, 2]
• Aerobic Exercise: [2, 3, 3, 2, 3, 2, 3, 3, 2, 2]
• Weight Training: [4, 4, 5, 5, 4, 5, 4, 5, 4, 5]

The psychologist wants to determine if there is a statistically significant difference in stress levels between these different types of exercise.

To conduct the ANOVA:

1. State the hypotheses:

• Null Hypothesis (H0): There is no difference in mean stress levels between the three types of exercise.
• Alternative Hypothesis (H1): There is a difference in mean stress levels between at least two of the types of exercise.

2. Calculate the ANOVA statistics:

• Compute the Sum of Squares Between (SSB), Sum of Squares Within (SSW), and Sum of Squares Total (SST).
• Calculate the Degrees of Freedom (dfB, dfW, dfT).
• Calculate the Mean Squares Between (MSB) and Mean Squares Within (MSW).
• Compute the F-statistic (F = MSB / MSW).

3. Check the p-value associated with the calculated F-statistic.

• If the p-value is less than the chosen significance level (often 0.05), then we reject the null hypothesis in favor of the alternative hypothesis. This suggests there is a statistically significant difference in mean stress levels between the three exercise types.

4. Post-hoc tests

• If we reject the null hypothesis, we conduct a post-hoc test to determine which specific groups’ means (exercise types) are different from each other.

Examples 2:

Suppose an agricultural scientist wants to compare the yield of three varieties of wheat. The scientist randomly selects four fields for each variety and plants them. After harvest, the yield from each field is measured in bushels. Here are the hypothetical yields:

The scientist wants to know if the differences in yields are due to the different varieties or just random variation.

Here’s how to apply the one-way ANOVA to this situation:

• Null Hypothesis (H0): The means of the three populations are equal.
• Alternative Hypothesis (H1): At least one population mean is different.
• Calculate the Degrees of Freedom (dfB for between groups, dfW for within groups, dfT for total).
• If the p-value is less than the chosen significance level (often 0.05), then we reject the null hypothesis in favor of the alternative hypothesis. This would suggest there is a statistically significant difference in mean yields among the three varieties.
• If we reject the null hypothesis, we conduct a post-hoc test to determine which specific groups’ means (wheat varieties) are different from each other.

How to Conduct ANOVA

Conducting an Analysis of Variance (ANOVA) involves several steps. Here’s a general guideline on how to perform it:

• Null Hypothesis (H0): The means of all groups are equal.
• Alternative Hypothesis (H1): At least one group mean is different from the others.
• The significance level (often denoted as α) is usually set at 0.05. This implies that you are willing to accept a 5% chance that you are wrong in rejecting the null hypothesis.
• Data should be collected for each group under study. Make sure that the data meet the assumptions of an ANOVA: normality, independence, and homogeneity of variances.
• Calculate the Degrees of Freedom (df) for each sum of squares (dfB, dfW, dfT).
• Compute the Mean Squares Between (MSB) and Mean Squares Within (MSW) by dividing the sum of squares by the corresponding degrees of freedom.
• Compute the F-statistic as the ratio of MSB to MSW.
• Determine the critical F-value from the F-distribution table using dfB and dfW.
• If the calculated F-statistic is greater than the critical F-value, reject the null hypothesis.
• If the p-value associated with the calculated F-statistic is smaller than the significance level (0.05 typically), you reject the null hypothesis.
• If you rejected the null hypothesis, you can conduct post-hoc tests (like Tukey’s HSD) to determine which specific groups’ means (if you have more than two groups) are different from each other.
• Regardless of the result, report your findings in a clear, understandable manner. This typically includes reporting the test statistic, p-value, and whether the null hypothesis was rejected.

When to use ANOVA

ANOVA (Analysis of Variance) is used when you have three or more groups and you want to compare their means to see if they are significantly different from each other. It is a statistical method that is used in a variety of research scenarios. Here are some examples of when you might use ANOVA:

• Comparing Groups : If you want to compare the performance of more than two groups, for example, testing the effectiveness of different teaching methods on student performance.
• Evaluating Interactions : In a two-way or factorial ANOVA, you can test for an interaction effect. This means you are not only interested in the effect of each individual factor, but also whether the effect of one factor depends on the level of another factor.
• Repeated Measures : If you have measured the same subjects under different conditions or at different time points, you can use repeated measures ANOVA to compare the means of these repeated measures while accounting for the correlation between measures from the same subject.
• Experimental Designs : ANOVA is often used in experimental research designs when subjects are randomly assigned to different conditions and the goal is to compare the means of the conditions.

Here are the assumptions that must be met to use ANOVA:

• Normality : The data should be approximately normally distributed.
• Homogeneity of Variances : The variances of the groups you are comparing should be roughly equal. This assumption can be tested using Levene’s test or Bartlett’s test.
• Independence : The observations should be independent of each other. This assumption is met if the data is collected appropriately with no related groups (e.g., twins, matched pairs, repeated measures).

Applications of ANOVA

The Analysis of Variance (ANOVA) is a powerful statistical technique that is used widely across various fields and industries. Here are some of its key applications:

Agriculture

ANOVA is commonly used in agricultural research to compare the effectiveness of different types of fertilizers, crop varieties, or farming methods. For example, an agricultural researcher could use ANOVA to determine if there are significant differences in the yields of several varieties of wheat under the same conditions.

Manufacturing and Quality Control

ANOVA is used to determine if different manufacturing processes or machines produce different levels of product quality. For instance, an engineer might use it to test whether there are differences in the strength of a product based on the machine that produced it.

Marketing Research

Marketers often use ANOVA to test the effectiveness of different advertising strategies. For example, a marketer could use ANOVA to determine whether different marketing messages have a significant impact on consumer purchase intentions.

Healthcare and Medicine

In medical research, ANOVA can be used to compare the effectiveness of different treatments or drugs. For example, a medical researcher could use ANOVA to test whether there are significant differences in recovery times for patients who receive different types of therapy.

ANOVA is used in educational research to compare the effectiveness of different teaching methods or educational interventions. For example, an educator could use it to test whether students perform significantly differently when taught with different teaching methods.

Psychology and Social Sciences

Psychologists and social scientists use ANOVA to compare group means on various psychological and social variables. For example, a psychologist could use it to determine if there are significant differences in stress levels among individuals in different occupations.

Biology and Environmental Sciences

Biologists and environmental scientists use ANOVA to compare different biological and environmental conditions. For example, an environmental scientist could use it to determine if there are significant differences in the levels of a pollutant in different bodies of water.

Advantages of ANOVA

Here are some advantages of using ANOVA:

Comparing Multiple Groups: One of the key advantages of ANOVA is the ability to compare the means of three or more groups. This makes it more powerful and flexible than the t-test, which is limited to comparing only two groups.

Control of Type I Error: When comparing multiple groups, the chances of making a Type I error (false positive) increases. One of the strengths of ANOVA is that it controls the Type I error rate across all comparisons. This is in contrast to performing multiple pairwise t-tests which can inflate the Type I error rate.

Testing Interactions: In factorial ANOVA, you can test not only the main effect of each factor, but also the interaction effect between factors. This can provide valuable insights into how different factors or variables interact with each other.

Handling Continuous and Categorical Variables: ANOVA can handle both continuous and categorical variables . The dependent variable is continuous and the independent variables are categorical.

Robustness: ANOVA is considered robust to violations of normality assumption when group sizes are equal. This means that even if your data do not perfectly meet the normality assumption, you might still get valid results.

Provides Detailed Analysis: ANOVA provides a detailed breakdown of variances and interactions between variables which can be useful in understanding the underlying factors affecting the outcome.

Capability to Handle Complex Experimental Designs: Advanced types of ANOVA (like repeated measures ANOVA, MANOVA, etc.) can handle more complex experimental designs, including those where measurements are taken on the same subjects over time, or when you want to analyze multiple dependent variables at once.

Disadvantages of ANOVA

Some limitations or disadvantages that are important to consider:

Assumptions: ANOVA relies on several assumptions including normality (the data follows a normal distribution), independence (the observations are independent of each other), and homogeneity of variances (the variances of the groups are roughly equal). If these assumptions are violated, the results of the ANOVA may not be valid.

Sensitivity to Outliers: ANOVA can be sensitive to outliers. A single extreme value in one group can affect the sum of squares and consequently influence the F-statistic and the overall result of the test.

Dichotomous Variables: ANOVA is not suitable for dichotomous variables (variables that can take only two values, like yes/no or male/female). It is used to compare the means of groups for a continuous dependent variable.

Lack of Specificity: Although ANOVA can tell you that there is a significant difference between groups, it doesn’t tell you which specific groups are significantly different from each other. You need to carry out further post-hoc tests (like Tukey’s HSD or Bonferroni) for these pairwise comparisons.

Complexity with Multiple Factors: When dealing with multiple factors and interactions in factorial ANOVA, interpretation can become complex. The presence of interaction effects can make main effects difficult to interpret.

Requires Larger Sample Sizes: To detect an effect of a certain size, ANOVA generally requires larger sample sizes than a t-test.

Equal Group Sizes: While not always a strict requirement, ANOVA is most powerful and its assumptions are most likely to be met when groups are of equal or similar sizes.

About the author

Muhammad Hassan

Researcher, Academic Writer, Web developer

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Statistics Made Easy

The Complete Guide: How to Report ANOVA Results

A one-way ANOVA is used to determine whether or not there is a statistically significant difference between the means of three or more independent groups.

When reporting the results of a one-way ANOVA, we always use the following general structure:

• A brief description of the independent and dependent variable.
• The overall F-value of the ANOVA and the corresponding p-value.
• The results of the post-hoc comparisons (if the p-value was statistically significant).

Here’s the exact wording we can use:

A one-way ANOVA was performed to compare the effect of [independent variable] on [dependent variable].   A one-way ANOVA revealed that there [was or was not] a statistically significant difference in [dependent variable] between at least two groups (F(between groups df, within groups df) = [F-value], p = [p-value]).   Tukey’s HSD Test for multiple comparisons found that the mean value of [dependent variable] was significantly different between [group name] and [group name] (p = [p-value], 95% C.I. = [lower, upper]).   There was no statistically significant difference between [group name] and [group name] (p=[p-value]).

The following example shows how to report the results of a one-way ANOVA in practice.

Example: Reporting the Results of a One-Way ANOVA

Suppose a researcher recruits 30 students to participate in a study. The students are randomly assigned to use one of three studying techniques for the next month to prepare for an exam. At the end of the month, all of the students take the same test.

The researcher then performs a one-way ANOVA to determine if there is a difference in mean exam scores between the three groups.

The following table shows the results of the one-way ANOVA along with the Tukey post-hoc multiple comparisons table:

Here is how to report the results of the one-way ANOVA:

A one-way ANOVA was performed to compare the effect of three different studying techniques on exam scores.   A one-way ANOVA revealed that there was a statistically significant difference in mean exam score between at least two groups (F(2, 27) = [4.545], p = 0.02).   Tukey’s HSD Test for multiple comparisons found that the mean value of exam score was significantly different between technique 1 and technique 2 (p = 0.024, 95% C.I. = [-14.48, -0.92]).   There was no statistically significant difference in mean exam scores between technique 1 and technique 3 (p=0.883) or between technique 2 and technique 3 (p=0.067).

Things to Keep in Mind

Here are a few things to keep in mind when reporting the results of a one-way ANOVA:

Use a descriptive statistics table.

It can be helpful to present a descriptive statistics table that shows the mean and standard deviation of values in each treatment group as well to give the reader a more complete picture of the data.

For example, SPSS produces the following descriptive statistics table that shows the mean and standard deviation of exam scores for students in each of the three study technique groups:

Only report post-hoc results if necessary.

If the overall p-value of the ANOVA is not statistically significant, then you will not conduct post-hoc multiple comparisons between groups. This means you obviously don’t have to report any post-hoc results in the final report.

If you do have to conduct post-hoc tests, the Tukey HSD test is the most commonly used one but occasionally you may use the Scheffe or Bonferroni test instead.

Round p-values when necessary.

As a general rule of thumb, the overall F value and any p-values in ANOVA results are rounded to either two or three decimal places for brevity.

No matter how many decimal places you choose to use, be sure to be consistent throughout the report.

Additional Resources

The following tutorials explain how to report other statistical tests and procedures in APA format:

How to Report Two-Way ANOVA Results (With Examples) How to Report Cronbach’s Alpha (With Examples) How to Report t-Test Results (With Examples) How to Report Chi-Square Results (With Examples) How to Report Pearson’s Correlation (With Examples) How to Report Regression Results (With Examples)

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• Published: 09 April 2024

Anger is eliminated with the disposal of a paper written because of provocation

• Yuta Kanaya 1 &
• Nobuyuki Kawai   ORCID: orcid.org/0000-0003-0372-1703 2   nAff1  

Scientific Reports volume  14 , Article number:  7490 ( 2024 ) Cite this article

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• Human behaviour

Anger suppression is important in our daily life, as its failure can sometimes lead to the breaking down of relationships in families. Thus, effective strategies to suppress or neutralise anger have been examined. This study shows that physical disposal of a piece of paper containing one’s written thoughts on the cause of a provocative event neutralises anger, while holding the paper did not. In this study, participants wrote brief opinions about social problems and received a handwritten, insulting comment consisting of low evaluations about their composition from a confederate. Then, the participants wrote the cause and their thoughts about the provocative event. Half of the participants (disposal group) disposed of the paper in the trash can (Experiment 1) or in the shredder (Experiment 2), while the other half (retention group) kept it in a file on the desk. All the participants showed an increased subjective rating of anger after receiving the insulting feedback. However, the subjective anger for the disposal group decreased as low as the baseline period, while that of the retention group was still higher than that in the baseline period in both experiments. We propose this method as a powerful and simple way to eliminate anger.

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Experiment 1

Introduction.

The need to control anger has been of importance for a long time in human societies, as inferred by a philosopher in Imperium Romanum who had already explored how to cease being angry 1 . However, it can still be challenging to suppress anger effectively. Frequent, unregulated anger often leads to violence towards children 2 , which has become an increasingly prevalent issue. One study found that the global estimate for children experiencing any form of violence (physical, sexual, emotional, or a combination) in the past year is one billion children aged 2–17 years 3 . The number of child abuse cases in Japan has reportedly doubled in the past decade 4 . Children learn about appropriate emotional expression and behaviour from their parents 5 , and children who have been maltreated may lack the opportunity to learn how to regulate anger. Consequently, these maltreated children may have difficulty controlling their own anger 6 , recognising anger in others 7 , and tend to exhibit externalizing behaviour problems 8 . These studies suggest that parental anger regulation issues negatively affect children’s emotional competence. Therefore, an effective way of reducing anger has been examined throughout the years 9 .

However, simply attempting to suppress anger is usually not effective 10 . Both cognitive reappraisal and distraction (i.e., thinking about something other than provocative comments) could reduce anger; however, distraction could suppress anger only for a transient period of time 11 . Cognitive reappraisal refers to the reinterpretation or modification of the meaning of an unpleasant situation. Although reappraisal is considered as an effective way to reduce anger 12 , it requires greater cognitive effort 13 , 14 . Therefore, reappraisal under stressful situations which require cognitive load was not found to be effective in reducing anger as compared to non-stressful situations 15 . Self-distancing, which may be responsible for the anger-reducing effect of reappraisal 12 is also considered as an effective way to reduce anger. Nevertheless, self-distancing or reflection on one’s provocation from a distance is often not feasible, especially in the heat of the moment 13 .

Failure to reduce anger can lead an individual to think about a provocative event repeatedly. Such ruminations are often produced in a self-immersed, experiential manner 16 . Self-immersed experiential rumination can lead to reliving past provocative events 17 , thus maintaining or even increasing subjective anger and vascular responses 18 .

However, among the types of ruminations, writing down a provocation event does not always maintain or increase anger; instead, anger is suppressed depending on the way of writing. For instance, anger was suppressed when participants wrote down the anger-inducing event in a detached, informational, ‘cool’ manner. However, their anger was not suppressed (and was maintained or even increased) when they failed to write down the event in an analytical manner, and wrote it down in a ‘hot’ (emotional) manner 12 . Somewhat relevant here is the expressive writing technique 19 , which is frequently used in emotion-focused psychotherapy treatment 20 . It is believed to be effective in suppressing anger in clinical settings. However, only one experimental study using this technique has been conducted, wherein it was found that there was a significant likelihood of reduced anger when sentences about the emotion were written in the past tense 21 . These studies suggest that anger may be successfully suppressed if individuals are able to separate their internal experience of provocative events from their sense of self 22 . Healy et al. 23 reported that negative self-referential statements (‘my life is pointless’), when presented in a defused format (‘I am having a thought that my life is pointless’), could decrease the emotional discomfort related to that statement.

These previous studies emphasised the cognitive processes (such as goals or valuations) that occur almost entirely inside individuals’ heads 24 . However, if we look at the literature more broadly, studies on emotion regulation (a situated cognitive approach) have demonstrated successful emotion control through dynamic interplay between the person and the situation 24 , 25 . From this situated cognition perspective, people perceive their environment in terms of the possibilities for the kinds of actions that they would pursue. These functional features of the environment (affordances) do not solely exist inside an individual’s mind but instead have a physical reality that exists in the individual’s relationship with the environment. For instance, people frequently use physical substances to modify their moods. People may take a hot shower when they feel lonely 26 , 27 or hold a teddy bear when they feel afraid 28 . Such access to physical objects can significantly modify individuals’ ability to manage their emotions.

In this study, we developed a new anger reduction strategy inspired by the situated cognition approach to emotion regulation 24 . Relevant to this approach, the notion of a grounded procedure of separation 29 also assumes that mental representations and functions are grounded in one’s own experiences and interactions with physical reality. For instance, if people want to take revenge through permanent removal (e.g. hatred for ex), they may destroy a related entity such that it is no longer recognisable (burn, melt, or tear related). In a related study, Briñol et al. 30 reported that writing down negative thoughts about a Mediterranean diet on a piece of paper and disposing of the paper in a trash can result in lower negative (more positive) evaluations of the diet, compared to a group that kept the paper in a booklet. These attitude changes may derive from the cognitive fusion that people often fuse with physical objects, such as jewellery, cars, and family heirlooms 31 . Such fused objects are valued more and are less likely to be abandoned because doing so means losing a part of themselves 32 , 33 . Specifically, throwing an object associated with negative emotions (anger) may result in losing the negative emotions (anger). However, to the best of our knowledge, no study has tested whether the disposal of anger-written paper can reduce or even eliminate anger.

Previous studies from a situated cognitive approach to anger management have changed the external environment of the individual in anger. Tool (object) use has received scant attention in these situated cognition approaches to anger management, except for a few studies, such as hitting a punching bag 34 and playing a video game 35 . This study examined a method in which the disposal of a paper (object) on which participants wrote down their descriptions or thoughts about a provocative event could neutralise anger. Participants threw the anger-written paper into a trash box in Experiment 1, and put the paper into a shredder in Experiment 2. If the action of disposal is crucial to modifying emotions, anger would be reduced only in participants in Experiment 1 but not in Experiment 2, as predicted by the grounded separation procedure 29 . Nevertheless, if anger was modified by the meaning of disposal, the subjective ratings of anger would be eliminated in both experiments. The disposal of the paper with the written descriptions would remove the psychological existence of anger for the provoked participants along with the disposal of paper by the dynamic interactions with the object 24 . This simple method of eliminating anger could potentially contribute to effective parental anger management toward their children.

Materials and method

Participants.

A total of 57 students (women = 21, mean age = 21.11, SD  = 1.05) from a local university participated in this experiment. The data from seven participants were excluded from the final analysis because they correctly guessed the purpose of the experiment and they did not express induced anger by insult (subjective ratings of anger were lower or the same compared to those of the baseline), as was the case in a previous study 36 . Our final analysis included 50 participants (women = 16, mean age = 21.10, SD  = 1.08). A sample size of 50 participants was determined by G*Power 3.1.9.4 37 using the a priori procedure for repeated measures ANOVA, within (periods)—between (disposal and retention) interaction with the parameters of 95% power, an expected effect size of 0.25 (defined as a medium effect by Cohen 38 ), alpha level of 0.05, a within-subjects measurement correlation of 0.5, and a nonsphericity correction ε of 1. The calculation suggested a sample size of 22 participants in each group. Based on these analyses, we concluded that the sample size was appropriate for this study.

Angry feelings were assessed with five adjective items: angry, bothered, annoyed, hostile, and irritated. These adjectives were previously used as measures of self-reported anger 39 . In this study, each response scale ranged from 1 (not at all) to 6 (extremely). As was the case in a previous study on anger 40 , scores on these five adjectives were averaged to form an anger experience composite, which was the score used in the analysis (Cronbach’s α = 0.90). We also used Positive and Negative Affect Schedule (PANAS) as a subjective scale to assess mainly negative feelings 38 . We used the Japanese version of the 6-point PANAS scale 41 .

In this experiment, participants' subjective emotional states were measured at three time points (baseline, post-provocation, and post-writing). The participants were told to write an essay on social problems (e.g., smoking in public) for which they would receive feedback from a doctoral student assessing the quality of the essay. They had seen the doctoral student before entering the experimental room. After the participants wrote the essay, they completed the PANAS and anger questionnaires for the baseline. The evaluation by the fictitious doctoral student was then provided to the participants. The evaluation included ratings of the essay on six characteristics using a 9-point scale (e.g. for intelligence, 1 = unintelligent, 9 = intelligent). All participants were given the following ratings: intelligence = 3, interest = 3, friendliness = 2, logic = 3, respectability = 4, and rationality = 3. Each essay was also provided with the following comment: ‘I cannot believe an educated person would think like this. I hope this person learns something while at the university’ 40 , 42 . All of these manipulations were successfully used in our previous study 40 . The participants were required to read the feedback ratings and comments silently for two minutes. Then, they filled out the subjective emotional questionnaires (PANAS and anger adjectives) for the post-provocation period.

Then, the participants were asked to write every thought of them on receiving the feedback and were given three minutes for this. The instruction was ‘Think about the event from your own perspective. Concentrate especially on the things that originally triggered the emotions and your reactions’. We added guide questions (‘Why were you feeling this way?’, ‘What made you feel this way?’) to induce analytical rumination. To allow the participants to write about their honest feelings, they were informed that the written paper would not be seen by anyone, including the experimenter. After writing, the participants were asked to review the sentences carefully for 30 s. For the retention group, the paper was turned over, put in a clear plastic folder, and placed on the right side of the desk. The participants in the disposal group rolled up the paper into a crumpled ball, stood up, threw the paper into the trash can held by the experimenter, and sat back in the chair. Finally, both groups of participants filled out the subjective emotional questionnaires (anger adjectives and PANAS) for the post-writing period. At the end of the experiment, all participants were debriefed and informed of the truth. They were also assured that the evaluations of their essays had been prepared in advance.

Data analyses

Angry feelings were analysed using a 2 (group: disposal or retention) × 3 (period: at baseline, post-provocation, and post-writing) ANOVA. All significance levels were set at p  < 0.05. We used the Greenhouse–Geisser correction when Mauchly’s test of sphericity was violated. When the interaction was significant, multiple comparisons using the Bonferroni correction method were used to assess the differences.

We also report Bayes factors (BFs) from the Bayesian repeated measures ANOVA in JASP 43 . For BFs, BF 10 values reflect the probability of an alternative relative to the null hypothesis. BFs greater than 3 indicate support for the hypotheses. A BF favouring the alternative over the null hypothesis (BF 10 ) offers strong evidence for the alternative hypothesis when it is over 10. Values less than 0.33 indicate support for the null hypothesis, and values between 0.33 and 3 indicate data insensitivity. We also reported 95% confidence intervals.

We aimed to examine (1) whether angry feelings resumed in the disposal group, and (2) whether angry feelings were different between the groups after the disposal or retention treatments. Our main interest was angry feelings, while we also verified PANAS scores using a 2 (group: disposal or retention) × 3 (period: at baseline, post-provocation, and post-writing) ANOVA.

Ethics statement

All participants were paid for their participation and had provided written informed consent in accordance with the procedures before participation. The study was approved by the Ethics Committee of the Department of Cognitive and Psychological Sciences at Nagoya University (201104-C-02–02). All methods were carried out in accordance with the ethical guidelines of the Declaration of Helsinki. All participants provided their written and informed consent prior to starting the study.

Anger experience

The left panel of Fig.  1 shows mean subjective ratings of anger for disposal and retention groups at three time points (baseline, post-provocation, and post-writing). Subjective ratings of anger of both groups increased at the post-provocation ( M disposal  = 3.34, SD  = 1.20, 95% CI [2.86, 3.82]; M retention  = 3.45, SD  = 1.11, 95% CI [3.00, 3.89]) from the baseline ( M disposal  = 1.59, SD  = 0.50, 95% CI [1.39, 1.79]; M retention  = 1.78, SD  = 0.71, 95% CI [1.50, 2.07]). Subjective ratings at the post-writing decreased from the post-provocation, however those of retention group were still higher than the baseline ( M retention  = 2.64, SD  = 0.95, 95% CI [2.26, 3.02]), while those of disposal group eliminated at the same level of the baseline ( M disposal  = 1.87, SD  = 0.71, 95% CI [1.59, 2.16]). A 2 (group: disposal or retention) × 3 (period: at baseline, post-provocation, and post-writing) mixed model analysis of variance (ANOVA) revealed a significant main effect of period [ F (2, 96) = 73.36, p  < 0.001, partial η 2  = 0.60, BF 10  > 100], while a main effect of group was not significant [ F (1, 48) = 3.21, p  > 0.05, partial η 2  = 0.06, BF 10  = 0.66]. The interaction between group and period was significant [ F (2, 96) = 3.12, p  < 0.05, partial η 2  = 0.06, BF 10  = 1.17]. Multiple comparisons with the Bonferroni method revealed that the subjective anger was significantly higher at the post-provocation than those at the baseline ( p  < 0.05), indicating that a provocative manipulation was exerted. Subjective ratings of anger post-writing decreased significantly, compared to post-provocation ( p  < 0.05). Importantly, however, subjective ratings of retention group at the post-writing period were still significantly higher than those of the baseline period ( p  < 0.05), whereas those of disposal group at the post-writing period eliminated to levels of the baseline period ( p  > 0.05). Subjective ratings of disposal group at the post-writing period were significantly lower than those of retention group ( p  < 0.01).

Self-reported anger during Experiment 1 (left) and Experiment 2 (right). Significant differences emerged at the end of time due to experimental manipulations. Possible values for anger range from 1 to 6. Each vertical line illustrates the 95% confidence intervals for each group.

Negative and positive affect

The negative affect subscale of the PANAS at post-provocation ( M disposal  = 3.10, SD  = 1.00, 95% CI [2.70, 3.49]; M retention  = 3.06, SD  = 1.03, 95% CI [2.64, 3.47]) was higher than at baseline ( M disposal  = 2.45, SD  = 0.66, 95% CI [2.18, 2.71]; M retention  = 2.50, SD  = 0.84, 95% CI [2.16, 2.83]) and post-writing ( M disposal  = 2.06, SD  = 0.65, 95% CI [1.80, 2.32]; M retention  = 2.39, SD  = 0.88, 95% CI [2.04, 2.73]). The 95% CIs of the disposal group overlapped a little bit between post-provocation [2.70, 3.49] and baseline periods [2.18, 2.71], and those of the retention group overlapped between both the post-provocation [2.64, 3.47] and baseline [2.16, 2.83]. The 95% CIs for the post-writing means partially overlapped between the groups. A 2 (group) × 3 (period) mixed ANOVA revealed a significant main effect of period [ F (2, 96) = 28.64, p  < 0.001, partial η 2  = 0.37, BF 10  > 100]. However, the main effect of group [ F (1, 48) = 0.29, p  > 0.05, partial η 2  = 0.01, BF 10  = 0.32] and the interaction between group and period were not significant [ F (2, 96) = 1.35, p  > 0.05, partial η 2  = 0.03, BF 10  = 0.31]. Multiple comparisons with the Bonferroni method revealed that the subjective negative affect post-provocation was significantly higher than at baseline and post-writing ( ps  < 0.05).

The PANAS positive affect subscale showed little variation at three periods ( M disposal  = 2.33, SD  = 0.80, 95% CI [2.01, 2.65]; M retention  = 2.32, SD  = 0.75, 95% CI [2.01, 2.62]), post-provocation ( M disposal  = 2.44, SD  = 0.76, 95% CI [2.13, 2.75]; M retention  = 2.42, SD  = 0.89, 95% CI [2.06, 2.78]), and post-writing ( M disposal  = 2.38, SD  = 0.87, 95% CI [2.03, 2.73]; M retention  = 2.27, SD  = 0.83, 95% CI [1.93, 2.60]). A 2 × 3 mixed ANOVA revealed that neither main effects nor interaction was significant ( Fs  < 0.90, ps  > 0.41, BF 10 s < 0.14).

This study examined whether writing about the provocative event and disposing of the paper into a trash can would suppress anger. The provocation treatments evoked anger in both the groups similarly. Nevertheless, the retention group still showed significantly higher anger compared to levels at the baseline period, while the disposal group completely eliminated their anger after the disposal of the anger-written paper. These results suggest that the disposal of the paper containing ruminated anger into the trash can neutralise anger. Our interpretation is that the act of throwing the paper with ruminated anger into the trash can produces a feeling similar to the psychological existence (anger) being discarded, leading to anger elimination, since the psychological entity (anger) was disposed along with the physical object (anger-written paper).

One may argue that it was not the disposal itself but the physical distance played a critical role in reducing anger. Since the paper was distanced from participants in the disposal group, whereas the paper in the retention group was located by them. Nevertheless, Zhang et al. 44 showed that engaging in an avoidance action rather than creating physical distance was critical for reversing the perceived effect of negative thoughts. In their study (Experiment 5), participants in avoidance action conditions either threw the ball to the opposite corner of the room (creating physical distance between themselves and the ball), or pretended to throw the ball (creating no distance between themselves and the ball). Participants in the no-avoidance action condition either carried the ball to the opposite corner of the room and left it there (creating physical distance between the self and the ball without involving a throwing action) or held the ball in their non-dominant hand (creating no distance). Participants in both avoidance action conditions reversed the negative thoughts, while participants in both no-avoidance conditions did not. Avoidance actions were crucial in their study. Therefore, the physical distance would not contribute to reduce anger in this study. However, disposal action might be the key to neutralising anger in this study. Nevertheless, we assume that the meaning (i.e. interpretation) of disposal is more important than the action itself. Other studies have also suggested that the meaning of an action is critical for determining its impact, not the action itself 30 , 45 . This study could not exclude throwing action's potential contribution to neutralising anger. Thus, we conducted another experiment to exclude the potential contribution of the throwing action as much as possible, confirm the effectiveness of the disposal method, and explore the variation in this method.

Experiment 2

Experiment 1 indicated that the disposal of a piece of paper containing the description of an anger-inducing experience into the trash can neutralise anger. However, it was unclear what aspect of the paper’s disposal neutralised anger. Although we interpreted the meaning of the action as critical to neutralising anger, the physical distance between the participant and the paper or the action itself (i.e. embodied cognition) might have played a critical role. We set up the second experiment: (1) to replicate the results of Experiment 1; (2) to exclude the embodied explanation as much as possible; and (3) to explore another version of the disposal method using a shredder on the desk. In this experiment, we asked participants to put the paper containing anger into the shredder instead of throwing it into the trash can which was kept at some distance from the participants. We also made a small change to the retention group. Participants of retention group put the paper into a clear box on the desk, and the disposal group put the paper into the shredder. Thus, the distance between the participants and the paper and the type of action were matched between the two groups. If the sensorimotor experience of throwing the paper was critical to neutralise anger, we would not be able to replicate the results of Experiment 1. Nevertheless, if the meaning of the disposal of a physical entity plays a critical role in reducing anger, we anticipated obtaining similar results. In line with our prediction, the attitude changed when the paper was transferred to a box labelled ‘trash can’, which indicated mentally discarding it, compared to a box labelled ‘safety box’ 46 , suggesting that the perceived meaning of actions, and not the actions per se, influence attitude change. Hence, we designed a new study to confirm whether the perceived meaning of action eliminates anger. We predicted that putting the paper in a shredder would reduce negative emotions (anger), as compared to keeping the paper.

A total of 48 participants (women = 24, mean age = 26.81, SD  = 9.42) were participated through worker dispatching company and a local university. There was no overlap between the participants of the two experiments. This sample size was determined using G*Power 3.1.9.4 37 using the a priori procedure for repeated measures ANOVA, within (periods)–between (disposal and retention) interaction with the parameters of 95% power, an expected effect size of 0.25 (defined as a medium effect by Cohen 38 ), alpha level of 0.05, a within-participants measurement correlation of 0.5, and a nonsphericity correction ε of 1. The calculation suggested a sample size of 22 participants in each group. Based on these analyses, we concluded that the sample size was appropriate for this study. As in Experiment 1, the data of two participants were excluded from the final analysis because they correctly guessed the purpose of the experiment and did not express anger by insult (subjective ratings of anger were lower or the same as those at the baseline). Our final analysis included 46 participants (women = 23, mean age = 26.39, SD  = 9.14).

As in Experiment 1, angry feelings were assessed using five adjectives: angry, bothered, annoyed, hostile, and irritated. Responses ranged from 1 (not at all) to 6 (extremely). Scores on these five adjectives will be averaged to form an anger experience composite, which is the score used in the analyses. We also used the Japanese version of the 6-point PANAS scale as a subjective scale to assess mainly negative feelings 40 , 41 .

For the disposal group, a dustbin-type shredder (ACCO Brands Japan Corp, GSHA26MB) was used. This shredder (30 cm × 10 cm × 28 cm) cuts paper into pieces of 2 mm × 14 mm on putting the paper in from the top. The lower part of the shredder holds a transparent dustbin, so that the pieces of paper can be observed from the outside. For the retention group, a hand-made clear plastic box (23 cm × 5 cm × 30 cm) was used. Paper can be placed from the top, as with the shredder. Furthermore, as with the lower part of the shredder, the box is also transparent so that the paper in the box can be observed from the outside.

This experiment followed the same method used in Experiment 1 with slight changes. The words “while at university” were removed from the provocative comment (‘I cannot believe an educated person would think like this. I hope this person learns something while at university’ 40 , 42 , because non-students participated in this study. The second change was the method of disposing or retaining the paper containing a description of the anger-inducing experience. After participants wrote down provocative events in an analytical manner, a transparent box or a transparent shredder bin was placed on the desk in front of them (Fig.  2 ), before they were asked to review the sentences carefully for 30 s. Then, participants were required to put the paper into the box, with the frontside of the paper facing them. Participants in the disposal group watched as the paper was cut in the shredder for five seconds. Participants in the retention group were required to enclose the paper in a clear file folder and place it in a transparent box showing their written sentences. Then, they observed the paper carefully for five seconds. Subsequently, the box was turned back to show the blank side of the paper. All participants rated their anger and provided responses to the PANAS after these treatments.

Pictures of experimental manipulations in Experiment 2. The disposal group (left) put the paper into the shredder, while the retention group (right) put the paper into the transparent box.

The right panel of Fig.  1 shows the mean subjective anger ratings for the disposal and retention groups at the three time points (baseline, post-provocation, and post-writing). This pattern of results is similar to that of Experiment 1. Subjective ratings of anger in both groups increased after provocation ( M disposal  = 3.14, SD  = 1.38, 95% CI [2.56, 3.72]; M retention  = 3.24, SD  = 1.04, 95% CI [2.80, 3.67]) from baseline ( M disposal  = 1.57, SD  = 0.75, 95% CI [1.25, 1.88]; M retention  = 1.64, SD  = 0.59, 95% CI [1.40, 1.89]). Subjective ratings at post-writing decreased from post-provocation. However, those of the retention group were still higher than those of the baseline ( M retention  = 2.75, SD  = 1.05, 95% CI [2.31, 3.19]), while those of the disposal group were eliminated at the same level as the baseline ( M disposal  = 1.98, SD  = 0.87, 95% CI [1.62, 2.35]). Only a small overlap (0.04) was observed in the 95% CI for the mean post-writing scores between the groups. A 2 (group: disposal or retention) × 3 (period: at baseline, post-provocation, and post-writing) mixed model ANOVA revealed a significant main effect of period [ F (2, 88) = 56.93, p  < 0.001, partial η 2  = 0.56, BF 10  > 100], while the main effect of group was not significant [ F (1, 44) = 1.68, p  > 0.05, partial η 2  = 0.04, BF 10  = 0.46]. The interaction between group and period was significant [ F (2, 88) = 3.49, p  < 0.05, partial η 2  = 0.07, BF 10  = 1.62]. Multiple comparisons with the Bonferroni method revealed that subjective anger was significantly higher at post-provocation than baseline ( p  < 0.05), indicating that provocative manipulation was exerted. Subjective ratings of anger at post-writing decreased significantly compared to post-provocation ( p  < 0.05). However, the subjective ratings of the retention group in the post-writing period were still maintained at the same level of anger as those of the post-provocation period ( p  > 0.05). Contrastingly, those of the disposal group in the post-writing period were significantly lower than those of the post-provocation period ( p  < 0.05).

Additionally, as was the result of Experiment1, the subjective ratings of the retention group in the post-writing period were significantly higher than those of the baseline period ( p  < 0.05). Those of the disposal group in the post-writing period were eliminated to the baseline period ( p  > 0.05). The subjective ratings of the disposal group in the post-writing period were significantly lower than those of the retention group ( p  < 0.05).

The negative affect subscale of the PANAS at post-provocation ( M disposal  = 3.34, SD  = 1.09, 95% CI [2.88, 3.79]; M retention  = 3.35, SD  = 0.89, 95% CI [2.98, 3.73]) was higher than at baseline ( M disposal  = 2.60, SD  = 0.78, 95% CI [2.27, 2.93]; M retention  = 2.73, SD  = 0.92, 95% CI [2.34, 3.11]) and post-writing ( M disposal  = 2.45, SD  = 0.96, 95% CI [2.05, 2.85]; M retention  = 2.57, SD  = 0.87, 95% CI [2.20, 2.93]). The 95% CIs of the disposal group overlapped a little bit between post-provocation [2.88, 3.79] and baseline periods [2.27, 2.93], and those of the retention group overlapped between both the post-provocation [2.98, 3.73] and baseline [2.34, 3.11]. A 2 (group) × 3 (period) mixed ANOVA revealed a significant main effect of period [ F (2, 88) = 20.19, p  < 0.01, partial η 2  = 0.68, BF 10  > 100]. However, the main effect of the group [ F (1, 44) = 0.15, p  > 0.05, partial η 2  = 0.06, BF 10  = 0.33] and the interaction between group and period were not significant [ F (2, 88) = 1.35, p  > 0.05, partial η 2  = 0.05, BF 10  = 0.13]. Multiple comparisons with the Bonferroni method revealed that the subjective negative affect post-provocation was significantly higher than at baseline and post-writing ( ps  < 0.05).

The positive affect subscale of the PANAS showed little variation at the three-time points ( M disposal  = 2.88, SD  = 1.03, 95% CI [2.44, 3.31]; M retention  = 2.57, SD  = 0.89), 95% CI [2.19, 2.94], post-provocation ( M disposal  = 2.49, SD  = 0.86, 95% CI [2.13, 2.85]; M retention  = 2.51, SD  = 0.94, 95% CI [2.12, 2.90]), and post-writing ( M disposal  = 2.49, SD  = 0.97, 95% CI [2.08, 2.89]; M retention  = 2.64, SD  = 1.02, 95% CI [2.21, 3.06]). A 2 × 3 mixed ANOVA revealed that neither the main effects nor interaction were significant ( Fs  < 2.28, ps  > 0.11, BF 10 s < 0.70).

The results were essentially the same as those of Experiment 1. The disposal group significantly reduced their anger after disposing of the anger-written paper into the shredder. The retention group showed significantly higher anger than the baseline period and disposal group. These results suggest that the results in Experiment 1 could be attributed neither to the physical distance between the participant and the paper nor to the action itself (i.e. embodied cognition). Specifically, Experiment 2 replicated the results of Experiment 1 and excluded the embodied explanation (the sensorimotor experience of throwing the paper) because the action of the disposal group was quite similar to that of the retention group in Experiment 2. The distance between participant and paper was the same in both groups, as the transparent box and shredder were placed on the desk.

General discussion

This study aimed to determine whether the disposal of anger-written papers could eliminate or at least reduce subjective anger. Disposal manipulation eliminated anger, either by throwing the paper into a trash can or placing it into the shredder. We propose that this anger reduction method is quite effective, so the subjective ratings of anger resumed as much as the baseline levels. We believe that this method can be used in daily life and especially for populations characterised by extreme levels of anger and aggression in their home. The use of this method may potentially contribute to emotion socialization, as parents are the primary model for their children.

These results indicate that the sensorimotor experience of throwing paper plays a small role in reducing subjective anger 44 . Instead, the meaning (interpretation) of disposal plays a critical role. These results are consistent with other studies which showed that the meaning of disposal was critical for determining its impact, not the action itself 30 , 45 . However, these results are partially inconsistent with those reported by Zhang et al. 44 . Their experiment tested whether certain behaviors could lower the perceived likelihood of bad luck, as is often the case with jinxes. Participants who threw a ball believed that a jinxed-negative outcome was less likely than those who held the ball. They demonstrated that engaging in an avoidant action rather than creating physical distance was critical for reversing the perceived effect of the jinx. The results of Experiment 1 in this study are consistent with their results. However, we demonstrated that neither avoidance action nor physical distance was crucial in reducing subjective anger.

Our results may be related to the phenomenon of ‘backward magical contagion’ 47 , which is the belief that actions taken on an object (e.g. hair) associated with an individual can affect the individuals themselves. Rozin et al. 48 discovered that individuals experience strong negative emotions when their personal objects are possessed by negative others (such as rapists or enemies). However, these emotions are reduced when the objects are destroyed, such as throwing them in a septic tank or burning them. The phenomenon of ‘magical contagion’ or ‘celebrity contagion’ refers to the belief that the ‘essence’ of an individual can be transferred to their possessions. This backward magical contagion operates in a reversed process, where manipulating an object associated with a person is thought to impact the individuals themselves. The current study's findings may be explained by the concept of backward magical contagion, which posits that negative emotions can be transferred from others to an individual through their possessions. This study did not involve the direct mediation of other individuals. The neutralization of subjective anger through the disposal of an object may be achieved by recognizing that the physical entity, such as a piece of paper, has been diminished, thus causing the original emotion to also disappear.

At least, however, some limitations regarding this disposal method should be addressed in future studies. First, the findings of this study are based on the assumption that participants identified their subjective anger with the paper. Thus, subjective anger had gone with the anger-written paper after its disposal. The participants were asked to review the sentences carefully for 30 s to enhance this identification between thought and paper. It is not clear whether this review process is necessary for identification.

Another limitation is that we did not test a digital device, such as a word processor or smartphone, but used only papers. We believe the present disposal method can be generalised to a digital device, whereas empirical data are limited only by physical entities, papers, trash cans, or shredders. Suppose the disposal method is proven to be effective in digital devices. In that case, it will be adopted in various situations, such as business meetings or daily conversations in schools, by writing and disposing of with a smartphone.

Furthermore, although the disposal method had a more significant effect so that the subjective ratings of anger were eliminated as much as the baseline levels, the effectiveness of this method was not directly compared to other anger reduction methods, such as self-distancing. Other methods may be as effective or even more effective than the present disposal method. Personality traits may modulate the effects of anger suppression, although this has not been examined in the techniques used in this or in other studies. Individuals with high (versus low) levels of trait anger tended to experience lapses in effortful control when exposed to anger-relevant stimuli 49 , 50 . As mentioned above, although cognitive reappraisal (the reinterpretation of the meaning of an unpleasant event) is considered an effective way to reduce anger 12 , it requires more significant cognitive effort 13 , 14 . Self-distancing is not feasible, particularly during the heat of the moment 13 . Conversely, the disposal method with low cognitive effort used in this study may be more effective for individuals with lower levels of trait self-control than for those with high trait self-control. Future research should examine whether personality traits moderate the relationship between the disposal method and the expected outcomes.

Individuals with higher levels of trait anger tended to have prolonged experiences of induced state anger 51 . However, experimental research on anger regulation strategies has predominantly emphasized the effectiveness of immediate control 10 , 11 , 12 , neglecting to investigate whether these strategies are equally effective in managing anger that persists over time. However, in everyday life, it is not always feasible to implement anger regulation strategies immediately after anger arises. Therefore, to ascertain its practical utility in real-world settings, it is imperative to examine whether the effectiveness of the disposal method varies with the duration of anger.

Moreover, it should be tested whether the disposal method can suppress subjective anger even if participants write down a provocation event in an experiential manner rather than in the analytic rumination manner used in this study. Previous studies suggest that anger rumination can maintain 52 or even increase 53 the original level of anger when participants wrote down a provocation event in an experiential rumination manner. As it may not be easy to write down analytically, especially in the heat of the moment, the disposal method will gain further strength if it is valid by experiential rumination.

It should be mentioned that although provocation was effective in both the subjective anger score and the PANAS negative score, the revealed emotion regulation strategy in this study seemed specific to anger (as no significant interaction effect for the PANAS negative score was observed). Kubo et al. 40 reported that the increase in the state of anger relevant to approach motivation (aggression) by provocation (measured using the STAXI and asymmetry of prefrontal brain activity) was reduced by an apology comment. However, an increase in the subjective scores of negative emotion (assessed using the PANAS) remained unchanged, regardless of the presence or absence of an apology comment. They proposed anger as not a unitary process but one that comprises multiple independent components (subjective anger and negative feelings). If the anger scale used in this study reflects the approach motivation component of anger as well as the STAXI, the disposal method appears to specifically suppress the components of anger’s approach motivation (aggression) and can be used to reduce aggression as a clinical technique.

Despite these limitations, this is the first study to be designed and used to conveniently eliminate subjective anger by interacting with physical entities. It offers a cost-effective and easy-to-use method to reduce anger by rumination about the provocative event, which otherwise lasts longer. Anyone with a pen and piece of paper can use this method. Suppose one maintains a diary or a personal log. In that case, they can write down a provocative event on the day on the memo pad, and throwing it into the trash can eliminate the provocative event. This action may help neutralize the negative emotions associated with the event, potentially protecting the children’s emotional socialization.

This study presents a new and convenient method for eliminating subjective anger. This method offers a cost-effective way to eliminate anger in various situations, including business meetings, childcare, and clinical applications. The building blocks of this method (e.g. applying it to a digital device or creating a specific application) could be useful in various daily situations as well as behavioural therapies. In particular, for someone who has difficulty suppressing their anger in their homes.

Data availability

The datasets used and analysed during the current study available from the corresponding author on reasonable request.

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This study was supported by JSPS KAKENHI Grant Numbers 21K18552 and 21H04421, by Aoyama Gakuin University grant for ‘Projection Science,’ and by JST SPRING, Grant Number JPMJSP2125.

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Nobuyuki Kawai

Present address: Department of Cognitive and Psychological Sciences, Nagoya University, Nagoya, 464-8601, Japan

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Department of Cognitive and Psychological Sciences, Nagoya University, Nagoya, 464-8601, Japan

Yuta Kanaya

Academy of Emerging Science, Chubu University, Kasugai City, 487-8501, Japan

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N.K.: Conceptualization, Methodology, Writing-Original draft preparation, Writing-Reviewing and Editing, Supervision, Validation. Y.K.: Data collection and curation, Writing-Original draft preparation Visualization, Investigation.

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Correspondence to Nobuyuki Kawai .

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Kanaya, Y., Kawai, N. Anger is eliminated with the disposal of a paper written because of provocation. Sci Rep 14 , 7490 (2024). https://doi.org/10.1038/s41598-024-57916-z

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DOI : https://doi.org/10.1038/s41598-024-57916-z

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