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Post Hoc Test: Types, Software, and Examples

Explore the importance of post hoc tests in statistical analysis within ANOVA. Learn how to interpret and apply them effectively in your research.

In statistical analysis, post-hoc tests, which are often necessary when you want to compare the differences between multiple groups in a study, are critical to obtaining reliable conclusions from research data. Following an initial study, such as ANOVA (Analysis of Variance), these tests allow researchers to assess the differences between groups. This article will examine post-hoc tests, their different types, and software possibilities, discuss data analysis approaches, and present valuable examples.

Statistical analysis illustrating the concept of post hoc tests and the potential impact of the type I error rate on the findings. data analysis techniques

Table of Contents

Understanding Post Hoc Tests

After an initial statistical analysis, Post hoc tests are used to discover individual differences between groups. These tests are helpful when dealing with several groups since they help determine whether specific pairs of groups differ considerably from one another.

What are Post Hoc Tests?

A post hoc test, derived from the Latin phrase "post hoc, ergo propter hoc" meaning "after this, therefore because of this," is a statistical analysis conducted after an initial test to explore the results further and determine which specific group or condition led to the observed differences. Post hoc tests are crucial in research to identify specific relationships between variables that were not initially apparent in the primary analysis. These tests help researchers avoid making false assumptions based on the initial findings and provide a more in-depth understanding of the data (Shahabadi & Uplane, 2015).

Importance of Post-Hoc Tests

Post-hoc tests are essential to prevent researchers from reaching inaccurate or misleading findings. Researchers may avoid the problem of presuming that all groups are equal, especially if they want to compare multiple groups within their study. When they are not by, undertaking these tests results in more accurate and nuanced findings.

Types of Tests

Several types of post hoc tests are commonly used in statistical analysis. Let's explore a few of them:

Tukey's Honestly Significant Difference (HSD)

Tukey's HSD test, which compares all possible pairs of means, is commonly used in ANOVA (Analysis of Variance). It determines whether groups have statistically significant differences by calculating the least considerable difference necessary to reject the null hypothesis; read more.

Bonferroni Correction

The Bonferroni adjustment is a conservative method for accounting for multiple comparisons by adjusting the significance threshold. It divides the desired alpha level by the number of comparisons, lowering the possibility of a Type I mistake. This adjustment is beneficial when performing a large number of post-hoc tests.

Scheffe's Method

Another methodology that uses an ANOVA to compare many groups is Scheffe's. It is more cautious than Tukey's HSD test. And making it appropriate for circumstances in which the assumption of variance homogeneity is broken.

Dunnett's Test

Dunnett's test is intended mainly for scenarios with a control group and numerous treatment groups. It compares each treatment group to the control group, looking for statistically significant differences.

Fisher's LSD (Least Significant Difference)

Fisher's LSD test analyzes individual group means to discover statistically significant differences. It is often employed when the assumption of equal variances is fulfilled.

Post Hoc Comparison table between different types of  Test

Post Hoc Test

Main Use


Key Features

Tukey's HSD (Honestly Significant Difference)

Compare all possible pairs of means

Assumes equal variances and normally distributed data

Controls familywise error rate (FWER), widely used for multiple comparisons

Dunnett's Test

Compare each treatment group to a control group

Assumes equal variances and normally distributed data

Compares multiple treatment groups against a single control group

Bonferroni Correction

Adjust significance level for multiple comparisons

None specific, but conservative

Divides desired alpha level by the number of comparisons to reduce Type I error risk

Scheffe's Method

Compare multiple group means

Assumes equal variances and normally distributed data

More conservative than Tukey's HSD, suitable when variances are not homogeneous

Games-Howell Test

Compare all possible pairs of means when variances are unequal

Does not assume equal variances

Robust to violations of equal variance assumption, useful when variances are different between groups

Fisher's LSD (Least Significant Difference)

Compare pairs of means

Assumes equal variances and normally distributed data

Higher power but does not control FWER well, useful for fewer comparisons, especially when you need to perform post hoc tests on a limited number of groups.

Dunn's Test

Non-parametric pairwise comparison after Kruskal-Wallis

No assumption of normality or equal variances

Uses rank sums to compare groups, suitable for non-parametric data

Holm-Bonferroni Method

Sequentially reject null hypotheses

None specific, more powerful than Bonferroni

Adjusts p-values in a stepwise manner, reducing the risk of Type I error while maintaining power

Nemenyi Test

Non-parametric pairwise comparison after Kruskal-Wallis

No assumption of normality or equal variances

Suitable for non-parametric data, controls Type I error for multiple comparisons

Software for Post Hoc Tests

Various statistical software packages provide tools for conducting post hoc tests. 

SPSS (Statistical Package for the Social Sciences)

SPSS is a popular statistical analysis software that includes post hoc tests. It has an easy-to-use interface and various options for running multiple post-hoc tests.

R (Programming Language)

R is a robust computer language frequently used in statistical analysis. When researchers need to perform post-hoc tests across multiple groups, they can use software such as multcomp and agricolae. Read more about R.

SAS (Statistical Analysis System)

SAS is a sophisticated statistical software package that includes substantial post-hoc testing features. Many scholars use it because of its user-friendly design and powerful features.

Excel (Spreadsheet Software)

Although Excel is most commonly associated with spreadsheet software, it also has some basic statistical capabilities. While it provides fewer extensive post-hoc testing possibilities than specialist statistical software, it is enough for primary studies.

Data Analysis Techniques for Post-Hoc Tests

Effect Size

The effect size measures the degree of the difference between groups, which is important when considering the confidence level for the results. Cohen's d, eta-squared (2), and omega-squared (2) are common effect size measurements. Understanding effect sizes aids researchers in interpreting the practical importance of variations discovered.

Confidence intervals

Confidence intervals provide a range of possible values for the actual population parameter. Researchers can use confidence intervals in post-hoc analyses to examine the precision of their estimations.


Visualizations like box and violin plots can help you comprehend group distinctions. These graphical representations give a clear overview of the data distribution, allowing you to see large deviations more quickly.

Latest Developments and Considerations

Some recent developments and considerations in the field of posthoc tests in R include:

Improved Visualization

Packages like ggpubr and rstatix now provide better visualization of ANOVA and posthoc test results, often directly on the boxplots or other relevant plots, which is essential when conducting multiple tests.

Robust Post-Hoc Tests

When the assumptions of ANOVA are violated, robust post-hoc tests like the Games-Howell test can be more appropriate. These tests are based on the number of groups and the type of data. The rstatix package provides easy access to these tests.

Controlling the False Discovery Rate (FDR)

Besides managing the FWER, some researchers prefer to control the FDR, which is the expected proportion of false positives among the rejected null hypotheses. The emmeans package provides options for FDR-controlling post hoc tests.

Bayesian Approaches

There is growing interest in Bayesian approaches to post-hoc tests, which can provide more nuanced interpretations of the differences between groups. The BayesFactor package is one option for Bayesian ANOVA and post-hoc analyses. Post-Hoc Tests in R Post-hoc tests are statistical analyses performed after an ANOVA or other omnibus test to determine which specific group means are significantly different from each other, based on the number of groups being compared.

These tests are necessary because ANOVA only tells you if there's a significant difference between groups but does not identify which groups are different. 

Post Hoc Test Examples

To illustrate the application of post hoc tests, let's consider a few examples:

Application of Post-Hoc Test in the Medical Field

In medical research, post hoc tests are frequently utilized to understand the outcomes of various studies. For example, a survey of musculoskeletal dysfunction in migraine patients employed post hoc tests such as the Bonferroni corrected posthoc t-tests and Dunn's Test to analyze the differences between headache-free participants and those with episodic and chronic migraines (Luedtke et al., 2017). Similarly, in evaluating antibacterial efficacy in root canal disinfection, Scheffe's post hoc test for pairwise comparisons followed one-way ANOVA to assess the inter-group differences (Mathew et al., 2022).

Application of Post-Hoc Test in the Psychology and Education

Moreover, post hoc analyses are not limited to medical studies but extend to diverse fields such as psychology and education. A survey of referred sensations in the orofacial region used Tukey post hoc tests to compare mechanical sensitivity and pain modulation between participants who experienced referred sensations and those who did not (Sago et al., 2023). Additionally, educational research developed a post hoc simulation program for computerized adaptive testing, demonstrating the applicability of post hoc analyses in optimizing testing strategies (Kalender, 2015).

Application of Post-hoc tests in the Meta-analyses and Systematic reviews

Furthermore, post hoc tests play a crucial role in meta-analyses and systematic reviews to enhance the understanding of aggregated data. Conducted a meta-analysis on B-type natriuretic peptide and N-terminal pro-B-type natriuretic peptide in heart failure diagnosis, highlighting the significance of avoiding arbitrary post hoc choices to boost test performance (Ewald et al., 2007). These examples underscore the importance of post hoc analyses in various research domains to extract meaningful insights from data that may need to be evident in the initial analyses.

Application of Post-Hoc Test in the Market Research

ANOVA demonstrates a substantial difference in a market research study comparing consumer preferences for three brands. Post hoc tests, such as Fisher's LSD, can determine which specific pairs of brands differ substantially.


Post hoc tests are essential tools in statistical analysis because they allow researchers to identify significant differences between groups. Researchers can derive meaningful and reliable conclusions from their data by knowing the many post hoc tests available, using suitable software, implementing effective data analysis procedures, and analyzing relevant instances.

Frequently Asked Questions (FAQs)

What is the Bonferroni test, and how does it control the error rate in ANOVA?

The Bonferroni test is a multiple comparison test used after ANOVA to control the family-wise error rate. Dividing the desired alpha (α) level by the number of comparisons reduces the probability of making Type I errors, ensuring more reliable results when comparing multiple groups.

What is ANOVA, and why is it used in statistical analysis?

ANOVA (Analysis of Variance) is a statistical test used to compare the means of three or more groups to determine whether they have statistically significant differences. It helps researchers test hypotheses about group differences and is widely used in various fields, including clinical trials and educational research.

How does one-way ANOVA differ from other types of ANOVA?

One-way ANOVA compares the means of three or more independent groups based on one independent variable. Unlike two-way ANOVA, which considers two independent variables, one-way ANOVA assesses whether significant differences exist among the groups' means for a single factor.

What are the null and alternative hypotheses in the context of ANOVA

In ANOVA, the null hypothesis states that the means of all groups are equal, implying no significant difference between the groups. The alternative hypothesis, on the other hand, suggests that at least one group mean is different, indicating a statistically significant result.

How do post-hoc tests control the error rate in multiple comparisons?

Post-hoc tests like Bonferroni and Tukey's HSD control the error rate by adjusting the significance level or test statistic. These adjustments help maintain the family-wise error rate at an acceptable level, ensuring that the findings of multiple comparisons are not due to chance alone.

What is the role of the null hypothesis in post-hoc tests after ANOVA?

The null hypothesis in post-hoc tests posits that there are no significant differences between the specific groups being compared. Post-hoc tests are performed to test this hypothesis and determine which group means differ significantly from each other, if any.

Why is multiple comparison important in ANOVA, and how are post-hoc tests used?

After finding an overall effect, multiple comparisons are crucial in ANOVA to identify which specific group differences are significant. Post-hoc tests, such as Tukey's HSD and Bonferroni correction, are used to make these comparisons, ensuring an accurate interpretation of the mean differences between groups.

How does variance affect the interpretation of ANOVA results and post-hoc tests?

Variance measures the spread of data within and between groups. In ANOVA, understanding variance is essential as it influences the test statistic and the significance of the results. Post-hoc tests further analyze these variances to pinpoint specific group differences.

What is the purpose of hypothesis testing in ANOVA and post-hoc tests?

Hypothesis testing in ANOVA evaluates whether the group means are equal, forming the basis for further post-hoc tests. These tests are conducted to determine specific differences between groups, enhancing the understanding of the data's underlying patterns.

How does the number of comparisons in post-hoc tests affect the statistical analysis?

The number of comparisons significantly impacts the error rate in statistical analysis, highlighting the need to consider the type I error rate. More comparisons increase the risk of Type I errors, making it crucial to use post-hoc tests like Bonferroni or Tukey's HSD to adjust the significance level and maintain accurate results.

What are some common post hoc tests used after ANOVA?

Common post hoc tests include Tukey's HSD, Bonferroni correction, Dunnett's test, and Scheffe's method. These tests compare group means and identify specific statistically significant differences after an initial ANOVA.

When should researchers use post-hoc tests in their analysis?

Researchers should use post-hoc tests when ANOVA indicates significant overall differences between groups. These tests help identify which specific group pairs have significant mean differences, providing detailed insights into the data.

What is an omnibus test, and how does it relate to post-hoc analysis?

An omnibus test, such as ANOVA, evaluates overall differences among group means without specifying which groups differ. When the omnibus test shows significant results, post-hoc tests are performed to explore specific group differences further.

What is a multiple comparison test, and why is it essential in ANOVA?

A multiple comparison test compares several group means simultaneously to determine significant differences. It is essential in ANOVA to control the error rate and ensure reliable findings when assessing multiple groups.

How do post-hoc tests function as statistical tests in research?

Post-hoc tests function as statistical tools to compare group means after finding significant results in an initial test like ANOVA. They are necessary when you need to perform post-hoc tests to identify differences. They help researchers draw more precise conclusions by identifying specific group differences.

What constitutes a statistically significant result in post-hoc tests?

A statistically significant result in post-hoc tests occurs when the p-value is below a predetermined alpha level, typically 0.05. This indicates that the observed differences between group means are unlikely due to chance.

Can post-hoc tests be used to compare only two groups?

While post-hoc tests are typically used for multiple groups, they can also compare two groups. However, simpler tests like the t-test are more commonly used for comparing two groups directly.

What role does the alpha (α) level play in post-hoc tests?

The alpha (α) level represents the threshold for statistical significance. In post-hoc tests, adjusting the alpha level controls the probability of Type I errors, ensuring that the results are not due to random variation.

How are post-hoc tests applied in clinical trials?

In clinical trials, post-hoc tests are used to compare the effectiveness of different treatments. After ANOVA shows overall differences, post-hoc tests identify which specific treatments differ significantly, aiding medical decision-making.

What are comparison tests, and how do they relate to post-hoc analysis?

Comparison tests, such as Tukey's HSD and Bonferroni correction, are statistical methods used in post-hoc analysis to compare group means. After finding significant overall effects, they help determine specific differences between groups, which is crucial when performing post hoc tests.

Where can I find more information on post-hoc tests and ANOVA

Statistics LibreTexts is an excellent resource for learning about post-hoc tests and ANOVA. It provides detailed explanations, examples, and tutorials on various statistical concepts and tests.

Which post-hoc tests are commonly used in statistical analysis?

Commonly used post-hoc tests include Tukey's HSD, Bonferroni correction, Dunnett's test, and Scheffe's method. These tests help researchers identify specific group differences after an initial ANOVA.

What are the different types of post-hoc tests available?

Different types of post-hoc tests include Tukey's HSD, Bonferroni correction, Scheffe's method, Dunnett's test, and the Games-Howell test. Each type has its specific applications and assumptions, which are based on the number of groups involved in the study.

Why is the use of post-hoc tests important in research?

The use of post-hoc tests is important to identify specific differences between groups after an initial significant finding. They provide detailed insights, helping researchers make accurate and reliable conclusions.

What is a range test, and how is it used in post-hoc analysis?

A range test, like Tukey's HSD, compares the range of means between groups to determine significant differences. It is commonly used in post-hoc analysis to control the family-wise error rate.

How does the number of tests affect the results of post-hoc analysis?

The number of tests affects the error rate in post-hoc analysis. More tests increase the risk of Type I errors, making it crucial to use adjustments like Bonferroni correction to maintain accuracy.

What is a test statistic, and how is it used in post-hoc tests?

A test statistic is a value calculated from sample data during a hypothesis test. Post-hoc tests help determine if the observed differences between group means are statistically significant.

What does it mean when ANOVA shows that group means are equal?

When ANOVA indicates that group means are equal, it means there is no statistically significant difference between the groups. This suggests that any observed differences are likely due to chance.

Why is a significance level of 0.05 commonly used in post-hoc tests?

A significance level of 0.05 is commonly used because it balances Type I and Type II errors. It indicates a 5% risk of concluding.

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  • Ewald, B., Ewald, D., Thakkinstian, A., & Attia, J. (2007). Meta‐analysis of b type natriuretic peptide and n‐terminal pro b natriuretic peptide in the diagnosis of clinical heart failure and population screening for left ventricular systolic dysfunction. Internal Medicine Journal, 38(2), 101-113.
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About the Author

Ph.D. Scholar | Certified Data Analyst | Blogger | Completed 5000+ data projects | Passionate about unravelling insights through data.

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