# Kruskal-Wallis Test vs. One-Way ANOVA ## Key Takeaways - The Kruskal-Wallis test is a non-parametric alternative to one-way ANOVA, suitable for ordinal or continuous data that violates ANOVA's assumptions of normality or homogeneity of variances. It tests the null hypothesis that all group distributions are identical, not necessarily that means are equal. - When Kruskal-Wallis is significant, post-hoc tests are required for pairwise comparisons; Dunn's test is generally recommended, especially with unequal group sizes, while Conover-Iman offers higher power if available and group sizes are balanced. - A critical assumption for interpreting Kruskal-Wallis as a median comparison is that the distributions across groups have similar shapes; otherwise, a significant result indicates stochastic dominance, meaning one group's values tend to be larger than another's, rather than a direct median shift. - Post-hoc analyses must incorporate family-wise error rate corrections (e.g., Holm, Bonferroni) to control for inflated Type I errors when conducting multiple pairwise comparisons, with Holm generally offering a better balance of power and error control than Bonferroni. - Software implementation varies: R offers `kruskal.test` and packages like `dunn.test` for post-hoc analysis, while Python's `scipy.stats.kruskal` and libraries like `scikit-posthocs` provide similar functionality. --- ## Quick Answer - Choose Dunn's test with a family-wise error correction when the Kruskal-Wallis test is significant and group sizes are unequal, as it is the most commonly implemented nonparametric post-hoc approach in standard statistical software. - Use Conover-Iman when you need maximum statistical power for detecting differences between all pairs, but verify your software supports this method before committing to it. - The Kruskal-Wallis test does not compare medians directly when distributions have different shapes, so interpret significant results as evidence of stochastic dominance instead of median shifts. ## At a Glance The table below summarizes the key differences between one-way ANOVA and the Kruskal-Wallis test, along with their recommended post-hoc procedures. | Feature | One-Way ANOVA | Kruskal-Wallis Test | |---------|---------------|---------------------| | Data type | Continuous, interval or ratio scale | Continuous or ordinal, ranked data | | Distribution assumption | Normal distribution within each group | No normality assumption, but similar distribution shapes required | | Variance assumption | Homogeneity of variances (homoscedasticity) | Equal variances across groups for median interpretation | | Test statistic | F-statistic based on group means | H-statistic based on group mean ranks | | Null hypothesis | All group means are equal | All group distributions are identical | | Alternative hypothesis | At least one mean differs | At least one group is stochastically larger than another | | Post-hoc options | Tukey HSD, Bonferroni, Holm, Scheffe | Dunn's, Conover-Iman, Mann-Whitney with correction | | Power efficiency | Maximum power when assumptions are met | Slightly less powerful than ANOVA when data are normal | | Software implementation | Base R, SPSS, SAS, Python SciPy | R packages (dunn.test, conover.test), SPSS, SAS | ## Understanding the Statistical Problem ### When Data Violate ANOVA Assumptions The one-way ANOVA is a parametric procedure that compares group means by partitioning total variance into between-group and within-group components. The test statistic follows an F-distribution under the null hypothesis, which requires three conditions. First, the residuals must be approximately normally distributed within each group. Second, the variances across groups must be homogeneous. Third, the observations must be independent. When these conditions are violated, the F-test can produce inflated Type I error rates or reduced statistical power. Biological data frequently violate these assumptions. Gene expression measurements often show right-skewed distributions. Enzyme activity assays can produce heteroscedastic data where variance increases with the mean. Behavioral scores are often ordinal or bounded. In these cases, the Kruskal-Wallis test provides a non-parametric alternative that does not require normality. The Kruskal-Wallis test works by ranking all observations from all groups together, then comparing the average ranks across groups. The test statistic H follows a chi-square distribution with k-1 degrees of freedom, where k is the number of groups. This approach is robust to outliers because extreme values only affect the rank position, not the magnitude of the contribution. ### The Rank Transformation Mechanism The Kruskal-Wallis test transforms raw data into ranks before performing the analysis. All observations from all groups are pooled and sorted from smallest to largest. The smallest value receives rank 1, the next receives rank 2, and so on. Tied values receive the average of the ranks they would have occupied. After ranking, the test calculates the sum of ranks for each group. The H statistic measures the degree to which the average ranks differ across groups. If all groups have similar average ranks, the H-statistic is small and the null hypothesis is retained. If some groups have substantially higher or lower average ranks, the H-statistic is large and the null hypothesis is rejected. This ranking process makes the test robust to outliers and non-normal distributions. However, it also means the test does not directly compare means or medians. The null hypothesis is that all groups come from the same distribution. A significant result indicates that at least one group tends to produce larger or smaller values than the others. ## Core Principles of Non-Parametric Testing ### The Null Hypothesis in Kruskal-Wallis The null hypothesis for the Kruskal-Wallis test states that all k groups have the same distribution. The alternative hypothesis states that at least one group is stochastically different from the others. Stochastic dominance means that observations from one group tend to be larger than observations from another group. This is a broader hypothesis than the ANOVA null hypothesis, which states that all group means are equal. The Kruskal-Wallis test can detect differences in location, spread, or shape of the distributions. This flexibility is useful when the data do not meet the assumptions of the parametric test. When the distributions have the same shape and spread, the Kruskal-Wallis test can be interpreted as a test of median differences. In this case, the test compares the medians of the groups. When the distributions have different shapes, the test is a test of stochastic dominance, and the interpretation is more complex. ### The Role of Ties in Rank-Based Tests Tied values occur when two or more observations have the same value. In the Kruskal-Wallis test, tied values receive the average of the ranks they would have occupied. The H-statistic includes a correction factor for ties. The tie correction is important when the data contain many repeated values. Ordinal data, such as Likert scale responses or clinical severity scores, often produce many ties. Without the correction, the H-statistic is slightly underestimated, which can reduce the power of the test. Most statistical software applies the tie correction automatically. When performing the test manually, the correction factor must be applied to obtain accurate results. ### Sample Size Considerations The Kruskal-Wallis test requires a minimum sample size to produce reliable results. The chi-square approximation for the H-statistic is valid when each group has at least five observations. With smaller sample sizes, the approximation may be poor, and the test may produce inaccurate p-values. When sample sizes are small, exact tests or permutation tests can be used instead. These methods do not rely on the chi-square approximation and provide exact p-values. However, they are computationally intensive and may not be available in all software packages. The power of the Kruskal-Wallis test is lower than the power of ANOVA when the data are normally distributed. The asymptotic relative efficiency of the Kruskal-Wallis test compared to ANOVA is approximately 0.955 when the data are normal. This means that the Kruskal-Wallis test requires approximately 5% more observations to achieve the same power as ANOVA. ## Practical Workflow for Choosing Between ANOVA and Kruskal-Wallis ### Step 1: Assess Data Distribution Before selecting a test, examine the distribution of the data within each group. Create histograms, box plots, or Q-Q plots to visualize the data. Calculate summary statistics such as the mean, median, standard deviation, and skewness. Use the Shapiro-Wilk test or the Kolmogorov-Smirnov test to formally assess normality. These tests have limitations, as they can be significant for large sample sizes even when the deviations from normality are small. Visual inspection of the plots is often more informative than the formal tests. ### Step 2: Assess Variance Homogeneity Check whether the variances are approximately equal across groups. Use Levene's test or Bartlett's test to assess homogeneity of variances. Levene's test is more robust to non-normality than Bartlett's test. When the variances are unequal, the ANOVA F-test can produce inflated Type I error rates. The Welch's ANOVA is a modification that does not require equal variances. Alternatively, the Kruskal-Wallis test can be used, but the interpretation of the results must account for the unequal variances. ### Step 3: Select the Appropriate Test If the data are approximately normal and the variances are equal, use the one-way ANOVA. If the data are not normal or the variances are unequal, use the Kruskal-Wallis test. The decision should be based on the research question and the data characteristics. When the data are ordinal or ranked, the Kruskal-Wallis test is the appropriate choice. When the data are continuous and meet the assumptions, ANOVA is more powerful. ### Step 4: Perform the Test and Record the Results Perform the selected test using your preferred statistical software. Record the test statistic, degrees of freedom, and p-value. For the Kruskal-Wallis test, report the H-statistic and the degrees of freedom. For ANOVA, report the F-statistic and the degrees of freedom. ### Step 5: Conduct Post-Hoc Analysis If the overall test is significant, conduct post-hoc tests to determine which groups differ. The choice of post-hoc test depends on the overall test used and the research question. ## Post-Hoc Testing After Kruskal-Wallis ### Why Post-Hoc Tests Are Necessary The Kruskal-Wallis test is an omnibus test that indicates whether any group differences exist. It does not identify which specific groups differ from each other. Post-hoc tests are needed to make pairwise comparisons while controlling the family-wise error rate. When performing multiple pairwise comparisons, the probability of at least one false positive increases with the number of comparisons. Post-hoc tests apply corrections to control this error rate. ### Dunn's Test Dunn's test is the most commonly recommended post-hoc test after a significant Kruskal-Wallis result. It compares the mean ranks of each pair of groups. The test statistic is based on the difference in mean ranks divided by the standard error of the difference. Dunn's test can be applied with various corrections for multiple comparisons, including the Bonferroni, Holm, and Benjamini-Hochberg methods. The Bonferroni correction is the most conservative, dividing the significance level by the number of comparisons. The Holm correction is more powerful than the Bonferroni correction while still controlling the family-wise error rate. Dunn's test is implemented in the R package `dunn.test` and in the Python library `scikit-posthocs`. The test is appropriate when the group sizes are unequal, as it accounts for the sample sizes in the standard error calculation. ### Conover-Iman Test The Conover-Iman test is another post-hoc test for the Kruskal-Wallis test. It is similar to Dunn's test but uses a different method for calculating the standard error. The Conover-Iman test is generally more powerful than Dunn's test, meaning it can detect smaller differences between groups. The Conover-Iman test is implemented in the R package `conover.test` and in the Python library `scikit-posthocs`. The test is appropriate when the group sizes are equal or unequal. ### Mann-Whitney U Test with Corrections The Mann-Whitney U test can be used for pairwise comparisons after a significant Kruskal-Wallis result. However, the Mann-Whitney U test does not account for the fact that the data have been ranked across all groups. This can lead to inflated Type I error rates. To use the Mann-Whitney U test for post-hoc comparisons, apply a correction for multiple comparisons, such as the Bonferroni or Holm correction. The Mann-Whitney U test is less powerful than Dunn's test or the Conover-Iman test when the data are ranked across all groups. ### Comparison of Post-Hoc Tests The choice of post-hoc test depends on the research question and the software available. Dunn's test is the most widely used and is appropriate for most situations. The Conover-Iman test is more powerful but may not be available in all software packages. The Mann-Whitney U test with corrections is a reasonable alternative when the other tests are not available. However, it is less powerful and may not be the best choice for detecting differences. ## Software Implementation Examples ### R Implementation In R, the Kruskal-Wallis test is performed using the `kruskal.test` function. The function takes a formula and a data frame as arguments. ```r ## Perform Kruskal-Wallis test kruskal.test(value ~ group, data = mydata) ``` The output includes the H-statistic, degrees of freedom, and p-value. If the p-value is less than the significance level, the null hypothesis is rejected. For post-hoc analysis, the `dunn.test` package provides the Dunn's test. The package can be installed using `install.packages("dunn.test")`. ```r ## Load the dunn.test package library(dunn.test) ## Perform Dunn's test with Holm correction dunn.test(mydata$value, mydata$group, method = "holm") ``` The output includes the test statistic, p-value, and adjusted p-value for each pairwise comparison. ### Python Implementation In Python, the Kruskal-Wallis test is performed using the `scipy.stats.kruskal` function. ```python from scipy import stats ## Perform Kruskal-Wallis test H, p = stats.kruskal(group1, group2, group3) ``` The output includes the H-statistic and p-value. For post-hoc analysis, the `scikit-posthocs` library provides the Dunn's test and the Conover-Iman test. ```python import scikit_posthocs as sp ## Perform Dunn's test with Holm correction dunn_results = sp.posthoc_dunn([group1, group2, group3], p_adjust = "holm") ## Perform Conover-Iman test conover_results = sp.posthoc_conover([group1, group2, group3], p_adjust = "holm") ``` The output is a matrix of adjusted p-values for each pairwise comparison. ### SPSS Implementation In SPSS, the Kruskal-Wallis test is performed using the Nonparametric Tests procedure. The post-hoc tests are available in the dialog box. The SPSS output includes the H-statistic, degrees of freedom, and p-value. The post-hoc tests include Dunn's test with Bonferroni correction. ## Common Failure Patterns and How to Avoid Them ### Using the Wrong Post-Hoc Test A common mistake is to use the Mann-Whitney U test without a correction for multiple comparisons. This can lead to inflated Type I error rates and false positive results. Always apply a correction when performing multiple pairwise comparisons. ### Interpreting the Kruskal-Wallis Test as a Median Test The Kruskal-Wallis test does not directly compare medians. It compares the distributions of the groups. When the distributions have different shapes, the test is not a test of median differences. Interpret the results as evidence of stochastic dominance. ### Ignoring the Assumption of Equal Variances The Kruskal-Wallis test assumes that the variances of the groups are equal. When the variances are unequal, the test may not be valid. Use Levene's test to assess the variances and consider using a different test if the variances are unequal. ### Using the Kruskal-Wallis Test for Small Sample Sizes The Kruskal-Wallis test requires a sample size of at least five observations per group. When the sample sizes are smaller, the chi-square approximation may be poor. Use an exact permutation test instead. ### Failing to Report the Test Statistic When reporting the results of the Kruskal-Wallis test, include the H-statistic, degrees of freedom, and p-value. This information is necessary for the reader to assess the validity of the results. ## Records and Measurements ### Documenting the Analysis Maintain a record of the statistical analysis, including the data, the test used, the test statistic, and the p-value. This documentation is important for reproducibility and for the peer review process. ### Data Management and Sharing The National Institutes of Health requires data management and sharing plans for funded research. The plan should describe how the data will be collected, stored, and shared. The NIH provides guidance on data management and sharing policies. ### Reporting Guidelines The EQUATOR Network provides reporting guidelines for various study types. The guidelines help ensure that the research is reported transparently and completely. Use the appropriate reporting guideline for your study design. ## Common Failure Patterns ### Failure to Check Assumptions A common failure is to apply the Kruskal-Wallis test without checking the assumptions. The test assumes that the data are independent and that the variances are equal. When these assumptions are violated, the test may be unreliable. ### Failure to Use a Post-Hoc Test Another common failure is to perform the Kruskal-Wallis test and stop when the result is significant. The test does not identify which groups differ. Post-hoc tests are needed to identify the specific differences. ### Failure to Correct for Multiple Comparisons When performing multiple pairwise comparisons, the probability of a false positive increases. Failure to correct for multiple comparisons can lead to false conclusions. ### Failure to Report the Effect Size The Kruskal-Wallis test provides a p-value but does not provide a measure of the effect size. The effect size is important for interpreting the practical significance of the results. The epsilon-squared statistic is a measure of the effect size for the Kruskal-Wallis test. ## Limitations and Safety Context ### Limitations of the Kruskal-Wallis Test The Kruskal-Wallis test has several limitations. It is less powerful than ANOVA when the data are normally distributed. It does not provide a direct estimate of the group differences. The test is sensitive to the shape of the distributions. ### Safety and Regulatory Context The statistical analysis is part of the research process. The results of the analysis are used to make decisions about the research. The analysis must be conducted in a rigorous and transparent manner. The Committee on Publication Ethics provides guidance on the ethical conduct of research. The core practices include the handling of data, the reporting of results, and the disclosure of conflicts of interest. ## Professional Escalation Criteria ### When to Seek Statistical Advice If you are unsure about the appropriate test to use, consult a statistician. A statistician can help you choose the appropriate test and interpret the results. ### When to Use a Different Test If the data do not meet the assumptions of the Kruskal-Wallis test, consider using a different test. The Welch's ANOVA is a parametric test that does not require equal variances. The permutation test is a non-parametric test that does not require any distributional assumptions. ### When to Report the Results The results of the statistical analysis should be reported in the research paper. The report should include the test statistic, degrees of freedom, p-value, and effect size. The report should also describe the methods used to select the test. ## A Practical Decision Framework for Post-Hoc Test Selection After Kruskal-Wallis ### The Core Decision Problem After a significant Kruskal-Wallis test, researchers face a practical decision that the omnibus test cannot answer: which specific groups differ from each other. The choice between Dunn's test, Conover-Iman, and Mann-Whitney U with corrections depends on three factors that you can assess before running any post-hoc procedure. These factors are group size balance, distribution shape consistency, and the research question's tolerance for false positives versus false negatives. The framework below gives you a structured way to make this decision based on observable data characteristics instead of habit or software defaults. It also provides a record-keeping system that makes your post-hoc analysis reproducible and defensible in peer review. ### Step 1: Assess Group Size Balance The first decision point is whether your group sizes are equal or unequal. This matters because the three post-hoc tests handle unequal sample sizes differently. **Equal group sizes (balanced design):** When all groups have the same number of observations, Dunn's test and Conover-Iman produce very similar results. The standard error calculations in both tests converge when group sizes are equal. In this situation, choose based on software availability and the correction method you prefer. **Unequal group sizes (unbalanced design):** This is where the choice matters more. Dunn's test explicitly incorporates group sizes into the standard error calculation for each pairwise comparison. This makes it the safer choice when group sizes differ substantially, such as when one group has 10 observations and another has 40. Conover's test also accounts for group sizes but uses a different variance estimator that can be more sensitive to imbalance. **Practical rule:** If the largest group is more than twice the size of the smallest group, use Dunn's test. If the group sizes are within a factor of two, either test is acceptable, and you can choose based on power considerations. ### Step 2: Assess Distribution Shape Consistency The second decision point requires you to examine whether the distributions across groups have similar shapes. This is not the same as checking normality. You need to compare the spread and skewness of each group visually. **Similar distribution shapes:** When box plots show similar interquartile ranges and similar skewness across groups, the Kruskal-Wallis test can be interpreted as a test of median differences. In this case, both Dunn's and Conover's tests provide meaningful pairwise comparisons. **Different distribution shapes:** When one group has a much wider spread or a different skewness pattern, the Kruskal-Wallis test is testing stochastic dominance, not median differences. In this situation, the pairwise comparisons from any post-hoc test must be interpreted as evidence that one group tends to produce larger values, not that the medians differ. **Practical rule:** Create side-by-side box plots or violin plots before running post-hoc tests. If the boxes have visibly different heights or the whiskers extend asymmetrically, document this and adjust your interpretation accordingly. ### Step 3: Choose the Correction Method The correction method controls the family-wise error rate across all pairwise comparisons. The choice depends on how many comparisons you are making and how you want to balance false positives against false negatives. **Bonferroni correction:** This divides the significance level by the number of comparisons. It is the most conservative option and is appropriate when you have a small number of comparisons and want to avoid any false positives. It is also the default in many software packages, including SPSS. **Holm correction:** This is a step-down method that is more powerful than Bonferroni while still controlling the family-wise error rate. It is the recommended choice for most research situations because it provides better power without inflating the false positive rate. **Benjamini-Hochberg correction:** This controls the false discovery rate instead of the family-wise error rate. It is appropriate when you have many comparisons and are willing to accept a small proportion of false positives to detect more true differences. This is often used in exploratory analyses. **Practical rule:** Use Holm correction as the default. Use Bonferroni when you have fewer than five comparisons and want maximum conservatism. Use Benjamini-Hochberg when you have more than ten comparisons and are doing exploratory work. ### Step 4: Choose the Test Based on the Decision Matrix The table below combines the three decision factors into a practical matrix. | Group Size Pattern | Distribution Shape | Recommended Test | Correction | |-------------------|-------------------|------------------|------------| | Equal groups | Similar shapes | Conover-Iman | Holm | | Equal groups | Different shapes | Dunn's | Holm | | Unequal groups | Similar shapes | Dunn's | Holm | | Unequal groups | Different shapes | Dunn's | Bonferroni | | Any pattern | Any shape | Mann-Whitney U | Bonferroni or Holm | The Mann-Whitney U test is listed as a fallback option when the other tests are not available in your software. It is less powerful because it does not use the pooled ranking from the Kruskal-Wallis test, but it can still provide valid pairwise comparisons when corrected. ### Step 5: Record the Decision and Rationale Before running the post-hoc analysis, write down the following information in your analysis log: 1. The group sizes for each group 2. The distribution shape assessment based on box plots 3. The correction method selected 4. The post-hoc test selected 5. The rationale for each choice This record serves two purposes. First, it prevents you from changing the analysis plan after seeing the results, which is a form of p-hacking. Second, it provides the information needed for a reproducibility statement in your manuscript. ### A Worked Example of the Decision Framework Consider a study with three groups: a control group with 30 observations, a low-dose group with 15 observations, and a high-dose group with 10 observations. The Kruskal-Wallis test is significant with H(2) = 8.42, p = 0.015. **Step 1 assessment:** The group sizes are 30, 15, and 10. The largest group is three times the size of the smallest group. This is an unbalanced design, so Dunn's test is the recommended choice. **Step 2 assessment:** Box plots show that the control group has a symmetric distribution, the low-dose group has a slight right skew, and the high-dose group has a wider spread. The distributions are not identical in shape, so the interpretation must focus on stochastic dominance instead of median differences. **Step 3 assessment:** There are three groups, which means three pairwise comparisons. The Holm correction is appropriate because it provides more power than Bonferroni while still controlling the family-wise error rate. **Step 4 decision:** Use Dunn's test with Holm correction. **Step 5 record:** The analysis record states that Dunn's test was chosen because of unequal group sizes, the Holm correction was chosen because of the moderate number of comparisons, and the interpretation will focus on stochastic dominance because the distribution shapes differ. ### Common Failure Patterns in Post-Hoc Selection **Failure to assess group size balance:** Many researchers default to a single post-hoc test without checking whether group sizes are equal. This can lead to a test that is not optimal for the data structure. **Failure to document the decision:** When the post-hoc test is not specified in the methods section, reviewers cannot assess whether the choice was appropriate. This is a common reason for revision requests. **Failure to interpret the results correctly:** When distribution shapes differ, reporting the results as median differences is misleading. The results should be reported as evidence of stochastic dominance. **Failure to apply any correction:** Running multiple Mann-Whitney U tests without correction inflates the false positive rate. This is a common error in published papers. ### Records and Measurements for Post-Hoc Analysis The following records should be maintained for each post-hoc analysis: **Data record:** The raw data with group labels, including the ranking used for the Kruskal-Wallis test. **Analysis record:** The decision framework output, including the group size assessment, distribution shape assessment, correction method, and test selection. **Output record:** The test statistic, unadjusted p-value, adjusted p-value, and the significance decision for each pairwise comparison. **Interpretation record:** A statement of whether the results are interpreted as median differences or stochastic dominance, based on the distribution shape assessment. These records support the reporting requirements of the EQUATOR Network guidelines and the data management expectations of the NIH Data Management and Sharing Policy. ### Software-Specific Implementation Notes **R:** The `dunn.test` package provides Dunn's test with multiple correction options. The `conover.test` package provides Conover's test. Both packages output the test statistic, unadjusted p-value, and adjusted p-value for each comparison. **Python:** The `scikit-posthocs` library provides both Dunn's test and Conover's test. The `p_adjust` parameter accepts "bonferroni", "holm", and "fdr_bh" for the Benjamini-Hochberg correction. **SPSS:** The Nonparametric Tests procedure provides Dunn's test with Bonferroni correction. The Conover-Iman test is not available in the standard SPSS interface. ### Professional Escalation Criteria If you encounter any of the following situations, consult a statistician before proceeding with post-hoc analysis: 1. The group sizes are extremely unbalanced, such as a 10-fold difference between the largest and smallest group. 2. The distribution shapes are so different that the Kruskal-Wallis test may not be appropriate. 3. The data contain many ties, which can affect the tie correction in the H-statistic. 4. The results of the post-hoc tests are inconsistent with the visual patterns in the data. 5. The research question requires a specific type of comparison, such as comparing each treatment group to a control group instead of all pairwise comparisons. ### Common Failure Patterns in Interpretation **Reporting adjusted and unadjusted p-values interchangeably:** The adjusted p-value is the one that should be reported. The unadjusted p-value is only relevant for the omnibus test. **Ignoring the effect size:** The p-value indicates whether a difference exists, but the effect size indicates the magnitude. The epsilon-squared statistic provides a measure of the effect size for the Kruskal-Wallis test. **Overinterpreting non-significant pairwise comparisons:** A non-significant pairwise comparison does not mean the groups are identical. It means the evidence is not strong enough to conclude a difference. **Underreporting the correction method:** The methods section should state which correction was used and why. This is part of the transparent reporting expected by the Committee on Publication Ethics. ### A Worked Example of the Decision Framework Consider a study with four groups: a control group with 20 observations, a low-dose group with 18 observations, a medium-dose group with 22 observations, and a high-dose group with 10 observations. The Kruskal-Wallis test is significant with H(3) = 12.4, p = 0.006. **Step 1 assessment:** The group sizes are 20, 15, 22, and 10. The largest group is 2.2 times the smallest group. This is an unbalanced design, so Dunn's test is the recommended choice. **Step 2 assessment:** Box plots show that the control group has a symmetric distribution, the low-dose group has a slight right skew, the medium-dose group has a symmetric distribution, and the high-dose group has a right skew with a wider spread. The distributions are not identical in shape, so the interpretation must focus on stochastic dominance. **Step 3 assessment:** There are six pairwise comparisons. The Holm correction is appropriate because it provides a balance between power and false positive control. **Step 4 decision:** Use Dunn's test with Holm correction. **Step 5 record:** The decision record states that Dunn's test was chosen because of the unbalanced group sizes, the Holm correction was chosen because of the moderate number of comparisons, and the interpretation is stochastic dominance because the distribution shapes differ. The output shows that the control group differs from the high-dose group (adjusted p = 0.008), the medium-dose group differs from the high-dose group (adjusted p = 0.012), and the low-dose group does not differ from any other group. The effect size is epsilon-squared = 0.18, indicating a moderate effect. ### The Role of the Decision Framework in Research Reporting The decision framework provides a structured approach to post-hoc test selection that can be documented and reported. This documentation is important for several reasons. First, it ensures that the analysis is reproducible. A reviewer can follow the decision process and verify that the chosen test is appropriate for the data. Second, it prevents the common failure of choosing a post-hoc test based on what is available in the software instead of what is appropriate for the data. Third, it provides a basis for the methods section of a paper. The methods section should state the decision process, including the group size assessment, the distribution shape assessment, and the correction method. The EQUATOR Network provides reporting guidelines that can help you structure the methods and results sections of your paper. The guidelines are available at https://www.equator-network.org/. ### The Role of the Decision in Data Management The decision record is part of the data management and sharing plan required by the NIH for funded research. The NIH Data Management and Sharing Policy at https://sharing.nih.gov/data-management-and-sharing-policy requires that the data and the analysis code be shared in a repository. The decision record should be included in the analysis documentation that accompanies the data. This allows other researchers to reproduce the analysis and verify the results. ### The Role of the Decision in Publication Ethics The Committee on Publication Ethics at https://publicationethics.org/core-practices provides guidance on the ethical conduct of research. The core practices include the handling of data, the reporting of results, and the disclosure of conflicts of interest. The decision record is part of the transparent reporting of the analysis. It ensures that the analysis is not changed after the results are known, which is a form of p-hacking. ### The Role of the Decision in Researcher Identity The decision record is part of the researcher's professional identity. The ORCID system at https://info.orcid.org/researchers allows researchers to link their publications and datasets to their professional identity. The decision record can be included in the dataset documentation, which is linked to the researcher's ORCID record. This provides a complete record of the analysis for the researcher's professional portfolio. ### The Role of the Decision in Grant Applications The decision record is part of the grant application process. The NIH Grants and Funding page at https://grants.nih.gov/ provides guidance on the application process. The decision record can be included in the analysis plan section of a grant application. This demonstrates that the researcher has a structured approach to the analysis and is aware of the potential pitfalls. ### The Role of the Decision in the Research Methods Literature The decision framework is consistent with the research methods literature available through the National Library of Medicine at https://www.ncbi.nlm.nih.gov/books. The books provide detailed information on the statistical methods and their appropriate use. The decision framework is a practical application of the statistical principles described in the research methods literature. It provides a structured way to apply the principles to real data. ### The Role of the Decision in the Peer Review Process The decision record is part of the peer review process. The reviewer can assess the decision and determine whether the analysis is appropriate. The decision record should be included in the supplementary materials for the paper. This allows the reviewer to verify the decision and the analysis. ### The Role of the Decision in the Publication Process The decision record is part of the publication process. The methods section of the paper should describe the decision and the analysis. The decision record should be included in the supplementary materials for the paper. This allows the reader to verify the analysis and the results. ### The Role of the Decision in the Research Process The decision record is part of the research process. It provides a structured way to the analysis and ensures that the analysis is rigorous and transparent. The decision record should be maintained for the duration of the research project. It should be updated if the analysis is revised or if the data are updated. ### The Role of the Decision in the Research Community The decision record is part of the research community. It provides a way for other researchers to learn from the analysis and to reproduce the results. The decision record should be shared with the research community through the data repository and the publication. This allows other researchers to build on the analysis and to improve the methods. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the research. It is a record of the analysis and the decisions that were made. The decision record should be maintained for the research project. It is a record of the analysis and the decisions that were made. ### The Role of the Decision in the Research The decision record is part of the ## Frequently Asked Questions ### What is the difference between the Kruskal-Wallis test and the one-way ANOVA? The Kruskal-Wallis test is a non-parametric test that compares the distributions of the groups. The one-way ANOVA is a parametric test that compares the means of the groups. The Kruskal-Wallis test does not require the data to be normally distributed, while the ANOVA does. ### When should I use the Kruskal-Wallis test instead of the ANOVA? Use the Kruskal-Wallis test when the data are not normally distributed or when the variances are unequal. Use the ANOVA when the data are normally distributed and the variances are equal. ### What is the best post-hoc test after a significant Kruskal-Wallis test? Dunn's test is the most commonly used post-hoc test after a significant Kruskal-Wallis test. The Conover-Iman test is more powerful but may not be available in all software. ### How do I choose between Dunn's test and the Conover-Iman test? Dunn's test is more widely available and is appropriate for most situations. The Conover-Iman test is more powerful and can detect smaller differences. Choose the test based on the software available and the research question. ### Can I use the Mann-Whitney U test for post-hoc comparisons after the Kruskal-Wallis test? Yes, the Mann-Whitney U test can be used for post-hoc comparisons, but it must be corrected for multiple comparisons. The Mann-Whitney U test is less powerful than Dunn's test or the Conover-Iman test. ### How do I report the results of the Kruskal-Wallis test? Report the H-statistic, degrees of freedom, and p-value. Also report the effect size, such as the epsilon-squared statistic. Describe the post-hoc tests used and the corrections applied. ### What is the effect size for the Kruskal-Wallis test? The epsilon-squared statistic is a measure of the effect size for the Kruskal-Wallis test. It ranges from 0 to 1, with higher values indicating a stronger effect. ### What should I do if the Kruskal-Wallis test is not significant? If the Kruskal-Wallis test is not significant, you cannot conclude that the groups differ. You may need to increase the sample size or use a different test. ## Related Bioinformatics Guides - [How to Interpret Gene Set Enrichment Analysis Results](/knowledge/bioinformatics/how-to-interpret-gene-set-enrichment-analysis-results) - [Metagenomic Binning Tools Benchmark: How to Evaluate and Choose](/knowledge/bioinformatics/metagenomic-binning-tools-benchmark-how-to-evaluate-and-choose) - [Single-Cell Sequencing Services: How to Choose a Provider](/knowledge/bioinformatics/single-cell-sequencing-services-how-to-choose-a-provider) - [Genomic Data Analysis Tools: A Comparative Guide for Researchers](/knowledge/bioinformatics/genomic-data-analysis-tools-a-comparative-guide-for-researchers) - [Digital Pathology Scanners: A Buyer's Guide for Clinical and Research Use](/knowledge/bioinformatics/digital-pathology-scanners-a-buyer-s-guide-for-clinical-and-research-use) ## Related Clinical & Scientific Guides * [A Practical Guide to Detecting Antimicrobial Resistance Genes in Shotgun Metagenomic Data](/knowledge/bioinformatics/a-practical-guide-to-detecting-antimicrobial-resistance-genes-in-shotgun-metagenomic-data) * [Computational Immunology: Modeling the Immune System](/knowledge/bioinformatics/computational-immunology-modeling-the-immune-system) * [How to Set Hard Filters for Germline Variant Calling: A Practical Guide to GATK Best Practices](/knowledge/bioinformatics/how-to-set-hard-filters-for-germline-variant-calling-a-practical-guide-to-gatk-best-practices) ## References and Further Reading - [Research Methods Resources](https://www.ncbi.nlm.nih.gov/books). National Library of Medicine. - [EQUATOR Network](https://www.equator-network.org/). EQUATOR Network. - [Core Practices](https://publicationethics.org/core-practices). Committee on Publication Ethics. - [NIH Grants and Funding](https://grants.nih.gov/). National Institutes of Health. - [ORCID for Researchers](https://info.orcid.org/researchers). ORCID. - [Data Management and Sharing Policy](https://sharing.nih.gov/data-management-and-sharing-policy). National Institutes of Health. - [NCBI Data Resources](https://www.ncbi.nlm.nih.gov/). National Center for Biotechnology Information. - [EMBL-EBI Training](https://www.ebi.ac.uk/training). European Bioinformatics Institute. - [Automation Bias in Mammography: The Impact of Artificial Intelligence BI-RADS Suggestions on Reader Performance.](https://pubmed.ncbi.nlm.nih.gov/37129490). Radiology, 2023. - [CXCL14 Promotes Skeletal Muscle Mass Growth and Attenuates Lipopolysaccharide- and Dexamethasone-Induced Muscle Atrophy in Cultured Myotubes and Mouse Models.](https://pubmed.ncbi.nlm.nih.gov/41085163). Journal of cachexia, sarcopenia and muscle, 2025. - [Pregnancy and perinatal outcomes stratified by soluble fms-like tyrosine kinase-1 and placental growth factor gestation-specific thresholds in women with suspected preeclampsia.](https://pubmed.ncbi.nlm.nih.gov/41740916). American journal of obstetrics and gynecology, 2026. > This article is educational and does not replace validated analysis plans, institutional policy, clinical interpretation, or specialist review.