Why do we apply post hoc tests in Anova?

As a statistical expert with a deep understanding of experimental design and analysis, I can provide insight into the role of post hoc tests in the context of ANOVA (Analysis of Variance). ANOVA is a statistical technique used to compare the means of three or more groups to determine if there are any statistically significant differences between them. It is a powerful tool for uncovering differences in group means, but it does not tell us which specific groups are different from each other. This is where post hoc tests come into play. Post hoc tests are additional analyses that are conducted after an ANOVA has shown a significant result. The primary purpose of these tests is to identify the specific group differences that contribute to the overall significant effect. Without post hoc tests, we would only know that there is a difference somewhere among the groups, but not which groups are contributing to this difference. Here are several reasons why post hoc tests are essential: 1. Clarification of Results: Post hoc tests provide clarity by pinpointing which groups are significantly different. This is crucial for interpreting the results of an experiment accurately. 2. Multiple Comparisons Issue: When you compare multiple groups, the chance of finding a significant difference by chance alone increases. Post hoc tests help to control for this by adjusting the significance level to account for the increased risk of Type I errors (false positives). 3. Practical Significance: While ANOVA can demonstrate a statistically significant difference, post hoc tests can help determine if the differences are practically meaningful. This is particularly important when the effect sizes are small. 4. Research Hypotheses: Post hoc tests can be used to test specific hypotheses about group differences that were not the primary focus of the ANOVA. This can lead to new insights and directions for future research. 5. Complex Designs: In more complex experimental designs, where interactions between factors are expected, post hoc tests can help to disentangle the sources of variation and identify the conditions under which effects occur. 6. Reporting Standards: Many scientific journals and reporting standards require that if an overall significant result is found, the researcher must also report which groups differ from each other. Post hoc tests facilitate this requirement. 7. Policy and Decision Making: In applied settings, knowing which groups differ can inform policy decisions, interventions, or the allocation of resources. It is important to note that post hoc tests should be chosen carefully based on the design of the study and the specific hypotheses being tested. There are various types of post hoc tests available, such as Tukey's HSD, Bonferroni correction, Scheffé's method, and others, each with its own assumptions and appropriate use cases. When interpreting the results of post hoc tests, it is also crucial to consider the context of the study, the effect sizes, and the practical implications of the findings. The goal is not just to find statistically significant differences but to understand the underlying patterns and relationships in the data. In conclusion, post hoc tests are a critical component of the ANOVA process. They allow researchers to move beyond a simple yes or no answer to the question of whether group means differ and instead provide a detailed understanding of where and how these differences occur.

Because post hoc tests are run to confirm where the differences occurred between groups, they should only be run when you have a shown an overall statistically significant difference in group means (i.e., a statistically significant one-way ANOVA result).