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Post Hoc Analysis

Post hoc analysis, or a posteriori analysis, generally refers to a type of statistical analysis that is conducted following the rejection of an omnibus null hypothesis. Post hoc analysis can be conducted for a variety of statistics including proportions and frequencies, but post hoc analysis is most commonly used for testing mean differences. This entry focuses primarily on post hoc analysis for investigating mean differences. Following are brief discussions of the omnibus test under a one-way analysis of variance (ANOVA), Tukey honestly significant difference test for pairwise contrasts, Scheffé’s procedure for pairwise or complex contrasts, other available post hoc procedures, and considerations for using post hoc analysis.

Omnibus Test

An omnibus test, from root omnis, meaning “for all,” is a kind of statistical test that simultaneously tests multiple null hypotheses. An omnibus test should be conducted when the researcher does not have specific planned, or a priori, comparisons of theoretical or applied interest. The one-way ANOVA is perhaps the most common and simplest example of an omnibus test. The omnibus test for the one-way ANOVA states that (or tests if) the group means are simultaneously equal to one another. Consider an experimental design with three groups, where the groups represent control, Treatment A, or Treatment B group. Under the null hypothesis (i.e., H0), the omnibus test states that all group means are equal (see Equation 1):

H0:μControl=μTreatment A=μTreatment B.

This hypothesis is tested using a preestablished Type I error rate (α), which is commonly set at .05 in the social sciences. If the null hypothesis is rejected, it is not evident how the null hypothesis is false. That is, the omnibus test does not indicate which means are different from each other. It is possible, for example, that the control group mean is different from both the Treatment A group mean and the Treatment B group mean, but the means of the treatment groups are not different from one another. Post hoc analysis is conducted to explore which means are different. In this sense, post hoc analysis can be considered an exploratory procedure.

Tukey Test for Pairwise Mean Comparisons

Following a rejected omnibus test, a researcher may opt to compare each group mean with every other group mean. These types of comparisons are called pairwise comparisons. The most common procedures for conducting all pairwise comparisons are the Tukey honest significant difference (HSD) test or Tukey–Kramer method. Tukey HSD test allows a researcher to make all pairwise comparisons among group means when each group has the same number of participants. The Tukey–Kramer method allows a researcher to make all pairwise comparisons among group means when the groups do not have the same number of participants. In practice, the groups frequently are not made up of the same number of participants.

From the earlier hypothetical example, the results of the omnibus ANOVA null hypothesis test are shown in Table 1. The group means and standard deviations were made up for this example. The mean and standard deviation, respectively, were 9.8 and 3.46 for the control group, 13.9 and 3.69 for Treatment A group, and 14.1 and 1.6 for Treatment B group.

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