Post-hoc analysis, also known as an a posteriori analysis, is a statistical technique used to explore the relationships between variables after the initial hypothesis testing has been conducted. It allows researchers to identify specific differences or patterns that were not initially predicted or hypothesized.
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Post-hoc analysis is commonly used in the context of ANOVA (Analysis of Variance) to identify which specific groups or conditions differ from one another after a significant overall effect has been found.
The purpose of post-hoc analysis is to provide more detailed information about the nature of the significant differences detected in the initial hypothesis testing.
Post-hoc tests, such as Tukey's Honest Significant Difference (HSD) or Bonferroni corrections, are used to adjust for the increased risk of Type I errors due to multiple comparisons.
Post-hoc analysis is often necessary when the initial hypothesis does not specify the exact nature of the expected differences, or when the researcher wants to explore additional relationships beyond the original hypotheses.
The choice of post-hoc test depends on factors such as the research design, the number of groups being compared, and the underlying assumptions of the statistical tests.
Review Questions
Explain the purpose of conducting post-hoc analysis in the context of the Chi-Square Test of Homogeneity (Section 11.4).
In the context of the Chi-Square Test of Homogeneity (Section 11.4), post-hoc analysis would be used to identify which specific groups or categories are driving the significant overall difference detected by the initial Chi-Square test. The post-hoc analysis allows the researcher to pinpoint the exact sources of the heterogeneity among the groups, providing more detailed information about the nature of the differences. This can be particularly useful when the initial hypothesis does not specify the expected pattern of differences, or when the researcher wants to explore additional relationships beyond the original research question.
Describe how post-hoc analysis can be used to compare the results of different Chi-Square tests, as discussed in Section 11.5 (Comparison of the Chi-Square Tests).
When comparing the results of different Chi-Square tests, as discussed in Section 11.5, post-hoc analysis can be used to determine which specific tests or comparisons are contributing to the overall significant difference. This is particularly important when multiple Chi-Square tests are conducted, as the increased number of comparisons can inflate the risk of Type I errors (false positives). Post-hoc tests, such as Bonferroni corrections, can be applied to adjust the significance level and provide a more accurate assessment of the differences between the various Chi-Square test results.
Analyze how post-hoc analysis can be utilized in the context of the Chi-Square Test of Independence, as described in Section 11.8 (Lab 2: Chi-Square Test of Independence).
In the context of the Chi-Square Test of Independence (Section 11.8), post-hoc analysis can be used to identify the specific cells or combinations of variables that are contributing to the significant overall association detected by the initial Chi-Square test. This can be particularly useful when the researcher wants to understand the nature of the relationship between the variables, rather than just the presence of an association. By conducting post-hoc analyses, the researcher can pinpoint the specific areas of the contingency table where the observed and expected frequencies differ the most, providing more detailed insights into the underlying relationships.
The process of using statistical methods to determine whether a particular claim or hypothesis about a population parameter is likely to be true or false.
The probability of rejecting the null hypothesis when it is true, also known as the Type I error rate.
Multiple Comparisons: The act of making several statistical comparisons simultaneously, which can increase the likelihood of finding significant differences by chance.