Tukey's Honestly Significant Difference (HSD) is a statistical method used in the context of one-way ANOVA to determine which specific group means are significantly different from each other. It is a post-hoc test that provides a way to identify where the differences lie after an ANOVA has shown that at least one group mean is significantly different from the others.
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Tukey's HSD is a conservative post-hoc test, meaning it controls the family-wise error rate and reduces the likelihood of finding false positive differences.
The test compares all possible pairs of group means and determines if the difference between any two means is large enough to be considered statistically significant.
Tukey's HSD is based on the Studentized Range distribution, which takes into account the number of groups and the sample size.
The test statistic used in Tukey's HSD is the Honestly Significant Difference (HSD), which is compared to the critical value to determine significance.
Tukey's HSD is commonly used when the one-way ANOVA has shown a significant overall effect, but the researcher wants to know which specific group means are different.
Review Questions
Explain the purpose of Tukey's HSD in the context of one-way ANOVA.
The purpose of Tukey's HSD is to determine which specific group means are significantly different from each other after a one-way ANOVA has shown that at least one group mean is significantly different from the others. Tukey's HSD is a post-hoc test that compares all possible pairs of group means and identifies which differences are large enough to be considered statistically significant. This allows the researcher to pinpoint the source of the significant overall effect found in the one-way ANOVA.
Describe how Tukey's HSD controls the family-wise error rate and what this means for the interpretation of results.
Tukey's HSD is considered a conservative post-hoc test because it controls the family-wise error rate. This means that it reduces the likelihood of finding false positive differences when making multiple comparisons between group means. By controlling the family-wise error rate, Tukey's HSD ensures that the overall probability of making a Type I error (incorrectly rejecting the null hypothesis) across all pairwise comparisons is maintained at the specified significance level (e.g., α = 0.05). This makes the results of Tukey's HSD more reliable and trustworthy when determining which group means are truly different from each other.
Analyze how the test statistic used in Tukey's HSD, the Honestly Significant Difference (HSD), is calculated and how it is used to determine statistical significance.
The test statistic used in Tukey's HSD is the Honestly Significant Difference (HSD), which is calculated based on the Studentized Range distribution. This distribution takes into account the number of groups and the sample size when determining the critical value for significance. The HSD is calculated as the difference between two group means divided by the standard error of the difference. This HSD value is then compared to the critical value from the Studentized Range distribution. If the HSD exceeds the critical value, the difference between the two group means is considered statistically significant. The use of the Studentized Range distribution in Tukey's HSD ensures that the family-wise error rate is controlled, making the test more conservative and reliable in identifying true differences between group means.