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To safely conduct post hoc comparisons without inflating the risk of making a Type I error, researchers use modified $$t$$-test procedures rather than standard $$t$$-tests. Techniques such as the Bonferroni procedure, Fisher's least significant difference (LSD) test, and Tukey's honestly significant difference (HSD) test are designed specifically for this purpose. While their mathematical details differ, their shared goal is to keep the overall risk of mistakenly rejecting a true null hypothesis at an acceptable level (typically close to $$5\%$$) across multiple comparisons.

Modified t-Test Procedures

The Bonferroni procedure is a specific modified $$t$$-test technique used by researchers to conduct post hoc comparisons. It is designed to safely compare pairs of group means following a statistically significant one-way ANOVA while keeping the overall risk of making a Type I error (mistakenly rejecting a true null hypothesis) to an acceptable level, typically close to $$5%$$, across all the multiple comparisons being made.

Bonferroni Procedure

Fisher's least significant difference (LSD) test is a modified $$t$$-test procedure utilized for post hoc comparisons after a significant one-way ANOVA result. Like other specialized techniques, its primary purpose is to evaluate differences between specific pairs of group means while mathematically controlling the compounded risk of making a Type I error (mistakenly rejecting a true null hypothesis) that occurs when conducting multiple standard $$t$$-tests.

Fisher's Least Significant Difference (LSD) Test

Tukey's honestly significant difference (HSD) test is a statistical procedure used for post hoc comparisons to determine which specific group means differ from one another following a significant one-way ANOVA. It serves as one of several modified $$t$$-test approaches designed specifically to limit the overall probability of committing a Type I error (mistakenly rejecting a true null hypothesis) when performing multiple pairwise comparisons.

Tukey's Honestly Significant Difference (HSD) Test

Why do researchers use modified t-test procedures, such as the Bonferroni procedure or Tukey's honestly significant difference (HSD) test, rather than standard t-tests when conducting post hoc comparisons?

If a researcher decides to use standard t-tests rather than modified t-test procedures for multiple post hoc comparisons, they will successfully keep their overall risk of mistakenly rejecting a true null hypothesis at an acceptable level.

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