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describe the equation and write it as a formula nd have it written in overleaf:

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Given the scope of this paper, we examine Tukey’s Honestly Significant Difference (HSD) test and the Holm-Bonferroni method for multiple comparisons. Both methods are used to address the problem of Type I errors when performing multiple hypothesis tests, but they approach this issue in different ways.

Tukey’s HSD test is a post-hoc analysis performed after an ANOVA indicates significant differences among group means. It aims to identify which specific means are significantly different from each other. The test uses the studentized range distribution, which considers the maximum difference between group means to control the family-wise error rate (FWER). The HSD test calculates the critical value for the smallest significant difference between group means (denoted as ( text{hsd} )). The formula for the HSD test when group sizes are equal is ( text{hsd} = q_{alpha,A} sqrt{frac{text{MSS}(A)}{S}} ), where ( q_{alpha,A} ) is the critical value from the studentized range distribution, ( text{MSS}(A) ) is the mean square error from the ANOVA, and ( S ) is the number of observations per group. For unequal group sizes, the formula is adjusted to ( text{hsd} = q_{alpha,A} sqrt{frac{1}{2} text{MSS}(A) left( frac{1}{S_a} + frac{1}{S_{a’}} right)} ), where ( S_a ) and ( S_{a’} ) are the sizes of groups ( a ) and ( a’ ) respectively.

The Holm-Bonferroni method is a sequentially rejective procedure used to control the family-wise error rate in multiple hypothesis testing. It is an improvement over the classic Bonferroni correction, providing more power while still controlling the FWER. The Holm-Bonferroni procedure works by ranking all individual p-values from the multiple tests in ascending order: ( p_1 leq p_2 leq cdots leq p_m ). Each ( p_i ) is then compared to ( alpha / (m – i + 1) ), where ( m ) is the total number of hypotheses tested, and ( alpha ) is the desired overall significance level. The null hypothesis ( H_i ) is rejected for the smallest ( p_i ) that satisfies ( p_i leq alpha / (m – i + 1) ), and all subsequent null hypotheses ( H_j ) (for ( j > i )) that also satisfy this condition are rejected. Mathematically, the adjusted significance levels can be represented as ( alpha_i’ = frac{alpha}{m – i + 1} ), where ( alpha_i’ ) is the adjusted significance level for the ( i )-th smallest p-value.

Both Tukey’s HSD and the Holm-Bonferroni methods are essential tools for multiple comparisons, yet they are applied in different contexts. Tukey’s HSD is particularly useful when dealing with pairwise comparisons after an ANOVA, providing a conservative approach to controlling Type I errors. On the other hand, the Holm-Bonferroni method is more flexible and powerful, applicable to a broader range of multiple testing scenarios. These methods ensure the robustness of statistical inferences in the presence of multiple comparisons, thereby enhancing the reliability of research findings across various fields.

bonforrini: https://www.bmj.com/content/310/6973/170.short

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