Bonferroni adjustment Bonferroni adjustment (equally weighted) – Reject H 0j with p i <α/m – Pr (at least 1 test is rejected) = Pr(⋃ i=1,…,m {reject H i }) ≤ Σ i=1,…,m Pr(reject H i ) ≤ Σ i=1,…,m (α/m) = α Bonferroni adjustment (unequally weighted) – Given nonnegative weights {w 1,w 2,…,w m } for the tests associated with the hypotheses {H 01,H 02,…,H 0m ) – Σw j =1 – Reject H 0j with p i <w j α Given p-value {p 1,p 2,…, p m } for the tests associated with the null hypotheses {H 01,H 02,…,H 0m } conservative and lack of power

Holm’s step-down procedure Step 1: If P [1] <α/m, reject H 0[1] and go to Step 2, else stop Step 2: If P [2] <α/(m-1), reject H 0[2] and go to Step 3, else stop … Step m: If P [m] <α,, reject H 0[m] and stop Let P [1] <P [2] <…<P [m] denote the ordered p-values corresponding to H 0[1],…,H 0[m]

Hocheberg’s step-up method Step 1: If P [m] <α, reject all H 0[j], j=1,…,m and stop, else go to Step 2 Step 2: If P [m-1] <α/2, reject H 0[j], j=1,…,m-1 and stop, else go to Step 3 … Step m: If P [1] <α/m,, reject H 0[j], j=1 and stop Let P [1] <P [2[ <…<P [m] denote the ordered p-values corresponding to H 0[1],…,H 0[m]

Bonferroni vs. Holm vs. Hochberg Ordered p-values Significance level Bonferroni (equally weighted) Holm (step-down) Hochberg (step-up)