Statistica Sinica 28 (2018), 339-362

HOCHBERG PROCEDURE UNDER

NEGATIVE DEPENDENCE

Jiangtao Gou and Ajit C. Tamhane

Hunter College and Northwestern University

Abstract: The Hochberg (1988) procedure is commonly used in practice to test
multiple hypotheses based on their *p*-values. It is a conservative step-up shortcut
to the closed procedure (Marcus, Peritz and Gabriel (1976)) based on the Simes
(1986) test. The Simes test is anti-conservative if the test statistics are negatively
dependent in a certain sense. So practitioners are reluctant to use the Hochberg
procedure under this condition and prefer to use the less powerful Holm (1979) procedure, which requires no dependence assumptions. But the Hochberg procedure
is conservative by construction, so we may conjecture that it will remain so under
certain types of negative dependence. In this paper we show that a slightly modified version of the Hochberg procedure controls the familywise type I error rate
(FWER) if the *p*-values follow a multivariate uniform distribution which is a mixture of bivariate components each of which is negative quadrant dependent (NQD)
(Lehmann (1966)) or positive dependent through stochastic ordering (PDS) (Block,
Savits and Shaked (1985)). By negative dependence we will mean this distribution
model, in particular, that its negatively dependent bivariate components are NQD.
Simulations suggest that conservatism of the Hochberg procedure is likely to be
true for more general negatively dependent distributions.

Key words and phrases: Familywise type I error rate, multiple comparisons, multivariate uniform distribution, negative/positive quadrant dependence, negative/positive dependence through stochastic ordering, simes test.