On the Use of the Shapiro‐Wilk Test in Two‐Stage Adaptive Inference for Paired Data from Moderate to Very Heavy Tailed Distributions

Abstract

Paired data arises in a wide variety of applications where often the underlying distribution of the paired differences is unknown. When the differences are normally distributed, the t-test is optimum. On the other hand, if the differences are not normal, the t-test can have substantially less power than the appropriate optimum test, which depends on the unknown distribution. In textbooks, when the normality of the differences is questionable, typically the non-parametric Wilcoxon signed rank test is suggested. An adaptive procedure that uses the Shapiro-Wilk test of normality to decide whether to use the t-test or the Wilcoxon signed rank test has been employed in several studies. Faced with data from heavy tails, the U.S. Environmental Protection Agency (EPA) introduced another approach: it applies both the sign and t-tests to the paired differences, the alternative hypothesis is accepted if either test is significant. This paper investigates the statistical properties of a currently used adaptive test, the EPA's method and suggests an alternative technique. The new procedure is easy to use and generally has higher empirical power, especially when the differences are heavy-tailed, than currently used methods.

Maximum Tests are Adaptive Permutation Tests

“…When this correlation is small the maximum test is often preferable to the MERT, in particular in case of ρ* ≤ 0.5. For ρ* ≥ 0.7 there was, however, virtually no difference in their powers (Freidlin et al, 1999Freidlin & Korn, 2002, Gastwirth & Freidlin, 2000, Neuhäuser et al, 2004, Zheng et al, 2002. Other linear combinations than the MERT are further alternatives to the maximum test, see e.g.…”

“…Instead, permutation tests may be preferable for several applications (Ludbrook & Dudley, 1998). Note that an approximation using the asymptotic distribution of a maximum statistic can be poor even when all univariate statistics are asymptotically normal (Freidlin & Korn, 2002).…”

“…Recently, MERTs were compared with the corresponding maximum tests (Freidlin et al, 1999(Freidlin et al, , 2003bFreidlin & Korn, 2002, Gastwirth & Freidlin, 2000 Hothorn, 1999, Neuhäuser et al, 2004, Zheng et al, 2002. Such a comparison depends on the minimum correlation ρ* between two of the univariate tests.…”

“…Bretz & Hothorn, 2001, Chung & Fraser, 1958, Freidlin & Korn, 2002, Freidlin et al, 1999, 2002, 2003bGastwirth & Freidlin, 2000;Hirotsu, 1986;Marozzi, 2004a, 2004b, Neuhäuser & Hothorn, 1999, Neuhäuser et al, 2000, 2004, Zheng et al, 2002. The approach has the advantage that neither a selector statistic nor the specification of which test should be performed for which values of the selector is needed.…”

“…Freidlin et al (2003a) proposed a test where the selector and the test statistics are asymptotically uncorrelated only. Furthermore, the concept of adaptive tests was applied to parametric tests .…”