Single Contrast Tests

Abstract

In Chap. 3, we discussed several testing procedures for testing the null hypothesis of no dose effect against ordered alternatives. Williams (Biometrics 27:103–117, 1971, 1972) proposed a step-down procedure to test for the dose effect. The tests are performed sequentially from the comparison between the isotonic mean of the highest dose and the sample mean of the control to the comparison between the isotonic mean of the lowest dose and the sample mean of the control. The procedure stops at the dose level where the null hypothesis of no dose effect is not rejected. Marcus (Biometrika 63:177–83, 1976) proposed a modification of the Williams procedure, in which the sample mean of the control was replaced by the isotonic mean of the control. The likelihood ratio test, discussed by Bartholomew (Biometrika 48:325–332, 1961), Barlow et al. (Statistical inference under order restriction. Wiley, New York), and Robertson et al. (Order restricted statistical inference. Wiley), uses the ratio between the variance calculated under the null hypothesis and the variance calculated under an ordered alternative. In the context of dose-response microarray data, Hu et al. (Bioinformatics 21(17):3524–3529, 2005) proposed a test statistic that was similar to Marcus’ statistic, but with the variance estimator calculated under the ordered alternative. Lin et al. (Stat Appl Genet Mol Biol 6(1), article 26, 2007) proposed a modification for Hu’s test statistic in which the degrees of freedom for the variance estimator are not fixed for all genes. In Chap. 7 we discussed different inference approaches for order-restricted alternatives and the directional inference for testing the isotonic regression and the multiplicity issue are discussed as well. We illustrate how to use the IsoGene package to compute the five test statistics discussed above and how to obtain the list of significant genes using the FDR correction.