Adjusted p-values for SGoF multiple test procedure

In the field of multiple comparison procedures, adjusted p-values are an important tool to evaluate the significance of a test statistic while taking the multiplicity into account. In this paper, we introduce adjusted p-values for the recently proposed Sequential Goodness-of-Fit (SGoF) multiple test procedure by letting the level of the test vary on the unit interval. This extends previous research on the SGoF method, which is a method of high interest when one aims to increase the statistical power in a multiple testing scenario. The adjusted p-value is the smallest level at which the SGoF procedure would still reject the given null hypothesis, while controlling for the multiplicity of tests. The main properties of the adjusted p-values are investigated. In particular, we show that they are a subset of the original p-values, being equal to 1 for p-values above a certain threshold. These are very useful properties from a numerical viewpoint, since they allow for a simplified method to compute the adjusted p-values. We introduce a modification of the SGoF method, termed majorant version, which rejects the null hypotheses with adjusted p-values below the level. This modification rejects more null hypotheses as the level increases, something which is not in general the case for the original SGoF. Adjusted p-values for the conservative version of the SGoF procedure, which estimates the variance without assuming that all the null hypotheses are true, are also included. The situation with ties among the p-values is discussed too. Several real data applications are investigated to illustrate the practical usage of adjusted p-values, ranging from a small to a large number of tests.