  This paper concerns the problem of applying the generalized goodness-of-fit
(gGOF) type tests for analyzing correlated data. The gGOF family broadly covers
the maximum-based testing procedures by ordered input $p$-values, such as the
false discovery rate procedure, the Kolmogorov-Smirnov type statistics, the
$\phi$-divergence family, etc. Data analysis framework and a novel $p$-value
calculation approach is developed under the Gaussian mean model and the
generalized linear model (GLM). We reveal the influence of data transformations
to the signal-to-noise ratio and the statistical power under both sparse and
dense signal patterns and various correlation structures. In particular, the
innovated transformation (IT), which is shown equivalent to the marginal
model-fitting under the GLM, is often preferred for detecting sparse signals in
correlated data. We propose a testing strategy called the digGOF, which
combines a double-adaptation procedure (i.e., adapting to both the statistic's
formula and the truncation scheme of the input $p$-values) and the IT within
the gGOF family. It features efficient computation and robust adaptation to the
family-retained advantages for given data. Relevant approaches are assessed by
extensive simulations and by genetic studies of Crohn's disease and amyotrophic
lateral sclerosis. Computations have been included into the R package SetTest
available on CRAN.
