Testing for Order-Restricted Hypotheses in Longitudinal Data

Abstract

In many biomedical studies, we are interested in comparing treatment effects with an inherent ordering. We propose a "quadratic score test" (QST) based on a quadratic inference function for detecting an order in treatment effects for correlated data. The quadratic inference function is similar to the negative of a log-likelihood, and it provides test statistics in the spirit of a "χ"-super-2-test for testing nested hypotheses as well as for assessing the goodness of fit of model assumptions. Under the null hypothesis of no order restriction, it is shown that the QST statistic has a Wald-type asymptotic representation and that the asymptotic distribution of the QST statistic is a weighted "χ"-super-2-distribution. Furthermore, an asymptotic distribution of the QST statistic under an arbitrary convex cone alternative is provided. The performance of the QST is investigated through Monte Carlo simulation experiments. Analysis of the polyposis data demonstrates that the QST outperforms the Wald test when data are highly correlated with a small sample size and there is a significant amount of missing data with a small number of clusters. The proposed test statistic accommodates both time-dependent and time-independent covariates in a model. Copyright 2006 Royal Statistical Society.

On methods for correcting for the look-elsewhere effect in searches for new physics

“…In order to simplify notation we drop the dependence of c PL ( θ ) on θ and write simply, c PL . The corresponding global p-value is P(C PL > c PL ); [14] prove that, under H 0 , C PL converges to the supremum of a mean zero Gaussian process as N → ∞. The approximation, p PL , of P(C PL > c PL ) is obtained through so-called tube formulae for Gaussian processes [20].…”

“…In order to overcome some of the limitations arising in multiple hypothesis testing, two promising methods have been recently proposed in physics literature. The first (henceforth PL) was introduced in 2005 [1] and refined in [14]. Its methodology relies on the Score function and is purported to be more powerful than the usual Likelihood Ratio Test (LRT) approach.…”

“…PL, on the other hand, involves the so-called tube formulae, where an approximation of P(supY t > c) is obtained as the ratio between the volume of a tube built around the manifold associated with supY t on the unit sphere, and the volume of the unit sphere itself. Although we describe both methods more fully in Section 4, we do not focus on their mathematical details, but rather emphasize their computational implementation; readers are directed to [1,2,14,15,[18][19][20] for technical development.…”

Iteratively Reweighted Generalized Least Squares for Estimation and Testing With Correlated Data: An Inference Function Framework

“…Existing research on the QIF (Qu et al 2000;Pilla and Loader 2005;Pilla 2005;Pilla et al 2006) has focused on estimation and testing. Pilla and Loader (2005) derived a generalization of the QIF test for general linear hypotheses involving correlated data while establishing a unified large-sample theory for the QIF.…”

Order‐restricted inference for multivariate longitudinal data with applications to the natural history of hearing loss

“…Most of these models restrict attention to the analysis of a single‐outcome variable, measured repeatedly over time. Shin et al , Peddada et al , Pilla et al , and Tan et al considered order restrictions in fixed‐effects models for repeated‐measures data. Silvapulle addressed mixed‐effects models, showing that some mixed linear models can be reduced to fixed‐effect models where the usual one‐sided tests for ordered hypothesis can be applied.…”