Follow-up testing in functional analysis of variance

Abstract

Sampling responses at a high time resolution is gaining popularity in pharmaceutical, epidemiological, environmental and biomedical studies. For example, investigators might expose subjects continuously to a certain treatment and make measurements throughout the entire duration of each exposure. An important goal of statistical analysis for a resulting longitudinal sequence is to evaluate the effect of the covariates, which may or may not be time dependent, on the outcomes of interest. Traditional parametric models, such as generalized linear models, nonlinear models, and mixed effects models, are all subject to potential model misspecification and may lead to erroneous conclusions in practice. In semiparametric models, a time-varying exposure might be represented by an arbitrary smooth function (the nonparametric part) and the remainder of the covariates are assumed to be fixed (the parametric part). The potential drawbacks of the semiparametric approach are uncertainty in the smoothing function interpretation, and ambiguity in the parametric test (a particular regression coefficient being zero in the presence of the other terms in the model). Functional linear models (FLM), or the so called structural nonparametric models, are used to model continuous responses per subject as a function of time-variant coefficients and a time-fixed covariate matrix. In recent years, extensive work has been done in the area of nonparametric estimation methods, however methods for hypothesis testing in the functional data setting are still undeveloped and greatly in demand. In this research we develop methods that address hypotheses testing problem in a special class of FLMs, namely the Functional Analysis of Variance (FANOVA). In the development of our methodology, we pay a special attention to the problem of multiplicity and correlation among tests. We discuss an application of the closure principle to the follow-up testing of the FANOVA hypotheses as well as computationally efficient shortcut arising from a combination of test statistics or p-values. We further develop our methods for pair-wise comparison of treatment levels with functional data and apply them to simulated as well as real data sets.