Hypothesis testing in functional linear regression models with Neyman's truncation and wavelet thresholding for longitudinal data

Abstract

Longitudinal data sets in biomedical research often consist of large numbers of repeated measures. In many cases, the trajectories do not look globally linear or polynomial, making it difficult to summarize the data or test hypotheses using standard longitudinal data analysis based on various linear models. An alternative approach is to apply the approaches of functional data analysis, which directly target the continuous nonlinear curves underlying discretely sampled repeated measures. For the purposes of data exploration, many functional data analysis strategies have been developed based on various schemes of smoothing, but fewer options are available for making causal inferences regarding predictor-outcome relationships, a common task seen in hypothesis-driven medical studies. To compare groups of curves, two testing strategies with good power have been proposed for high-dimensional analysis of variance: the Fourier-based adaptive Neyman test and the wavelet-based thresholding test. Using a smoking cessation clinical trial data set, this paper demonstrates how to extend the strategies for hypothesis testing into the framework of functional linear regression models (FLRMs) with continuous functional responses and categorical or continuous scalar predictors. The analysis procedure consists of three steps: first, apply the Fourier or wavelet transform to the original repeated measures; then fit a multivariate linear model in the transformed domain; and finally, test the regression coefficients using either adaptive Neyman or thresholding statistics. Since a FLRM can be viewed as a natural extension of the traditional multiple linear regression model, the development of this model and computational tools should enhance the capacity of medical statistics for longitudinal data.

Wavelet-based genetic association analysis of functional phenotypes arising from high-throughput sequencing assays

“…In particular, Morris et al (2008) presented a framework for identifying locations within a region that show significant effects of covariates. Other relevant work on wavelet methods for regression analysis of functional data include Abramovich and Angelini (2006), Antoniadis and Sapatinas (2007), Fan and Lin (1998), Yang and Nie (2008), Zhao and Wu (2008). Previous applications of wavelet-based methods in genomics include Clement et al (2012), Day et al (2007), Mitra and Song (2012), Spencer et al (2006), Wu et al (2010), Zhang et al (2008).…”

A quasiF‐test for functional linear models with functional covariates and its application to longitudinal data

“…In this paper, we assumed that missing values are ignorable in the sense that we can use the observed values to obtain unbiased estimates. Figure 1 shows the average BDI scores by treatments at baseline (week 0) and during treatments (weeks [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. It is suggestive that all treatments were associated with reduction of depression level in the first 2 to 3 weeks, with treatments CM and CBT C CM appearing more effective than the other two.…”

“…Linear models are frequently used for interpreting or predicting responses by a set of covariates or predictors. Many authors have studied the form of functional linear models: Y.t / D Xˇ.t /C .t /, where the responses are functions and the covariates are scalar vectors (see, for example, [6][7][8][9][10][11][12][13]). Ramsay and Silverman [6] laid out some general ideas on estimation and provided preliminary methods for inference.…”

“…Shen and Faraway [9] proposed a functional F -test for comparing nested functional linear models, and Shen and Xu [10] considered diagnostics for such models. Yang et al [11] and Yang and Nie [12] used such models to analyze a longitudinal data set from a clinical trial. Zhang and Chen [13] proposed an L 2 -norm-based test statistic and studied its asymptotic properties.…”