Penalized Generalized Estimating Equations for High‐Dimensional Longitudinal Data Analysis

Abstract

We consider the penalized generalized estimating equations (GEEs) for analyzing longitudinal data with high-dimensional covariates, which often arise in microarray experiments and large-scale health studies. Existing high-dimensional regression procedures often assume independent data and rely on the likelihood function. Construction of a feasible joint likelihood function for high-dimensional longitudinal data is challenging, particularly for correlated discrete outcome data. The penalized GEE procedure only requires specifying the first two marginal moments and a working correlation structure. We establish the asymptotic theory in a high-dimensional framework where the number of covariates p(n) increases as the number of clusters n increases, and p(n) can reach the same order as n. One important feature of the new procedure is that the consistency of model selection holds even if the working correlation structure is misspecified. We evaluate the performance of the proposed method using Monte Carlo simulations and demonstrate its application using a yeast cell-cycle gene expression data set.

Covariance estimators for Generalized Estimating Equations (GEE) in longitudinal analysis with small samples

“…However, there are still several limitations for this work. First, the modifications discussed here for variance estimation are directly focusing on the "sandwich" variance estimator, but some other methods were also proposed but not covered here (i.e., improving the efficiency and robustness of parameter estimates) [28,29,32,33,34]; Second, our recommendation on the appropriate sample sizes for each estimator for preserving Type I error is obtained through limited simulation studies under general set-ups (i.e., equal cluster sizes); however, this guideline may not be always applicable, for instance, the cases with unequal cluster sizes; Third, we only evaluate the Type I error, but the Type II error or the power warrants further investigations. It is also worth pointing out that the selection of an appropriate modification method relies on various aspects of the real application (i.e., study design or intra-subject correlation) [2,4,35].…”

Ultrahigh dimensional time course feature selection

“…A key advantage of the GEE approach is that, when p is of order o ( n 1/3 ), it yields a consistent estimator even with mis-specified working correlation structures (Wang, 2011). But it fails when the dimensionality p greatly exceeds the number of subjects n , even if regularized methods are used (Wang et al, 2012; Xu et al, 2013). This brings up a high demand of screening methods that can quickly reduce p .…”

“…Conditions (C1) and (C2) are analogous to conditions (A1), (A4) of Wang et al (2012) for generalized estimating equations. Condition (C3) has been assumed in Wang et al (2012), Zhu et al (2011), and Li et al (2012).…”

“…Condition (C3) has been assumed in Wang et al (2012), Zhu et al (2011), and Li et al (2012). This condition could be relaxed by the following moment condition: For each 1 ≤ i ≤ n and 1 ≤ k ≤ m , there exists a positive constant t 0 such that $$\underset{1\le j\le p_n}{\text{max}}Efalse\{\text{exp}false(t{X}_{ijk}false)false\}<\infty ,$$ for all 0 < t < t 0 .…”

“…These methods have included, for example, bridge-, LASSO- and SCAD-penalized generalized estimating equations (GEE) (Fu, 2003; Wang et al, 2012), penalized joint log likelihoods for mixed-effects models with continuous responses (Bondell et al, 2010), and a two-stage shrinkage approach (Xu et al, 2013). However, they all stipulate that the number of covariates p grows to infinity at a polynomial rate o ( n α ) for some 0 ≤ α < 4/3.…”