Likelihood Methods for Regression Models with Expensive Variables Missing by Design

Abstract

In some applications involving regression the values of certain variables are missing by design for some individuals. For example, in two-stage studies (Zhao and Lipsitz, 1992), data on "cheaper" variables are collected on a random sample of individuals in stage I, and then "expensive" variables are measured for a subsample of these in stage II. So the "expensive" variables are missing by design at stage I. Both estimating function and likelihood methods have been proposed for cases where either covariates or responses are missing. We extend the semiparametric maximum likelihood (SPML) method for missing covariate problems (e.g. Chen, 2004; Ibrahim et al., 2005; Zhang and Rockette, 2005, 2007) to deal with more general cases where covariates and/or responses are missing by design, and show that profile likelihood ratio tests and interval estimation are easily implemented. Simulation studies are provided to examine the performance of the likelihood methods and to compare their efficiencies with estimating function methods for problems involving (a) a missing covariate and (b) a missing response variable. We illustrate the ease of implementation of SPML and demonstrate its high efficiency.

Design and relative efficiency in two-phase studies

“…They considered the expensive response and the expensive covariate problems. Following Zhao et al (2009) in this research, we discuss design strategies for both the expensive response model and the expensive covariate model, which are different from those in Thomas (2007). We consider both analytic and numerical approaches.…”

“…Lawless et al, 1999). Zhao et al (2009) described a semiparametric (MLE) method for regression models with expensive variables missing in twophase designs. They considered the expensive response and the expensive covariate problems.…”

“…For example, let S be a response variable, D be an expensive covariate with missing values and X be a fully observed covariate. Chatterjee et al (2003) and Zhao et al (2009) considered estimating the joint conditional distribution PrðS,D9XÞ ¼ PrðS9D,XÞPrðD9XÞ, given the fully observed covariate X, where estimation of the marginal distribution of X was not required. This is more efficient than estimating the joint distribution PrðS,D,XÞ ¼ PrðS9D,XÞPrðX9DÞPrðDÞ as in Morara et al (2007) in general.…”

Rejoinder

“…Likelihoodrelated methods are convenient and efficient in many settings (e.g. Kalbfleisch & Lawless, 1988;Chen & Little, 1999;Chatterjee et al, 2003 Zhao et al, 2009), and more generally, a wide variety of "augmented" estimating functions (Robins et al, 1995) exist.…”

Discussions

“…Likelihood‐related methods are convenient and efficient in many settings (e.g. Kalbfleisch & Lawless, 1988; Lawless et al , 1999; Chen & Little, 1999; Chatterjee et al , 2003; Zhang & Rockette, 2005; McLeish & Struthers, 2006; Zhao et al , 2009), and more generally, a wide variety of “augmented” estimating functions (Robins et al , 1995) exist.…”