Modeling the random effects covariance matrix for longitudinal data with covariates measurement error

Abstract

Longitudinal data occur frequently in practice such as medical studies and life sciences. Generalized linear mixed models (GLMMs) are commonly used to analyze such data. It is typically assumed that the random effects covariance matrix is constant among subjects in these models. In many situations, however, the correlation structure may differ among subjects and ignoring this heterogeneity can lead to biases in model parameters estimate. Recently, Lee et al developed a heterogeneous random effects covariance matrix for GLMMs for error-free covariates. Covariates measured with error also happen frequently in the longitudinal data set-up (eg, blood pressure and cholesterol level). Ignoring this issue in the data may produce bias in model parameters estimate and lead to wrong conclusions. In this paper, we propose an approach to properly model the random effects covariance matrix based on covariates in the class of GLMMs, where we also have covariates measured with error. The resulting parameters from the decomposition of random effects covariance matrix have a sensible interpretation and can be easily modeled without the concern of positive definiteness of the resulting estimator. The performance of the proposed approach is evaluated through simulation studies, which show that the proposed method performs very well in terms of bias, mean squared error, and coverage rate. An application of the proposed method is also provided using a longitudinal data from Manitoba follow-up study.

A Time-Heterogeneous D-Vine Copula Model for Unbalanced and Unequally Spaced Longitudinal Data

“…It is clear from Web Figure 3 (see Web Appendix C) that the response variable exhibits a relationship with covariates for different visit times such as baseline, visit 1, visit 2, and so on. The MFUS data set has been recently analyzed in Hoque and Torabi (2018) using a mixed model approach. Hence, we employ the LMM to fit the data using the function lme from the R library nlme (Pinheiro et al.…”

Spatial models for non‐Gaussian data with covariate measurement error

“…In the presence of ME, one is unable to observe , , but is observed as a surrogate for X i through Z i = X i + U i , where X i has a normal distribution with mean μ x and variance–covariance matrix , is a random vector from a normal distribution with mean 0 k and variance–covariance matrix with I k as the identity matrix of dimension k , and is known (Hoque & Torabi, ; Torabi, ; Torabi, Datta, & Rao, ). We can then write where is a diagonal matrix with the given elements on the diagonal.…”

“…x I k , U i = U (s i ) is a random vector from a normal distribution with mean 0 k and variance-covariance matrix 2 u I k with I k as the identity matrix of dimension k, and 2 u is known (Hoque & Torabi, 2018;Torabi, 2012;Torabi, Datta, & Rao, 2009). We can then write (…”

Exploratory assessment of treatment‐dependent random‐effects distribution using gradient functions

“…8,14,15 Several models flexible enough to accommodate this dependence are available. [16][17][18][19] However, it is ideal to consider in advance whether such complex models are indeed required. Furthermore, dependence of the random-effects distribution on covariates can occur concurrently with violation of the normality assumption.…”

“…The dependence on covariates is usually ignored in the modeling unless valid evidence is available; however, when fitting generalized linear mixed‐effects models, the estimates of fixed effects may be sensitive to the assumption that random effects do not depend on the covariates 8,14,15 . Several models flexible enough to accommodate this dependence are available 16‐19 . However, it is ideal to consider in advance whether such complex models are indeed required.…”