title: On estimating and testing associations between random coefficients from multivariate generalized linear mixed models of longitudinal outcomes.
abstract: Different types of outcomes (e.g. binary, count, continuous) can be simultaneously modeled with multivariate generalized linear mixed models by assuming: (1) same or different link functions, (2) same or different conditional distributions, and (3) conditional independence given random subject effects. Others have used this approach for determining simple associations between subject-specific parameters (e.g. correlations between slopes). We demonstrate how more complex associations (e.g. partial regression coefficients between slopes adjusting for intercepts, time lags of maximum correlation) can be estimated. Reparameterizing the model to directly estimate coefficients allows us to compare standard errors based on the inverse of the Hessian matrix with more usual standard errors approximated by the delta method; a mathematical proof demonstrates their equivalence when the gradient vector approaches zero. Reparameterization also allows us to evaluate significance of coefficients with likelihood ratio tests and to compare this approach with more usual Wald-type t-tests and Fisher's z transformations. Simulations indicate that the delta method and inverse Hessian standard errors are nearly equivalent and consistently overestimate the true standard error. Only the likelihood ratio test based on the reparameterized model has an acceptable type I error rate and is therefore recommended for testing associations between stochastic parameters. Online supplementary materials include our medical data example, annotated code, and simulation details.
authors: 1 Department of Psychiatry, School of Medicine, University of Colorado Anschutz Medical Center, Aurora, USA.|2 Department of Biostatistics and Informatics, Colorado School of Public Health, University of Colorado Anschutz Medical Center, Aurora, USA.
authors: Susan K
authors: SK
authors: Mikulich-Gilbertson
authors: 2 Department of Biostatistics and Informatics, Colorado School of Public Health, University of Colorado Anschutz Medical Center, Aurora, USA.
authors: Brandie D
authors: BD
authors: Wagner
authors: 1 Department of Psychiatry, School of Medicine, University of Colorado Anschutz Medical Center, Aurora, USA.
authors: Paula D
authors: PD
authors: Riggs
authors: 2 Department of Biostatistics and Informatics, Colorado School of Public Health, University of Colorado Anschutz Medical Center, Aurora, USA.
authors: Gary O
authors: GO
authors: Zerbe
mesh_terms: D000293:Adolescent; D000428:Alcohol Drinking / Q000453:epidemiology; D001289:Attention Deficit Disorder with Hyperactivity / Q000150:complications / Q000453:epidemiology; D015928:Cognitive Behavioral Therapy; D006801:Humans; D016014:Linear Models; D008137:Longitudinal Studies; D008385:Marijuana Smoking / Q000453:epidemiology / Q000517:prevention & control / Q000523:psychology; D012107:Research Design; D013269:Stochastic Processes
keywords: Fisher’s z transformation; Joint modeling; delta method; likelihood ratio tests; random coefficient associations; stochastic parameter associations