A marginalized multilevel model for bivariate longitudinal binary data

Abstract

pagesThis thesis study considers analysis of bivariate longitudinal binary data. We propose a model based on marginalized multilevel model framework. The proposed model consists of two levels such that the first level associates the marginal mean of responses with covariates through a logistic regression model and the second level includes subject/time specific random intercepts within a probit regression model. The covariance matrix of multiple correlated time-specific random intercepts for each subject is assumed to represent the within-subject association. The subject-specific random effects covariance matrix is further decomposed into its dependence and variance components through modified Cholesky decomposition method to handle possible computational and statistical problems that may be associated with its highdimensionality. Then the unconstrained version of resulting parameters are modelled in terms of covariates with low-dimensional regression parameters, which provides better explanations related to dependence and variance parameters and a reduction in the number of parameters to be estimated in random effects covariance matrix to avoid possible identifiability problems. Marginal correlations between responses of subjects and within the responses of a subject are derived through a Taylor series-based approximation. Data cloning computational algorithm is used to compute the maximum likelihood estimates of the parameters in the proposed model and their standard errors. The validity of the proposed model is assessed through a Monte Carlo simulation study under different scenarios, and results are observed to be at acceptable level. v Lastly, the proposed model is illustrated through Mother's Stress and Children's Morbidity study data, where both population-averaged and subject-specific interpretations are drawn through Emprical Bayes estimation of random effects.

Comments on ‘Assessing correlation of clustered mixed outcomes from a multivariate generalized linear mixed model’

“…As pointed out by McCulloch [2] and Inan [3], when the link function associating binary responses with covariates is the logit link function and the distribution for the random effects is the normal distribution within generalized linear mixed models, closed-form expression for marginal variance, covariance, and correlation functions cannot be obtained. , we can show that these functions can be approximated through a first-order Taylor series expansion.…”

“…This was due to the good collaboration of identity and log link functions with normally distributed random effects within MGLMMs framework.We would like to bring to the authors' attention the binary responses which are also commonly observed in clustered data studies. As pointed out by McCulloch [2] and Inan [3], when the link function associating binary responses with covariates is the logit link function and the distribution for the random effects is the normal distribution within generalized linear mixed models, closed-form expression for marginal variance, covariance, and correlation functions cannot be obtained. Nonetheless, following Goldstein and Rasbash [4], Vangeneugden et al [5], Vangeneugden et al [6], and Inan [2], we can show that these functions can be approximated through a first-order Taylor series expansion.To illustrate, we follow the conventional notation used in Chen and Wehrly [1] and assume that Y ijt , which is the binary outcome of subject i at measurement t for method j, is associated with some set of covariates x ijt = (x 1,ijt , … , x p j ,ijt ) T through the logit link function.…”

Comments on “Longitudinal beta‐binomial modeling using GEE for overdispersed binomial data”

“…Since GEE approach requires the first 2 marginal moments to be specified, Wu et al derived the marginal mean, variance, and covariance using the formulas of the conditional mean, variance, and covariance. Following Inan, it is easy to show that the marginal mean . However, as given in the appendix of Wu et al, the expression is remained unsolved in the formulas of the marginal variance and covariance.…”

“…The expressions E ( μ i j (1− μ i j )) and V a r ( μ i j ) in Equation and C o v ( μ i j , μ i k ) in Equation do not have closed‐form solutions (Inan), but they can be approximated by first‐order Taylor series as follows: where ϕ (.) is the probability density function of standard normal distribution.…”

“…where E(W ij |u i ) = ij = Φ(X T ij + u i ) and E(W ik |u i ) = ik = Φ(X T ik + u i ). The expressions E( ij (1− ij )) and Var( ij ) in Equation 2 and Cov( ij , ik ) in Equation 4 do not have closed-form solutions (Inan 2 ), but they can be approximated by first-order Taylor series as follows:…”