Latent Gaussian copula models for longitudinal binary data

Abstract

Longitudinal binary data arise commonly in a variety of fields including public health, biomedicine, finance, agriculture and social science, among many others. In longitudinal binary studies, the aims are to assess the association of longitudinal binary response with certain covariates of interest, and to quantify the within-subject correlations for longitudinal binary responses. In the literature, various methods were developed to model longitudinal binary data but little work was done to account for the fact that the correlation coefficients of correlated binary responses have the so-called Fréchet–Hoeffding bounds. Ignoring this fact can lead to incorrect statistical inferences for longitudinal binary data. In this paper, based on latent Gaussian copula a new statistical modeling method is proposed to model the mean and within-subject correlation structures, simultaneously, for longitudinal binary data. Specifically, the mean structure is modeled by a semiparametric regression model, and the within-subject correlation coefficients are modeled through introducing a latent Gaussian copula model with certain latent correlation structures characterized by some parameters. Generalized estimating equations are then proposed to estimate the parameters in the mean and latent correlation structures, and consistency and asymptotic normality of the resulting parameter estimators are established. The proposed model and method ensure that the estimated correlation coefficients must satisfy the Fréchet–Hoeffding bounds for longitudinal binary data. Simulation studies show that the proposed method has a stable numerical performance. A practical data set is analyzed using the proposed method for illustration.