An Efficient Estimation Method for Longitudinal Data Using Bayesian Conditional Transformation Models

Bayesian conditional transformation models (BCTMs) address the direct estimation of the conditional distribution function of a random variable $Y$ conditional on a set of explanatory $\mathrm{variables}X$ . The BCTMs infer the conditional distribution by applying a transformation function of $Y$ given $X=x$ towards a baseline distribution free of parameters to be estimated. The benefit of these models is that the explanatory variables $X=x$ impact the whole conditional distribution of $Y$ given $X=x$ instead of only the mean, variance, kurtosis, or skewness. The transformation functions are an essential part of the model, and they range from loss-complex and low-parameterized functions to complex relationships between explanatory variables and response variables represented by nonlinear functions. The general construction of the BCTM class explores monotonic B-splines for parameterizing the transformation function. Smoothness and regularization are accomplished through an adequate prior distribution for the parameters. We proposed a new estimation procedure for the BCTM based on the integrated nested Laplace approximation, which is tested through a simulation study. Also, two longitudinal studies using real data are considered. The first application is a cardiovascular study and compares our proposed algorithm, named integrated Laplace with Bayesian conditional transformation models (ILBCTM), with the original Markov chain Monte Carlo-based algorithm for BCTM. We obtained similar results with a shorter computational time. The second application considers the ILBCTM in a study of the mortality rate of bronchial and lung cancer in Brazil.