Antedependence (AD) models are useful for modeling nonstationary covariance structures for longitudinal data. A limitation of these models is that they are discrete; that is, they do not recognize an underlying continuous correlation structure over a time range of interest. In addition, they are problematic for sparse data, as they rely on the particular, possibly random, measurement times obtained and involve a large number of parameters when the number of unique measurement times is large. This situation creates difficulties in carrying out available numerical methods for maximum likelihood (ML) estimation. In this research, we define a continuous AD model based on a ’non-stationarity function’. We discuss the interpretation of this function and special cases. In addition, we present a novel approach to estimation for this model using nonlinear least squares. We examine properties of this method in simulation studies, and show that it does as well as ML for balanced data, but also allows valid estimation in sparse data situations where ML breaks down. We also consider the use of the continuous AD covariance structure in the general linear model and provide a generalized least squares method to estimate the mean structure. We apply the above methods to data from the Multi Center AIDS Cohort Study (MACS). Finally, we discuss implications and issues involving study design.
According to the simulation studies, The proposed new approach using nonlinear least squares (NLLS) for estimation of correlation parameters in the continuous 1st order ante-dependence model did better compared to the MLE approach in terms of bias, and MSE, for small samples. As the sample size increased both approaches were similar in terms of bias and MSE. The proposed new approach estimated the underlying non-stationary correlation structure with minimal bias in all scenarios of sparse longitudinal data, including the scenario of complete longitudinal data, across all sample sizes.