Semi‐parametric latent process model for longitudinal ordinal data: Application to cognitive decline

Ordinal and quantitative discrete data are frequent in biomedical and neuropsychological studies. We propose a semi-parametric model for the analysis of the change over time of such data in longitudinal studies. A threshold model is defined where the outcome value depends on the current value of an underlying Gaussian latent process. The latent process model is a Gaussian linear mixed model with a non-parametric function of time, f(t), to model the expected change over time. This model includes random-effects and a stochastic error process to flexibly handle correlation between repeated measures. The function f(t) and all the model parameters are estimated by penalized likelihood using a cubic-spline approximation for f(t). The smoothing parameter is estimated by an approximate cross-validation criterion. Confidence bands may be computed for the estimated curves for the latent process and, using a Monte Carlo approach, for the outcome in its natural scale. The method is applied to the Paquid cohort data to compare the time-course over 14 years of two cognitive scores in a sample of 350 future Alzheimer patients and in a matched sample of healthy subjects.