Penalized likelihood approach to estimate a smooth mean curve on longitudinal data

Abstract

This paper aims to propose a penalized likelihood approach to estimate a smooth mean curve for the evolution with time of a Gaussian variable taking into account the correlation structure of longitudinal data. The model is an extension of the mixed effects linear model including an unspecified function of time f(t). The estimator (circumflex)f(t) is defined as the solution of the maximization of the penalized likelihood and is approximated on a basis of cubic M-spline with a reduced number of knots. We present modifications of four criteria (cross-validation, generalized cross-validation, T of Rice, Akaike's criterion) to estimate the smoothing parameter when data are correlated; these four criteria gave very similar results in the simulation study. The simulation study showed also the superiority of the Bayesian confidence bands of the mean curve over the frequentist ones. We develop empirical Bayes estimates of subject-specific deviations. This approach was applied to study the progression of CD4+ lymphocyte counts in a cohort of HIV patients treated with protease inhibitors.

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

“…When is fixed, pl is maximized by using an approximation off (t) on a basis of cubic M-splines with a limited number K of knots [18]. Typically, K is between 5 and 20.…”

“…The non-parametric alternative defines the estimator of the function f (t) as the solution of the maximization of the penalized log-likelihood which leads to a natural cubic-spline estimator forf (t) with one knot for each distinct time of measurement [15--17]. As the latter is computationally prohibitive when the number of distinct time points is large, an hybrid approach has emerged where a moderate number of knots is used (5 to 20 in most cases) and parameters are estimated using penalized likelihood [13,18,19]. This approach is computationally tractable and remains non-parametric in spirit as the number of knots must be large enough so that penalized likelihood estimations are unchanged when the number of knots increases.…”

“…However, the smoothing parameter estimation is a difficult issue. Some authors estimate the smoothing parameter by cross-validation (CV) [18], whereas others take advantage of the rewriting of spline models as particular random effect models where the smoothing parameter becomes a variance parameter estimated by restricted maximum likelihood [19]. Unfortunately, the rewriting of spline models as random effect models holds only for linear model.…”

“…In this paper, we propose a semiparametric cumulative probit model for ordinal or quantitative discrete longitudinal data using a smoothing spline function estimated by penalized likelihood and a flexible correlation structure. This model is an extension of a semiparametric model for the Gaussian longitudinal data [18] using a threshold approach with a Gaussian latent process. This work is motivated by the study of the pre-diagnosis cognitive decline in Alzheimer's disease (AD).…”

Regularized Finite Mixture Models for Probability Trajectories

“…Specifically, polynomial methods are widely used in longitudinal modeling, and spline methods have been previously used in latent class time series analysis (Winsberg and De Soete, 1999) and in mixed effects longitudinal analysis (Jacqmin-Gadda, 2002). For regression models it has been shown that penalized least squares estimation with (2) is equivalent to fitting the trajectories with a basis of cubic splines (Wahba 1990).…”