Performance in population models for count data, part II: a new SAEM algorithm

Abstract

Analysis of count data from clinical trials using mixed effect analysis has recently become widely used. However, algorithms available for the parameter estimation, including LAPLACE and Gaussian quadrature (GQ), are associated with certain limitations, including bias in parameter estimates and the long analysis runtime. The stochastic approximation expectation maximization (SAEM) algorithm has proven to be a very efficient and powerful tool in the analysis of continuous data. The aim of this study was to implement and investigate the performance of a new SAEM algorithm for application to count data. A new SAEM algorithm was implemented in MATLAB for estimation of both, parameters and the Fisher information matrix. Stochastic Monte Carlo simulations followed by re-estimation were performed according to scenarios used in previous studies (part I) to investigate properties of alternative algorithms (Plan et al., 2008, Abstr 1372 [ http://wwwpage-meetingorg/?abstract=1372 ]). A single scenario was used to explore six probability distribution models. For parameter estimation, the relative bias was less than 0.92% and 4.13% for fixed and random effects, for all models studied including ones accounting for over- or under-dispersion. Empirical and estimated relative standard errors were similar, with distance between them being <1.7% for all explored scenarios. The longest CPU time was 95 s for parameter estimation and 56 s for SE estimation. The SAEM algorithm was extended for analysis of count data. It provides accurate estimates of both, parameters and standard errors. The estimation is significantly faster compared to LAPLACE and GQ. The algorithm is implemented in Monolix 3.1, (beta-version available in July 2009).

Figure 1

Distribution of relative estimation error (REE) for all parameters (left panel) and standard errors (right panel) across all the models. The errors (y axes) are given as percentages (%). (Abbreviations: PS = Poisson, PMAK = Poisson with Markovian Features, PMIX = Poisson with a mixture distribution for individual observations, ZIP = Zero-inflated Poisson, GP = Generalized Poisson, and NB = Negative binomial model)

Figure 2

Visual predictive check of variance versus mean on normal (upper panel) and log (lower panel) scale. Observed data are given as scatter points, while prediction intervals are represented as red lines. VPC is shown for the true (left panel) and for a mis-specified zero-inflated Poisson models, when variance used for simulations is equal to 1.5 the true variance (right panel).

Figure 3

Comparison of NPDEs and normal distribution with zero mean and variance of one, represented as histogram (upper panel) and q-q plot (lower panel). NPDEs are shown for the true (left panel) and for a mis-specified zero-inflated Poisson models, when variance used for simulations is equal to 1.5 times the true variance (right panel).