A new SAEM algorithm for ordered-categorical and count data models: implementation and evaluation
Marc Lavielle (1) and Radojka M. Savic (2)
(1) INRIA Saclay and University Paris 11, (2) INSERM U738 and Université Paris 7
Objectives: Analysis of categorical and count data from clinical trials using mixed effect analysis has recently become the method of choice. However, algorithms available for parameter estimation, including LAPLACE and Gaussian quadrature, are associated with certain limitations, including bias in parameter estimates. This is a consequence of multiple approximations of the likelihood integral, whose impact is amplified when the proportion of response categories are skewed- for ordinal data, and when models accounting for under- or over dispersion of individual variance compared to the mean are applied, in case of count data. Additionally, the more quadrature points used to approximate likelihood integral, the longer the analysis runtime [1, 2]. The SAEM algorithm has proven to be a very efficient and powerful tool in the analysis of continuous data [3]. The aim of this study was to implement and investigate the performance of a new SAEM algorithm for discrete data.
Methods: 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 scenarios similar to those used in previous studies to investigate properties of other algorithms were employed. For ordered categorical data, the proportional –odds model was explored using six different scenarios with varying parameter values. For count data, a single scenario was used to explore six different probability distribution models, (i.e., Poisson, Zero-inflated Poisson, Generalized Poisson, Poisson with Markovian Features, Poisson with a mixture distribution for individual observations and Negative binomial models). Performance of the algorithm was assessed by computing the relative bias (RB), root mean square error, and assessing the CPU time of the analysis. The accuracy of standard errors (SE) estimates was assessed as an absolute distance (AD) between actual and empirical relative SEs.
Results: For proportional-odds model, RB was < 8.13 % for all scenarios explored, including ones with skewed distributions of response categories. For count data models, RB was < 4.13 % for all models studied including ones accounting for over- or under-dispersion. Estimates of standard errors were close to the empirical SEs, with AD < 5.8% , for all explored scenarios. The longest CPU time out of all studied models was for the analysis of the Negative binomial model taking 40s for parameter estimation and 37s for SE estimation.
Conclusions: The SAEM algorithm was extended for analysis of ordered categorical and count data with extensions to the Hidden Markov Model. It provides accurate estimates of both, parameters and standard errors. The estimation is significantly faster compared to other algorithms. The algorithm will be implemented in Monolix 3.1, (beta-version available in July 2008).
References:
[1] Jonsson, S., Kjellsson, M.C.and Karlsson, M.O., Estimating bias in population parameters for some models for repeated measures ordinal data using NONMEM and NLMIXED. J Pharmacokinet Pharmacodyn, 2004. 31(4): p. 299-320.
[2]Plan, E.L., et al. Maximum Likelihood Approximations: Performance in Population Models for Count Data. in PAGE. 2008.
[3]Kuhn E., Lavielle M., Maximum likelihood estimation in nonlinear mixed effects models. Computational Statistics and Data Analysis, 2005. vol. 49, No. 4, pp 1020-1038.
