Multinomial logistic functions in Markov-chain models for modeling sleep architecture: internal validation based on VPCs
Mezzalana E (1), Bizzotto R (1), Sparacino G (1), Zamuner S (2)
(1) Dept. of Information Engineering, University of Padova, Italy; (2) Clinical Pharmacology/Modeling&Simulation, GlaxoSmithKline, Verona, Italy.
Objectives: Simulation based diagnostics are increasingly used to illustrate model properties , especially in the context of categorical data. A multilogit Mixed-Effect Markov-chain model has been recently proposed for describing sleep architecture in primary insomnia patients [2,3]. The aim of the present work was to perform the internal validation of this model based on visual predictive checks (VPC) for categorical data .
Methods: Data were obtained from the first-night of polysomnographic (PSG) recordings from 116 subjects diagnosed with primary insomnia and belonging to the placebo arm of a PSG-parallel study. Each individual sequence of sleep stages (awake, stage1, stage2, slow-wave sleep and REM) was supposed to obey to a Markov-chain and each transition probability was modeled through a piecewise linear multinomial logistic function depending on time [2,3]. VPCs were performed to evaluate potential model misspecification and model robustness (uncertainty of model parameters estimates). Each VPC was based on 100 datasets simulated from point estimates of model parameters. Two types of VPCs were performed: a) the first one is based on simulated and observed data; b) the second one depends on model parameters re-estimated from the simulated data. In particular, a) transition frequencies and fractions of observations for each stage were calculated from raw data considering ten intervals of the night having equal width and compared to the 95% confidence interval derived from simulated datasets on the same intervals. Subsequently, b) typical transition probabilities and the 5% and 95% percentiles of individual transition probabilities distribution along the night were computed from model parameter estimates and compared to their corresponding simulation based 95% CI.
Results: The VPCs showed in general a good agreement between the statistics derived from raw and simulated data. Most of the transitions among sleep stages were well characterized in terms of parameters accuracy and precision. A slightly higher uncertainty was observed for the transitions from slow-wave sleep and REM sleep. This result is likely due to the limited occurrences of these transitions.
Conclusions: Model-based diagnostics are essential in the process of model development and validation: their application may become complex in case of categorical data. The model adequacy of our multinomial Markov-chain model of sleep architecture [3,4] was addressed by performing VPCs on transition\stages frequencies and model parameters. Overall, the application of VPCs allowed to conclude that parameters estimates were unbiased and precise and did not suggest model misspecification.
 M. O. Karlsson and R. M. Savic. Diagnosing Model Diagnostics, Clinical Pharmacology & Therapeutics 82:17–20 (2007)
 R. Bizzotto, S. Zamuner, G. De Nicolao, M. O. Karlsson, R.Gomeni. Multinomial logistic estimation of Markov-chain models for modeling sleep architecture in primary insomnia patients, J Pharmacokinet Pharmacodyn, DOI: 10.1007/s10928-009-9148-2 (2010)
 R. Bizzotto, S. Zamuner, G. De Nicolao, R. Gomeni, A. C. Hooker, M. O. Karlsson. Multinomial logistic functions in Markov-chain models for modeling sleep architecture: internal validation based on VPCs. PAGE 19 (2010)
 M. Bergstrand, A. C. Hooker, M. O. Karlsson. Visual Predictive Checks for Censored and Categorical data, PAGE 18 (2009) Abstr 1604 [www.page-meeting.org/?abstract=1604]