A Bayesian Multivariate Model for Repeated Measures of Correlated Data
Varun Goel (1), Timothy E Hanson (2), Brian Corrigan (3), Raymond Miller (3), Richard C Brundage (1)
(1) Department of Experimental and Clinical Pharmacology, University of Minnesota, Minneapolis, MN, (2) Department of Biostatistics, University of Minnesota, Minneapolis, MN; (3) Pfizer Inc., New London, CT
Objectives: Most often in proof-of-concept trials repeated measures of efficacy and safety outcomes are collected, for example a crossover trial with varying doses. These outcomes are often correlated. The objective of this analysis was to develop an exposure response analysis for a vector of correlated observations, each arising from a member of the exponential family of distributions. PD 0200390 is a ligand of the a2d subunit of the voltage-gated calcium channel, being investigated for the treatment of primary insomnia and non-restorative sleep. Wake after sleep onset (WASO) and number of awakenings (NAASO-2) are the measures of efficacy while ease of awakening (AFS) and behavior following wakefulness (BFW) are measures of residual effects. The distribution of WASO, NAASO-2, are assumed to be Lognormal and Poisson; AFS and BFW are assumed Bernoulli.
Methods: Six dose levels (5mg-75mg), and placebo data were available for 126 patients with primary insomnia from two phase II double blind, randomized, placebo controlled, crossover studies. Hierarchical non-linear dose response models were developed for observation vector of WASO, NAASO-2, AFS and BFW in WinBUGS. The observation vector components were assumed independent conditional on the latent-subject-specific vector of random effects drawn from a multivariate normal distribution1. Comparison between multivariate and individual models was assessed by posterior predictive checks, deviance information criteria (DIC), conditional predictive ordinate,2,3 and values of log pseudo marginal likelihood2,3 (LPML).
Results: Dose response relationships for WASO, NAASO-2 were described by inhibitory Emax and for the logistic models for AFS, BFW as linear. Significant correlations were observed between baselines and ED50s for WASO and NAASO-2, and additive random effects for AFS and BFW. DIC and LPML values indicated multivariate model as significantly better at fitting and prediction of correlated outcomes.
Conclusions: Simultaneous PK PD modeling is often encountered in the Pharmacometrics literature, however simultaneous modeling of correlated safety and efficacy endpoints is rare. In this work we model correlations by introducing multivariate distributions for random effects. Knowledge of this correlation helps in understanding of drug's action and reduces uncertainty in future simulations for dose and sample size selection.4
References:
[1]Gueorguieva R. (2001). A multivariate generalized linear mixed model for joint modeling of clustered outcomes in the exponential family; Statistical Modeling 1, 177.
[2]Geisser S. and Eddy William. (1979). A predictive approach to Model Selection. Journal of American Statistical Association 74, 365.
[3]Gelfand A.E. and Dey D.K. (1994) Bayesian Model Choice: Asymptotics and Exact Calculations. Journal of Royal Statistical Society B 56, 501-514.
[4]Goel V., Miller R., Ito K., French J., Zhao Q. and Corrigan B., (2009) Development of Stochastic Multi-Attribute Decision Based Clinical Utility For Phase III Dose Selection. Clinical Pharmacology and Therapeutics 85, S62.