Assessment of bias in model parameter estimates of continuous time Markov models for categorical data
Klas J. Petersson (1), Brigitte D. Lacroix (1,2), Lena E. Friberg (1)
1.Department of Pharmaceutical Biosciences, Uppsala University, Sweden. 2. Pharmacometrics, Global Exploratory Development, UCB Pharma SA, Belgium
Objectives: Models with Markovian elements where the estimated parameters are rate constants describing the flow of probability over time [1,2] are a fairly new way of modeling categorical data with high correlation between consecutive observations. These models generally require fewer parameters than ordinary Markov models and do not assume equally spaced observations; there is also less need to know the exact time of transition. When modeling ordered categorical or repeated time to event data and the number of observations is few in one category or only a few individuals have multiple events the LAPLACE method in NONMEM has been prone to bias [3]. The objective was therefore to assess bias in continuous time Markov models and later the Type I error
Methods: Two models of the continuous time type Markov model for categorical data formed the basis for the evaluation. The first model, which described EPS events of antipsychotic drugs, had only one parameter for inter-individual variability (IIV) [PAGE 2011 ref] while six IIV parameters were included in the second model characterizing ACR response [1]. To assess the influence of sparse data in a category and IIV magnitude on bias six scenarios were tested where data were simulated using; 1) parameters estimated from the original data , 2) parameters resulting in few observations of highest category (<5 observations which is <1% of all data), and 3-6) as 1) & 2) with IIV increased/decreased fourfold(resulting in that less than 10 individuals had more than one transition ). One-hundred simulations and re-estimations were performed to assess the bias using PsN and the LAPLACIAN estimation method in NONMEM 7.
Results: The EPS model showed a mean of the parameters’ absolute bias of 20% (Range: 8 - 34%) for the non-sparse data and 40% (3.8 – 135%) for the sparse scenario. Bias was highest in IIV estimates and rate constants associated with the most sparse observation type. Bias decreased with the higher IIV magnitude; mean 10% (2.7-19%). For the larger ACR model, with much longer runtimes, preliminary results were pointing in the same direction as the model for the EPS data. Low IIV or omission of IIV in the model would occasionally yield datasets where one or more population parameters were not estimable.
Conclusions: Biased parameter estimates were found also for continuous time Markov models and increased when the observed data distributions became more skewed. As expected, increased IIV made the number of transitions increase and the bias decreased.
References:
[1] Lacroix B. D., et al. Simultaneous modeling of the three ACR improvement thresholds – 20, 50 and 70% - in rheumatoid arthritis patients treated with certolizumab pegol PAGE 19 (2010) Abstr 1811 [www.page-meeting.org/?abstract=1811]
[2] Bergstrand, M., et al., Mechanistic modeling of a magnetic marker monitoring study linking gastrointestinal tablet transit, in vivo drug release, and pharmacokinetics. Clinical pharmacology and therapeutics, 2009. 86(1): p. 77-83.
[3] Jönsson S., et al., Estimating bias in population parameters for some models for repeated measures ordinal data using NONMEM and NLMIXED. J Pharmacokinet Pharmacodyn. 2004 Aug;31(4):299-320.
