Parameter Estimation in Bivariate Mixed Hidden Markov Models
Ari Brekkan (1), Mats O. Karlsson (1), Siv Jönsson (1), Elodie L. Plan (1)
(1) Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden
Objectives: Observable stochastic processes may depend on underlying hidden processes that can be modelled with mixed hidden Markov models (MHMMSs), allowing for inferences about the hidden state of individuals at any time-point given a set of observations (1), and can be implements in NONMEM (NM), as described by Plan et al. (2). Here we aim to i) extend the work in (2) by developing a bivariate MHMM (BV-MHMM) and ii) explore parameter estimation in a BV-MHMM in terms of accuract and precision.
Methods: A novel 2-state BV-MHMM with simultaneous calculation of the likelihood for 2 variables was developed based on (2). Of the potentially correlated continuous variables (through parameter ρ), one was more variable than the other and subject to time decay. An IIV term (ω12) and a drug effect (DE) were introduced on 1 of the 2 transition probability parameters. A simulation study with ρ set to either 0 or 0.33 and different combinations of ω12 and DE was set up, generating data for 500 individuals, each with 60 observations per variable. Stochastic simulations and estimations (n=100) were used to determine the bias and imprecision of parameters. Estimation was done in NM with SAEM/IMP and mu-parameterization.
Results: : Data simulated with correlation (ρ=0.33) and fitted without (ρ fixed to 0) yielded imprecise and biased estimates for all parameters (average absolute bias [AB] and average root mean squared error [RMSE] increased by 2881% and 16%, respectively, compared to when ρ is estimated), including DE (AB and RMSE increase of 138% and 12%, respectively) and ω 12 (largely unaffected). Further, the objective function value (OFV) increased by an average of 3197 points. Simulating without correlation (ρ=0) but estimating it resulted in estimates of ρ that were close to 0, but OFV increased by an average of 66 points, but fell within the 90% confidence interval of the OFV from runs with ρ fixed to 0. This was not translated into a meaningful increase in average AB (14% decrease) or average RMSE (1% increase) of all parameters.
Conclusions: A BV-MHMM capable of describing the relationship between 2 hidden states and 2 observed continuous variables was developed. The model was able to estimate correlations between the variables. When the correlation was not considered although it was present, parameter estimation was biased and imprecise and goodness-of-fit markedly impacted.
References:
[1] Altman RM. Mixed Hidden Markov Models. J Am Stat Assoc. 2007 Mar 1;102(477):201–10.
[2] Plan, EL., Nyberg, J., Bauer, RJ., Karlsson, MO. Handling Underlying Discrete Variables with Mixed Hidden Markov Models in NONMEM. Presented at PAGE 24. 2015. Abstr 3625. www.page-meeting.org/?abstract=3625
Acknowledgements: Joakim Nyberg and Robert Bauer are thanked for their valuable input.
