Population optimal design with correlation using Markov Models
Joakim Nyberg and Andrew C. Hooker
Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden
Objectives: Incorporating correlation for continuous data in optimal design has shown to be important [1]. Therefore; different ways of incorporating correlation into discrete type data are investigated using Markov models.
Methods: A Dichotomous model with a Markov element on the baseline effect of the drug was used. One elementary design with 200 individuals and 20 observations per individual were used. The observations were split into a first dose (10 first obs.) and a second dose (10 last obs.) Four parameters (3 fixed effects and 1 random effect) are included in the model. The Markov element is constructed so that the probability of having a response at time i (DVi) increases if the previous observation (DVi-1) was a response.
Three Fisher information matrix (FIM) calculations are investigated: S1) The FIMM given the complete Model with Markov Element, S2) The FIMS=FIM1+FIM2 when splitting the Model into two models, FIM1 from model1 were DVi-1 was a response and FIM2 from model2 were DVi-1 was a non-response and S3) FIMD with a dichotomous model without Markov elements were the probability of having a response versus dose was the same as in the Markov model. S1 had to be implemented with a high order approximation of the likelihood, i.e. a Laplace (Lap) approximation and/or Monte Carlo (MC) integration technique [2], but with S2 and S3 an analytic approximation (AA) without correlation of the random effects could also be used [3].
Optimal designs for the two doses were found using a discrete grid (25*25) between 0-1 units. PopED 2.11 [4] was used for all computations.
Results: For S1, the standard by which other methods should be compared, the optimal doses were (0, 0.25). With AA; S2 and S3 had optimal doses (0, 0.38) and (0, 0.50) respectively while both Lap and MC approximations had optimal doses (0, 0.42) and (0, 0.63). S2 shows more similarities with the optimal design from S1 while using a dichotomous model (S3) appears less similar.
Conclusion: Optimal designs were different if Markov elements were included in the model. For S1; the slower Lap or MC methods had to be used, while, the faster AA could be used in the S2 and S3 calculations. S2 appear to better approximate the S1 method. With the proposed methods a first order Markov Model was used, but any Markov order is possible within this framework of calculating FIM. Moreover, models without any known link functions could be used, with the drawback of an increased calculation time using the Lap or MC methods.
References:
1. A. C. Hooker, J. Nyberg, R. Höglund, M. Bergstrand and M.O. Karlsson. Autocorrelation reduces sample time clustering in optimal design. PAGE 18 (2009) Abstr 1655 [www.page-meeting.org/?abstract=1655]
2. J. Nyberg, M. O. Karlsson and A. C. Hooker. Population optimal experimental design for discrete type data. PAGE 18 (2009) Abstr 1468 [www.page-meeting.org/?abstract=1468].
3. Longford N.T. (1994). Logistic regression with random coefficients. Computational Statistics & Data Analysis 17, 1-15.
4. PopED, version 2.11 (2011) http://poped.sf.net/.