Mixture models in NONMEM - how to find the individual probability of belonging to a specific mixture, and why this can be useful information
Carlsson, Kristin C., Radojka M. Savic, Andrew Hooker, Mats O. Karlsson.
Division of Pharmacokinetics and Drug Therapy, Uppsala University, Sweden
Background: Mixture models are used to describe populations with bi- or multimodal distributions. In the POSTHOC step, NONMEM will assign the individual patients to the mixture (subpopulation) with the highest probability. NONMEM reports the assigned mixture (MIXEST), but not the probability. The probabilities can be calculated from the individual objective function value (IOFV) [1] and the total probability in the population of belonging to the respective mixtures.
Method: IOFV values can be obtained from NONMEM by using a script that is called by $CONTR in NONMEM. In cases where the likelihood option is used in $ESTIMATION, $ CONTR can not be used, and IOFV values have to be obtained by rerunning the model with final parameter estimates and MAXEVAL=0 for each patient. The probability of belonging to a specific mixture is calculated as follows:
OFV = - 2 ln (L) => ILmix1 = e(-IOFVmix1/2)
IPmix1 = (IL mix1 * Ppop,mix1) / ((IL mix1 * Ppop,mix1) + (IL mix2 * Ppop,mix2))
where ILmix1 is the individual likelihood for mixture 1 and IPmix1 is the individual probability of belonging to mixture 1. The sum of IP for all mixtures is 1. Ppop,mix1 is the population probability for belonging to mixture 1, estimated in NONMEM.
To investigate the use of IPmix, a six-category proportional odds model for clomethiazole sedation in stroke patients (n=1545) described previously was used [2]. The model has a mixture describing two subpopulations of patients, those without (mixture 1: 20%) or with (mixture 2: 80 %) stroke induced sedation. IPmix was calculated for the patients in this data set.
Results: The 50, 75, 90 and 99th percentile of IPmix1 were 0.05, 0.28, 0.64 and 0.90, indicating that mixture assignment is associated with considerable uncertainty. If calculated in real-time, IPmix1 for a patient will change as more data comes in. Therapeutic decisions could then be more informatively based on IPmix1, rather than the dichotomous assignment of MIXEST. IPmix1 shows, in contrast to the MIXEST estimates, no shrinkage to the larger mixture when data are sparse. It is therefore more suitable for investigating relations with potential covariates.
Conclusion: The individual probability of belonging to a specific mixture can be calculated from individual objective function values in NONMEM. This distribution of probabilities can be of use if the assigned mixture is to be used further, e.g. in diagnostics, simulations and in individualized therapy.
References:
[1] S. Sadray et al. Likelihood-based diagnostics for influential individuals in nonlinear mixed effects model selection. Pharmaceutical Research (1999),9,8, 1260 - 1265.
[2] P.- H.Zingmark et al. Clomethiazole pharmacokinetics and sedation in stroke patients. Br J Clin Pharmacol (2003),56,173 -183.