Correction of the likelihood function as an alternative for imputing missing covariates
Wojciech Krzyzanski (1), An Vermeulen (2)
(1) University at Buffalo, Buffalo, USA, (2) Janssen Research and Development, a division of Janssen Pharmaceutica NV, Beerse, Belgium.
Objectives: Missing covariates in a population PKPD model are typically imputed in order to obtain a full data set for population analysis [1]. The major drawback of imputation is the creation of artificial data that might not reflect the actual covariate distribution. We applied the maximum likelihood method for missing covariates [2,3], detailing how to implement this method in NONMEM based on a case example, where the likelihood function is corrected for missing data and imputation is not required.
Methods: Simulated plasma concentration data for N=80 subjects, based on the one compartment model after an IV bolus dose were used for testing the new method. Body weight (BW) was the only covariate related to CL and V according to power functions with exponents SCL=0.75 and SV=1. BW was log-normally distributed with mean log(65.8) and 10% CV. Individual CL and V were log-normally distributed. The plasma concentrations were log-transformed. For missing data, BWs of 20 subjects were excluded from the original dataset. The distribution of the remaining BWs was similar to the original one with logBW~N(log(65.5),0.18). The likelihood function was corrected by allowing the BW for subjects with a missing covariate to be drawn from the log-normal distribution N(log(65.5),0.18). An additional data set was generated where the missing BWs were imputed with the mean BW of 65.5 kg. The original model was fitted to the data sets with all (ALL) and imputed covariates (IMPUT), whereas the model with the corrected likelihood function was applied to the data set with missing BWs (MISS). Parameters were estimated using the FOCE method as implemented in NONMEM 7.3 [4].
Results: The estimates of typical values for CL, V, SCL, and SV were close to the original ones with relative absolute error less than 10% in all cases. The estimates of BSV for CL and V were less than 11% for the ALL and MISS models, whereas imputing BW resulted in 64% error (0.1 vs. 0.164) in the estimate of variance of V. The eta-shrinkages for CL and V were less than 17% for all models. The objective function values were -960.9 (ALL), -929.7 (MISS), and -913.1 (IMPUT).
Conclusions: Information about the distribution of the covariates can be used to correct the likelihood function for the probability of missing a covariate, as was described before [2,3]. This approach results in accurate estimates of the population parameters that are not different from the estimates obtained by the model with all covariates. Imputation of missing covariates with the mean value can result in biased estimates of BSV, if the distribution of the covariates is wide.
References:
[1] Johansson AM, Karlsson MO. Multiple imputation of missing covariates in NONMEM and evaluation of the method’s sensitivity to η-shrinkage. AAPS J. 15:1035-1042 (2013).
[2] Karlsson et al. Assumption testing in population pharmacokinetic models: illustrated with an analysis of moxonidine data from congestive heart failure patients. J Pharmacokinet Biopharm 26:207-246 (1998).
[3] Johansson AM, Karlsson MO. Comparison of methods for handling missing covariate data. AAPS J. 15:1232-1241 (2013).
[4] Beal SL, Sheiner LB, Boeckmann AJ & Bauer RJ (Eds.) NONMEM Users Guides. 1989-2011. Icon Development Solutions, Ellicott City, Maryland, USA.
