Which matrix is the most reliable to judge the inclusion of covariates: reduction of unexplained parameter variability, increase in explained parameter variability or change in OFV?

When performing a population pharmacokinetic (PK) modeling analysis covariates, such as weight, gender etc. might be included into the model to explain part of the parameter variability. Parameter variability can be defined as the sum of unexplained (random) parameter variability (UPV) and explained (predictable) parameter variability (EPV). If inclusion of a covariate on a certain parameter is considered significant, the following measures are expected to change accordingly: a drop in the objective function value (ΔOFV>3.84 for df=1 for nested models), a decrease of the UPV and an increase in the EPV.

Explore the reliability of the different matrices: OFV change, decrease in UPV and increase in EPV to diagnose significant covariate-parameter relationships of different strengths and different nature. 

Stochastic simulations and estimations (SSEs) were performed in NONMEM in connection with PsN (Nsim=100) using FOCE+I to explore the relationships between the decrease in UPV, the increase in EPV and the change in OFV for models with continuous and categorical covariates, different degrees (weak to strong) of parameter-covariate relationships and different covariate models. Additionally, changes in the relative standard error (RSE) of the covariate parameter were calculated.  A 1-compartment intravenous bolus PK model was used for simulation following a single dose in 200 subjects with 5 observations per subject.  The PV on clearance and volume were described according to a log-normal distribution model with mean zero and variance of 0.09 (30%) without correlation. A combined residual error model was assumed. The continuous covariates were simulated simultaneously in NONMEM from a normal distribution with mean zero and a standard deviation of 1. Categorical covariates were simulated from a random uniform distribution with an equal probability for the 2 categories. Alternative designs with less subjects (n=50), less samples/subject (s=2) higher PV on CL of 0.2025 (45%) were tested as well.Furthermore, three published ‘real’ data sets, which include covariate relationships, were investigated (1-3). Here the total PV on CL and V, EPV and UPV were calculated in the final published model. Then reduced models were produced with excluded the covariates one at the time until no covariate was included in the model. The change in OFV compared to the full model, the change in UPV and EPV were calculated.

For both the simulated examples and the real data examples it was shown that the reduction in UPV equaled the increase in EPV. Furthermore, it could be shown that the larger the drop in the OFV, the larger the decrease in UPV and increase in EPV.  In the simulated examples it could be shown that this follows a linear trend with the strength of the parameter-covariate relationship. The RSE on the covariate parameter reduced in order from weaker to stronger parameter-covariate relationship.
The same results were valid for the real data examples chosen. Except for one case (1), where one covariate (creatinine on CL) was included with the addition of a new parameter (additional degree of freedom) and one covariate (total body weight on V) was included without including an additional parameter into the model. Here we found that inclusion of weight on V reduced the UPV and increased the EPV by nearly 60%, the drop in OFV however, was only half compared to including creatinine on CL which only resulted in a 37% reduction of UPV and increase of EPV.
In the simulated examples it could also be shown that the power to include a covariate relationship reduced when fewer subjects were included followed by fewer subjects and less samples and less subjects and higher PV on CL.

It was found that the three matrices are highly correlated and perform equally well indicating the significant inclusion of a covariate, when included with additional parameters/degrees of freedom. However, in cases were no additional parameters are included into the model to describe the relationship and when correlation between two parameters exist before covariates are added; questions in regards to the matrix which guides the modeler to the most influential parameter-covariate relationship are still unclear.

1. Karlsson MO, et al. J Pharmacokinet Biopharm. 1998 Apr;26(2):207-46.
2. Karlsson MO, Sheiner LB. J Pharmacokinet Biopharm. 1993 Dec;21(6):735-50.
3. Bruno R, et al. J Pharmacokinet Biopharm. 1992 Dec;20(6):653-69.