Conditional weighted residuals, an improved model diagnostic for the FO/FOCE methods.
Hooker, Andrew, Christine E. Staatz and Mats O. Karlsson.
Division of Pharmacokinetics and Drug Therapy, Dept. of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden.
Objectives: Weighted residuals (WRES) are a common diagnostic tool in population model building and evaluation, generally used to test for model misspecification. WRES are calculated using the first order (FO) approximation, however, a majority of population model analyses have shifted to the more complex first-order with conditional estimation (FOCE) approximation to the true model. The FO method linearizes the model about the population mean of the random model parameters whereas the FOCE approximation conditions the linearization of the model around each individual’s empirical Bayes (post-hoc) estimates, allowing use of hypothesis testing in model building and often resulting in less biased model parameter estimates. Utilization of FO based WRES during model development under the FOCE method may lead to misguided model development and evaluation. We present a new diagnostic tool, the conditional weighted residuals (CWRES), which are calculated based on the FOCE approximation.
Methods: CWRES for an individual are calculated as
CWRES = [y – E(f)]/Cov(y)1/2
E(f) = f(θ,ηph) - L· ηph
Cov(y) = L ΩL´ + diag(HΣH´)
Where y is a vector of data from an individual, E(f) is a vector of expected values from the model evaluated at an individual’s empirical Bayes estimates ηph, Cov(y) is the covariance of the data conditional on the model and L is the derivative of the model with respect to the population random effects η evaluated at ηph (WRES are evaluated at η=0). CWRES are computed using verbatim code in NONMEM and a post processing step using code in either Matlab or R (available by request).
Results: Using simulations in a wide variety of models we find that the CWRES behave as theoretically expected (normal distribution, N(0,1)). In contrast, the WRES have distributions that greatly deviate from the expected, falsely indicating model misspecification when the model is correct. We also find that, if differences exist between the WRES and CWRES after FO estimation, FOCE will generally give better model parameter estimates whereas when no major differences exist, FO and FOCE parameter estimates are similar. In three real data examples [1,2,3] with good model fit characteristics but WRES distributions that are not N(0,1) we find that the CWRES show markedly better distributions. Simulations from the final models and subsequent calculations of WRES and CWRES confirm poor properties of WRES for these models.
Conclusions: Utilization of CWRES could improve model development and evaluation and give a more accurate picture of if and when a model is misspecified when using the FOCE approximation. CWRES can also indicate if the FOCE estimation method will improve the results of an FO model fit to data or not.
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