A strategy for residual error modeling incorporating both scedasticity of variance and distribution shape
Anne-Gaëlle Dosne (1), Ron J. Keizer (1), Martin Bergstrand (1), Mats O. Karlsson (1)
(1) Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden
Objectives: Implement a new error modeling strategy including dynamic transform both sides [1,2] with dynamic scedasticity (dTBS) and t-distributed residuals in NONMEM. Evaluate these methods with respect to their type 1 error rate when using likelihood ratio test for model building, practical estimation properties and improvement in fit to real data.
Background: Maximum likelihood estimation in non-linear mixed effects modeling is based on the assumption of normally distributed residuals. Violations of this assumption can cause bias in parameter estimates and invalidate the likelihood ratio tests (LRT). In this work, two error models relaxing the normality assumption are presented: (1) a t-distributed residual error model to account for heavy tailed residuals and (2) a power error model (yobs=ypred+ypred ζ ε) combined with a Box-Cox transformation of both dependent variable and model prediction (dTBS). Estimating shape and scedasticity dTBS parameters λ and ζ can correct for skewness in the residual error distribution and allow for non-linear relationships between the residual error magnitude and individual predictions.
Methods: The investigated error models were evaluated over a range of published PK and PD models. The results were evaluated with respect to improvement in model fit (OFV) and simulation properties compared to the published model. The type I error rate associated with additional residual error parameters was performed through stochastic simulation and estimation (SSE).
Results: Nominal type I error rates were not inflated when estimating dTBS or t-distribution parameters. Estimation of dTBS parameters was successful in all real data examples and lead to reasonable λ and ζ estimates. The OFV drop was significant in all cases and graphical improvement was observed in the distribution of IWRES-NPDE [3]. Estimation of dTBS parameters lead to changes in other parameter estimates as well as related parameter precision. Implementation of the t-distribution was successful in all real data examples and improved model fit in 75% of cases. Parameter estimates and precision were only slightly changed.
Conclusions: The use of dTBS and/or t-distribution models for the residual error provides a more flexible framework to characterize the distribution of the residual error. Both methods can improve model fit and relax modeling assumptions.
Acknowledgements: This research was performed as part of the DDMoRe project.
References:
[1] Frame B. Within Subject Random Effect Transformations with NONMEM VI, Wolverine Pharmacometrics Corporation, Nov 2009.
[2] Caroll RJ, Ruppert D. Transformation and Weighting in Regression, Chapman and Hall, 1998.
[3] Keizer RJ et al., PAGE 2012, abstract 2538.
