Semiparametric distributions with estimated shape parameters: Implementation and Evaluation
K. Petersson, E. Hanze, R.M. Savic, M.O. Karlsson
Div. of Pharmacokinetics and Drug Therapy, Dept of Pharmaceutical Biosciences, Faculty of Pharmacy, Uppsala University, Sweden
Introduction: In parametric population analysis interindividual random effects are assumed to be normally distributed and PK and PD parameters a fixed, often exponential, transformation of this distribution. It is possible to also estimate parameters that relates to the shape of the transformed distribution [1], but this has not become common practise, possibly because of implementation difficulties of the previously suggested transformation functions. We therefore investigate two transformations that can be easily implemented
Objectives: The aims of this work were: (i) to evaluate if such transformations can improve model fit; (ii) to assess the actual significance level for inclusion of these transformations into the models [2] and (iii) to investigate simulation properties of the models after transformation inclusion.
Methods: The analysis was done using NONMEM VI. Two transformations were evaluated, the Box-Cox and the Logit transformation which are mathematical formulas with one and two parameters respectively. These parameters were estimated simultaneously from the data along with other parameters as fixed effects (thetas) in NONMEM. The transformations were applied to parameter distributions (etas) in 27 already existing PK and PD models, both to a single parameter and to multiple parameters within the same model. Monte Carlo simulation studies were performed to assess cut-off values for statistical significance for both transformations. Simulation properties were studied through numerical predictive checks [3].
Results: New transformations significantly improved the model fit in 14 models out of 27 with drops in OFV ranging from 4 to 239. The cut-off values (i.e. drop in OFV) for significant inclusion of the Logit and Box-Cox transformations were 7 and 4 (p=0.05). Thus nominal and actual significance levels agree. Improved models with significant transformations included showed similar simulation properties as the original models.
Conclusions: A novel method for parameter distribution estimation is introduced, which allows for estimation of flexible semi-parametric shapes. Transformations are easy to implement and powerful to detect deviations from normality, thus model fit may be improved.
References:
[1] Davidian M., Gallant R. A. Smooth Nonparametric Maximum Likelihood Estimation for Population Pharmacokinetics, with Application to Quinidine. J Pharmacokinet Biopharm 1992; 20(5):529-56
[2] Wahlby U, Jonsson EN, Karlsson MO. Assessment of actual significance levels for covariate effects in NONMEM. J Pharmacokinet Pharmacodyn. 2001 Jun;28(3):231-52.
[3] Wilkins J, Karlsson MO, Jonsson EN Patterns and power for the visual predictive check. PAGE 15 (2006) Abstr 1029 [www.page-meeting.org/?abstract=1029]
