An Improved Framework for Shrinkage Computations in NLME Population Models
Robert H. Leary (1), Michael Dunlavey (1), Kevin Feng (1), Shuhua Hu (1)
(1) Certara/Pharsight
Objectives: Shrinkage evaluation is based on comparing the variance over subjects of empirical Bayesian estimates (EBEs) of a random effect ETA(I) to the corresponding parameter OMEGA(I,I). ETA shrinkage refers to the fact this variance is smaller than OMEGA(I,I), while EPS shrinkage refers to the variance of residuals evaluated at ETAs set equal to EBE values being smaller than the corresponding residual error parameter [1]. The most common form of EBE in current use is the mode (MAP estimate) of the empirical Bayesian distribution (EBD) which is the central focus of any conditional method such as FOCEI. However, in EM-based methods, the mean rather than the mode of the EBD plays the central role. We investigate the properties of EBD mean- vs mode-based shrinkages.
Methods: The methods used to derive the properties of EBD mean-based shrinkages are based both on a)theoretical analysis of the fixed point conditions that hold at convergence of EM methods and b) numerical examples of mode- vs mean-based shrinkages based on applying the ELS FOCE, QRPEM, and nonparametric algorithms in Phoenix NLME™ to several simulated models.
Results:
a)The results in [2] are claimed by the authors to suggest that “on average, there is a 1:1 relationship” between shrinkage and relative estimation error for EBEs of PK parameters. In fact, this relationship becomes an easily provable theorem in the mean-based case.
b)In the mean-based case, there is a relationship between EPS shrinkage and the ETA shrinkages which can be used to assign the relative amounts of intra- and inter-individual variability. In the special case of homoscedastic linear models, this amounts to an exact division of the total degrees of freedom (number of observations) into EPS and ETA related parts.
c) Mean based shrinkage calculations can easily and naturally be extended to nonparametric population modeling.
d) The linear regression used in EM methods to update covariate parameters is based on response variables defined by the EBD means. Thus we may expect investigation of prospective linear covariate relationships to be more informative, particularly in the high shrinkage case, than using responses based on EBD modes. This in fact is born out in several test examples..
Conclusions: Shrinkage computations based on means rather than modes of the EBD are more amenable to exact theoretical analysis and additionally offer practical advantages.
References:
[1] Savic RM, Karlsson MO. Importance of shrinkage in empirical Bayes estimates for diagnostics:problems and solutions. AAPS J. 2009; 11(3):558-69.
[2] Xu SX, Yuan M, Karlsson MO, Dunne A, Nandy P, Vermeulen A. Shrinkage in nonlinear mixed effects population models: quantification, influencing factors and impact. AAPS J. 2012; 14(4):927-36.
