Shrinkage in nonlinear mixed-effects population models: quantification, influencing factors, and impact

Abstract

Shrinkage of empirical Bayes estimates (EBEs) of posterior individual parameters in mixed-effects models has been shown to obscure the apparent correlations among random effects and relationships between random effects and covariates. Empirical quantification equations have been widely used for population pharmacokinetic/pharmacodynamic models. The objectives of this manuscript were (1) to compare the empirical equations with theoretically derived equations, (2) to investigate and confirm the influencing factor on shrinkage, and (3) to evaluate the impact of shrinkage on estimation errors of EBEs using Monte Carlo simulations. A mathematical derivation was first provided for the shrinkage in nonlinear mixed effects model. Using a linear mixed model, the simulation results demonstrated that the shrinkage estimated from the empirical equations matched those based on the theoretically derived equations. Simulations with a two-compartment pharmacokinetic model verified that shrinkage has a reversed relationship with the relative ratio of interindividual variability to residual variability. Fewer numbers of observations per subject were associated with higher amount of shrinkage, consistent with findings from previous research. The influence of sampling times appeared to be larger when fewer PK samples were collected for each individual. As expected, sample size has very limited impact on shrinkage of the PK parameters of the two-compartment model. Assessment of estimation error suggested an average 1:1 relationship between shrinkage and median estimation error of EBEs.

Fig. 1

Comparison of the variance-based derived and empirical shrinkage equations for a linear random-intercept model. Model-estimated interindividual and residual variability were used in the equations

Fig. 2

The relationship between shrinkage and residual variability at IIV = 10% and 40% for CL and Vc (a) and the relationship between shrinkage and ratio of interindividual variability to residual variability (b); in b, the boxplots with black and red median dots represent simulations for IIV = 10% and 40%, respectively. The empirical variance-based shrinkage was used

Fig. 3

The relationship between shrinkage (empirical variance-based) and number of observations per subject for CL and Vc

Fig. 4

The influence of sampling times (in hours) on shrinkage of CL and Vc when observations per subject are two (a), four (b), or eight (c). S2-1: (0.25, 24), S2-2: (0.5, 20), S2-3: (2, 16), and S2-4: (4, 12); S4-1: (0.25, 1, 16, 24), S4-2: (0.5, 2, 12, 20), S4-3: (1, 4, 8, 12), and S4-4: (2, 4, 6, 8); and S8-1: (0.25, 0.5, 1, 2, 12, 16, 20, 24), S8-2: (0.5, 1, 2, 4, 10, 12, 16, 20), S8-3: (1, 2, 4, 6, 8, 10, 12, 16), and S8-4: (0.5, 2, 4, 6, 8, 12, 18, 24)

Fig. 5

The relationship between shrinkage (empirical variance-based) and sample size for CL and Vc

Fig. 6

Impact of shrinkage (empirical variance-based) on median estimation error of subject-level parameter values for different PK parameters