**Application relevant shrinkage metrics**

Martin Bergstrand

Pharmetheus, Uppsala, Sweden

**Background: **There is a general concern that high η-shrinkage [1] may bias applications such as sequential PKPD analysis and sub-group comparisons [2]. This work proposes to instead evaluate the shrinkage in the secondary parameter relevant to a particular application.

**Objectives:** To illustrate how relevant shrinkage metrics can be derived for secondary parameters such as AUC and C_{max} and how that shrinkage may affect a sequential PKPD analysis.

**Methods:** A simulation/re-estimation study was set up based on a hypothetical study with 4 equally sized sub-groups (100 subjects) with partly different study designs. All groups were administered 11 repeated p.o. doses once daily. Group A was also administered an i.v. dose on a separate occasion. Group A and B had semi-rich PK sampling with 16 and 11 PK samples, respectively. Group C and D had sparse PK sampling with 2 and 1 trough PK samples, respectively. A 1-compartment PK model (Θ_{CL} = 2.5 L/h, Θ_{Vc} = 100 L, Θ_{ka} = 1.5 h^{-1}, Θ_{F} = 0.5, Ω_{CL} = 0.1, Ω_{Vc} = 0.2, Ω_{ka} = 0.4, Ω_{F }= 0.15 (logit)) was assumed for simulating plasma exposure. Allometric scaling of CL and Vc was included in the model and a uniform weight range from 40 to 100 kg was simulated for the study population. Continuous drug response variables (R1, R2 and R3) was simulated assuming a linear relationship to AUC_{ss}, C_{max,ss} and C_{max,dose1}, respectively (ER_{slope} = 1, ER_{intercept} = 0). Three scenarios (CS) were simulated for this setup, (CS1) five different dose levels (100, 200, 400, 600 and 800 mg) was studied for with equal allocations to each dose level, (CS2) all subjects were administered the 400 mg dose, and (CS3) as CS2 but without the weight covariate in the PK model used for re-estimation. Shrinkage for the secondary parameters (SP) was calculated based on the standard deviation for the log of the post-hoc parameter estimates and corresponding simulated (x 500) parameters with the estimated model parameters: Shrinkage-SP = 100*(1 - sd(log(SP_{post-hoc})) / sd(log(SP_{simulated}))).

**Results: **The simulation/re-estimation study showed that: **(1)** The typically reported overall η-shrinkage may in fact differ a lot between different subgroups in an analysis population. The difference in η-shrinkage depends primarily on the number of observations and the timing of these (in other cases it may depend on factors such as dose and/or dosing regimen). **(2)** For all three case studies the shrinkage for AUC_{ss} , C_{max,ss }and C_{max,dose1 }was low (≤15%) for group A, B and C resulting in a relatively unbiased estimates for the exposure response parameters. This despite η-shrinkage for Vc, ka and F being as high as 23%, 96% and 56% for group C. **(3)** In the case of a single dose level, no weight covariate in the model and a very sparse PK sampling (CS3-D) the shrinkage in AUC_{ss} was 18% and resulted in relatively unbiased estimates of ER-slope (1.05, CI_{95} 0.78-1.32). For C_{max,ss }the shrinkage was higher (40%) but still did not substantially bias the ER-slope estimate (1.25, CI_{95} 0.89- 1.61). For C_{max,dose1 }the shrinkage was very high (75%) and resulted in a biased estimate for ER-slope (2.19, CI_{95} 1.39-3.00).**(4)** Higher shrinkage in the exposure parameters generally resulted in higher uncertainty for the exposure response parameters (high shrinkage is correlated with high uncertainty for the individual exposure estimate). In extreme cases it induced a bias towards a steeper exposure response relationship.

**Conclusions:** η-shrinkage is not a good metric to judge the validity of a sequential PKPD analysis approach. Shrinkage for the exposure parameter of interest is a better metric. In case of high shrinkage for such exposure parameters (>25%) a simulation/re-estimation approach can be applied to more thoroughly evaluate if and how the shrinkage is expected to affect a sequential assessment of exposure-response.

**References:**

[1] Savic RM, Karlsson MO. Importance of shrinkage in empirical bayes estimates for diagnostics: problems and solutions. AAPS J. 2009 Sep;11(3):558-69.

[2] Xu XS, 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 Dec;14(4):927-36.