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Nonparametric Methods: Detecting Departures from Normality in Population PK/PD Studies

This article explores the use of nonparametric methods in population PK/PD studies to detect departures from normality in the distribution of ETA. Various nonparametric methods are discussed, including NP-NONMEM, NPML, NPEM, and SNP. Simulation studies are used to compare the performance of these methods against parametric estimation assuming normality.

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Nonparametric Methods: Detecting Departures from Normality in Population PK/PD Studies

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  1. Nonparametric (NP) methods: When using them? Which method to choose? Julie ANTIC and advisors: D. Concordet, M. Chenel, C.M. Laffont, D. Chafaï

  2. A too restrictive normality assumption • Usual population PK/PD studies assume normality of ETA • But the true distribution of ETA may be more complex! Parametric estimation (normal) True distribution bimodal asymmetric heavy-tailed ETA

  3. How to detect departures from normality? • If ETA-shrinkage is low Parametric estimation (normal) Empirical Bayes Estimates (EBEs) True distribution ETA

  4. How to detect departures from normality? • But if ETA-shrinkage is high, EBEs can be misleading [Karlsson & Savic, 2007] Parametric estimation (normal) Empirical Bayes Estimates (EBEs) True distribution ETA

  5. A possible solution: NP methods NP method = estimates an increasing number of parameters with N (N= number of individuals in the sample) → for large samples, a lot of distributions are available! → no restrictive assumption on ETA distribution

  6. Several NP methods • Some discrete NP: - NP-NONMEM [Boeckmann & al., 2006] - NPML [Mallet, 1986] - NPEM [Schumitzky, 1991] - others: NP adaptative grid, extended grid… • Some continuous NP: - SNP [Davidian & al., 1993] - others: splines, kernels… frequencies support points

  7. Discrete NP Without assumption on ETA distribution, the MLE is (MLE = the maximum likelihood estimator) • discrete with at most N support points [Lindsay, 1983] → the likelihood is explicit ! • consistent[Pfanzagl, 1990] frequencies support points

  8. How to compute the discrete NP-MLE? frequencies NP-NONMEM [Boeckmann & al., 2006] • support points = EBEs • frequencies maximize the likelihood NPML [Mallet, 1986] and NPEM algorithm [Schumitzky, 1991] • increase the likelihood at each iteration • by modification of support points + frequencies • here implemented - using NP-NONMEM as starting point - in C++ - more details in [Antic, 2009] support points

  9. Smooth NP (SNP) SNP [Davidian & al., 1993] • = the MLE over a set of smooth distribution with density = polynomial² × normal density • examples • the degree of the polynomial increases with N • consistent [Gallant & al., 1987] density(ETA) = (1)²×exp(-0.5×ETA²)/√(2×PI) density(ETA) = (0.2+ETA)²×exp(-0.5×ETA²)/√(2×PI) density(ETA) = (0.3-0.4×ETA-0.6×ETA²)²×exp(-0.5×ETA²)/√(2×PI) density(ETA) = (0.9+0.06×ETA+0.06×ETA²+0.06×ETA3)²×exp(-0.5×ETA²)/√(2×PI) Normal distribution Asymmetric distribution Bimodal distribution Multimodal distribution

  10. Comparison of NP methods • several simulation studies:

  11. Details on the PK scenari Slow-metabolisers sub-population volume volume clearance clearance

  12. Details on the PK/PD scenario Non-responder sub-population baseline + disease progression(linear with time) baseline baseline + disease progression – effect (Emax model with effect compartment) 1 year time Effect at 100 days for a median AUC

  13. Simulation studies strategy • Strategy: for each scenari, repeat 100 times Dataset simulation with non-normal ETA Parametric estimation assuming normal ETA → estimation of residual variance , EBEs SNP nlmix code [Davidian & al., 1993] NP-NONMEM fixed NONMEM VI [Boeckmann & al., 2006] NPML (after NP-NONMEM) fixed implemented in C++[Antic & al., 2009] NPEM (after NP-NONMEM) fixed implemented in C++[Antic & al., 2009]

  14. Comparison of NP methods T1-distance True distribution Estimated distribution • T1 distance Estimated cumulative distribution function True cumulative distribution function ETA • Graphical inspection of marginal distributions Mean of estimated distributions

  15. ETA-shrinkage ~ 9%; PK IV bolus EBEs NP-NONMEM NPML (after NP-NONMEM) NPEM (after NP-NONMEM) SNP T1-distance Parametric EBEs and NP methods are roughly equivalent All methods seem consistent 0 N 50 100 200 300 400

  16. ETA-shrinkage ~ 9%; PK IV bolus clearance clearance clearance clearance clearance clearance N=200 TRUE EBEs NP-NONMEM NPML (after NP-NONMEM) All methods generally allow suspecting a departure from normality NPEM (after NP-NONMEM) SNP

  17. ETA-shrinkage ~ 34%; PK IV bolus EBEs NP-NONMEM NPML (after NP-NONMEM) NPEM (after NP-NONMEM) SNP T1-distance Parametric EBEs consistency is very slow! Only slight differences between NP methods N 50 100 200 300 400

  18. ETA-shrinkage ~ 34%; PK IV bolus; EBEs seem misleading clearance clearance clearance clearance clearance clearance N=200 TRUE EBEs No clear difference between NP methods NPML (after NP-NONMEM) NP-NONMEM NPEM (after NP-NONMEM) SNP

  19. ETA-shrinkage ~ 31%; PK oral EBEs NP-NONMEM NPML (after NP-NONMEM) NPEM (after NP-NONMEM) SNP T1-distance EBEs seem not consistent! NP-NONMEM is not as good as the other NP methods N 50 100 200 300 400

  20. ETA-shrinkage ~ 31%; PK oral; EBEs seem misleading clearance clearance NP-NONMEM seems biased clearance clearance clearance clearance N=300 EBEs TRUE NPML (after NP-NONMEM) NP-NONMEM SNP NPEM (after NP-NONMEM)

  21. ETA-shrinkage > 40%; PK/PD NP-NONMEM and NPML poorly detected the subpopulation Only NPEM and SNP appear to detect the non-responder sub-population EBEs NEVER detect the non-responder subpopulation TRUE EBEs 25% 25% Drug effect Drug effect NPML (after NP-NONMEM) 25% NP-NONMEM 25% Drug effect Drug effect 25% 25% NPEM (after NP-NONMEM) SNP Drug effect Drug effect

  22. Conclusion • EBEs are misleading when ETA-shrinkage is high (>30%) • NP methods appeared to be a good solution (with reasonable computation times) • Our recommendations: - use NP-NONMEM - easy to implement in NONMEM - quite fast to compute + a more advanced NP method (especially if ETA-shrinkage > 40%): ex. NPEM, SNP…

  23. To learn more on NP, go and see: • poster 107 [Comets, Antic & Savic] • poster 105 [Baverel, Savic & Karlsson] • poster 133 [Goutelle, Bourguignon, Bleyzac & al.] • poster 29 [Jelliffe, Schumitzky, Bayard & al.] • MM USC-PACK software demonstration [Jelliffe, Schumitzky, Bayard, & al.] Thanks for your attention.

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