Nonparametric (NP) methods:When using them? Which method to choose?

Julie ANTICand advisors: D. Concordet, M. Chenel, C.M. Laffont, D. Chafaï

A too restrictive normality assumption

Parametric estimation (normal)

• Usual population PK/PD studies assume normality of ETA

• But the true distribution of ETA may be more complex!

bimodalasymmetricheavy-tailedTrue distribution

ETA

Parametric estimation (normal)

• If ETA-shrinkage is low

How to detect departures from normality?

Empirical Bayes Estimates (EBEs)

ETA

True distribution

How to detect departures from normality?

• But if ETA-shrinkage is high,

EBEs can be misleading [Karlsson & Savic, 2007]

ETA

Parametric estimation (normal)

True distribution

Empirical Bayes Estimates (EBEs)

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

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…

support points

frequencies

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]

Discrete NP

support points

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]

How to compute the discrete NP-MLE?

frequencies

support points

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 distributionAsymmetric distributionBimodal distributionMultimodal distribution

• several simulation studies:

Type of data Number of ETAs ETA-shrinkage

PK 2 ~ 9%

PK 2 ~ 34%

PK 3 ~ 31%

PK/PD 5 > 40%

Comparison of NP methods

Details on the PK scenariRoute of administration IV oral

Inspired from Population PK of Phenobarbital

[Grasela & al., 1985] [Yukawa & al., 2005]

Structural model 1 compartment with 1st order elimination

and absorption

Error model proportional

Random effects’ distribution

Number of individuals 50, 100, 200, 300, 400

Individual information not sparse

(ni ~ 2.1)

sparse

(ni ~ 1.3)

sparse

(ni ~ 2.3)

Eta-shrinkage (clearance)[Karlsson & al., 2007]

~9% ~34% ~31%

volume

clearance

volume

clearance

Slow-metabolisers sub-population

Details on the PK/PD scenario

Inspired from Population PK/PD of Gliclazide [Frey & al., 2005]

Structural model

Error model Homoscedastic

Random effects’ distribution 25% of non-responders + 75% of responders

Experimental design N = 634; ni~ 8.3

Eta-shrinkage (ETAs related to effect) [Karlsson & al., 2007]

[ ~42% : ~86% ]

fast

pla

sma

glu

cose

fast

pla

sma

glu

cose

fast

pla

sma

glu

cose

fast

pla

sma

glu

cose

time 1 year

baseline

baseline + disease progression (linear with time)

baseline + disease progression – effect

(Emax model with effect compartment)

Effect at 100 days for a median AUC

0%

25%

Non-responder sub-population

• Strategy: for each scenari, repeat 100 times

Simulation studies strategy

Dataset simulation with non-normal ETA

Parametric estimation assuming normal ETA→ estimation of residual variance , EBEs

2̂

NP-NONMEMfixed

NONMEM VI [Boeckmann & al., 2006]

SNP

nlmix code [Davidian & al., 1993]

NPML (after NP-NONMEM)fixed

implemented in C++ [Antic & al., 2009]

NPEM (after NP-NONMEM)fixed

implemented in C++ [Antic & al., 2009]

2̂

2̂ 2̂

Estimated cumulative distribution functionTrue cumulative distribution functionT1-distance

Comparison of NP methods

ETA

• T1 distance

• Graphical inspection of marginal distributions

Mean of estimated distributions

True distribution Estimated distribution

0

EBEs

NP-NONMEM

NPML (after NP-NONMEM)

NPEM (after NP-NONMEM)

SNP

N50 100 200 300 400

T1-distance

ETA-shrinkage ~ 9%; PK IV bolus

Parametric EBEs and NP methods are roughly equivalent

All methods seem consistent

ETA-shrinkage ~ 9%; PK IV bolus

TRUE

NP-NONMEM

NPEM

(after NP-NONMEM)

EBEs

NPML

(after NP-NONMEM)

SNP

All methods generally allow suspecting a departure from normality

clearance clearance

clearanceclearance

clearance clearance

N=200

EBEs

NP-NONMEM

NPML (after NP-NONMEM)

NPEM (after NP-NONMEM)

SNP

N50 100 200 300 400

T1-distance

ETA-shrinkage ~ 34%; PK IV bolus

Parametric EBEs consistency is very slow!

Only slight differences between NP methods

TRUE

NP-NONMEM

NPEM

(after NP-NONMEM)

EBEs

NPML

(after NP-NONMEM)

SNP

ETA-shrinkage ~ 34%; PK IV bolus; N=200

clearance clearance

clearanceclearance

clearance clearance

EBEs seem misleading

No clear difference between NP methods

EBEs

NP-NONMEM

NPML (after NP-NONMEM)

NPEM (after NP-NONMEM)

SNP

N50 100 200 300 400

T1-distance

ETA-shrinkage ~ 31%; PK oral

NP-NONMEM is not as good as the other NP methods

EBEs seem not consistent!

NP-NONMEM

NPEM

(after NP-NONMEM)

TRUE EBEs

NPML

(after NP-NONMEM)

SNP

ETA-shrinkage ~ 31%; PK oral;N=300 EBEs seem misleading

NP-NONMEM seems biasedclearance clearance

clearanceclearance

clearance clearance

TRUE

NP-NONMEM

NPEM

(after NP-NONMEM)

EBEs

NPML

(after NP-NONMEM)

SNP

ETA-shrinkage > 40%; PK/PD EBEs NEVER detect the non-responder subpopulation

NP-NONMEM and NPML poorly detected the subpopulation

Only NPEM and SNP appear to detect the non-responder sub-population

25% 25%

25% 25%

25% 25%

Drug effect Drug effect

Drug effect Drug effect

Drug effect Drug effect

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…

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.