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

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Nonparametric (NP) methods:

When using them? Which method to choose?

Julie ANTIC

and advisors: D. Concordet, M. Chenel, C.M. Laffont, D. Chafaï

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

How to detect departures from normality?

• If ETA-shrinkage is low

Parametric estimation (normal)

Empirical Bayes Estimates (EBEs)

True distribution

ETA

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

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…

frequencies

support points

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

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

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

Comparison of NP methods

• several simulation studies:

Details on the PK scenari

Slow-metabolisers sub-population

volume

volume

clearance

clearance

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

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]

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

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

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

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

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

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

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)

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

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.