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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. Chafa ï. A too restrictive normality assumption. • Usual population PK/PD studies assume normality of ETA.

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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

Nonparametric (NP) methods:

When using them? Which method to choose?

Julie ANTIC

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


Nonparametric np methods when using them which method to choose julie antic

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


Nonparametric np methods when using them which method to choose julie antic

How to detect departures from normality?

• If ETA-shrinkage is low

Parametric estimation (normal)

Empirical Bayes Estimates (EBEs)

True distribution

ETA


Nonparametric np methods when using them which method to choose julie antic

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


Nonparametric np methods when using them which method to choose julie antic

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


Nonparametric np methods when using them which method to choose julie antic

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


Nonparametric np methods when using them which method to choose julie antic

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


Nonparametric np methods when using them which method to choose julie antic

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


Nonparametric np methods when using them which method to choose julie antic

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


Nonparametric np methods when using them which method to choose julie antic

Comparison of NP methods

• several simulation studies:


Nonparametric np methods when using them which method to choose julie antic

Details on the PK scenari

Slow-metabolisers sub-population

volume

volume

clearance

clearance


Nonparametric np methods when using them which method to choose julie antic

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


Nonparametric np methods when using them which method to choose julie antic

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]


Nonparametric np methods when using them which method to choose julie antic

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


Nonparametric np methods when using them which method to choose julie antic

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


Nonparametric np methods when using them which method to choose julie antic

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


Nonparametric np methods when using them which method to choose julie antic

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


Nonparametric np methods when using them which method to choose julie antic

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


Nonparametric np methods when using them which method to choose julie antic

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


Nonparametric np methods when using them which method to choose julie antic

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)


Nonparametric np methods when using them which method to choose julie antic

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


Nonparametric np methods when using them which method to choose julie antic

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…


Nonparametric np methods when using them which method to choose julie antic

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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