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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. - PowerPoint PPT Presentation
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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.