Two bootstrapping routines for obtaining uncertainty measurement around the nonparametric distribution obtained in NONMEM VI

1
Two bootstrapping routines for obtaining
uncertainty measurement around the nonparametric
distribution obtained in NONMEM VI Paul G.
Baverel 1, Radojka M. Savic 2 and Mats O.
Karlsson 1 1 Division of Pharmacokinetics
Drug Therapy, Uppsala University, Sweden 2 INSERM
U738, University Paris Diderot Paris 7, Paris,
France
Background
Results
N Quantifying uncertainty in parameter estimates
is essential to support decision making
throughout model building process. N Despite
providing enhanced estimates of parameter
distribution, nonparametric algorithms
do not yet supply uncertainty metrics.

• Overall, the trend and the magnitude of the 95
  CI derived with the full and
• the simplified nonparametric bootstrapping
  methods (N100 and N500)
• matched the true 95 CI in all
  distributional cases and regardless of
• individual numbers in original data.
• The simplified version induced slightly less
  bias in quantifying uncertainty
• (prediction errors of 95 CI width) than the
  full version. This is expected as
• the former methodology derives uncertainty
  from the original data.

Objectives
UNDERLYING BIMODAL CLEARANCE DISTRIBUTION
200 IDs
50 IDs
Prediction errors
Materials and Methods
N Six informative datasets of 50 or 200
individuals were simulated from an IV bolus
PK model in which CL and V conformed to various
underlying distributional shapes
(log-normal, bimodal, heavy-tailed). Residual
variability was set to 10 CV.
UNDERLYING HEAVY-TAILED CLEARANCE DISTRIBUTION
(200 IDs)
N100
Prediction errors
N500
N Re-estimation was conducted assuming normality
under FOCE, and NPDs were estimated by
applying FOCE-NONP method in NONMEM VI.

• Two different permutation methods automated in
  PsN 1 were developed to quantify uncertainty
  around NPD (95 confidence interval) and
  nonparametric estimates (SEs and
  variance-covariance matrix)
• The full method 2 relies on N bootstraps of
  the original data and a re-analysis of both the
  preceding parametric as well as the
  nonparametric step
• The simplified method relies on N bootstrap
  samples of the vectors of individual
  probabilities associated with each unique
  support point of the NPD

Figure 2. On the left 95 confidence intervals
obtained based on the full and simplified
nonparametric bootstrapping methodologies in case
of various underlying distributions of CL. The
true 95 CI around the true parameter
distribution is also represented for comparison,
as well as the parametric cumulative density
function. On the right Prediction errors of the
95 CI width are displayed for each quartile of
parameter distribution, the true uncertainty
being taken as reference.
SE 100 Stochastic Simulations followed by Estimations (SSE) 100 Stochastic Simulations followed by Estimations (SSE) 100 Stochastic Simulations followed by Estimations (SSE) Simplified nonparametric bootstrap version
SE True empirical FOCE Asymptotic (COV) FOCE True empirical FOCE-NONP (N100) FOCE-NONP
T CL 0.022 0.022 0.021 0.021
T V 0.024 0.022 0.024 0.02
O CL 0.01 0.009 0.01 0.008
O CL,V 0.007 0.006
O V 0.011 0.01 0.011 0.008

• N matrices MN (JxJ) of individual probabilities
  
• Row entries J individuals
• Column entries J support points

Table 1. SEs of parameter estimates obtained
from 100 stochastic simulations followed by
estimations given the true model under FOCE,
FOCE-NONP and the analytical solution under FOCE
(COVARIANCE) in NONMEM VI. The simplified
methodology was applied to each simulated
dataset SEs were computed and average SEs were
reported for comparison with SSE.
? N sets NPDnewN defined at J support points of
NPDN
N SEs obtained with the simplified methodology
matched the ones obtained by SSE.
Conclusion
N Two novel bootstrapping routines intended for
nonparametric estimation methods are
proposed. Their evaluation with a simple PK model
in the case of informative sampling design
was performed when applying FOCE- NONP in
NONMEM VI but it is easily transposable to other
nonparametric applications. N These tools
can be used for diagnostic purpose to help
detecting misspecifications with respect
to the distribution of random effects. N From
the sampling distribution obtained, standard
uncertainty metrics, such as standard
errors and correlation matrix can be derived in
case reporting uncertainty is intended.
? A single set of NPDnewN defined at J support
points of NPD
Figure 1 Sequential steps of the operating
procedure of both the full and simplified
nonparametric bootstrapping methods intended for
nonparametric estimation methods.
N The true uncertainty was derived by standard
nonparametric bootstrapping (N1000) 3 of
the true individual parameters and used as
reference for qualitative and quantitative
assessment of the uncertainty measurements
derived from both techniques.
References 1. Perl-speaks-NONMEM (PsN
software) L. Lindbom, M. Karlsson, N. Jonsson.
http//psn.sourceforge.net 2. Savic RM,
Baverel PG, Karlsson MO. A novel bootstrap method
for obtaining uncertainty around the
nonparametric distribution. PAGE 18 (2008)
Abstract 1390. 3. Efron B. Bootstrap
methods another look at the jackknife. Ann Stat
1979 71-26.