more on parametric and nonparametric population modeling: a brief summary roger jelliffe, m.d. usc...
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More on Parametric and Nonparametric Population Modeling: a brief Summary
Roger Jelliffe, M.D.USC Lab of Applied Pharmacokinetics
See also Clin PK, Bustad A, Terziivanov D, Leary R, Port R, Schumitzky A, and Jelliffe R: Parametric and Nonparametric Population Methods: Their Comparative Performance in Analysing a Clinical Data Set and Two Monte Carlo Simulation Studies. Clin. Pharmacokinet., 45: 365-383, 2006.
InTER-Individual Variability• The variability between subjects in a
population.• Usually a single number (SD, CV%) in
parametric population models• But there may be specific subpopulation
groups• eg, fast, slow metabolizers, etc. • How describe all this with one
number?• What will you DO with it?
InTRA-Individual Variability
• The variability within an individual subject.• Assay error pattern, plus• Errors in Recording times of samples• Errors in Dosage Amounts given• Errors in Recording Dosage times• Structural Model Mis-specification• Unrecognized changes in parameter values
during data analysis.• How describe all this with one number?• How describe interoccasional variability only
with one number? • What will you DO with these numbers?
Nonparametric Population Models (1)
• Get the entire ML distribution, a DiscreteJoint Density: one param set per subject, + its prob.
• Shape of distribution not determined by some equation, only by the data itself.
• Multiple indiv models, up to one per subject.• Can discover, locate, unsuspected
subpopulations.• Get F from intermixed IV+PO dosage.
Nonparametric Population Models (2)
• The multiple models permit multiple predictions.
• Can predict precision of goal achievement by a dosage
regimen.• Behavior is consistent.• Use IIV +/or assay SD, stated ranges.
What is the IDEAL Pop Model?
• The correct structural PK/PD Model.
• The collection of each subject’s exactly known parameter values for that model.
• Therefore, multiple individual models, one for each subject.
• Usual statistical summaries can also be obtained, but usually will lose info.
• How best approach this ideal? NP!
NPEM can find sub-populations that can be missed by parametric techniques
True two-parameter densitySmoothed empirical density of20 samples from true density
NPEM vs. parametric methods, cont’d
Best parametric representation using normality assumption
Smoothed NPEM results
The Clinical Population - 17 patients, 1000mg Amikacin IM qd for 6 days
• Seventeen patients• 1000 mg Amikacin IM qd for 5 doses• 8-10 levels per patient,
usually 4-5 on day 1-2,
and 4-5 on day 5-6,• Microbiological assay,
• SD = 0.12834 + 0.045645 x Conc• Ccr range - 40-80 ml/min/1.73 M2
Getting the Intra-individual variability
IIV = Gamma x (assay error SD polynomial)
so,
IIV = Gamma x (0.12834 + 0.045645 x Conc)
Gamma = 3.7
Amikacin - Parameterization as Ka, Vs, and Ks
IT2B NPEM NPAG
With Med/CV% Ka 1.352/4.55 1.363/20.42 1.333/21.24
Vs .2591/13.86 .2488/17.44 .2537/17.38
Ks .003273/14.83 .003371/15.53 .003183/15.76
Amikacin - Log Likelihood, Ka, Vs, and Ks, with and without gamma
IT2B NPEM NPAG
No
Log - Lik -809.996 -755.111 -748.295
With
Log - Lik -389.548 -374.790 -374.326
Estimates from Pop Medians, Ka, Vs, Ks parameterization, no /
• IT2B NPEM NPAG
• r2 = .814/.814 .876/.879 .877/.880
• ME = .979/-.575 -.584/-.751 -0.367/.169
• MSE = 55.47/48.69 28.96/29.01 29.06/29.70
ConclusionsAll parameter values pretty similarLess variation seen with IT2BBut log likelihood the least
NPEM, NPAG more likely param distribsNo spuriously high param correlationsNPAG most likely param distributionsNPEM, NPAG best suited for MM dosage
NPEM, NPAG are consistent, precise.
New - Non-parametric adaptive grid algorithm (NPAG)
• Initiate by solving the ML problem on a small grid
• Refine the grid around the solution by adding perturbations in each coordinate at each support point from optimal solution at previous stage
• Solve the ML problem on the refined grid (this is a small but numerically sensitive problem)
• Iterate solve-refine-solve cycle until convergence, using decreasing perturbations
• Best of both worlds - improved solution quality with far less computational effort!
103
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109
-800
-750
-700
-650
-600
-550
-500
-450
-400Adaptive grid greatly improves NPEM performance
Number of grid points
Log
likel
ihoo
d
Adaptive GridNPEM
NPAG outperforms NPEM by a large factor
CPU TIME MEMORY LOG-LIK (HRS) (MB)
NPEM: 2037 10000 -433.1NPAG: 1.7 6 -433.0
NPEM run was made at SDSC on 256 processors of Blue Horizon, an IBM SP parallel supercomputer that was then the most powerful non-classified computer in the world
NPAG run was made on a single 833 MHz Dell PC
Leary – A Simulation Study• One compartment model h(V,K) = e-Kt/V with unit
intravenous bolus dose at t=0
• Five parameters in N(m,S): mV=1.1, mK=1.0 sV=0.25, sK =0.25, r= –0.6, 0.0, and +0.6
• 1000+ replications to evaluate bias and efficiency
• N=25, 50, 100, 200, 400, 800 sample sizes
• Two levels (moderately data poor) with 10% observational error
NPAG and P-EM are consistent (true value of mV = 1.1)
0 100 200 300 400 500 600 700 800 900 10001.07
1.08
1.09
1.1
1.11
1.12
1.13
N - number of subjects
Ave
rag
e of
100
0 in
dep
end
ent
estim
ates
of
mea
n o
f V NPAG
P-EMIT2B
Consistency of estimators of mK
(true value of mK = 1.0)
0 100 200 300 400 500 600 700 800 900 10000.99
0.995
1
1.005
1.01
1.015
1.02
1.025
1.03
N - number of subjects
Ave
rag
e o
f 1000 in
dep
en
den
t est
imate
s of
mean
of
K
NPAGP-EMIT2B
Consistency of estimators of sK
(true value of sK=0.25)
0 100 200 300 400 500 600 700 800 900 10000.15
0.2
0.25
0.3
N - number of subjects
Ave
rag
e o
f 1000 in
dep
en
den
t est
imate
s of
std
. d
ev.
of
K
NPAGP-EMIT2B
Consistency of estimators of V-K correlation coefficient (true value r = -0.6)
0 100 200 300 400 500 600 700 800 900 1000-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
N - number of subjects
Ave
rage e
stim
ate
d co
rrela
tion
in 1
000 s
imul
atio
ns
NPAGP-EMIT2B
Consequence #1 of using F.O.C.E approximation– loss of consistency
• small (1-2%) bias for mV, mK
• moderate (20 – 30%) bias for sV, sK
• severe bias for correlations
true value average estimate
-0.6 +0.2
0.0 +0.6
+0.6 +0.85
Statistical efficiencies of NPAG and PEM are much higher than IT2B
0 100 200 300 400 500 600 700 800 900 10000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
N - number of subjects
Eff
icie
ncy
rela
tive t
o d
ata
ric
h lim
it
NPAGP-EMIT2B
Asymptotic stochastic convergence rate of IT2B is 1/N1/4 vs. 1/N1/2 for NPAG and P-EM
101
102
103
10-2
10-1
N - number of subjects
Sta
nd
ard
devi
atio
n o
f est
imato
r of
mean
of
V
NPAGP-EMIT2B
Approximate likelihoods can destroy statistical efficiency
0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
2
4
6
8
10
12
14
16
histogram (blue) of NONMEM FOestimators
histogram (white) of PEM estimators
NONMEM FOCE does better, but still has less than 40% efficiency
relative to exact ML methods
0.05 0.055 0.06 0.065 0.07 0.075 0.080
1
2
3
4
5
6
7
8
9
10
red: NPAG blue: NONMEM-FOCE
Consequences of usingF.O. and F.O.C.E approximations
versus exact likelihoods
• Loss of consistency• Severe loss of statistical efficiency• Severe reduction of asymptotic
convergence rate : • need 16 X the number of subjects to
reduce the SD of IT2B estimator by factor of 2,
• vs. 4 X for NPAG and PEM, as theory says