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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-800

-750

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-400Adaptive grid greatly improves NPEM performance

Number of grid points

Log

likel

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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

800 Normally distributed (K,V) points, correlation = +0.6

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

V

K

800 normal points give 70 NPAG support points

0 0.5 1 1.5 2 2.50

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K

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

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N - number of subjects

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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

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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

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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

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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

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Asymptotic stochastic convergence rate of IT2B is 1/N1/4 vs. 1/N1/2 for NPAG and P-EM

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Approximate likelihoods can destroy statistical efficiency

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

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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

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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

Efficiency and Relative Error

• Estimator Rel Efficiency Rel Error

• Direct observation 100.0% 1.00

• PEM 75.4% 1.33

• NPAG 61.4% 1.63

• NONMEM FOCE 29.0% 3.45

• IT2B FOCE 25.3% 3.95

• NONMEM FO 0.9% 111.11