Slide no 1 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Dealing with censored data in linear and non-linear models Wan Hui Ong Clausen Birgitte Biilmann

Slide no 1 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •

Dealing with censored data in linear and non-linear models

Wan Hui Ong Clausen

Birgitte Biilmann Rønn

DSBS/FMS 26 Apr 2006


Overview

• Background

• Model

• Estimation

• Implementation

• Examples

• Simulation

• Conclusion


Censored PK data

•PK data: Pharmacokinetic data• Concentration of drug/preparation over time

• Disposition of the drug/preparation

•Example 1: Biphasic insulin• Three subcutaneous injections a day

• Concentrations measured over 24 hours


Biphasic insulin concentration over time – three subcutaneous injections

Censored at 13pmol/l


Censored PD data

•PD data: Pharmacodynamic data• Effect of the drug/preparation

• Measurements of the effect over time

•Example 2: Dose-response trial with inhaled insulin• 5 dose levels given in iso-glycaemic clamp

• Glucose infusion rate measured over 10 hours


Cumulated glucose infusion rate versus dose

aerx-1560/current - 25APR2006 - plot_indi.sas/presentation/plot_indi_gir_aerx_outlier.cgm

log(

AUC

)

-1

0

1

2

3

4

5

6

7

8

9

log(dose)

-4 -3 -2 -1 0


Censored GIR observations

•The method (manual clamp) might not be sufficiently sensitive, when the ’true’ glucose need is very low

•AUC(0-10h)GIR valued 0 are instead included in the analysis as being less than a treshold value, (e.g. 3.5).


Analysis with censored data:

• ’Usual’ solution:• Treat observations as

missing

• Problem:• Biased estimate of mean

• Biased estimated of variance

• Simple solution:• Obtain original data

when possible

μcensoredc

σ

σcensored

μ


Model with normal distributed error

Linear or non-linear mean structure and general covariance structure:

where Yi is the observation vector for subject i, β is the vector of fixed parameters, bi is the vector of random effects, bi~N(0,Ψ) mutually independent and independent of εi, the residual error vector, εi~N(0,Σ).

,iiii )b(fY


Marginal likelihood function for fixed effects parameters

with full data:

iy

iii

iqi

Ti

yp

iiT

ii

dbbbfy

dbbbbfybfy

l

i

i

),0,()),,(,(

)2(

2/exp

)2(

2/)),(()),((exp½

1

½

1


Marginal likelihood function for fixed effects parameters

with censored data:

iiiCy

Cyii

dbbbfC

bfyl

ij

ij

),0,()),,(,(

)),,(,(


Approximate likelihood inference

•The intergral can rarely be solved explicitly• for repeated measurements• For non-linear mean function (in the random

effects)

• Intergral approximations must be used• Laplace approximation or Adaptive Gaussian

quadrature

See eg.Wolfinger, R.D. (93) Laplace’s approximation for nonlinear mixed effects models, Biometrica 80:791-795, Davidian,M., Giltinan, D.M. (95) Nonlinear Models for Repeated Measurements Data. London: Chapman & Hall, Pinheiro, J.C., Bates, D.M. (1995). Approximations to the log-likelihood function in nonlinear mixed-effects model. J.Computat.Graph.Statist. 4:12-35, or Vonesh, E.F. Chinchilli, V.M. (97). Linear and Nonlinear Models for the Analysis of Repeated Measurements. New York: Marcel Decker, Inc.


Cumulated glucose infusion rate-after dosing with 5 different doses

• Primary interest: regression on log(dose)

• 6 out of 13 subjects recieving the lowest dose level are non-responders wrt GIR

• Treshold C=3.5

aerx-1560/current - 25APR2006 - plot_indi.sas/presentation/plot_indi_gir_aerx_outlier.cgm

log(A

UC)

-1

0

1

2

3

4

5

6

7

8

9

log(dose)

-4 -3 -2 -1 0


Cumulated glucose infusion rate

Linear mixed model:

with intercept α, slope β, random subject effect, Ui~N(0,ω2) and residual εij~N(0,σ2

dose) with variace depending on dose level

ijiijij UdoseAUC )log()log(


Estimation with PRC NLMIXED in SAS

proc nlmixed data=PDdata;

parms intercept=9 slope=1 vlow=13 vnlow=0.1 s1randsubj=-1.9 s1s2=-2 s2randsubj=-1.4;

if (treatment=1) then randsubj = rand1;

else randsubj = rand2;

m = intercept + slope*logdose + randsubj;

if (low_dose=0) then ll = -(lauc-m)**2/(2*vnlow) - 0.5*log(2*3.14159*vnlow);

if (low_dose=1) then do;

if cens=0 then ll = -(lauc-m)**2/(2*vlow) - 0.5*log(2*3.14159*vlow);

if cens=1 then ll = log(probnorm((3.5-m)/sqrt(vlow)));

end;

model lauc ~ general(ll);

random rand1 rand2 ~ normal([0,0],[exp(s1randsubj), exp(s1s2),

exp(s2randsubj)]) subject=subj_id;

run;


Estimates -from analysis of log(AUC(0-10h)GIR)

AUCGIR

Intercept Slope CV:

Between subjects

CV:

Higher doses

CV:

Low dose

(REML)

Imputed values

8.92

[8.62; 9.22]

1.15

[1.01; 1.28]

38% 41% 372%

(ML)

Imputed values

8.92

[8.63; 9.22]

1.15

[1.02;1.28]

37% 40% 372%

(ML)

Censored values

8.94

[8.63; 9.25]

1.16

[1.03;1.30]

37% 40% 167%


Example: PK data


Biphasic insulin concentration over time – three subcutaneous injections

70 out of 873 serum insulin concentrations were reported as < LLoQ at 13pmol/l


PK Example: Compartment model

j

jsfsjj

sIKD)α-1()t-t(δ

dt

dI

jsfsjfjpfjj

jfIKIKα D)tt(δ

dt

dI

IKV

IK

dt

dIxp

i

3

1jjfjpf


Nonlinear PK models Two-level random effects model

Level 1: between-subject variations on all parameters, diagonal variance structure

Level 2: For Kpf, j,iipfijpf bbKlnKln

Vary Fixed effects, estimates (log-scale)

Between subject

variance

Between injection (within subject)

variance

Variance (residuals)

Kpf (min-1) 0.0087 (-4.7496)

0.59662 0.39482 25.18302

Kfs (min-1) 0.0056 (-5.1916)

2.10942

Kxp (min-1) 0.0190 (-3.9610)

0.57262

Vi (L Kg-1) 0.9584 (-0.0425)

0.45222

Clausen W.H.O., De Gaetano A. & Vølund A. (2005) Pharmacokinetics of Biphasic Insulin Aspart Administered by Multiple Subcutaneous Injections: Importance of Within-subject Variation. Research report 09/05


Does this approximate approach leads to better estimates?


Simulation study: Theophylline data


Simulation study: First-order open-compartment model

Central compartment

V=Cl/Ke

KeKa

D: DoseKa: Absorption rateKe: Elimination rateCl: Clearance

)ee()K-Cl(K

KDKc tKtK

ea

eat

ae


Simulation study – cont’

• 1000 simulations

• 12 subjects

• 10 concentrations at

t = 0, 0.25, 0.5, 1, 2, 3.5, 7, 9, 12, 24h

• Dose = 4.5mg

• lKa = 0.5, lCl = -3, lKe = -2.5

• lKa and lCl are allowed to vary randomly, bi ~ N(0, ψ), where ψ is diagonal, 0.36 and 0.04 respectively

• 36% of the simulated data <LLoQ (3mg/l)


Mean estimates – Laplacian Approx___________________________________________________________

lKa lCl lKe ψlKa ψlCl σ2

___________________________________________________________

True value 0.500 -3.000 -2.500 0.360 0.040 0.490

Full data 0.498 -3.016 -2.505 0.280 0.036 0.480

LLoQ=3mg/l

Suggested

method 0.498 -3.015 -2.504 0.279 0.036 0.475

Omit data 0.661 -3.154 -2.680 0.254 0.029 0.442

___________________________________________________________Clausen, W.H.O., Tabanera, R., Dalgaard, P. (2005) Solvng the bias problem for censored

pharmacokinetic data. Research report 05/05 University of Copenhagen.


Mean estimates – AGQ (5 abscissae)___________________________________________________________

lKa lCl lKe ψlKa ψlCl σ2

___________________________________________________________

True value 0.500 -3.000 -2.500 0.360 0.040 0.490

Full data 0.495 -3.011 -2.498 0.286 0.037 0.476

LLoQ=3mg/l

Suggested

method 0.491 -3.008 -2.492 0.287 0.037 0.471

Omit data 0.635 -3.142 -2.661 0.266 0.030 0.432

___________________________________________________________Clausen, W.H.O., Tabanera, R., Dalgaard, P. (2005) Solvng the bias problem for censored

pharmacokinetic data. Research report 05/05 University of Copenhagen.


Conclusion

• Models with closed-form representation• The method could be applied using PROC NLMIXED

available in SAS

• Models without closed-form representation • a differential equation solver is necessary

• With censored data, the same approach can be applied – need some programming work

• The results from simulation study shows that bias introduced by left censoring is almost fully removed.

Documents

Slide no 1 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Dealing with censored data in linear and non-linear models Wan Hui Ong Clausen Birgitte Biilmann