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
Slide no 2 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
Overview
• Background
• Model
• Estimation
• Implementation
• Examples
• Simulation
• Conclusion
Slide no 3 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
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
Slide no 4 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
Biphasic insulin concentration over time – three subcutaneous injections
Censored at 13pmol/l
Slide no 5 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
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
Slide no 6 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
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
Slide no 7 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
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).
Slide no 8 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
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
Slide no 9 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
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
Slide no 10 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
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
Slide no 11 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
Marginal likelihood function for fixed effects parameters
with censored data:
iiiCy
Cyii
dbbbfC
bfyl
ij
ij
),0,()),,(,(
)),,(,(
Slide no 12 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
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.
Slide no 13 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
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
Slide no 14 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
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(
Slide no 15 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
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;
Slide no 16 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
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%
Slide no 17 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
Example: PK data
Slide no 18 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
Biphasic insulin concentration over time – three subcutaneous injections
70 out of 873 serum insulin concentrations were reported as < LLoQ at 13pmol/l
Slide no 19 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
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
Slide no 20 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
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
Slide no 21 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
Does this approximate approach leads to better estimates?
Slide no 22 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
Simulation study: Theophylline data
Slide no 23 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
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
Slide no 24 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
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)
Slide no 25 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
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.
Slide no 26 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
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.
Slide no 27 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
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.