Download Dealing with censored data in linear and non

Dealing with censored data in
linear and non-linear models
Wan Hui Ong Clausen
Birgitte Biilmann Rønn
DSBS/FMS 26 Apr 2006
Slide no 1 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
Overview
• Background
• Model
• Estimation
• Implementation
• Examples
• Simulation
• Conclusion
Slide no 2 • 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 3 • 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 4 • 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 5 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
Cumulated glucose infusion rate
versus dose
9
8
7
log(AUC)
6
5
4
3
2
1
0
-1
-4
-3
-2
log(dose)
-1
0
aerx -1560/c urrent - 25A P R 2006 - plot_indi.s as /pres entation/plot_indi_gir_aerx _outlier.c gm
Slide no 6 • 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 7 • 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
Slide no 8 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
μ μcensored
c
σ
σcensored
Model with normal distributed error
Linear or non-linear mean structure and general
covariance structure:
Yi  fi( bi)  i,
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).
Slide no 9 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
Marginal likelihood function for fixed
effects parameters
with full data:


exp  ( yi  f (  , bi ))T  1 ( yi  f (  , bi )) / 2 exp  bi  1bi / 2
l  

dbi
p
½
q
½
(2 ) 
(2 ) 
yi
    ( yi , f (  , bi ), )   (bi ,0,  )dbi
yi
Slide no 10 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
T
Marginal likelihood function for fixed
effects parameters
with censored data:
l     ( yi , f (  , bi ), )
yij C
  (C , f (  , bi ), )   (bi ,0,  )dbi
yij C
Slide no 11 • 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 12 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
Cumulated glucose infusion rate
-after dosing with 5 different doses
• Primary interest:
9
regression on
log(dose)
recieving the lowest
dose level are nonresponders wrt GIR
• Treshold C=3.5
7
6
log(AUC)
• 6 out of 13 subjects
8
5
4
3
2
1
0
-1
-4
-3
-2
log(dose)
-1
0
aerx-1560/current - 25A P R 2006 - plot_indi.sas/presentation/plot_indi_gir_aerx_outlier.cgm
Slide no 13 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
Cumulated glucose infusion rate
Linear mixed model:
log( AUCij )      log( doseij )  U i   ij
with intercept α, slope β, random subject effect,
Ui~N(0,ω2) and residual εij~N(0,σ2dose) with
variace depending on dose level
Slide no 14 • 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 15 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
Estimates
-from analysis of log(AUC(0-10h)GIR)
Intercept
Slope
AUCGIR
(REML)
8.92
1.15
Imputed
values
[8.62;
9.22]
[1.01;
1.28]
(ML)
8.92
1.15
Imputed
values
[8.63;
9.22]
[1.02;
1.28]
(ML)
8.94
1.16
Censored [8.63;
values
9.25]
CV:
CV:
CV:
Between
subjects
Higher
doses
Low dose
38%
41%
372%
37%
40%
372%
37%
40%
167%
[1.03;
1.30]
Slide no 16 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
Example: PK data
Slide no 17 • 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 18 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
PK Example: Compartment model
dIs j
dt
dI f j
dt
 δ( t - t j )  (1 - α)D j  K fs Is j
 δ( t  t j )  α D j  K pf j I f j  K fs I s j
3
K pf j I f j
dI 
j1

 K xp I
dt
Vi
Slide no 19 • 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, ln K pf ij  ln K pf  bi  bi, j
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
Vary
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 20 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
Does this approximate approach leads
to better estimates?
Slide no 21 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
Simulation study: Theophylline data
Slide no 22 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
Simulation study:
First-order open-compartment model
Ka
Central
compartment
Ke
V=Cl/Ke
DK a K e
ct 
(e  K e t  e  K a t )
Cl(K a - K e )
Slide no 23 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
D:
Ka:
Ke:
Cl:
Dose
Absorption rate
Elimination rate
Clearance
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 24 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
Mean estimates – Laplacian Approx
___________________________________________________________
lKa
lCl
lKe
ψlK
a
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
0.498
-3.015
-2.504
0.279
0.036
0.475
LLoQ=3mg/l
Suggested
method
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 25 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •
Mean estimates – AGQ (5 abscissae)
___________________________________________________________
lKa
lCl
lKe
ψlK
a
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
0.491
-3.008
-2.492
0.287
0.037
0.471
LLoQ=3mg/l
Suggested
method
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 26 • 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.
Slide no 27 • Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 •