The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
Pseudo-likelihood Estimation for a Combined
Model for Time-to-event Data
Achmad Efendi2 Geert Molenberghs1,2 Geert Verbeke2,1
1
2
I-BIOSTAT, Universiteit Hasselt, Belgium
I-BIOSTAT, Katholieke Universiteit Leuven, Belgium
IAP Workshop 4
Leuven, November 18-19, 2010
1 / 29
The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
Outline
1
The combined model
2
Estimation
3
Application
4
Simulation study
5
Discussion
6
Goodness-of-fit
7
Broader Goodness-of-fit Picture
2 / 29
The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
Introduction
The need of extending the exponential family; for two
reasons 1) overdispersion, and 2) hierarchical structure.
Hence combined models needed.
Molenberghs et al (2010) defined broad class of
generalized linear models accommodating both.
The focus will be on combined model for time-to-event
cases, in particular the estimation strategy.
4 / 29
The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
Weibull- and exponential-type models
The general Weibull model for repeated measures, with both
gamma and normal random effects can be expressed as
f (y i |θ i , bi ) =
ni
Y
ρ
λρθij yijρ−1 ex ij ξ +z ij bi e−λyij θij e
0
0
x 0ij ξ +z 0ij bi
j=1
f (θ i ) =
f (bi ) =
ni
Y
1
α −1
θij j e−θij /βj ,
αj
β Γ(αj )
j=1 j
1
e− 2 bi
q/2
1/2
(2π) |D|
1
0
D −1 bi
.
5 / 29
The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
Weibull-type model (Cont’d)
A few observations:
The gamma random effects are independent.
Setting ρ = 1 leads to the special case of an exponential
time-to-event distribution.
The classical gamma frailty model and the Weibull-based
GLMM follow as special cases.
S-called “strong conjugacy” applies.
6 / 29
The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
Partial marginalization
Defined by Molenberghs et al (2010), used by integrating over
the gamma random effects only. For Weibull case:
f (yij |bi ) =
λκij eµij ρyijρ−1 αj βj
(1 + λκij eµij βj yijρ )αj +1
When censoring applies:
Z +∞
f (yij |bi )dyij =
f (Cij |bi ) =
Cij
1
(1 + λκij eµij Cijρ )αj
Integrating the normal random effects using SAS Proc
NLMIXED –> marginal model.
8 / 29
The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
Pseudo-likelihood estimation
The log of the pseudo-likelihood is defined as
p` =
N X
X
(s)
δs ln fs (y i ; ξ)
i=1 s∈S
Renard, Molenberghs, and Geys (2004) refer to pairwise
likelihood. The contribution of the ith cluster to the log
pseudo-likelihood then specializes to
X
p`i =
ln f (yij , yik )
j<k
if it contains more than one observation. Otherwise p`i = f (yi1 ).
9 / 29
The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
Pseudo-likelihood estimation (Cont’d)
The need of corrected standard error estimates
–> Sandwich estimator.
Maximizing the likelihood function f (y i |θ i , bi ) produces a
consistent and asymptotically normal estimator e
ξ 0 , (Arnold and
Strauss 1991, Geys, Molenberghs, and Ryan 1999, Aerts et al
2002).
√
N(ξ̃ N − ξ 0 ) converges in distribution to
Np [0, I0 (ξ 0 )−1 I1 (ξ 0 )I0 (ξ 0 )−1 ]
10 / 29
The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
Recurrent asthma attack in children
Duchateau and Janssen (2007)
Occurring in children aged between 6 to 24 months
Children randomized to placebo or drug, and asthma
events that developed over time are recorded in a diary
Recorded time is the time at risk for a particular event
230 patients; 1770 observations
12 / 29
The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
Table: Asthma data for the first two patients
Patient ID Drug Begin End Status
1
0
0
15
1
1
0
22 90
1
1
0
96 325
1
1
0
329 332
1
1
0
338 369
1
1
0
370 412
1
1
0
418 422
1
1
0
426 474
1
1
0
477 526
1
1
0
530 600
0
2
1
0 180
1
2
1
189 267
1
2
1
273 581
1
2
1
582 600
0
13 / 29
The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
Analysis of asthma study
In this asthma study analysis:
Consider an exponential model, a model of the form (1)
with ρ = 1, and further a predictor of the form:
κij = κ0 + bi + ξ1 Ti
bi ∼ N(0, d)
The combined model was conveniently fitted by full
likelihood as well as pairwise likelihood
Model fitting done using SAS Proc NLMIXED and Macro
14 / 29
The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
Table: Asthma Study. Estimates and s.e. for coefficients
Exp
Exp-gamma
Par. Estimate (s.e.) Estimate (s.e.)
ξ0 -3.3709 (0.0772) -3.9782 (15.354)
ξ1 -0.0726 (0.0475) -0.0755 (0.0605)
√λ 0.8140 (0.0149) 1.0490 (16.106)
d
—
—
γ
—
3.3192 (0.3885)
18,693
18,715
Exp-normal
Combined
Effect
Par. Estimate (s.e.) Estimate (s.e.)
Intercept
ξ0 -3.8095 (0.1028) -3.9923 (20.337)
Treatment
ξ1 -0.0825 (0.0731) -0.0887 (0.0842)
Shape par. √λ 0.8882 (0.0180) 0.8130 (16.535)
Sd. of re
d 0.4097 (0.0386) 0.4720 (0.0416)
Gamma par. γ
—
6.8414 (1.7146)
−2log-lik
18,611
18,629
Effect
Intercept
Treatment
Shape par.
Sd. of re
Gamma par.
−2log-lik
15 / 29
The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
Table: Asthma Data. Combined model (naive; robust s.e.)
Effect
Without cens.
Intercept
Treatment
Shape par.
Gamma par.
Sd of re
With cens.
Intercept
Treatment
Shape par.
Gamma par.
Sd of re
Full lik.
Par. Estimates (s.e.)
Pseudo-lik.
Estimates (s.e.)
ξ0
ξ1
λ
√γ
d
-3.9923 (20.337) -3.4862 (6.2316; 0.0856)
-0.0887 (0.0842) -0.1060 (0.0203; 0.0953)
0.8130 (16.534) 0.8272 (5.1551; 0.0049)
6.8414 (1.7146) 6.7758 (0.6648; 1.1875)
0.4720 (0.0416) 0.3958 (0.0202; 0.0383)
ξ0
ξ1
λ
√γ
d
-4.0195 (28.663)
-0.1115 (0.0996)
0.7882 (22.592)
3.5633 (0.6281)
0.5620 (0.0506)
-3.6233 (0.4998; 0.09381)
-0.1269 (0.0221; 0.10571)
0.9189 (0.4590; 0.00003)
4.5882 (0.3627; 0.71248)
0.4443 (0.0211; 0.03906)
16 / 29
The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
Table: Wald test for treatment effect’s assessment in combined model
Model
Without censoring full likelihood
Without censoring pseudo-likelihood
With censoring full likelihood
With censoring pseudo-likelihood
Z value
-1.0534
-1.1123
-1.1205
-1.1292
p value
0.1461
0.1330
0.1312
0.1294
17 / 29
The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
Alternating Imputation Posterior Algorithm
AIP algorithm used for fitting multilevel combined models:
Having a constant value for one random effect parameter
Fit the t-th nested sub-model using offset vector from other
sub-model.
Sampling the model parameters from an approximation.
Sampling vector of random effects.
Repeating step 2-4 until N times, e.g. N=200.
Averaging the parameter estimates, as well as calculating
the standard errors.
18 / 29
The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
Table: Comet Assay data:Weibull model with two normal random
effects
Parameter
Beta0
Beta1
Beta2
Beta3
Beta4
Lambda
RE1
RE2
Bayesian
Estimate (sd)
-2.470 (0.0845)
-2.817 (0.1014)
-3.062 (0.1021)
-3.284 (0.1028)
-1.797 (0.1228)
1.419 (0.0188)
448.0 (550.70)
21.19 (5.5460)
Full likelihood
Estimate (sd)
-2.4712 (0.0773)
-2.8142 (0.0908)
-3.0400 (0.0917)
-3.2871 (0.0927)
-1.7894 (0.1075)
1.4173 (0.0189)
-0.1629 (0.0276)
0.2192 (0.0248)
Pseudo-likelihood
Estimate (sd, scorrect)
-2.4087 (0.0083; 0.5621)
-2.7642 (0.0109; 0.3770)
-3.1243 (0.0112; 0.4152)
-3.1833 (0.0116; 0.4492)
-1.7878 (0.0106; 0.2438)
1.3907 (0.0030; 0.2316)
-0.1663 (0.0064; 0.0566)
-0.1370 (0.0074; 0.0566)
19 / 29
The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
Simulation Setting
Generate dataset with true parameters were set equal to
the estimates obtained from the analysis done.
Three different sample sizes were considered : 50, 100,
and 200 subjects.
Number of observation within a subject generated via
normal distribution with µ=12 and σ 2 =4.
Bernoulli random variable with φ= 0.9, 0.75, and 0.5 to
generate censoring covariates.
For each setting, 500 datasets were generated and the
combined model was fitted.
As a measure of consistency, the Mahalanobis distance is
used.
q
DM (ξˆn ) = (ξˆn − ξ 0 )T S −1 (ξˆn − ξ 0 )
(1)
21 / 29
40
30
40
50
15
Mahalanobis distance
Pseudo lik.
Full lik.
10
0
150
Sample size
200
20
30
40
50
Pseudo lik.
Full lik.
0
Mahalanobis distance
10
5
100
10
Censoring Percentage
5
15
20
Censoring Percentage
0
Mahalanobis distance
15
0
10
Pseudo lik.
Full lik.
50
10
Mahalanobis distance
10
5
50
15
30
10
20
Censoring Percentage
5
10
Pseudo lik.
Full lik.
5
15
Pseudo lik.
Full lik.
0
5
10
Mahalanobis distance
Pseudo lik.
Full lik.
0
Mahalanobis distance
15
The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
50
100
150
Sample size
200
50
100
150
200
Sample size
Figure 1: Mahalanobis distance for different sample sizes and
censored observation percentages
22 / 29
The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
Discussion
The combined model is conveniently fitted using both full
and pairwise likelihood; in the setting of without and with
censoring
Findings were about computational benefit: reaching
convergence, Geys, Molenberghs, and Ryan (1999) and
the robustness against different starting values, where
pseudo-likelihood performs better than full likelihood.
Other estimation method for combined multilevel models?
Effect of misspecification of random effects distribution?
24 / 29
The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
Goodness-of-fit
Combined model generalizes (a) generalized linear mixed
model
Combined model generalized (b) frailty model
It can be conveniently used as a goodness-of-fit tool
In turn, its own goodness-of-fit tools require further study
26 / 29
The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
Goodness-of-fit
Verbeke et al: Goodness-of-fit tools for linear mixed
models
Verbeke, Molenberghs et al: Goodness-of-fit tools for
incomplete data
Molenberghs et al: Goodness-of-fit tools for categorical
data
28 / 29
The combined model Estimation Application Simulation study Discussion Goodness-of-fit Broader Goodness-of-fit P
References
Geys, H., Molenberghs, G. and Ryan, L. (1999).
Pseudo-likelihood Modelling of Multivariate Outcomes in
Developmental Toxicology. Journal of the American
Statistical Association, 94: 34–45.
Molenberghs, G., Verbeke, G., Demetrio C.G.B., and Vieira,
A. (2010). A family of generalized linear models for
repeated measures with normal and conjugate random
effects. Statistical Science, In Press.
Molenberghs, G., Verbeke, G., Efendi, A., Braekers, R., and
Demetrio, C.G.B. (2010b). A Combined Gamma Frailty and
Normal Random-effects Model for Repeated,
Overdispersed Time-to-event Data. In preparation.
29 / 29