Cox Model With Intermitten and
Error-Prone Covariate Observation
Yury Gubman
PhD thesis in Statistics
Supervisors:
Prof. David Zucker, Prof. Orly Manor
Introduction

Regression analysis of right-censored survival data
commonly arises in many fields, especially in medical
science.

The most popular survival data regression model is the
Cox (1972) proportional hazard model, in which the
hazard function for an individual with covariate vector X
is modeled as
 (t | X )  0 (t ) exp( T X )
Introduction

In many cases, the covariate X is not measured exactly.
Instead of X, we observe a surrogate measure W, which is
subject to error. Measurement error in the covariates has
three main effects:
a)
b)
c)

causes bias in parameter estimation for statistical models;
leads to a loss of power, sometimes profound, for detecting
covariate effects;
masks the features of the data, making graphical model analysis
difficult.
In addition, although in theory the covariate is a
continuous process in time, in practice measurements are
taken only over a discrete grid of timepoints (every 6
months, once a year, etc.). This discrepancy may also lead
to a bias.
Introduction

The classical measurement error model for individual i and
measurement j can be written as:
Wij  X i   ij

where measurement error term is independent of Xi with zero
mean given Xi. Independence across i is also assumed.
Let n be the number of individuals and J the number of
observations (we assume that J is the same for all individuals),
and suppose the measurements of the surrogate covariate are
taken at timepoints  1 ,..., J .
Introduction

Tsiatis and Davidian (2004) have suggested a so-called
joint model, where models for event time distribution and
longitudinal data depend on a common set of latent
random variables.

Andersen and Liestol (2003) proposed a simple approach
based on the regression calibration idea. The bias due to
observing the longitudinal covariate process over a
discrete grid is handled by introducing additional variables
to the standard Cox model, while measurement error is
treated using an external procedure.

Most of the proposed approaches are limited to some
special cases or/and specific distribution assumptions.
Proposed Method


We propose an approach that is of intermediate complexity
relative to the simple approach of Andersen and Liestol and
the joint modeling approach.
We assume additive model for measurement error. The error
term is independent across i and j, and independent of all other
random variables in the model. We do not assume a specific
parametric distribution of εij .

We assume a working parametric model F for the conditional
distribution Wi ( t2 ) | { Wi ( t1 )  w,Yi ( t2 )  1 } , t2 > t1.

Note that we do not assume that the data is actually
distributed according to F, but rather than F is a close enough
approximation to yield reasonable estimators of Cox
coefficients.
Proposed Method

Assume that Moment Generating Function (MGF) of F
exists and is well-defined.

To cover a variety of cases, flexible distributions may be
used, such as the Semi-Nonparametric distribution (SNP)
of Gallant and Nychka, 1987.
Proposed Method
To start with, assume that Wij  X i (no measurement
error). Define: N( t )  I ( Ti  t , i  1 ), where Ti - event time,
δi is the event indicator.

 (t )  E[dN(t ) | {W (u), u  t}, Y (t )  1]
It follows, that hazard function may be represented as:

~
 ( t )  E [ dN( t ) | { W ( u ),u  t },Y ( t )  1 ] 
E [ E [ dN( t ) | { W ( u ),u  t },Y ( t )  1 ] | W ( u ),u  t },Y ( t )  1 ]] 
E [ e W ( t ) | W ( u ),u  t },Y ( t )  1 ]
where Y(t) is an indicator to be at risk at time t.
Proposed Method

The above expression may be approximated by the
moment generating function (MGF) of F. It follows that
approximated hazard is given by:
* ( t )  0 ( t )EeW ( t )|{ W (  ( t )),Y ( t )1 }
 0 MGFF (  , ( t ),Wi (  ( t )),t )
i

i
i
where  ( t ) is a parameters vector of F estimated at t,
 (t ) is a nearest timepoint before t at which Wi is availible.
We need to evaluate  at every timepoint t. However, we
can see Wi only at  1 ,..., J .
Proposed Method

The discrete grid problem is treated by introducing
working models for each central moment mk:
g k (mk (t1 , t2 , w))   k1   k 2t1   k 3t12   k 4 (t 2  t1 )   k 5 (t 2  t1 ) 2
  k 6 w   k 7 w2   k 8 (t2  t1 ) w   k 9 Slopehist (t1 )
  kT Z (t1 , t 2 , w)


In the above expression, gk is some function (chosen by
numerical reasons), and Slopehist is a slope of the
historical data (before t1).
θk are estimated using all available data at the observed
timepoints τq, τp (τq<τp, conditioning on being at risk at
τq). OLS technique is applied.
Proposed Method



Using estimated coefficients of the working model in the
previous slide:

Solve:

Given these estimated moments, the formulas for F moments
ˆ (W ( (t )), (t ),t ) may by calculated for
can be backsolved and 
i
every t.
mˆ k ( q , p , w))  g k1 (ˆkT Z ( q , p ,W ( q ))
Given the above, the MGF is defined at every point, and
the hazard * (t | W (s)) can be calculated for every t > s.
The estimator for  is obtained using the Cox partial
likelihood, incorporating the proposed hazard function.
Proposed Method

Note that:
EeWi (t )  Ee [ Xi (t )i (t )]  EeX i (t ) Ee i (t )  MGFX i (t )MGF i (t )

Cox partial likelihood is given in this case by:
MGFWi (Ti ) (  i )
n
LCox  
i 1
n
 Y (T )MGF
j 1
i 1
 Y (T )MGF
j
i
X j (Ti )
n
 Y (T )MGF
j 1

( )

(  )MGF (Ti ) (  )
MGFX i (Ti ) (  )
n
i 1
W j (Ti )
n
j 1

i
MGFX i (Ti ) (  ) MGF (Ti ) (  )
n

j

j
i
X j (Ti )
( )
Variance of ˆ is estimated using Weighted Bootstrap
approach.
Simulation Study


Data simulation is based on the setting of Andersen and Liestol
(2003), patterned after a clinical trial studying the effect of
prednisone treatment versus placebo on survival with liver
cirrhosis (Christensen, 1986).
The true data is simulated from the model:
X i (t )  t  Ai  U i (t )
where t denotes a common trend of the form ~t , Ai
represents initial variation between individuals, and U i (t ) is a
stochastic process representing changes in the covariate over
time.

We assume the Ai ~ N (0,  A2 ) and the measurement error  ij (tik )
2
is normal with zero mean and variance  ME
.
Simulation Study





Following Andersen and Liestol (2003), we take Ui(t) to be
either a Brownian motion (BM) process or an Ornstein–
Uhlenbeck (OU) process with correlation parameter
0.282.
The trend ~ term was taken to be a linear function with
initial level of 72 and a decrease of 5 units per year.
In the paper, Cox regression parameter is: β=-0.04.
Sample size is 300, and 12 observations are available, every
half a year (total trial length is 6 years).
Failure times were simulated from a Weibull hazard. Each
result was obtained based on 500 simulation runs.
Results
Estimation of β, W(t) is measured without error
Parameters
Process
BM
OU

0
-0.04
-0.06
-0.08
-0.04
-0.06
-0.08
-0.04
-0.06
-0.08
-0.04
-0.06
-0.08
-0.04
-0.06
-0.08
-0.04
-0.06
-0.08
2.3
2.3
2.3
1
1
1
0.2
0.2
0.2
2.3
2.3
2.3
1
1
1
0.2
0.2
0.2
Proposed Method
with SNP
approximation
std (ˆ )
ˆ
-0.0425 0.0080
-0.0627 0.0093
-0.0832 0.0116
-0.0419 0.0099
-0.0631 0.0126
-0.0840 0.0157
-0.0409 0.0144
-0.0621 0.0177
-0.0831 0.0211
-0.0398 0.0088
-0.0530 0.0105
-0.0628 0.0119
-0.0411 0.0104
-0.0569 0.0122
-0.0676 0.0134
-0.0423 0.0190
-0.0626 0.0208
-0.0776 0.0213
Andersen and
Liestol’s method
ˆ
-0.0078
-0.0263
-0.0422
-0.0098
-0.0291
-0.0489
-0.0280
-0.0448
-0.0620
-0.0043
-0.0220
-0.0365
-0.0160
-0.0281
-0.0442
-0.0286
-0.0430
-0.0610
std (ˆ )
0.0044
0.0067
0.0068
0.0053
0.0088
0.0092
0.0127
0.0186
0.0210
0.0051
0.0081
0.0082
0.0063
0.0107
0.0109
0.0157
0.0211
0.0216
LVCF (naive
method)
ˆ
-0.0066
-0.0235
-0.0351
-0.0136
-0.0311
-0.0449
-0.0257
-0.0422
-0.0611
-0.0029
-0.0202
-0.0306
-0.0112
-0.0292
-0.0393
-0.0248
-0.0447
-0.0564
std (ˆ )
0.0035
0.0044
0.0047
0.0049
0.0065
0.0063
0.0101
0.0120
0.0136
0.0045
0.0061
0.0058
0.0056
0.0071
0.0075
0.0118
0.0147
0.0147
Proportion
of events
0.58
0.57
0.56
0.28
0.27
0.27
0.09
0.09
0.08
0.58
0.56
0.56
0.34
0.29
0.28
0.09
0.09
0.09
Results
Estimation of β, W(t) is measured with error, 0  2.3
Parameters
Process
BM
OU

 2
-0.04
-0.04
-0.06
-0.06
-0.08
-0.08
-0.04
-0.04
-0.06
-0.06
-0.08
-0.08
50
100
50
100
50
100
50
100
50
100
50
100
Proposed Method
with SNP
approximation
std (ˆ )
ˆ
-0.0401
0.0083
-0.0386
0.0088
-0.0579
0.0101
-0.0551
0.0109
-0.0743
0.0118
-0.0694
0.0125
-0.0371
0.0089
-0.0352
0.0093
-0.0490
0.0107
-0.0464
0.0114
-0.0575
0.0114
-0.0544
0.0127
Proportion
of events
0.58
0.58
0.57
0.57
0.56
0.56
0.58
0.58
0.56
0.56
0.55
0.55
Results
Variance estimation by Weighted Bootstrap
Proposed method with SNP approximation,   0.04, 0  2.3
Process
BM
OU
 2
0
213
0
213
std emp (ˆ ) std Boot (ˆ ) Pcover
ˆ
-0.0425
0.0080
0.0076 0.9480
-0.0386
0.0128
0.0100 0.8840
-0.0398
0.0088
0.0087 0.9350
-0.0349
0.0108
0.0114 0.8720
Conclusions

We propose a new semiparametric estimation approach
for the Cox regression model when covariates are
measured intermittently and with error. The intermittent
measurement issue is handled by modeling the
parameters of the distribution of the covariate among
individuals still at risk as a function of time. The relative
risk is then computed using the MGF.

The accuracy of the proposed estimators depends
critically on the form of the OLS working model for inbetween times. Increasing the accuracy of estimates by
assuming more flexible interpolation model is a topic for
future research.
Conclusions

In a simulation study we found that in most cases the
proposed method provides reasonable estimates for the
Cox regression parameter and its standard deviations.

Because the SNP model covers a range of distributional
shapes, the method can be applied in a range of settings.

The computational burden is moderate – less than one
minute for one run for the SNP - based procedure.