TIlE EM ALGORITHM FOR CENSORED DATA
H. Schneider*
and L. Weissfeld**
*Free University of Berlin, Germany
**Department of Biostatistics, University of North Carolina at Chapel Hill
ABSTRACT
Three methods for applying the FM algorithm to censored data are
considered, the Buckley-James (1979), a proposed simpler nonparametric
method and a normal model for censored data.
A new estimator for the
variance of y in the Buckley-.James model is proposed and simulations
comparing the three methods are descrihcd.
To illustrate the use of
these methods they ;:lre appUed to the Stanford heart transplant data.
Key Words:
Censored data; f)VI algorithm; Linear regrcssJOn; Kaplan-
Meier estimator.
I.
Introduction
There have been many methods proposed for handling regression problems
in which the dependent variable may be censored.
While several of these
methods aSSlUlle an underlying distributional form, that is, normal, Weibull
or exponential,
others are of a nonparametric nature and requ.ire minimal
assumptions about the unspecified distribution.
One of these techniques, developed by Cox (1972), aSSlUlles that the hazard
function A(y,X) = f(y,x)/(l-F(y,x)) has the form
A(y,X) = Ao (y)exS
where AOCY) is the underlying hazard.
He then used conditional arguments
to form a partial likelihood function (Cox, 1972, 1975), independent of
~
AO(y), for the estimation of S.
We will focus on three methods, which are based on the linear model
T
y=x6+E:,
Two of the methods considered here are nonparametric in that the distribution
of Y is unspecified while the third assumes that y is normally distributed.
These methods rely on the expectation maximization (EM) algorithm, introduced
by
D~npster,
Laird and Rubin (1977), which is broadly applicable for computing
estimates from incomplete data and is used for the estimation of B and the
variance of y by all three methods considered here.
Although their presentation
was based on a parametric model, the EM algorithm has been applied to
nonparametric models (Buckley and James, 1979).
In section 2 we discuss each of
these methods and propose a new estimator of the variance of y for the BuckleyJames method.
In section 3 we present results from a simulation study
-2-
which compares the performance of the three methods.
Finally, in section 4,
we present an example using data from the Stanford Heart transplant study.
2.
Estimation Techniques for Linear Regression with Censored Data
We consider the linear model
Y.
J
where the
E
j
T
= XS
J
(1)
j=l, ... , n
+ E.
J
are independent and identically distributed with the distribution
function F which has mean zero and finite variance.
The covariate vector x.T
J
is k dimensional with x . =: 1.
oJ
We assume here that the observed times are generated by random censorship.
Let the life times Yl"'"
Yri be i.i.d. with distribution function F as specified
above and let the censoring times C ' ... , C be i.i.d. with distribution
n
1
function G and further, under the assumption of random censorship, let C1 '···, en
be independent of Y1 ,···, Yn '
We observe
Z. = min ry , c. )
J
J J
(2)
and
{
1
0. =
.J
, 0
C.
J - J
if
y. <
if
y. >
J
(3)
C.
J
with
(4) .
For Type I censorship, a special case of r;mdom censorship, the C.
J
constants.
'5
are given
~
-3-
Let X be the design matrix and define the vector y*(S) by
y-J;(S) = o.y. + (1-0.) E[Y.
J
J J
J
J
I Y.J > C.,J
x~SJ.
J
(5)
Then the EM est:unate of B is the solution to
(6)
This solution requires an iterative procedure, since E(Y -, Y > C ' X}G)
j
j
j
is unknown and has to be estimated.
The variance
2
0
can be estimated by
7
m 0-
.
=
A
A
xS)
(y*(S) -
T
A
A
XB)
~ (y*(S) -
where m is an appropriate constant and
~
(7)
is a diagonal matrix.
immediately evident what to substitute in (7) for
" 2
y~(S)
J
It is not
when y. is censored.
J
Aitkin (1981), who considers the case where F is normal, uses the maximum
likelihood approach to derive the form
0.y.2 + (1-0.) Ery.2
J
J J
J
I y.J
with m = n and 6 = I, the identity matrix.
>
C· xTSJ
J' J
(8)
He suggests using the bias
corrected estimate
'2 = 0 2 nu
a
n -k
o
(9)
----'.~
u
Schrnee and Hahn (1979), who also consider the normal model suggest
y* (S) 2
j
wi th
III
= n- k and
<5. Y. 2 + (1- <5.) 13 ry.
J J
J
.J
=
~
= I.
I y.J
Ji T. SJ2
J' J
> C
(10)
But wlfortunate1y, thi s approach results in an
underestimation of cr 2 which becomes severe for moderate and heavy censoring.
-4-
Consequently, the estimate S suffers from poor estimation of
0
2.
Since the
results from the maximum likelihood estimator are better for moderate censoring we shall not report the results for the Schmee and Hahn estimator.
Buckley and James (1979) consider the nonparametric case where the underlying distribution F is unspecified.
1bey suggest that the censored observations be ignored for the estimation of 0 2 and use
(11)
but introduce the correction
" 2
°BJ
(12)
We propose a nonparametric estimator of
Aitken (1981) in equation (8).
su£gest that this estimator of
0
analogous to that proposed by
In fact, the simulation studies we performed
0
has a smaller bias and mean square error
(MSE) than the estimator proposed by Buckley and James.
In order to apply the EM algorithm to equation (6) for the estimation
of B we have to establish appropriate estimators for E[Y·
J
I Y"J >
T
C., Xi R1
J
•
.
In the parametric case it is easy to find explicit expressions for this
expectation and replace it by an appropriate estimate.
where F
In the normal model,
we obtain lAitkin (1981)),
= ¢,
E[Y.
J
I y.J
>
C.
x"i.'" Coil =
J,Jj"
x~ S
+ II W((C' . _AT J) /
J'J J
(j ')\
and
E[y.2jy. > C.,x~Sl=(x'~S)2 +0 2 + O(C.+x~S) W(cC.-x'fr3)/o~.
J
J
J
J
J
JJ
JJ
(14)
where W(u) = ¢(u) / (1- ~1 (u)) is the hazard rate of the normal distribution.
In order to obtain an estimate for the latter two expectations we replace
G and
0
by their current estimates.
-5-
Buckley and James, who consider the case where the underlying distribution
function F is unknown, estimate F using the product limit estimator (Kaplan and
Meier, 1958),
'"
A
.. <
where eel) < e(2) < •
eCn )
are the ordered values of the residuals
is the number at risk at
e.CO,B)
= z.-x~~,n(.)
~
~
1
~
number of failures at e(i)'
I Y.J
estimate E[Y.
J
> C.,
J
eCi )-
and d
Ci
) denotes the
The Kaplan-Meier estimator is then used to
x~SJ in the following manner. Uncensored observations
J
are replaced by
T
A
A
E[Y.
J
Y. >
J
c.,
J
T
x_.S]
=y. = x. S
J
J
J
+
U
L
K
T'
A
w. (SHy -). 3)
~K
K
K
(15)
where the sum is over the set of uncensored residuals and
W.
'l-K
(S)
= "\r
'"
'"
TA
A
v K (S)/(l - F(C.-x,3)}
when e.(o,S)
< e (o,S)
K
~
'l'l-
l0
(16)
otherwise
where v CS) is the mass of the Kaplan-Meier estimator at the uncensored points.
K
In this case the Kaplan-Meier estimator
F(e,~)
assigns the remaining mass to
the largest residual if it is censored.
We propose to estimate the variance
0
2 by applying the above reasoning to
the computation of
(17)
Replacing the
"'2
no
y~
J
(S) by their estimated expectations we have
n
=
I
j=l
A
E[Y;
2
(6)] +
(xJi3) 2
A
TA
2E[Y~(B)]x.~
J
J
(18)
-6-
and from (15) we see that for censored observations
(19)
and
substituting into equation (18) we have
a"2
1 n {
T" 2
I
6.(y.-x.8) + (1-6.)
n j =1 J J J
J
= -
IKu
w· "
(8)
JK
T' 2}
(YK-XK8) .
This estimator, unlike the Buckley-James estimator, uses the information in
both the censored and uncensored observations for the estimation of o.
To
reduce bias we applied the bias correction
(20)
It is easy to see that the Buckley-James procedure becomes quite complicated,
particularly when a large number of observations are tied.
Therefore, we also
consider a modified method which is very easy to implement.
We propose
estimating the expectation with
(21)
where l is the number of F(J:)
and uncensored residuals c.
> Cj'
= y~(B)
1.·~
This IS a simple average of all censored
'["
- x.G
1,
exceeding e. =
J
c·J -
T~
xJ~'
This
method, which is in spirit of the EM algorithm, replaces censored observations
by their expectations and treats them as uncensored observations.
-7-
To estimate the covariance matrix of B in the normal model we calculate
the inverse of the Fisher information matrix (Amemiya (1973)), which would be
an approximation for the random censorship case.
However, for the BJ estimator
there are no theoretical results and Buckley and James suggest using the
approximation
•
(22)
where {:,
diag (8..) and
'l.-J
XU has clements nu-1 \'.
8. x... Equation (22) can
L.
J 1.-J
J
also be used to approximate the covariance matrix of the simplified estimator.
Simulations
3.
To gain some insight into the difference between the Buckley-James (BJ) ,
simplified nonparametric (NP) and maximum likelihood
studies were carried out.
methods simulation
In each case 1000 samples of size 50 were drawn
with the covariates evenly spaced at 2i, i=l, ... , 50.
•
~~)
Independent life times
(y.) and censoring times (c.) were generated for each covariate and Z. =
1.-
1.-
min(y., c.) was modelled.
1.-
1.-
1.-
The Beta distribution
f(x) = 13 (~,q)
(x-a) p-1 (b-x) q-l
(b-a)p+q-1
was used to generate the censoring times because a wide variety of censoring
patterns could be represented by varying the parameters, p and q, of the
distribution.
Symmetrical distributions are generated when p=q and various
degrees and patterns of asymmetry can be generated by allowing p to be
unequal to q.
Each of the tables presents the average percent of censoring,
means of the parameters, MSE of the parameters and the percentage of runs
which did not converge.
This percentage was computed using the total
number of samples required to achieve 1000 convergent samples.
-8-
Table 1 presents the results for a nonnally distributed lifetime with
equal censoring, that is, the percentage of censoring is the same at each design point.
This case is of interest because the BJ estimator of
0
is thought
to be well-behaved under this assumption; however, we found that in general
the BJ estimator had the largest bias and MSE of all estimators of a considered.
The best estimator of
by equation (20).
0
was the new estimator which is defined
This estimator uses the infonnation from the BJ results
and makes full use of the censored data so that in general it has a smaller
bias and MSb than the other estimators and, in particular, this estimator
always behaves better than the BJ estimator. Both the BJ and NP
underestimated
0
estima~ors
but the NP estimator was less biased than the BJ and was
often less biased than the ML estimator of
0.
The results for all three
methods of estimating 60 and 61 were very similar with all methods being
biased for the estimation of 6. While the ML estimator performed better for
o
the estimation of 60 under heavy cellsoring (p=l, q=2), the BJ estimator was
superior for the estimation of 61 in this case. The percentage of nonconvergent samples was similar in most of the cases with the ML method
having the most problems in the case of heavy censoring.
Table 2 presents the results for a nonnally distributed lifetime with
increasing
c~nsoring.
tends to overestimate
underestimate
0.
estimators of
0
In general the simulations show that the ML estimate
0
while the BJ, the NP and the new estimators tend to
The NP estimator also tends to underestimate 61 ; however,
it has the smallest MSE in all the cases considered. Both the NP and ML
tend to be better than the BJ estimator; however, the new
estimator is generally superior to all the others in terms of bias and MSE.
The results for the estimation of 6
alld 61 tend to be very similar with
the ML estimator doing very poorly in the case of heavy censoring (p=l, q=2).
0
-9-
Once again all the methods are biased for the estimation of So with the
BJ performing the worst.
As before, although the NP estimator of Sl generally
has a smaller MSE it is always biased downward.
.
Table 3 contains simulations where a beta distribution was used to
generate both the lifetime and censoring distributions.
The simulations
were performed to explore the robustness of the ML method to departures
from normality and to compare the behavior of the parametric and nonparametric
methods in this setting. In general, the new estimator of a had the smallest
MSE and bias.
In this instance the BJ estimator of a behaved very poorly.
For the symmetrical (p=q) lifetime distributions the ML estimator of Sl tended
to perform somewhat better than either of the other two, however, it was not
consistently superior to the other estimators.
e
The BJ estimator of Bo tended
to be the least biased of the methods and often had the smallest MSE. For
the case of the asymmetric lifetime distributions the BJ estimator of 61 was
the least biased of the three estimators but had the largest MSE, with the
same results holding for 6.
Thus, the violation of the distributional
o
assumption, although it does have some impact, did not seem to seriously
affect the bias and MSE of the ML estimator.
TABLE 1
Equal Censoring for a Normally Distributed Lifettme
So=O , Sl=0.2
censoring
distribution
beta
p
q
%
censoring
ML
NP
1
1
50
2
2
50
3
3
50
2
0.5
14
BJ
ML
NP
BJ
ML
NP
BJ
ML
NP
BJ
t.~
NP
1
0.5
30
2
1
28
3
1
17
1
2
71
9
0.5
1
BJ
ML
NP
BJ
ML
NP
BJ
tvrr..
NP
BJ
ML
NP
BJ
ML
NP
9
1
2
9
2
6
BJ
ML
NP
e
BJ
MSE
%
So
0.117
-0.639
-0.025
0.013
-0.939
-0.156
-0.190
-1. 241
-0.379
-0.032
-0.113
-0.054
0.322
-0.001
0.283
0.210
-0.119
0.152
0.186
0.023
0.153
-0.529
-1. 841
-0.195
0.066
0.037
0.055
0.231
0.191
0.203
-0.178
-0.264
-0.238
* in bJ'ac]~cts nc\, cst im;ltor for
(J
MSE
Sl
13.95
14.19
13.98
13.24
13.42
13.55
11.11
12.44
12.29
0.199
0.200
0.199
0.199
0.199
0.198
0.203
0.202
0.202
0.201
0.202
0.201
0.195
0.196
0.195
0.197
0.198
0.197
0.197
0.198
0.196
0.206
0.197
0.196
8.60
8.67
8.63
11.73
11.67
11.75
10.83
10.40
10.79
9.81
9.70
9.80
19.65
23.03
22.14
8.07
8.00
IL03
9.39
9.26
9.31
8.49
8.28
R.45
0.200
0.201
0.201
0.196
0.197
0.196
0.203
0.203
0.203
e
(x 100)
0.400
0.397
0.398
0.364
0.350
0.372
0.324
0.326
0.349
0.247
0.253
0.249
0.344
0.346
0.345
0.319
0.312
0.319
0.291
0.287
0.290
0.586
0.571
0.615
0.229
0.228
0.228
0.274
0.271
0.273
0.238
0.236
0.236
a
10.48
9.79
9.38(9.76)*
10.50
9.61
8.91(9.60)
10.50
9.53
8.61 (9. 53)
10.21
9.63
9~57(9.90)
10.28
9.74
9.60(9.88)
10.28
9.58
9.32(9.84)
10.22
9.46
9.32(9.85)
10.96
10.10
·9.13(9.46)
10.16
9.76
9.76(9.92)
10.17
9.63
9.61(9.90)
10.21
9.37
9.31(9.~7)
MSE
2.92
2.18
2.27(2.13)*
2.86
2.17
2.87(2.08)
2.84
2.12
3.47(2.16)
1. 38
1. 27
1.31(1.14)
1. 79
1.45
1. 51 (1. 41)
1.85
1.49
1. 74(1.47)
1. 52
1.40
1.57(1.30)
6.32
4.59
4.20(3.63)
1. 21
1.03
1. 03(1. 05)
1. 24
1.09
1.10(1.04)
1. 25
1.32
1.39(1. (13)
- - -_._P
nonconvergent
samples
1
5
5
1
6
5
1
7
6
0
3
0.2
0
4
1
0
4
2
0
4
1
26
10
12
,0
0
0
0
1
0
0
2
0.1
e--
,.
t
e
•
LE 2
'"
e
Increasing Censoring for a Normally Distributed Lifetime
80 =0 , 8(:0. 2
censoring
distribution
beta
_
%
censoring
P----.il
ML
NP
1
1
66
2
2
69
BJ
lvlL
NP
BJ
~lL
NP
3
3
72
BJ
ML
NP
2
0.5
30
1
0.5
46
2
1
48
3
1
37
1
2
81
9
0.5
9
9
1
14
BJ
ML
NP
BJ
ML
NP
BJ
ML
NP
BJ
ML
NP
BJ
ML
NP
BJ
ML
NP
MSE
%
BJ
80
MSE
81
(x100)
-0.025
-0.960
-0.220
-0.200
-1.457
-0.648
-0.391
-2.090
-1. 079
0.003
-0.155
-0.109
0.292
-0.114
0.173
0.124
-0.365
-0.059
0.177
-0.080
0.038
16.6
17.2
17.4
15.7
16.9
16.9
0.202
0.190
0.200
0.202
0.182
0.200
0.592
0.582
0.615
0.573
0.544
0.622
14.1
16.4
15.7
9.1
9.1
9.2
0.205
0.184
0.205
0.201
0.198
0.201
0.590
0.522
0.658
0.286
0.284
0.292
12.9
12.5
12.7
11.8
11.3
11.8
10.5
10.1
10.4
>100
31. 5
29.6
8.1
8.1
8.0
9.5
9.3
9.4
0.195
0.191
0.194
0.429
0.413
0.421
0.200
0.192
0.199
0.198
0.192
0.197
0.225
0.179
0.200
0.201
0.199
0.200
0.196
0.195
0.196
0.392
0.372
0.398
0.346
0.330
0.346
20.40
0.82
1.061
0.234
0.234
0.237
0.283
0.280
0.287
10.38
9.34
8.84(9.70)
14.00
10.14
8.84 (9 .19)
10.20
9.18
9.08(9.81)
10.24
9.11
8.90(9.79)
8.9
8.8
8.9
0.203
0.198
0.203
0.275
0.267
0.282
10.31
9.07
8.63(9.70)
0.202
-3.071
-1. 498
0.047
0.047
0.013
0.228
0.156
0.156
-0.173
286
- .310
* in brackets new estimator for l)
ML
NP
BJ
9
2
25
-a·
a
10.81
9.65
9.00(9.52)*
11.01
9.75
8.63(9.22)
11.19
9.79
8.35(9.06)
10.32
9.33
9.02(9.77)
10.44
9.46
9.12(9.75)
10.50
9.49
8.88(9.67)
MSE
5.25
3.42
3.89(3.10)*
6.57
3.82
4.84(3.90)
7.49
3.96
5.83(3.86)
1.94
1.77
2.22(1.50)
2.64
2.03
2.41(1.91)
2.91
2.15
3.04(2.1D)
2.21
1.91
2.67(1.67)
>100
9.58
8.56(6.32)
1. 38
1.59
1. 75(1.15)
1. 52
1. 75
2.16(1. 23)
1.72
2.00
2.95(1.41)
nonconvergent
samples
2
8
9
4
g
10
6
12
12
0
4
2
0
5
4
0
4
6
0
5
3
40
40
35
0
3
1
0
4
2
0
3
2
TABLE 3
Increasing Censoring for a Beta Distributed Lifetime
81=0.2
lifetime
distribution
beta
p
q
p
q
-"--
ML
%
censoring
distribution
beta
%
censoring
0.5
0.5
1
1
64
/oiL
NP
BJ
0.5
0.5
9
0.5
31
ML
NP
BJ
2
2
9
0.5
15
2
2
3
1
39
NP
BJ
~[.
NP
BJ
ML
NP
1
1
u.S
0.5
1
3
1
42
3
1
55
1
2
1
50
0.5
3
1
1
31
2
3
1
1
57
0.5
3
3
1
7
2
3
3
1
27
3
2
9
1).5
7.-1
BJ
~[.
NP
-
BJ
ML
NP
BJ
ML
NP
BJ
1,[.
NP
BJ
ML
NP
BJ
ML
NP
BJ
ML
!l!'
BJ
e
* in hrackets new estimator for
4
0
bias
80
-1.08
-4.88
-1.94
1.83
-0.59
-0.42
0.25
0.06
0.08
0.23
-0.51
-0.24
0.74
-0.93
-0.39
1.52
-1.68
-0.77
0.44
-1.64
-0.62
-0.27
-0.30
-0.05
-0.13
-1. 25
-0.28
0.00
0.02
0.04
0.10
-0.17
0.00
0.12
-f).f)7
-0.10
MSE
81
MSE
(x100)
55.68
63.43
52.91
54.91
43.41
43.24
16.61
16.44
16.40
19.5
17.6
18.8
35.7
29.5
31.1
58.11
43.68
45.56
37.20
31.64
32.54
7.22
7.16
7.57
16.88
15.98
16.20
6.92
7.07
7.12
12.64
11.97
12.62
13.26
12.44
12.70
0.199
0.187
0.204
0.198
0.194
0.204
0.202
0.197
0.200
0.200
0.193
0.202
0.191
0.186
0.198
0.183
0.184
0.201
0.193
0.186
0.199
0.201
0.199
0.200
0.200
0.197
0.202
0.198
0.199
0.200
0.198
0.197
0.200
0.206
0.195
0.20:
1.596
1.253
1.927
1.490
1.616
1. 767
0.502
0.506
0.524
0.595
0.538
0.622
1.043
0.931
1.105
1.524
1.404
1. 761
1.102
0.949
1.138
0.200
0.206
0.213
0.539
0.471
0.525
0.196
0.203
0.207
0.392
0.374
0.402
0.433
0.427
0.455
e
Bias
1.09
-3.00
-8.59(-1.73)·
4.04
-2.15
-4.99(-.11)
0.80
-1.24
-1.62(-.14)
0.87
-1.06
-2.26(-.29)
1.69
-1.62
-4.23(-.59)
2.92
-2.17
-7.22(- ..85)
1.50
-1.60
-4.51(-.79)
-1.11
-2.11
-2.38(-.51)
0.15
1.18
-2.02( - .45)
-0.39
-1.39
-1.49(-.24)
0.19
-1. 25
-1. 77( -.25)
1.23
-0.70
-1.7,0(-. OS)
a
MSE
16.17
16.49
79.84(10.06)·
23.18
8.13
27.23(3.84)
2.78
2.66
3.65(1.49)
4.06
3.01
6.70(2.15)
9.01
5.97
20.2(3.63)
18.34
9.59
55.71(5.47)
9.26
6.25
23.18(4.43)
3.89
6.67
7.98(4.21)
3.77
3.79
6.37(3.60)
2.28
3.54
3.86(2.49)
2.08
2.85
4.31(1.87)
3.87
nOllconvergent
sample
3
11
12
0
3
1
0
2
1
0
3
-+
0
6
6
0
7
6
0
9
7
0
1
0.3
0
4
6
0
1
0.4
0
2
1
0
1.86
I
2.96(1. 40)
1
e
-13-
4.
Stanford Heart Transplant Example
In order to further explore the relationship between the BJ,
estimators and to see how our proposed estimator of
0-
M[~
and NP
for the BJ method
perfOl1ned in a set of data, we analyzed the Stanford heart transplant data
•
given in Miller and Halperin(1982).
as a function of age for survival
We modelled survival time (log 10)
tim~s
greater than 10 days.
These results,
given in table 4, follow the pattern of the previous sinmlations with the
new estimator (20) giving a larger estimate of
0-
than the BJ estimator.
HL estimate of the slope B is almost identical with the BJ estimate.
The
NP estimator results in a somewhat smaller estimate of Bo and B1 than the
ML and BJ estimators.
..
tt
MLE
BJ
NP
Bo
B1
0
3.826
3.745
3.645
-.02453
-.02434
-.0229
.890
.655
.751
Table 4:
(.761)*
Estimators for the heart transplant data.
*new estimator (20) of (J
The
-145.
Conclusions
Typically, all three methods for linear regression with censored data
were very similar with no particular estimator of slope appearing superior.
However, we did find that the new estimator of
0
enhanced the BJ procedure
and would cause any tests of significance for this model to be less anticonservative so that we would recommend the use of this new estimator.
\Ve
also found that the normal model tended to be fairly robust against departures
from normality and in general performed quite well.
Helmut Schneider's work was supported by the Deutsche Forschungsgelneinschaft
and the Air Force Office of Scientific Research.
Lisa Weissfeld's work was partially supported by National Heart, Lung and
Blood Institute contract NOI-HV-12243-L.
•
REFERENCES
AITKEN, M. (1981). A note on the regression analysis of censored data.
Technometrics, 23, 161-163.
•
AMENIYA, T. (1973). Regrcssion a"~llysis when thc dC1)C,ltlcjlt v;Jri:lhlc
truncated normal. Econometrica, 41, 997-1016 .
BUCKLEY, J. and JAMES, J. (1979).
Biometrika, 66, 429-436.
COX, D.R. (1975).
J S
Linear regression with censored data.
Partial likelihood.
Biometrika, 62, 269-76.
COX, D.R. (1972). Regression models and life tables (with discussion).
~. Roy. Statist. Soc., B34, 187-202.
DEMPSTER, A.P., LAIRD, N.M. and RUBIN, D.B. (1977). Max:iJTIum likelihood from
incomplete data via the EM algorithm (with discussion). ~. Roy. Statist.
_Soc., B39, 1-22.
MILLER, R. and HALPERIN, J. (1982).
Biometrika, 69, 521-531.
Regression with censored data.
and HAJW, G.J. (1979). A simple method for regression analysis
with censored data. Technometrics, 21, 417-432 .
S~ffiE,J.
...