RESTRICTED ALTERNATIVES TESTS IN COMPETING RISK ANALYSES
by
Cicilia Yuko Wada
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1847T
March 1988
RESTRICTED ALTERNATIVES TESTS IN COMPETING RISK ANALYSES
by
Cicilia Yuko Wada
A dissertation submitted to the faculty of the University of
North Carolina at Chapel Hill in partial fulfillment of the
requirements for the degree of Doctor of Philosophy in the
Department of Biostatistics.
Chapel Hill, 1988
Reader
ABSTRACT
CICILIA YUKO WADA.
Restricted Alternatives Tests in Competing Risk Analyses.
(Under the direction of Dr. P.K. SEN)
The objective of the present study is to derive a union-intersection test statistic for
subhypotheses against restricted alternatives in a competing risk situation with k causes
of failure and p concomitant variables. Asymptotic properties of the proposed tests are
investigated and compared to those of the usual score test.
For
the
Cox
regression
model involving p
concomitant
variables
and
two
treatments, testing of the subhypotheses against orthant restricted alternatives based on
partial likelihood is considered.
Roy's union-intersection principle is applied and the
Kuhn-Tucker-Lagrange minimization technique is utilized for the derivation of the
proposed test.
The distribution of the proposed test under the null and under "local"
alternatives is studied.
The asymptotic relative efficiency of the proposed test with
respect to the classical score test is investigated.
The above derivation is extended to the comparison of r+l subsamples by
considering tests of subhypotheses against restricted alternatives, such as the orthant
alternative,
the ordered
orthant
alternative and finally
the ordered
alternative.
Asymptotic properties of these tests are also studied.
The derivation of the tests using the same approach
is also considered when the
logistic model and a dependent parametric model are utilized to assess hypotheses of no
difference between the two treatments
treatment is superior to the a.nother.
against the restricted alterna.tive that one
A numerical illustration of the Cox regression model in a competing risk situation
with two causes of failure, two treatments and with two covariates is performed. MonteCarlo sim ulation
is utilized to show the power superiority of the proposed test
compared to the score test.
iii
ACKNOWLEDGEMENTS
I wish to express my deepest gratitude to Dr. Pranab Kumar Sen, advisor of this
dissertation.
I am grateful for his firm quidance, for his patience, and for his valuable
support and encouragement in difficult times.
I express my deep appreciation to Dr. Timothy Morgan, for his help, comments
and suggestions.
gratitude.
I
His interest and solicitude will be always be remenbered
with
also thank Dr. Dana Quade, for his kindness and patience in helping to
edit this dissertation.
I extend my thanks to other members of my commitee, Drs.
Shrikant Bangdiwala, C. Edward Davis and Sherman James.
I am indebted to the University of Sao Paulo State (UNESP) and
Nacional de Desenvolvimente Cientifico e Tecnologico
In
to Conselho
Brazil (CNPq) for financial
support during the first years of my doctoral program.
I thank my family: my parents, Hisae and Wasaji Tsuji, and especially my sister
Mitie Tsuji, for their love, care and support.
I appreciate the friendship and care given
to me by my friends from Brazil and those from here during the years I spent here.
Finally, I wish to thank Ms. Ans Janssens for helping me to format this
dissertation.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS
1.
INTRODUCTION AND LITERATURE REVIEW
1.1.
Introduction
.
1
1.2.
Basic Formulation and Notation in Competing Risk
Problems
.
3
Review of Competing Risk Models
.
7
Parametric Models..........................................................
8
1.3.1.1. The Likelihood Function....................................
1.3.1.2. The Maximum Likelihood Estimators................
13
Semi parametric Approach......................
15
1.3
1.3.1.
1.3.2.
1.4.
1.5.
2.
11
Review of Restricted Alternatives and Union-Intersection
Tests
.
18
Proposed Research
.
23
RESTRICTED ALTERNATIVES TESTS IN A PROPORTIONAL HAZARD
MODEL WITH COMPETING RISK DATA
2.1.
Introduction
.
24
2.2.
The Union-Intersection Test Based on Partial
Maximum Likelihood Estimators
.
26
..
.
27
Asymptotic Distribution of the Union-Intersection
Test Statistic Ii
..
51
Local Asymptotic Power and Efficiency of Ii
..
54
2.2.1.
2.2.2.
2.3
2.4.
Partial Maximum Likelihood Estimators
The Union-Intersection Test
41
vi
3.
RESTRICTED ALTERNATIVES TEST IN A PARAMETRIC MODEL
AND IN A LOGISTIC MODEL WITH COMPETING RISK
DATA
3.1.
Introduction
3.2.
Union-Intersection Test for a Parametric Model
in Competing Risk Situation
.
61
3.2.1.
3.2.2.
..
.
64
75
..
84
.
60
•
3.3.
Union-Intersection Test for a Logistic Model
in Competing Risk Situation
3.3.1.
3.3.2.
4.
Notation and Assumptions
The Union-Intersection Test
.
.
86
92
ORDERED (ORTHANT) ALTERNATIVE TESTS IN A PROPORTIONAL
HAZARD MODEL WITH COMPETING RISK DATA
4.1.
Introduction
4.2.
4.3.
4.4.
..
98
The Problem and Notation
.
98
The Union-Intersection Test for Orthant (Ordered)
Alternative I~
.
102
The Union-Intersection Test for Ordered Alternative I;
..
104
.
104
4.4.1.
4.4.2.
4.4.3.
5.
The Problem and Assumptions
The Union-Intersection Test.
I;
Union-Intersection Test Statistic
Derivation
Asymptotic Distribution of the Statistic
Local Asymptotic Power and Efficiency
I;
..
..
113
118
NUMERICAL ILLUSTRATION AND FURTHER RESEARCH
5.1.
Introduction
.
123
5.2.
Numerical Illustration
.
123
5.3.
Results
.
125
5.4.
Suggestions for Further Research
.
137
REFERENCES
141
vii
e·
CHAPTER I
INTRODUCTION AND LITERATURE REVIEW
U
Introduction
A considerable amount of statistical research has focused on problems concerning
the failure or deaths of individuals from anyone of the several potential causes.
Such
problems occur in many areas, such as Epidemiology, Demography, Clinical Trials and
Industry and are called Competing risk problems.
Data for such problems usually
consist of the cause of failure or death and the time of the failure for the study unit.
Other information may also be available, as in the case of additional variables describing
different conditions in which the experiment was conducted (e.g. exposure levels, treatment, etc.).
These additional variables are called concomitant variables or covariates.
One example for this situation was given in Hoel (1972), in a study of mortality an
experiment in which two groups of mice, one kept in a germfree environment and
another in a conventional laboratory environment, receiving a certain dose of radiation.
The cause of death was determined by autopsy and could be: tymic lymphoma (C 1 ),
reticulum cell sarcoma (C z ) and all other causes (C 3 ). The available data for each mice
are the failure and a concomitant variable which is the indicator of group. The objective
in such situation may be to estimate the survival rates for each cause as well as for all
causes and to perform comparisons among groups.
A second example which was
presented by Holt (1978), illustrates the extension of the competing risk situation to
paired data (multivariate failure data): in a study of smoking discor-dance in pairs of
twins two causes of death were recorded, vascular disease (C 1 ) and other (C z ).
The
available data for each twin are the failure times, the causes of death and the vector of
concomitant variables indicating the smoking habit and whether or not the pair is
2
monozygotic. The objective was to assess whether or not the smoking effects on the two
risks of deaths differ for monozygotic and dizygotic twins. Lagakos (1977) studied data
from
a lung cancer clinical trial being conducted at that time by the Eastern
Cooperative Oncology Group.
The data contain failure times of 194 patients with
squamous cell carcinoma as well as the cause of failure C I , local spread of disease, and
cause
C2,
metastatic
spread
of
disease.
Three
covariates
were
considered:
Zl = performance status (ambulatory, not ambulatory), Z2=treatment (A and B) and
Z3 = age in years. The objective was to assess the differences between treatments A and
B.
In all the examples presented above, the comparison of the two groups could be
denoted by the hypothesis H o: 0=0 and the alternative hypothesis Hal: 0
represents the difference between the first and the second groups.
i= 0,
where 0
Now, the problem is
to test the null hypothesis of no difference between the groups against the alternative
that the first treatment is better than the second, that is, against the restricted
alternative H a2 : 0
> O.
If we consider the r+1 sample problem involving a control and r treatment groups,
then, 0=(0 1 , O2 ,
... ,
0,,) is the vector of the unknown parameters, where OJ represents
the j-th treatment effect over the control.
Against the null hypothesis of no treatment
effect, that is, H o: 0=0, we may be interested in the alternative none of the treatments
is inferior to the control, that is,
H~l): Oj~O,j =1,2, ... , r with at least one strict inequality,
A second hypothesis of interest may be the ordering of the treatment effects when H o
may not hold, that is,
H (2)
a : 0 I ~ ...
' h
I
"
~0
'" WIt
at east
one
strIct
lDequ al'Ity,
e·
3
A third hypothesis of interest may be the ordered orthant alternative, that is,
(3)
H" : 0
~ 0 1 ~'"
~Or,
with at least one strict inequality.
\\'e are interested in testing the hypothesis of no treatment effect against restricted
alternatives stated above and comparing the efficiency of such tests against the test with
unrestricted alternatives, in a competing risk situation with concomitant variables.
1.2. Basic Formulation and Notation in Competing Risk Problems
Initially we introduce a notation which is primarily due to Prentice et al (1978).
Suppose a subject from a population is susceptible to k causes of failure, denoted by
C1, C2 ,
...
,C£,. A sample of size n from this population is subjected to an experiment.
For each subject, survival experience can be represented by (T,I,z), where T is a random
variable corresponding to the observed failure time, I is a random index defined by
if failure is due to C i , i=l, " .,k
otherwise
and Z'=(Zl' Z2, ... , Zm) is the vector of concomitant variables.
The overall survival
function is represented by
P(T~t;z)
(2.1)
and the overall hazard function by
~T(t;z)
= lim
~t ...... O
P{t <T<t+~tl T>t; z} _ fT(t; z)
~t
- STet; z)
(2.2)
where fT(tj z) is the probability density function of T. Cause specific hazard functions
4
are defined by
.
= 11m
Llt-O
P{ t <T<t+Llt, I=il T>tj z}
Llt
(2.3)
for i= 1, 2, ... , k. The function >';(t; z) is the instantaneous failure rate from cause C i at
the time t, given the vector of concomitant variables z, in the presence of all other types
of failure.
•
The overall hazard function (2.1) can be expressed in terms of cause-specific
hazard functions as
k
L
(2.4)
Ai(t; z)
i=1
and the overall survival function (2.1) can be written as
J
k
t
STet; z)
=
exp {-
AT(u; z) du }
=.Il
exp {-
1=1
o
J
t
k
A;(U; z) du }
0
Il
Gi(tj z)
i=1
(2.5)
\\' here
t
G;(tj z) = exp {-
J
A;(U; z) du }
(2.6)
o
represents a distribution associated with cause C i alone (Elandt-Johnson, 1980).
Alternatively, competing risk problems have also been formulated using the concept
of latent failure times (Cox, 1959, Moeschberger and David, 1971 and Gail, 1975).
Following this concept, presumably at birth subject is endowed with a set of k latent
failure times, denoted by Xl' X 2 , ""XC each one corresponding to k acting causes C 1 ,
C2,
"',
C c ' In this formulation, the joint distribution of the latent survival times, called
multiple decrement function, is given by
(2.7)
e·
5
where O<t;< 00, i=1, 2, ... , k, S(O, 0, ...,0)=1 and S(oo, ...,00)=0. Since (Xl' X 2 ,
... ,
Xk) can not be observed jointly, the same holds for the multiple decrement function
S(t l , t 2 ,
... ,
t k ). The observable quantity is
(2.8)
the overall survival distribution, already given in (2.1), can be expressed as
STet; z) =
P{T~t}
= P
k
{n
(Xi> t)}= Set, t, ... , t)
i=1
(2.9)
and the cause-specific hazard functions given in (2.3) can be written as
A;(t;Z)
= lim
~t-+O
P{t
<T<t+~t,
I=il T>t; z}
~t
(2.10)
for i=l, 2, .. ", k.
Elandt-Johnson (1980) called this hazard function the "crude hazard
rate". The overall hazard function given in (2.2) can also be written as
_ [ 8log STet)] _ [ aST(t)l
AT(t) at
- at
J
(2.11)
The marginal distribution of the latent failure times Xi is given by
Si(t) = P (X;> t) = S(O, 0, ... , 0, t;, 0, ... , 0)
(2.12)
6
and its corresponding hazard function (net hazard rate) is defined by
(2.13)
for i= 1, 2, ... , k. Gail (1975) has interpreted the marginal survival distribution given in
(2.12) as the survival distribution for cause C i alone.
When the Xi are independent, the joint survival distribution of
Xl' X 2 ,
... ,
X k can be expressed by
(2.14)
and the overall survival distribution by
STet) =
k
n
Si(t).
i=l
(2.15)
Furthermore, when the Xi are independent,
(2.16)
and
(2.17)
Si(t).
Elandt-Johnson (1975) called (2.16) the "identity of forces of mortality".
Note that
when (2.16) holds, the Xi are not necessarily independent (Gail, 1975); however, the
events {Xi> t}, i=l, 2, ... , k are independent (Elandt-Johnson, 1980).
Suppose now that we have r+1 subpopulations subject to k competing causes of
failures.
The survival experience for a subject from the j-th subpopulation, j=l, 2, ... ,
r+1, can be represented by (T,I,z, 6), where TE(O,oo) is the observed survival time, 6 is
a censoring indicator which is one if the observation is not censored a.nd zero if censored,
I indicate the cause of failure, i = 1, 2, . 00' k, a.nd z is a (p xl) vector of cova.ria.tes. The
cause-specific hazard function can be defined as in
(2.3) or as in (2.10) adding the
7
subscript j, that is, can be represented as Aii(t), for i=l, 2, ... , k and j=l, 2, ... , r+1.
We may formulate the model as in the latent time formulation, postulating the existence
of hypothetical survival times Xi for each C i under the actual study conditions.
Then
{X;j(I)' l~i ~ k, l~j ~ r+1, 1=1, 2, ... , nJ denote the survival times from on~ of r+1
independent random samples subject to k
failure time is defined by T i(l) =min (X1i(I)'
competing causes of failures.
X 2i (I)' ... , Xki(I))'
The observed
As shown by Prentice et
al (1978), observations on (T,I,z, 6) permit only the estimation of those cause-specific
functions without further assumptions.
These in turn permit the estimation of related
quantities, such as, the crude survival probability, the probability that an individual will
survive up to age t and die from cause C;,
P{T
> t,
I
= il J = j)
7-
A;j(u) exp {- A(u; z)du}
(2.18)
t
where
(2.19)
A(t; z)
and the crude survival function Gii(t; z) defined in (2.6).
The functions Qii(t) and
Gij(t) are the basic distributions for cause C i (Elandt-Johnson, 1980). Certainly the
overall survival distribution ST(t) defined in (2.5) is also estimable.
1.3. Review of Competing Risk Models
In this section we present a review of some parametric, semiparametric and
non parametric models for a specific class of the competing risk problems. The principal
issues that competing risk model has been aimed at studying are (following Prentice et
aI., 1980):
1.
Inference on the effects of treatments, exposure or other variables on specific types
of failures;
2.
Study of the interrelation among failure types; and,
8
3.
Estimation of failure rates for certain types of failure given the "removal" of some
or all other failure types.
We intend
to present a review of the pertinent material focusing on the first class of
problems, in parametric and semiparametric approaches, in subsections (1.3.1) to
(1.3.2).
Before the review of competing risk models, we need to review some concepts
useful in the study of likelihood.
..
Some specific problems can arise when mortality with
human populations is dealt in an experimental type setting, where the patients could be
alive at the end of the study or could be lost to observation for some reason (e.g. change
of address, disinterest, etc.).
In these cases, mortality data are incomplete since the
time of failure of those individuals is unknown. In the terminology of reliability studies,
those subjects which are alive at the end of the study are said to have "censored" or
"truncated" lifetimes. This censoring scheme is called Type I censoring or truncation. In
animal studies or clinical trials, patients may enter the study at different times.
Censoring may occur in the following forms: loss to follow-up, dropout or termination of
the study.
The censoring times, that is, the lengths of time until one of the censoring
forms above, are random. This censoring scheme is called random censoring and Type I
is a particular case of the random censoring scheme. In this type of censoring it is usual
to assume that time of failure and censoring are independent (independent random
censorship model), which in many situations is not a realistic assumption. Another form
of incomplete data can arise when a preassigned number of failures determine the end of
the study.
The subjects who remain alive are said to have Type II censoring. This led
to increased concern with models that allow for the incorporation of censoring and
concomitant variables in the analysis of single risk survival data as well as of competing
risk data.
~
Parametric Models
Parametric models for competing risk problems assume the underlying distribution
e·
9
of failure times S(t 1 , t 2 ,
... ,
t k ) known.
One of the problems here in competing risk situation consists then in the estimation
and inference about the parameters of the underlying distribution from the distribution
of the identified minimum (T, I). Given the joint probability distribution of (T,I),
P ;(t) = P(T<t) = P (X;< t,X;< Xj,i=j) =
J'[00J...J00
f( tl''''' t k
o
t1
,;
)!! dt ] dt;
r
(3.1.1)
r-1
estimable from the data, the joint survival distribution S(t 1 , t 2 ,
""",
t k ) can not be
estimated nor its marginals, without the assumption of the independence of failure times
(Berman,1963, Tsiatis,1975, Elandt-Johnson, 1979 among others). With the knowledge
of the form of parametric model, it is possible to estimate all the parameters given the
distribution of (T,I) defined above (3.1.1). Some particular results concerning the study
of identifiability of the parameters of S(t 1 ,
.. ",
t A,) given the distribution of (T,I) are
known, for the case of dependent failure times.
Nadas (1971), Basu and Ghosh (1978,
1980) have proved that the distribution of (T,I) identifies the parameters of normal
distribution and bivariate exponential distribution of Marshall and Olkin (1967),
bivariate
distribution
of
Gumbel
(1960)
and
bivariate
Weibull
distribution.
Moeschberger and David (1971), Moeschberger (1974), David and Moeschberger (1978)
and more recently Moeschberger and Klein (1985) have presented some studies dealing
with parametric independent and dependent models in competing risk situations.
Another type of problem arises when concomitant variables are present and the
underlying survival distribution of failure times is assumed to be known, which is
equivalent to specifying the form of .Ao;(t), the hazard function of the underlying
survival when covariates are ignored. In general, the hazard function in competing risk
problems can be written as
.A;(tj z) = .Ao;(t) c(Zj fJ)
i=l, 2, ..., k
(3.1.2)
10
where
13
is the vector of parameters corresponding to Z'=(Zl' Z2' ... , Zp) vector of
covariates, c(z; /3) is any function of Z and /3 so that c(z; 0)= 1.
The covariables and
their corresponding parameters can be introduced in the likelihood and estimation and
inference can proceed as usual.
been considered extensively.
However, parametric models with covariates have not
Lagakos (1977) has extended
Glasser's exponential model
(1967) for competing risk problems, allowing for type I censoring.
The cause-specific
hazard function was defined by
>.;(t) = exp {O';
where
0';
and
/3;=(13;1>
/3;2' ... ,
13;p)
+ /3;
z}
1= 1,2, ..., k
(3.1.3)
are the parameters to be estimated; a full likelihood
was derived for estimating and testing the parameters.
Kalbfleisch and Prentice (1980)
have suggested an "accelerated failure times model" in which the covariate Z accelerates
or decelerates the time of failure with hazard function given by
= 1,2, ... , k
In particular, when
13;n13 j =O,
(3.1.4)
i:;tj, the above model can be written as
\ {
"'0;
-/3~z} -/3~z
tee
1,2, ..., k
(3.1.5)
In this case, due to the factorization of the likelihood function, survival analysis for a
single risk can be used to estimate the /3;, i=l, 2, ..., k.
Farewell and Prentice (1977)
consider the exponential, Weibull, and log-normal distributions as special cases of the
model.
e·
11
1.3.1.1 The Likelihood Function
Suppose that the underlying distribution S(t 1 , t 2 ,
')'2' ... , ')'p)
is assumed to be known.
... ,
tj;) with p parameters ')"=(-Yl'
The hazard function A;(t; ')')
in (2.3) and the
survival function for cause G;(t) in (2.6) are estimable from the data.
Likelihood
methods are employed for the estimation of the r parameters of the underlying
distribution.
David and Moeschberger (1971, 1978) have provided a formal justification of the
likelihood function, in the case of independent failure times and when the data are
complete, that is, without censoring.
When n individuals are observed and d; of them
fail due to C;, the likelihood function can be written as
(3.1.1.1)
where tier) denote the time to failure of the r-th subject who fails from C;. For the case
of incomplete data, they formulated the likelihood incorporating the data from Type I
censoring if his lifetime T,=min (Xli' X 21 ,
censoring time
TI'
... ,
X kl ), 1=1, 2, ... , n is greater than the
For the case of type II censoring, only a preassigned number of
failures are observed (m
< n).
The remaining (n-m) subjects are considered under type
II censoring. They have assumed that all subjects were under observation for the same
length of time.
Let d; denote the number of subjects who fail due to C; and m=E d;
the total number of failures; s denotes the total number of survivals such that n=m+s,
where n is the total number of observed subjects. Let tier) denote the time to failure of
the subject with the r-th lifetime among those who fail due to C;, i=1, 2, ... , k and r=1,
2, ... , n;.
The likelihood function allowing for type I and type II censoring can be
written as
(3.1.1.2)
12
where
tiel)
< t i (2) < ... < ti(d i )' i=1, 2, ... , k, ti(j) <
Ti(r)'
denotes the censoring times of the s survivors and
of the subjects whose failure time is tier)"
T
r=1, 2, ... , d i ,
T(u)'
u=1, 2, ... , s
i( r) denotes the fixed censoring times
Using (2.5) the likelihood function (3.1.1.1)
and (3.1.1.2) can be written, respectively, as
(3.1.1.3)
(3.1.1.4)
Alternatively, Prentice et al (1978), assuming the cause-specific hazard functions
Ai(t j
,
Zj)'
j=1, 2, ... , n for each individual are known, and consequently the overall
survival function, arrived at a similar likelihood function, under an independent
censoring mechanism (which includes type I and type II censoring):
L3
Q(
j~1 {[
Ai(t;(r); 1';)J
A;(ti(r); 1';)J
Oj
./1
Oj
S(t j ;
Zj) }
tj
exp { -
1=1
J
Ai(u; z(u» du }
(3.1.1.5)
0
where
if the j-th individual fails
if censored.
David and Moeschberger (1978) as well as Prentice et al (1978) have pointed out
13
the likelihood factorization which allows one to perform a separate maximization of each
of L j , i=l, 2, ... , k when each Aj(tj 1'j) has a different set of parameters, that is,
1'jn1'j=0, iij·
Also, factorization as in (3.1.1.3) and (3.1.1.4) justifies the common
procedure of treating failure from other causes than C j as well as live withdrawals as
censored.
1.3.1.2. The Maximum Likelihood Estimators
Suppose that we have n independent observations vectors on x,
Xl' X2' ... , Xn
from
a distribution with probability density function f(x; 9), where 9 is the vector of unknown
parameters lying in a k-dimensional parameter space
n and
X
is a p-dimensional random
vector with values over a region R independent of O. Suppose further that 0 is
partitioned as 0=(0 1 ,9 2 )', where 91 is px1 and 9 2 is (k-p)x1. The likelihood function
for 9 is defined as
k
I1
L( 9)
(3.1.2.1)
f(x; 9)
i=l
Let
8
be a point of
n
at which L(9) is maximized;
8
is called a maximum likelihood
estimate (m.l.e) of O. It is common to work with log L(O), which is also maximized at
and in many cases
8 can
8,
be readily found by solving the likelihood equations
Uj(O)=O (i=l, 2, ... , k), where
U· = 8log L(0)
•
a
i = 1,2, ... , k
() j
The Uj(O)'s are called scores and the kx1 vector U(0)=(U 1 U 2 )
vector. The matrix
1S
called the score
14
with entries
I .. (0) = E ( _8
')
2
log L(O»)
80 j Oj
i, j
1,2, ... , k
is called the Fisher information matrix.
If the likelihood L(O) satisfies the regularity conditions (assumptions A1-A4), some
large sample properties of 0 hold. The regularity conditions are:
AI. For almost all xER and for all OEO.
mog f(x; 0) 8 2 10g f(x; 0)
d 8 3 10g f(x; 0)
80 r
'
80 r 80.
an
80 r 88.88 t
exist for i, j, t=l, 2, ... , k.
A2. For almost all xER and for every OEO,
where Fr(x) and Fr.(x) are integrable over R, r,s=l, ... , k.
These assumptions permit certain interchanges of order of differentiation and
integration.
A3. For every OEO, the matrix 1(0)=[ Ir.(O)j r, s=l, 2, ..., k] with
r.
g fl 810g fl
1 ( IIn) -_ E 0 [810
80r 0 80.
is positive definite with finite determinant.
A4. For almost all xER and for all OeO,
8310g f
I
I80 r 80.80,O
I< Hr.,(x),
where there exists a positive real number M such that
OJl
15
for all 8en and r,s=1, 2, ... , k.
Condition A4 can be replaced by the following condition:
Under the regularity conditions above,
0 is
a consistent estimator of 8. In addition,
several other asymptotic results hold that lead to useful inference procedures.
First,
U(O) is asymptotically N k [ 0, 1(0)]. Second, 1(9) is a consistent estimator of 1(0). As
result, under the hypothesis H o: 0=0 0
the score test statistic
X(k)" Note that this test does not require the calculation of
subset of the OJ'S can also be obtained.
U'(Oo) I(Oor l U(Oo)
is
O. Tests and estimates for a
Suppose that we want to test the following
m.l.e. of O2 , obtained by maximizing L(OlO' O2 ). Then, under H o: 8 1 =8 10 , 8 2 unspecified,
Since in only a few situations can m.l.e. 's be found analytically, iterative methods
are required for obtaining
O.
Newton-Raphson iteration and the method of scoring are
the most commonly used.
1.3.2. Semiparametric Approach
The most attractive model which incorporates concomitant variables in hazard
function was proposed by Cox (1972) for a single risk survival problem. It was called
the proportional hazard model and it assumes that
~(tj
z) =
~o(t)
exp(p'z)
(3.2.1)
16
where the underlying hazard function Ao(t) is assumed to be the same for all subjects in
the study but not assumed to be known, z is the (px1) vector of covariates, and /3 is the
vector of regression parameters. In Cox's formulation, the vector z can be time
dependent in which case should be denoted by z(t).
He proposed a "partial likelihood"
approach in order to estimate the coefficients of the regression variables and studied the
asymptotic properties of /3.
hypothesis H o: /3=0.
He proposed to use the score statistic test for assessing the
An important feature of
Cox's formulation is the possibility of
estimation of the underlying survival distribution 5 0 (t) by the estimation of Ao(t) and /3.
Cox (1972), Kalbfleisch and Prentice (1973), Breslow (1975) and more recently Tsiatis
(1981) and Link (1984) have suggested methods of estimation of the underlying survival
function.
Asymptotic properties of the estimates have been studied
(1978,1981), Bailey (1983) and Link (1984).
by Tsiatis
Tsiatis gives a proof of asymptotic
normality of /3 and Ao(t)= J Ao(u) du using stochastic processes in a random sampling
and random censoring setting. The estimates of the asymptotic variances calculated
from the limiting processes are the same as obtained by himself in 1978 using a heuristic
model.
Bailey utilized a projection technique due to Hajek leading to an asymptotic
representation of the score function as a sum of independent (but identically distributed)
random variables.
For proving asymptotic normality of the cumulative hazard function
Ao(t), he utilized an asymptotic representation of the estimator as the sum of an
independent increment process and another term which is the product of
fixed function of time.
(jJ - (3) and a
Finally, Link has proposed a smooth estimator of Breslow
estimator of the cumulative hazard function and used a heuristic argument to derive its
asymptotic variance.
No formal study of asymptotic normality was provided for this
estimator.
Extension to competing risk problems has been proposed by Holt (1978) and by
Kalbfleisch and Prentice (1978).
subject with covariates
The cause-specific hazard functions at time t for a
Z'=(Zl' Z2' ... ,
zp) was formulated as
17
1= 1,2, ... , k
~
here >'Oi(t)
(3.2.2)
0 are the hazards for C i if covariates are ignored and may vary for
different causes C i .
Using the conditional argument, that is, computing the product of
conditional probabilities for each cause C i at time ti(j) conditional on the risk set
R{ t,(j)}, the partial likelihood can be written as
k
di
TI TI
(3.2.3)
i=l j=l
where ti(l)' ... , ti(d ) are the d i ordered observed survival times for cause C i .
i
likelihood allow z to be time dependent.
estimation
of f3 i 's
can
be obtained
This
Due to the factorization into k components,
by applying standard asymptotic likelihood
techniques in k factors, assuming no common parameters among the f3 i ' sexist.
When >'Oi(t)
~O
do not vary for different causes, the cause-specific hazard function
can be written as
>'i(tj z)=>'o(t) exp{f3;z}
1= 1,2, ... , k
(3.2.4)
and the Kalbfleisch and Prentice (1973) invariant argument of marginal distribution of
ranks of observations leads to the marginal likelihood
n
ex:
TI
1=1
(3.2.5)
~
hER{t i }
The likelihoods L 1(13) and L 2 (f3) are identical when the risks a.re assumed to be
independent.
18
1.4. Restricted Alternatives and the Union-Intersection Test.
Suppose we want to test a hypothesis against a restricted alternative, that is,
(4.1 )
where
no
su bset of
and
n.
nl
are subsets of the parameter space
n,
such that
nOun l
is a proper
The test statistic derived for testing the hypothesis above is usually more
powerful than the statistic derived for the test with unrestricted alternative
when actually wEn l . Barlow et al (1972) present a thorough discussion of various
parametric and non parametric tests in univariate setups.
In the multivariate setup, Kudo (1963) derived the likelihood ratio statistic for
testing the hypothesis that the mean vector of a p-variate normal distribution is null
against an orthant restriction when the covariance matrix is known.
If Xl' X 2 ,
... ,
X n are independent, identically distributed N pep, E), then the hypothesis above can be
formulated as H o: p=O against He: P? O. Kudo has shown that the LR test rejects the
hypothesis Ho in favor of He above for large values of
(4.2)
where p* is the maximum likelihood estimator of p in the region 0 1 ={p: P? OJ. Kudo
outlined an algorithm for calculating Q2, Let J be any subset of P={l, 2, ..., p} and
its complement. For each
J,0~J~P, let
J
19
and
denote the corresponding parttions of X and E.
set J,
In addition, we need to define for each
0~ J ~ P,
E
,=E(
(jj:j)
0 0) J]
E
,(E,
Uj)
U
I
j )
)-1 E ,
U
(4.3)
j)'
XU:j) = XJ - E Uj) (E Uj)
, )-1 X
U)·
I
I
I
( 4.4)
I
Kudo has shown that
Q 2 - '"
0~r~p
N X'
U:j')
(E
Uj:j')
)-l X
U:j')
I(X-
U:/)
>0
(~
,,uU'j')
where I(A) represents the indicator function for the set A.
)-1 X-
(i')
< 0)
,
(45)
.
Under H o : /J=O, the
distribution of Q2 in (4.5) reduces to
(4.6)
where for each j, O~j ~p,
X;
represents a chi-squared random variable with j degree of
freedom and w j is the sum of (f) orthant probabilities of the normal random vectors
[XU:j'): -
(E
U' j ' l l XU) for all subsets J with j elements.
Nuesch (1966) simplified the derivation of Q2 and its distribution under the null
hypothesis, by
reducing it from
programming problem.
a
maximum likelihood problem
to a
quadratic
He also developed the LR test statistic and its distribution for
H o: /J=O against H o : /J~ 0 when the covariance matrix E=(7'2Eo, where Eo is completely
specified and (7'2 is unknown.
For this particular case, the LR test rejects H o: /J=O in
20
favor of H a : J.l? 0 for large values of
(4.7)
where Q2 is the statistic in (4.5) with Eo replacing E. Under H o: Jl=O, the distribution
of Q~ reduces to
(4.8)
where ;3(s,t) represents a Beta random variable with parameters sand t (;3(O,t)=O).
Chatterjee and De (1972) first utilized the union-intersection (UI) principle to
derive a bivariate linear rank test and also Chin chilli and Sen (1981a), using the UI
principle, derive asymptotically distribution-free tests for a class of restricted alternative
problems in the general linear multivariate model.
The union-intersection principle of test construction suggested by Roy (1953) is
intended for multivariate situations. Suppose that the composite null hypothesis H o and
the composite alternative H a can be decomposed as
(4.9)
where f is an index set such that reasonable tests for H ov vs Hall are available.
Specifically, suppose that the acceptance region for H ov against Hall is All, liEf.
The
union-intersection principle dictates that H o be accepted against H a over A=n All and
liEf
rejected on its complement.
The UI test for H o vs H a depends both upon the
decomposition of the null and alternative hypothesis and upon the family of tests for
component hypothese.
For example, if X-N (Jl, E), then a'X-N 1 (a'Jl, a'Ea) for every
aER P and the hypothesis Ho:Jl=Jlo has natural decomposition
e-
21
n
Ho=
{Ho(a): a'l1= a'l1o}·
aER P
The original multivariate hypothesis of 11=0 is true if and only if a' l1=a' 110 for all nonnull a.
Acceptance is equivalent to accepting all univariate hypothese for varying a.
Since the univariate acceptance region is t 2(a) ~ t~/2,N-1' the multivariate acceptance
region is the intersection
na [t 2(a) ~ t a2 I')_, N-1], which is the same as specifying that maxa
Therefore, using the VI principle it is necessary to maximize some function.
For
tests against ordered alternatives the function is nonlinear and must be maximized
su bject to inequality constraints.
The nonlinear programming maximization methods
may be applied here in order to obtain the maximum of that function.
In nonlinear
programming terminology the problem is as follows.
Determine the vector AEA such that h(A)=inf h(A), where
AEA
i)
h(A) is a scalar valued function of A, and
ii)
A={AER P ; hl(A) ~O, hz(A)=O}, where h 1 and h z are r 1 and rz vector
valued functions of A.
This is known as minimization problem with constraints. h is called the objective
function and hi and h z are called the constraint functions. Another important function
which assists in solving the minimization problem is the Lagrangean function
(4.10)
where t 1 and t 2 are rl and r2 vectors.
A Kuhn-Tucker-Lagrange (KTL) point is any point (A*, t 1 , t 2 ) which satisfies the
system of conditions
22
ii) h l (") ~ 0
iii) h 2 C") = 0
iv)
ti
hlC") = 0
(4.11)
Sen (1984) has proposed VI principle test statistics for a class of su bhypothese
testing against restricted alternatives for the Cox regression model. Sen considered the
general regression model due to Cox (1972), involving both design variables (controlled
and non-stochastic) and concomitant variables (stochastic).
The cause-specific hazard
function was defined as
(4.12)
where ha(t), the hazard rate for c j =0,
Zj
=0, is an unknown, arbitrary nonnegative
function, /3 is the vector of unknown parameters relating to the design-effects and 'Y is
the vector of unknown parameters relating to the effect of concomitant variables.
For
the null hypothesis H a: /3=0, the following alternatives have been considered: Hal:
/3/?0, j=l, ... , r, H a2 :
/31~'"
~/3r
and H a3 : 0~/31~'"
~/3r,
all the alternative with at
least one strict inequality. The partial likelihood was considered and utilizing the VI
principle he proposed the following test
L 1N
= sup {a " UN /(a' -V 11.2a) 1/2 : aEA }
with result similar to the statistic given in (4.5).
(4.13)
23
1.5. Proposed Research
The focus of this dissertation is on the development of statistical tests with
restricted alternatives by applying the VI principle in the context of the competing risk
situation.
Comparisons to tests with unrestricted
alternatives are also a focus of this
study.
In Chapter II Cox's semi parametric model is formulated in a two-sample competing
risk setting with two causes of failures, in the presence of concomitant variables. The VI
test is derived
for the null hypothesis of no difference between the two groups against
the alternative hypothesis that the survival experience in one group is greater than in
the second group.
The distribution of the test statistic under the null and alternative
hypothesis is studied.
Efficiency is investigated by the comparison with a test statistic
with a unrestricted alternative.
In Chapter III the parametric bivariate exponential model of Sarkar (1987) is
considered as well as a logistic model in a two-sample competing risk situation with k
causes of failure.
VI tests are derived for the hypothesis of no difference between the
two groups.
In Chapter IV ordered alternatives are considered in a similar context as in
Chapter II, that is, with the Cox regression model but r+ 1 su bpopulations subject to k
competing risks.
Again, tests statistics are derived using the VI principle and
distributions under the null and local alternatives are investigated.
Also, the power of
the test is analyzed and compared with the power for the unrestricted test.
Finally, in Chapter V, results of a numerical aplication concerning the performance
of the VI test with respect to the score test are presented. Here we present some final
comments and a few suggestions for further research.
CHAPTER II
RESTRICTED ALTERNATIVES TESTS IN A PROPORTIONAL HAZARD
MODEL WITH COMPETING RISK DATA
2.1. Introduction
In this Chapter, we consider the partial likelihood approach for the comparison of
the survival distributions derived from two independent subpopulations in a competing
risk situation with two causes of failure.
A Union-Intersection test based on partial
maximum likelihood estimators is derived for a restricted alternative test and its
asymptotic distributions under the null and alternative hypothesis are studied. Also the
efficiency of the proposed test with respect to the one for unrestricted alternative is
studied.
The survival experience for an individual to k competing causes of failure is represented by the bivariate random variable (T, I), where TE(O, 00) is the observed survival
time and IE{I, 2, ... ,k} indicates which of the k potential causes actually caused the
failure.
The
cause-specific
hazard
function
for
AOi(t) exp
(P: z)
a
subject
with
covariate
Z'=(Zl' z2' ... , zp) is taken to be
Ai (tl z)
(1.1 )
where the underlying hazard function AOi(t) may be different for different causes and the
regression parameters are different for each lifetime, that is, if P~ =(P i1 ,
the set of the parameters associated with cause C i , then
Pi
.. "P i p)
n P ='
j
denotes
for i:¢:j and
iJ=1,2, ...,k.
Let til
<
til
<... <
t idi denote the d i observed failures of type i and Zi(l) be the
e-
25
regression function for the subject that fail at
tiel)'
The partial likelihood can be written
as in Holt (1978) as
n
L(f3) =
where
k
n~E
)
exp (13'
iZi(l)
k
n n
1=1 i=1 {
E
uER{ti(l)}
}Oi(l)
. (13' )
exp iZi
(1.2)
Oi(l) is an indicator function which assumes the value 1 if the I-th subject
1=1
fails from C i and zero otherwise.
Suppose now we have two subpopulations and we are interested in testing the
equality of survival distributions from the two subpopulations.
We have a set
f3: j =(f3 ij1 , f3 ij2 , ... , f3 ij P) of vectors of parameters for each subject who fails from C j
and comes from the jth population. Let f3j=(f3~j' f3~j)' j=l, 2 represent the 2p-vector
of the parameters corresponding to the jth population.
In order to compare two
su bpopulations, the test may be on the parameters 13 i j ' specifically of
( 1.3)
Ho: 131
against the alternative
(1.4 )
We may also use the following reparameterization
(1.5)
where
tJ.'
( 1.6)
•
Then the hypothesis (1.4) and the alternative (1.5) can be stated as
H n: tJ. = 0
(1.7)
and
Ha
:
tJ.
1=
0
(1.8)
26
considering /3 2 =7 as a nuisance parameter vector.
In order to derive the union-intersection test for the hypothesis and the alternative
above, we need to check whether or not the regularity conditions of the maximum
likelihood estimator are satisfied when partial maximum likelihood is utilized.
The
regularity conditions are discussed in Section 2.2.1, whereas the derivation of the
statistic is presented in Section 2.2.2.
In Section 2.3 we study the asymptotic
distribution of the union-intersection test for the null and alternative hypothesis above.
Finally, in Section 2.4 we present the study of local asymptotic power and efficiency.
2.2. The Union-Intersection Test Based Q..!l Partial Maximum Likelihood Estimators
The construction of the likelihood is one the central issues in statistical inference.
The partial likelihood has been discussed in many studies, especially after Cox (1972).
Consider a random variable Y having probability density function fl/(Y; 0), where
0=(¢,'1).
Suppose that Y is transformed to a new random variable (v, w) by a
transformation not depending on the unknown parameters.
Also, suppose that we are
interested in inference about ¢ treating '1 as a nuisance parameter vector.
The
factorization of the full likelihood leads us to a marginal and conditional likelihood
fl/(Y; 0) = fl/(v; ¢). f w /tI(w; '1)
The existence of such a factorization is of key importance to the nuisance
problem:
besides
the
simplicity
of the
analysis
achieved
by
the
parameter
reduction
dimensionality with many nuisance parameters there is a gain in robustness.
of
In this
particular case, the likelihood suggested by Cox (1972) is the first term of the full
likelihood below, in a situation with hazard function
L(/3, '1) =
n
n
[~i(tl z) ]
1=1
ST (t" z,)
I
~i(tlz)
as given in (1.1).
27
}b
i
)
(/)
n
exp (/3 '
iZi(/)
=
,
x
1=1 i=l {
L
exp(/3i zu)
UER{ti(l)}
k
n n
x
fI i1 {GiO(t)
1=1 i=l
x
[
L
uER{ti(l)}
exp
(/3~zu)
J
bi
(/)}
(2.1 )
where the vector of nuisance parameters '1={Ao i (t), i=l, 2, ... , k}. Therefore, in Cox's
partial likelihood there is no need to estimate the underlying hazard function Ai (t) and
the corresponding survival functions GOi(t).
\Vith the reparameterization in (1.5), the
cause specific hazard function (1.1) may be written as
(2.2)
where w i(/) is equal to 1 if the l-th subject comes from the first population and is equal
to zero if comes from the second population, /3~=(-Y:, Lli) i=l, 2, /3 i
Ll ip ), Ll ij =(t. ijO Ll ij1 Ll ij2 ).
n /3;=0, for i:;Z!:j,
The partial likelihood (1.3) can be written as
(2.3)
where
(2.4)
where I(Ti(/)?t i (/»
is a indicator function which assumes the value 1 if the l-th subject
survived until time t (belong to the risk set R{t i (/)} and 0 otherwise.
28
For 1=1, 2, ... , n we assume the covariates z are independent and identically
distributed with Jl=E(z) and u=E(x-Jl) (x-Jl)', and u is positive definite with det(u)
<00.
2.2.1 Partial Maximum Likelihood Estimators.
Consider the partial likelihood defined in (2.3).
as a function of
of l' and
~
P,
i3
If L( 1',~) has a unique maximum
is the partial maximum likelihood estimator (p.m.l.e). The p.m.l.e
can be found by solving the system of simultaneous equations
(2.1.1)
and
(2.1.2)
Cox treats the partial likelihood as an ordinary likelihood function for the purposes
of inference on (3. Then, the usual assumptions of the M.L.E might be satisfied also for
the p.m.l.e. They are studied below.
1.
For the l-th subject and i-th cause, the log of the conditional probability of failure
given the risk set is
log
Pi(I)C"Y,~j Z, t) = { C"Y~+ ~:
wi(l»
zi(l)-
log
f:
1=1
I(Ti(I)~ti(I»
exp
(P:
Zi(l»}
(2.1.3)
which is a continuous function in
PeP and differentiable. Thus, first and second partial
derivatives exist. The existence of the third derivatives can be replaced by the following
e·
29
condition:
lim
E{
<--+00
sup
IP*-PI«
I
8 2 log p/(p*)
8 2 log p/(P)
8P78p'7
8P i 8P:
I}=
0
Let us consider a p x p matrix whose elements are
2
8 log Pl(P)
81: 8 1:
(2.1.4)
We can write the last product term as
(2.1.5)
30
The pq-th element of this matrix can be written as
Then
(2.1.6)
Then
02l og Pl(f3*)
02l og Pl(f3)
or;or';
OriOr';
exp( r: +~:w;(l») z;(1)
- f: I(Ti(1)~t;(I») exp(r:+~/iwi(1»)Zi(l)
1=1
}
31
and
exp C~:
f:
1=1
ICT;(l)~
t;(l))
W;(l) z;(/))
)
expC(:+~: W;(l))
z;(/)
Then
lim
(-+00
{
sup
11*-71«
I
02log P1CP*)
07~07~
02log p/CP)
02log P/Cp*)
02log p/CP)
07i07~
07;07:
07;07:
I}
= 0
and
E{ (-+00
lim
sup
17*-71«
I
I}
= 0
32
The same can be verified for all
E{
lim
sup
£ ---- 0
113*-PI <£
the condition that E { sug
I
13 E 13. Note that for
I xri
xri
- xnl } =
1 }
<
lim
£----0
E{
sup
If3*-PI<£
I xri
- xnl }
suffices. Therefore, if
00
(2.1.7)
with kEI+, then
lim
£ ---- 00
2.
E{
sup
o2l og p/(f3*)
o2l og p/(f3)
o,~o,~
O'i O,;
1,*-,1<£
I}
= 0
(2.1.8)
e·
In order to exchange integral and derivatives signs we need the conditions
I ~~~/) I ~ T(z)
where
J T(z) dz
a
of3r
Pi(/)
Sin ce ---.,.-:--,...:.
<
00
and
OIog Pi(l)
f3
r
J h(z) dz
and
<
00.
x Pi(l)' with Pi(l) given in (2.1.3), we have for 'ir E 13
_
- Pi(l)
{
_it zi(r) exp(,: + .6.: wiu) Ziu
Zi(/)
}
"L.J exp ("V~+.6./.
I , . w.tu ) z.ttl
U
(2.1.9)
33
Then, we have (2.1.9)
8Pi(l)
I
81ir
~
exp(1~+~: wi(I») Zi(l)
it exp(-y~+~: wi(I») Zi(l)
= T(z)
In order that T(z) is integrable, we need to have
(2.1.10)
Now, since
(2.1.11)
we have
From (2.1.6) we know that
2
8 log Pi(l)
81i 81
1
(2.1.12)
34
Then, from (2.1.9) and (2.1.12)
.'
and H(z) is integrable if
(2.1.13)
3.
The regularity condition (A 3 ) concerns the variance-covariance matrix I(P)
it
IS
required that
E I ( oPr
Olog Pi(l») (Olog
PiCI»)1
oPs
<
is satisfied. From (2.1.11) and
o Pi(l)
oPr
Olog Pi(l)
Pr
x PiCI)
we have for
Taking the expected value of both sides, we have that
.
( 02 Pi (I»)
SlDce E O-YiO-yj
=E (
(J Pi(l»)
{J-Yi
= O. Then, by (2.1.12)
00
e-
35
E
Pi(l»)
a1i
_ (8l0g
1
(8lo g
Pi(l»)1 <00.
a'
1j
From the study of the regularity conditions on the MLE from the partial likelihood
function as given in (2.4), we verify that there is a need for additional assumptions
which can be stated as
Consider the efficient scores vector
DLl
=
•
[.l
{"ii
8log L(1,Ll) ]
8Ll i
A
-0 ,1-1
_-
u-
(2.1.14)
and the matrix
ICPl
where
= [
]
(2.1.15)
36
(2.1.16)
is the (n,m)-th element of I.),"Y' n,m=l, 2, . ,', p, i=l, 2.
(2.1.17)
is the (n,m)-th element of I LlLl and finally
(2.1.18)
is the (n,m)-tb element of I
A
"Yu
or I
A
U"Y
,n,m=l, 2, ..., p, i=l, 2. Since Ll j n"Y.,=0 for
1
e-
37
iii' and 'Y; n'Yi/=0 for i i i ' .
The submatrices I'Y'Y' I'Y~' I~'Y and I~~ are
diagonal matrices with I~1' I~~, I~1 and
regularity conditions and H o:
1.
~=o
I~~, i=l, 2
on the diagonal.
block
Under
, the following properties of 10 hold.
There exists a vector 1 of solutions of the likelihood equations (2.1.1) and (2.1.2)
which converges in probability to 'Y. Also, we have that for any estimator {30 of {3
that 1({3) converges in probability to 1({3), that is
- 1({3).
2.
{Ii (1 -'Y) has a limiting multivariate normal distribution with mean 0 and
variance-covariance matrix j 'Y1'
Consider now a sequence of local alternative K n : ~n=~ =tn for some aER
2P
.
In
order to derive the asymptotic properties of 1, the p.m.l.e of 1 under K n , we need to
verify contiguity of a sequence of local alternative {K n } to the null hypothesis.
concept of contiguity is due to LeCam (1960).
The
A very detailed discussion of the
contiguity is contained in Chapter VI of Hajek and Sidak (1967). Basically, if {P n } and
{Qn} are two sequences of (absolutely continuous) probability measure on measure
space (X n , .An, Pn) such that for any sequence of events AnE.A n , {Pn(.An)-O} {Qn(.An)-O}, then the sequence of measures {Qn} is said to be contiguous to {P n }.
Consider the likelihood functions L(7,~o) and L(1, ~o+
-In).
The log of the likelihood
ratio is
log L = log L({3a) - log L({3o)
where
{3'a
Log L({3a)
(7', ~~ +
A)
and {3~ = (1: ~~).
38
Let (0
a) = a o where 0 is 2p x 1 vector of zeros, a is a 2p x 1 vector and
and
Then we can write the log of the likelihood as log L n =a~ U~
Since U~ "" N(O, 1(,80)) and V~
-+
-
+
~ a~ V~ a o
+
ope 1).
1(,80) by Slutzky's theorem, we conclude that
Let a~ 1(,80) ao=u, we have
eUsing Corollary VI. 1. 2 in Hajek and Sidak, we conclude that the densities f(x, ,8a) are
contiguous to the densities f(x, ,80)' By Le Cam's third Lemma, we conclude that under
(2.1.19)
where Pa=('1, a) is the p.m.l.e. of,8 under K n .
To establish the relationship between ;- and '1 consider now the vector of the
partial scores under H o
where
u1
=[ :vi
1
mog
L(P)]
81
A_O-,1-1
u-
=0
39
u
~
Expanding the scores
U in
=[ Vi
_1
Olog L(
O~
pn
JLl=O,y =1
a Taylor series about the true parameter value
P=( /, 0) we have
U(.8) = U(P) + Vi(.8 - P) U(P*)'
where
IP*-PI < 1.8-PI.
(0
.
-
slDce I (P)= -
Now, we can write the expression above as
U~)=
(o210g
L(P)
aProPs
)
(U r
U~)
-
Vi(1-/
0) I(P) + opel)
. Then
and
(2.1.20)
Then
since E(UO)=O and V (Uo)=I(P). Consider now the vector of partial scores under H a
iJ = ( iJ / iJ ~)
where
40
v
~
Expanding the scores
V
=[
_1
Blog L(P)]
{Ii
B~
~=
tn,r =i'
in a Taylor series about the true parameter value
P=( i,
In)
we have
v(in = U(P) + {Ii(P
where
-P) U(p*)'
1,8*-,81 < IP-PI. Now, we can write the expression above as
(0
V ~)= (U~ U~)
- {Ii ( i'-i 0)
1(,8) + op(l)
e-
Since lJi=O
U~ = {Ii(1'-i) Iii
+
op(l)
{Ii(i'-i)= I:Yi U~
+
op(l)
Now expanding the true scores UOin a Taylor series about
(2.1.21)
pa we have
(2.1.22)
41
Since U(po) ..... N(O, i(p)) it follows that
U(P") ..... N (ao i(p), i(p))
where aO=(O a). Then
Now from (2.1.30) we have
(2.1.22)
(2.1.23)
Using (2.1.22) in (2.1.20) we have
Using (2.1.21) in the expression above we have
(2.1.24)
2.2.2. The Union-Intersection Test
In this section we derive the union-intersection test for the hypothesis in (1.7) H o:
~=O
against the restricted alternative H a :
~
>
0, that is based on partial maximum
likelihood estimators and the application of Roy's union-intersection principle. Define
42
where a i =(ail' ai2, ...,aiP)' i=l, 2 such that
homogeneous and for each ~ E 0 let ~ =
O(a)
a~O.
tn. For a fixed
Assume that 0 is positively
aE R +
2p
construct
= { ~: ~~O, ~ = tn}
(2.2.1 )
and consider the alternative hypothesis
*.A _ a
H a·u.-fIi
(2.2.2)
If we define the set A such that
A= { a: a
~O }
(2.2.3)
we have
Oc
since
n is
U O(a)
aEA
positively homogeneous. Thus according to the union- intersection principle
Ii = sup
2 log 11
(2.2.4)
aEA
where
L(t, ~)
L(t, 0)
where
t
is the p.rn.l.e of "1 given ~=
tn, t
is the p.rn.l.e of "1 given ~=O and
(2.2.5)
L(.;.) is
the likelihood function as defined in (2.3). Then
"
('
,wi(l»
~exp "1i + ai "1'D Zi(l)}
~exp ("1~ +a;
,w;(I)
"1'D )Zi(l)
(2.2.6)
e-
43
Since from (2.1.25)
_r::: ( •
.)
'In 1i- 1i
= - (Ii11 )-1 Ii16ai
we have
"
w exp1- 0,
i Zi(l)
-log u
Expanding exp [
-
~ I~l(I~;rl
order around zero, we have
+
a~ Wii?J
zi(l)
in a Taylor series up to the second
44
e1
--
, '"
-0,
a;
LJ exp"Y;
Z;(I) W;(I) z;(l)
u
{Ii
,
1
+ -~
~n
a; I
1
-2n
Lu
0
eXP1; 'zi(l)
a-v, (I;"Y"Y )-1'"
LJ
u
i
-0,
Z;(I) exp"Y; z;(l)
",·0,
LJ exp"Y i Z;(I)
u
2 Z;(l) ai
a'i '"
LJ exp (-0,
"Y; Z;(I) ) W;(I)
u
Lu
0
eXP1; 'z;(/)
'LJ
"
u
- 0, zi(/)
exp"Y;
45
[it exp(
+~
2n
(
-a~ I~1(I11)"1 + a'i W i(l) )Zi(I)J2
[it exp(
2
-2:
. 1 a~
1=
-~
r?'Zi(I»
r?'Zi(I)J
.
I~ ...,
2
2:
2n i=l
n
a'·
I
2:
6(1)
1=1 •
{L..'" exp (,0,
1;
u
2:
u
Zi(l)
)
2
,
wi(l)z;(l) Z;(l)
0
expr; 'z;(I)
46
"
~
_ u
• 0,
"
• 0,
)
Z;(I) exp ( -y;
z;(I) ) '~
Z;(I) w;(I) exp ( -y;
Z;(I)}.
u
[
it eXP1;
+
op(1).
+
opel).
.
(1' )-11'
l2
0
-y-y
a~
ll-y'
'Zi(l)J
e-
(2.2.6)
SInce 0-y=0 and Illll:-y= I llll -
Ill-y rf-y l-yll'
Then the union-intersection test In
(2.2.4) requires the maximization of 2 log 11 over the set A.
The problem of determining the maximum of 21og1 1 =2 a' OIl - Illll:-y
+
opel)
which is a convex quadratic form in a, over the set A is equivalent to finding the
- inf( - 2a'O II + I llll:-Y) over A. In non-linear programming terminology, the problem is
to determine the vector aEA such that
47
h(a*) = - inf {h(a), aEA}
(2.2.8)
where h(a) is a scalar-valued function of a and
where hi (a)
and
h 2 (a)
are functions of a.
The Kuhn-Tucker-Lagrange (KTL)
minimization technique can be applied here in order to minimize h(a), considering
hi (a)= -a~O and the equality constraint missing.
The solutions for the Lagrange
function
L(a,td = h(a)
+
t~hl(a)
where t l is a 2p vector values,
IS
= -2 a' U ~ + a'
I~~:1 a -t~a
(2.2.9)
any point (a*, ti) which satisfies the system of
conditions:
ii) a 2: 0
iii) a' t l = 0
. ) 0 L(a, t l
oa
IV
)
(2.2.10)
From iv), we have
and
a't l
Then
a'
fJ ~ =
-2 a'
fJ ~ +
2 a' I~~:1 a = 0 (from iii)
a' I ~~:1 a. Therefore
h(a) = -2 a' U ~ + a' I~~:1 a = - a' I~~:1 a
48
1* = - inf h(a) =
a' I~~:"Y a
From (iv) we have
(2.2.11)
From (i) and (ii) we see that both t 1 and a as given above are non-negative, yet their
inner product is zero by (iii) in (2.2.10). This is possible if for some 0
and aU)
> 0,
~
J ~ A,
t(i)
that is, we partition a and t 1 as a=(aU)' a(J))' and t=(t U )' to»)"
=0
Also
for each a we partition U ~ and I ~~:"Y as
and
eI~~:"Y =
We denote by
1
-
r~~:"Y U~,
(2.2.12)
where
(2.2.13)
and
49
Then from (2.2.11) and using (2.2.12) and (2.2.13) we have
If we let t j =0, aj
>
°
and a] =0, we have
°
+ 21 r-n t J =
1
a] =
W-,~(])
- ,
W
~(j)
+ 21 r-ji
(2.2.14)
and
1Ii]~~:"Y( InLl~:"Y)-1 t] > °
(2.2.15)
From (2.2.14) we have
t~) = -2
rnji \V'Au(j)- >
°
(2.2.16)
and from (2.2.15) we have
- ,
W
~(j)
1
+2
r-j1
Ii]
(
j ~~:"Y
In
W ~(j)
jj
-
Ll~:"Y
jj
)-1 t]
-1
-
r (r ) w ~(]) =
-
*
W ~(j)
>
°
(2.2.17)
Kudo (1963) has shown that the collection of all 2 r sets, r=2(p+1)
-*
W ~(j)~ 0,
o~
J~
A={1, 2, ... , r=2p} is a disjoint and exhaustive partitioning of r-th dimensional
Euclidean space, and for a fixed a, (a*, ti) is a KTL point, where
50
(2.2.18)
and
(2.2.19)
Then
h(a*) = - a*' I~~:1 a*
= -[
_*
W
~(j) OJ I~~:1
l
W ~(j)
0
* J=
so that the union-intersection test statistic for the hypothesis (2.3.1) can be written as
In terms of
U~,
the statistic test (2.3.20) can be written as
or
as from (2.2.17)
(2.2.21 )
51
where
(2.2.22)
and from (2.2.11)
2.3. Asvmptotic Distribution of the Union-Intersection Test Statistic l~
In this section, we study the asymptotic distribution of the UI test l~ under H o:
Ll=O and under Ha : a=O.
First we want to show the following.
I. Under {H o : Ll=O} there exist c
>
0 such that
*
P{ll ~ cl H o}=
2p
L:
.=0
2
w. Ph. ~ c}
2p
where the w, are nonnegative quantities with
variable with k degree of freedom and
X6 =
L:
,=0
w,=l and
(2.3.1)
X;
is a Chi-squared random
1 with probability 1. The distribution above
is referred as chi-bar distribution. The weights w, can be computed as
w,
where
L:
=
L:
{j}
P {W~(
<»
1
01
denotes the sum over all sets J,
H o}. p{r~~ W'A(~) ~
0~J ~
{j}
is N'j(O,
(I~Ll:'Yrl) under H o and W Ll(}) is
Proof: From (2.2.20) we have for c> 0,
~ 1
11
01
H o}
(2.3.2)
- *LlU)
P, such that J has s elements, W
N I ] (0,
r l1
), under H o.
52
" p{ (
P{l~ ~ c}
* ,
w- *) ~c,
W- ~(') Iii
~~.
L.,.,
0~J~ P
" 1 ~(')
'
-*
W ~("»
0 ,
(2.3.3)
as the distribution of l~.
I(r;; W
I( W~(j)'
~(])< 0)
First, we need to show that the events I(W~(j»
are independent under H o.
I~~:I W~(j)) and
I(
W~(j»
The first independence can be
0) and
Second, we need independence of
0).
shown as follows.
The r-variate normal density
function for W ~ with mean p and dispersion matrix rl~:/' say g(W ~, p, Il~:/) can
be factored into
ewhere 0~J~P={1, 2, ... , r}. Then
Then,
{W~(j»
o}
and
{r"
W~O)<
o}
are independent, regardless whether or not
E(W~) =0.
The second part can be shown by using Lemma 3.2 of Kudo (1963).
Kudo has
shown that under H o' if X is an a-variate, null mean normal random vector with
covariance ~, that P (X/~-lX5 x,
Bx~ 0)=P(X/~-lX5 x) P( Bx~ 0), where
B is a
(bx a) matrix of full rank b (~a). If we take B to be the identity matrix, then the two
events
I(W~(j)' I~~:7 W~(j)~c) and I(W~(j»
Thus,
from
the
independence
above,
0) are independent under H o.
we
conclude
that
the
events
53
independent, under He.
From (2.2.11) we have that under He'
null mean and dispersion matrix I1~:1'
W~
is r-variate normally distributed with
-*
From (2.2.21), we can see that W ~(j)
IS
normally distributed with null mean and dispersion matrix (I~~:1rl. It follows that
W~(j)' I~~:1 W~(j) is
xi j
where
Sj
is the degree of freedom and s is the dimension of
set J, J ~ P.
Then (2.3.3) can be written as
P{lts c}
Regrouping the 2 r terms (r=2p) in terms of the cardinality k (0 S k S 2p), we obtain
P{ltS c} =
The second part of this section studies the distribution of It under the set of local
alternatives H a : ~=
In.
II. Under {H a : ~=
In} there exist c> 0 such that
We want to show that
lim P{lt S c}
(2.3.4)
where
Nk (X,
jj,
E)
IS
the k-variate normal distribution function with mean
jj
and dispersion
matrix E,
(2.3.5)
and
54
(2.3.6)
Proof: From (2.2.20), we have for c>O,
(2.3.7)
as the distribution of It under Ha:A=
..tn.
Following the same arguments used to show
the distribution under H a , we can see that the events I(W!(j»
0) and I(rnW A(1)<O)
are independent regardless of whether or not E{W!cn} is non null. In fact, we can see
from (2.2.21) that W!cn -- N.;(aj, (
I~A:,rl), since W A -- N r (a, rlA:,) under
H a . Thus, (2.3.7) becomes
*
P{II~cIHa}=
l:
0~J ~
P
* ,
*
-*
} {-
j j . WAC·)~c,WAC·»O
P { WAC·)I
P r-:-:wAC-:)<O
) AA . , )
)
J))
}
(2.3.8)
Therefore, the probabilities in (2.3.8) are expressed as follows.
J d N.;(t
CIU)
where C lU ) is given in (2.3.5).
p{ r n WA()< o} =
J
C2 U)
d N,;(t 2 ; a J
; (r n
r
l
)
l ;
aj;
(I~A:,rl)
55
where C 2 (i) is given in (2.3.6).
2.4. Local Asvmptotic Power and Efficiency
In this section, we present our study of local asymptotic power of the unionintersection test for the hypothesis given in (1.7).
expressed as
a function of a and examined when
the two tests may be
The power function /3(0) can be
a tend to zero.
The comparison of
performed in terms of the slopes of their power functions at a
point a=O.
I. The power function for the UI test of H o :
/31
(a) =
a:
+
a
'"' P {-W ~(i)
*
{ 0~J~
P
L...-
I~~:1' t
f
x
~=O
1
I
against H a :
~
*
> 0 is given by
-*
jj
- ~(j)~c, W ~(j»
I~~:1'
W
d N./t 1 ; 0;
(I~~:)-l)}
0
}
(2.4.1)
C1 (i)
where the expression in the brackets is a 2p-vector of non-negative elements and a: is the
level of significance of the test. In the particular case of J = P and J =0, the power can
be expressed as
p,(a) = a
+
a[
rL~"1
(2.4.2)
where the expression in the square brackets is a 2p vector of non-negative elements and
C(j) ={ W~:
W~ >O}.
From the distribution of the UI test under the alternative, given
in (2.3.4), the power of the test can be expressed as
56
Expanding PICa) in a Taylor series around a=O, we have
Plea) = PI(a) I<1=0+ a
8 PICa) I
8 Plea) I
8a
<1=0+ o(lal) = a +a
8a
<1=0+ o(lal)
where
8 PICa)
8
=
a
L
0~J ~
8
x 8a
* ,
P
{{ jj
PH
W ~(.) I~~.
a
*
* ,
-*
W ~(.)~ c, W ~(.»
-
·r)
)
*
-*
jj
P Ha { W ~(j) I~~:r
W
~(j)~ c, W ~(j»
J
0
0
}
}}
(2.4.3)
The derivatives of the probabilities above can be calculated as:
e(2.4.4)
which is a 2p vector whose s j elements are zero, where s j is the dimension of J, that is,
and
8~c
)
P Ha { f
n W ~())>
J
O} = -
C2 (j)
f
n
Ct 2 - aj) d N'j(t 2 ; aj ; (f jj
r
l
)
= -f-jj
aa.
*
*
-*
jj
ii) 8 P Ha { W ~(j)' I~~:r
W
~(j)~ c, W ~(j»
0 }}
(2.4.5)
57
which is also a 2p vector whose s" elements are zero, where s" is the dimension of
1
1
J,
that
IS,
-*~(j) , I~~:7
jj
-*
-*
}
fJ P
{ W
W ~(j)~ c, W ~Ul> 0 = 0
fJa" H a
1
and
* ,
*
-*
}
fJ
{ii
fJaj
fJaj P Ha W ~(j) I~~:7 W ~(j)~ c, W ~Ul> 0 x fJaj
=
J
I~~:7(tl
-
aj)
d N.j(t l ;
aj;
(I~~:7rl))
Cl(j)
J
I~~:7
t l d N./t l ;
aj;
(I~~:7rl)
Cl(j)
j
,
(
where a *j =aj - I ~~:1'
In
~~:7 )-1
a,. Then
" {
- 0~~ P P Ho
{-W ~U)
* , I~~:7 W ~U)~
* c, W- ~(j»
* 0}
jj
-
J
C2 (j)
x
I~~:7 J
C1U)
x r~~
11
t l d N./ t l ; 0;
(I~~:7rl)}
58
x
I !:i.!:i.:-y
ii
J
t 1 d N , i (t 1; 0
;
(Ii!
A:-v)-l)}
u.u.,
C1U)
which is non-negative, since
W!(j»O, I~!:i.:-Y
is p.d. and -
f"
W !:i.e;)
>0. We would
like to compare the test with restricted alternative with the test with unrestricted
alternative, that is, with Ha:!:i. =1= O. For the test with unrestricted alternative, the test
statistic will be
Q
(2.4.6)
which under H o is a X 2 with 2p degrees of freedom and the power function will be a noncentral chi-squared with 2p degrees of freedom and non-centrality parameter (}=a'
e·
I!:i.l:i.:, a, that is,
PH a
where r=2p.
}=
AA W A >
{ W-,A I u.u.:,
u.
u. -
X
When H o is true, P2«(})=0'=p{ X;
e
00 ()' {2
X +"
-~() I: -,-, P
, = a 2 s.
r., >-
x
}
~ x}. In order to compare the two
sets, we need to express P2«(}) as a function of a or as a function of (). Thus, we expand
P2 «(})
around (}=O in a Taylor series
The derivative of the power function is given by
59
which is the density function of a non-central chi-squared with r+2 degrees of freedom
and non-centrality parameter
(J.
When
(J=O
we have
which is non-negative since it is a p.d.f. Thus,
(2.4.7)
=a' I AA : a, a quadratic form in a. Since a' I AA : a~k, where
r
r
alIAA:ral
Then,
k=chmaxI AA:.." (Rao, page 50, 1f2.1), we have
a'a
~k for every a.
is a function of
(J
l
ja'IAA:.."al ~ O(laI2)= o(lal). The power function {J2 can be expressed as
(2.4.8)
Now, for the comparison of the power functions given in (2.4.1) and (2.4.8), we can
calculate
For larger values of a the quantity ~k2(o(1))=0 and (Jl(a)-{J2(a) is positive. Therefore,
the test statistic l~ defined in (3.2.20) is more powerful than Q defined in (2.4.6).
CHAPTER III
RESTRICTED ALTERNATIVES TEST IN A PARAMETRIC MODEL
AND IN A LOGISTIC MODEL WITH COMPETING RISK DATA
3.1. Introduction
In this Chapter we derive the union-intersection test for comparison of survival
distributions from two independent subpopulations in a competing risk situation with
two causes of failure using two different approaches: the first, when a parametric model
is known, and the second, using the logistic model.
Since independence of causes of failures is not realistic in many competing risk
situation we may consider dependent causes of failure, that is, the theoretical times of an
individual failing from one cause may be correlated with the theoretical times of the
same individual failing from the other cause. Thus, a dependent parametric model may
be considered.
The exponential survival distribution has found considerable acceptance
in survival analysis for a single cause of failure.
Hence it would be desirable to work
with some bivariate distribution whose marginals are exponential within the context of
dependent competing cause of failure.
The bivariate exponential of Sarkar (1987)
appear to meet the requirements cited above, that is, it is a dependent model with
exponential marginals. Furthermore, it is an absolutely continuous distribution, which is
adequate for the situation where failure do not occur for both causes simultaneously.
With data of type (T ,I), where T is the random variable corresponding to the
observed failure time and I is the random index indicating failure type, it may be
possible to estimate all the parameters of the dependent underlying distribution.
However, the estimated value of the parameter in a dependent model measuring
the
dependence between the hypothetical times Xi' i=l, 2, ... , k, should not be taken as an
e·
61
indicator of dependence. The reason for this lies in the fact that the cause-specific hazard
function from a dependent model may arise an independent model with the same causespecific hazard function (Prentice et al., 1980).
The logistic linear model has been an alternative to the proportional hazard models,
ill
the analysis of the data from prospective studies. The data from those studies consist
basically of values on a dichotomized dependent variable, failure or not, and of values on
concomitant variables observed at entry time.
In a competing risk situation, the cause
of failure among the k causes is also observed. Others data may be incorporated to the
model: entry time y, follow-up time
T.
This information is useful in situations when
subjects enter the study at different times or withdrawal before its termination.
The
concomitant variables can be time dependent.
Some studies have appeared comparing logistic linear models and proportional
hazard models
ill
prospective studies; among others, Green and Symons(1983) and
Elandt-Johnson (1983). It appears that the two models are equivalent for studies when
the disease is rare and/or follow-up time is short.
For cohort data, when all individuals enter the study at the same time and there
are no withdrawals before the termination of the fixed time t, no knowledge of the
underlying survival distribution is needed. When individuals enter the study at different
times or censoring occurs before the termination, the underlying parametric survival
distribution may be specified.
3.2. Union-Intersection Test for !!: Parametric Model
In this section we derive a union-intersection test for the comparison of
two
parametric distributions within Sarkar's family of the absolutely continuous bivariate
exponential distribution, in a competing risk situation.
The absolutely continuous bivariate exponential distribution of Sarkar (1987) has a
survival distribution of the form
62
if 0
< t l <t 2
= exp{- PI+'\12)td
if
tl
>
t 2>
{1- [A('\2 t l)}"Y[AP2 t 2)]1+"Y}
0
(2.1)
,\
where '\1 > 0, '\2> 0 and '\12~ 0, "Y=~ and A(z)=l - exp( -z) for z> O. Note that
1+2
Xl and X 2 are independent when '\12 =0. The marginals are exponential,
e·
and the density of min(X l , X 2)=T is exponential with parameter '\, that is,
(2.4 )
The hazard functions are
since
63
and
ST(t)=exp {-A t}.
The probability distribution of the time of failure for cause C; are
J
J
a
a
t
Qi(t) =
t
A;(u) ST(u) du =
(1+ 1 ) A;exp{ -A u} du
(2.6)
A·
A·
= (1+ 1 ) A' (exp {-A t}) = A +'A (1 - exp {-At}), i= 1,2
1
2
and the proportions of failure due to C; are
A;
.
1 2
~,l=,.
"I
+ "2
(2.7)
The conditional survival function due to cause C; in the presence of each other cause is
also exponential with parameter A
S*(t) -_ Q~(oo) *- Q~(t)
Q; (00)
,
1, 2
= exp{
- \"t }'
,I =
with density given by ~(t)=A exp{ -At}, i=l, 2.
(2.8)
Since S~(t)=ST(t)=exp{-At}, the
overall hazard rate and the hazard functions are proportional, that is, A;<t)=C;' where
O<c; < 1. Thus, the distribution associated with cause C; alone (Elandt-Johnson, 1980) is
64
a fractional power (power is c;) of ST'
G;(t)
= {ST(t) }c;
{>.A;} .
(2.9)
= exp - Al +A t ,1= 1, 2.
2
The unconditional probability of failure from C; in an interval of time [t;,
t;+~]
is given
.'
by
(2.10)
and the likelihood for O'=(Al' A2' A12 ) can be written as
L(O,t)
_
-
TIn
1=1
{~}6;(1)
A +A
1
exp
e·
{_ \ I}
At
2
(2.11)
where A=A 1 +A 2 +A 12 and 6;(1) is the indicator which assumes the value 1 if cause of
failure for the l-th subject is C; and zero otherwise.
3.2.1. The Problem and Assumptions.
For the comparison of survival distributions for two independent subpopulations,
the statistical formulation of the hypothesis involve the equality of the parameters of the
corresponding distributions, or in a reparameterized form, the difference between the
parameters corresponding to the two distributions.
That is, if the underlying
distribution is the absolutely continuous bivariate exponential of Sarkar, the hypothesis
of interest is
.
H o·
\(1)_ \(2) \(1)_ \(2) \(1)_ \(2)
"'1 - "'1 ,"'2 - "'2 ,"'12 - "'12
(2.1.1)
65
which may be equivalent to
(2.1.2)
where ~=(~1' ~2' ~12) is the vector of differences between the parameters of the
distributions of the two subpopulations, that is,
A.(1)'
s=1, 2, 12,
where the superscripts (1) and (2) indicate the subpopulations.
The alternative of
interest is
(2.1.3)
The parameter vector is O=(Al' A2, A12 ,
~1' ~2' ~12)'
For testing H o in (2.1.2) against
the restricted alternative in (2.1.3), A=(A1' A2, A12 ) will be considered a vector of the
nuisance parameters. The likelihood function (2.11) may be written as
n
II
f( t l ; 0)
(2.1.4)
1=1
where
WI
is a indicator function which assumes the value 1 if the subject comes from
subpopulation 1 and zero otherwise. The log of the likelihood (2.1.4) can be written as
66
(2.1.5)
The maXImum likelihood estimators of A and
~
are the solutions of the system of
.
8Iog L(O, t)
8Iog L(O, t)
.
equatIOns
OA
=0 and
o~
=0, respectIvely.
For the study, the regularity conditions for the large sample derivations of MLE
must be satisfied. They are assumptions A1-A3 stated in (2.13)-(2.15), in Chapter II.
1.
The first derivatives of log f(t/; 0) exist, where f(t/;O) are the unconditional
probabilities of failing due to C; for the I-th subject (2.1.4). They are:
67
Oln f(t,;lJ)
0D. 2
= w,
Oln f(t,;lJ)
0>'2
Oln f(t,;O)
0D. 12
= w,
Oln f(t,;O)
0>'12
The second derivatives also exist and they are:
021 n f(t,;O) _ 021n f(t,;O) _ 021n f(t,;lJ)
0>'1 0>'12 0>'2 0 >'12 - 0>'12 0 >'12
(2.1.6)
68
w,
w,
(A+w,~)
(A+W,~)2
69
w/
8 2ln f(t,;O)
8 2ln f(t1iO)
8A28~12 = 8A18~12
_ 8 2ln f(t/iO)
8A128~1
8 2ln f(t1;O) _ 8 2ln f(t/iO)
8A128~2 - 8A128~12
It is easy to verify that the third derivatives exist for all O,.EO, r=l, 2, 12.
Then
assumption Al holds.
2. We may write f(t/iO) from (2.11) as
where a/=TI
i=1
Then, for O,E
iC
{(A+W/~)(Ai+W/~i)
}6 /) does
(Al +A2)+w/(~1 +~2)
(J,
not depend on t and
b/=(A+w/~).
70
of(t,;O)
00.
oa,
{
}
{
}
ob,
= 00. exp - b,t, + exp - b,t, a, (00) t,
Also, for OrEO, OrEO, we have
+
OC,
oOr t, exp {-b,t,
}
od,
+ oOr exp {-b,t,}
= e, tfexp{ -b,t,} + f, t, exp{ -b,t,} + g, exp{ -b, t,}
_
ob, _ od, OC,
_ od,
where e,--c, oOr' f,-d, oOr + oOr and g'-oOr' Then,
For all 1=1, 2, ..., b,=A+W,~=Pl +A2+A12)
>
O.
and tf exp{ -b,t,} are integrable on (0, 00) and so are
assumption A 2 holds.
Then, exp{ -b,t,}, t, exp{ -b,t,}
J get,)
dt, and
J h(t,)
dt,. Thus
71
3.
Consider now the first derivatives given in (2.16).
For each subject who fails
from the ith cause, we can write those derivatives as
8ln
.1 i)
t" (t1i B)
oB,
_
-
f(B)-
C
°i(l)
,
t1
°i(l)
f(B.)
[1 - 6 \B )J
i( 1)
,
where feB,) is a function of the parameters. Then, for a positive TJ,
since t 1 is exponentially distributed, all its moments are finite and the last term of the
inequality above is positive and finite, that is,
2
Ji)
t· ( t 1; B)
E 81 n oB,
I
1
+TJ
< K<
00
r
11
lOr a
B
B
•E .
Under the regularity conditions above, consistency and asymptotic normality of
M.L.E. are verified. The M.L.E. of
A=(.AI'
A2 , A12 ), the vector of nuisance parameters,
is calculated under H o, solving the following equations.
(2.1.8)
+ 02(1{
A2+~la2 + A+~la - (.AI +A2) +~l(al +a >] - t 1}Ia= 0
2
(2.1.9)
72
~~~ L(t , j9) la= 0 = 1~1 { 61(1{ A+~,a]
(1{ A+~,a]
+ 62
- t , }Ia= O'
(2.1.10)
n
2 n
Let ni=L: 6i(l) indicate the number of subjects who die due to C i , i=1,2, L: L: 6i(l)=n
1=1
i=11=1
-0
-0 -0 -0
-0
-0
-0
-0
.
and A =(,\1' A2' A12) be the vector of MLE of ,\ and A =A 1 +A2+ A12' ConsIder the
score functions
-0
-0
-0
-0
U a = (U aI' U a2' U a12)
where
- 0
_
U ~1 -
OIog
0)..1
. )
L(t/,9
I~= O,A=5.
(2.1.11)
- 0
_
U ~1 -
8Iog
0)..2
. )
L(t/,9
I~= O,A=
e-
5.
(2.1.12)
and
I
- 0 _ olog
. )
U ~1- OA12 L(t , ,9 a= O,A= 5.
(2.1.13)
and the sample variance-covariance matrix
_ [1,\,\ I,\a]
I( 9) =
6X6
I
where I A'\ is a 3 x 3 8ubmatrix with elements
a,\
I
~~
(2.1.14)
I, ,
"2"12
=1"
"1"12
=1,
,
"12"12
,I AA isa3x3submatrixwithelements
llll
74
and finally
II~
and I ~I are 3 x 3 submatrices with elements given below.
75
Under the regularity conditions, we have the following properties of
-yO,
under R o.
1.
There exists a vector ~oof solutions to the likelihood equations (2.19)-(2.21), which
converges in probability to -y.
2.
fTi('x° ->.) has a limiting multivariate normal distribution with mean 0 and
variance- covariance I~>..
3.2.2. The Union-Intersection Test
We derive the union-intersection test for the hypothesis
(2.2.1)
against
R~: ~l> 0, ~2> 0, ~12> 0
Define no={~: ~=O} and nl={~En C R
a=(a p a2' a 12 ) such that
~En
and
~#
0, let
a
a~O.
~={D'
+3 }
such that n=nOUn 1.
(2.2.2)
Define also
Assume that n is positively homogeneous and for each
For a fixed aER+3 , construct the parameter subspace
(2.2.3)
76
and construct the alternative hypothesis
(2.2.4)
If we define the set A such that
(2.2.5 )
and due to positive homogeneity of
n,
ncu n(a)
aEA
Then, according to the VI principle,
1*= sup
log I,
(2.2.6)
aEA
where
L()'c>",!!,,)
'{i1
1=
vectors of MLE under
n1 and no,
L(~O, 0)
(2.2.7)
respectively. Then we can write the log of (2.2.7) as
e·
77
(2.2.8)
Using matrix notation
-
-/
Ai = A C il i= 1, 2, 12,
where
(2.2.9)
(2.2.10)
and
where
(2.2.11)
78
Similar notation holds for ai' i=l, 2, 12 and a. Then, the log of I given in (2.2.8) can be
written as
2
log I =
I:
n
I:
2
n
{
(\(1)
i= 1 1= 1
+
(Xa!t
\0'
log I = .I: I:
1=1 1=1
+ log
{ log
{
0i( /)
\ 0'
("
-
(
{"
wla')C.
fD'
-0
A Ci
+
(X a '+ wI 8o')C
fD
log
-0
A C
1 'I
r1
w/ ')C
-fDa ~AJU+fD8o i
log----'-----=--"""o- - - - - ' - - - A Ci
1 , I r1 + wI ')C
fD a ~AJU fD a
-0
A C
\0'
1 'I
r,l
1 ( A - fD a ~A U
log ( 1 + _r::
- 0
.,n
A C
w, a ')C )
+ {ii
79
1 (-0'
1a, lA' I"-1 + WI'
-O')}
-Vi
.x -Vi
~A
Via -.x C
AA
Using Taylor's expansion up to the second term for 10g(1+x), we can write the
expression above as
2
n { 6
{I
B' C. - -1 ( B' C.)2
10 g I _-.2:
-Vi 2:
.(1)
-0'
2n -0' 1=1 1=1
.x C.
.x C.
+
1 B' C
-{Ii-o'
.x C
1 ( B' C)2
1 B' D
1 ( B' D)2}
1
,
}
-2n 5,0'c
- Vi 5,0'D -2n 5,0'D
- Vi B Ct/
where
B"
=a
w, -
l
a , I ~.x r~~ = B'1
-
B'2'
(2.2.12)
1-1
B "1 =a wI an d B'2 = a , 1~.x.x.x'
(2.2.13)
Rearranging the expression above, we have
log
I=
t t
i=l 1=1
{6.%)
~n
[Bi, C. + Bi, C - Bi, DJ 5,0 C.
5,0 C
5,0 D
.~
~n
Bi Ct l }
80
(2.2.14)
Each term of the expression above is calculated below. The first term can be calculated
as:
~
~
L.J
L.J
i=ll=l
{Oi(l)
--
Vi
[Bi
Ci + Bi C - Bi DJ - -1 B'1 C t l }
-~O'Ci
~O'C
~O'D
Vi
(2.2.15)
using (2.2.9)-(2.2.13).
Using the score functions given in (2.13)- (2.1.15) the first term
(2.2.14) above can be summarized as
~
~
L.J
L.J
i=l 1=1
{Oi(l)
-;=:::'l
n
[Bia' CC
-
~
i
i
DJ
+ Bi
- a' C _ Bi
- a'
~ C
~ D
_
1
~
'l
n
B'
1
Ct}
I
81
(2.2.16)
The second term of log I in (2.2.14) can be developed as
+1=1
Ln {6.r::n(I) b z
Z
'j
[1
=0 -
A
1]
~
Al +A z
(2.2.17)
where B 2 given in
(2.2.12) can be written also as B;=(b 1 b z b 1z )'.
expression above using the score functions (2.1.13)- (2.1.15), we have
Rearranging the
82
The third term of log I in (2.2.13) is calculated below.
(2.2.19)
Rearranging the expression above and using the elements of (2.1.14), we have
(2.2.20)
Using (2.1.11)-(2.1.13) and also (2.2.13)-(2.2.14) the fourth term of log I in (2.2.13) can
be written as above.
83
(2.2.21)
Finally, the fifth term of log I in (2.2.13) can be calculated similarly to the third term
and expressed as
(2.2.22)
Then, using (2.2.16), (2.2.18), (2.2.20), (2.2.21) and (2.2.22), the log of I from (2.2.13)
can be expressed as
84
= a' V ~ +
~a' I~~:"Y a +
(2.2.23)
op(l).
The derivation of the VI test statistic 1* as defined in (2.2.6) with log I as above in
(2.2.23), can be proceed in the same lines as in Chapter II, by utilization of the KuhnTucker-Lagrange minimization technique.
The test statistic for H o defined in (2.2.1)
against the orthant alternative defined in (2.2.2) can be expressed by (2.2.20), in
Chapter II.
3.3. linion-Intersection Test for
~
Logistic Model in Competing Risk Situation.
In this section we study the comparison of two survival distributions from two
independent subpopulations in a competing risk situation with two causes of failures and
with a logistic model using union-intersection test.
\Ve consider cohort data, where all n subjects enter the study at the same time (y)
and all are followed over a fixed period of time t. The response variable, failure or not,
is observed and also the cause of failure, say C., i=l, 2, ... , k.
Also, a vector of
concomitant variables z' =(zl' ... , zp), which may be considered as predicting factors of
failure and is observed at entry times for each subject and may be also time dependent.
Following Elandt-Johnson (1983), let us denote the conditional probability of
failure from cause C. over the period t in the presence of all causes as Q7(tIY,z) and
k
L
Q7(tIY,z) = FT(tIY,z) =1- ST(tIY,z),
(3.1 )
i=l
where
ST(tIY,z) is the conditional probability of surviving this period, for a subject
whose vector of concomitant variables z was observed at time of entry_ A logistic model
over a period t can be defined as
e-
85
( )
Iog Qt(tly,z)
ST(t\y,z) = 1i t
where 1~ =("(i1' "Yi2' ... , "Yip) and P~ = (l3 i 1'
i = 1, 2, ..., k.
+
O'i Y
+
P'
i
(3.2)
Z
Pi2 , .. .,f3 ip ) are the parameter vectors for
Note that "Y may be a function of the period t and z may be time
dependent. From (3.1) we have
k
ST(tIY,z)= 1 -
L
(3.3)
Qt(tIY,z)
i=1
and from (3.2),
Qt(tIY,z)
{( )
ST(tIY,z) = exp 1i t
+
O'i Y
+
P' }
i
Z
Thus,
Qt(tly,z)=ST(tIY,z) expbi(t)
+
O'i Y
+
P~ z}.
(3.4)
Using (3.4) in (3.3), we have
k
L
ST(tIY,z)= 1 - ST(tIY,z)
exp{ "Yi(t)
+
O'i Y
+
P~ z}
i=1
or
ST(tIY,z) = --.--k-------"'-1----1
+L
exp{ 1i(t)
+ O'i Y + P~
(3.5)
z}
i=1
Thus, using (3.5) in (3.4),
Qt(tIY,z)= eXPbi(t)
+
O'i Y
+ P~
z}
k
1
+L
1
exp{ 1i(t) + O'i Y + P~ z}
i=l
The likelihood functions for n subjects in the study may be written as
(3.6)
86
{n
=n
eXPh'i(t)+O'iY+P;Z}[l+f: eXP{'Yi(t)+O'iY+P;Z}]-l}
1=1 i=l
i=l
(3.7)
where6 i (l) is 1 if the l-th subject fails from C i and 0, otherwise.
3.3.1. Notation and Assumptions
Consider a follow-up study in which the time of entry is the same for all su bjects
and they are followed for length of time t. The response variable (dead, alive) and the
cause of death (the cause of interest or other causes) are then observed. The vector of
concomitant variables z is then observed at entry time and considered here not
dependent on time.
Also, a particular variable
WI
indicate which one the two
subpopulations the l-th subject belongs to. Thus, the logistic model (3.1) can be written
as
(3.1.1)
where
and
p7
indicate the difference between the subpopulation
1 and
2,
as
(3.1.2)
are interested in testing
H o:
~
= 0
(3.1.3)
against the alternative
(3.1.4)
e-
87
considering 1 as a vector of nuisance parameters.
The likelihood function (3.7) can be
written as
ti [1 + i=1
t exp{ 1: + wI~DxI J-
x
1
(3.1.5)
1=1
Since L(1, ~)=
n
n
1=1
f(-y,~,
XI)' we can write f(1, Ll, XI) as
and
(3.1.7)
In order to estimate the parameters by the maximum likelihood method and derive
the large sample test for the hypothesis (3.1.3), some regularity conditions have to hold.
1. The first derivatives of log f(-y,
Olog f( 1,
~,
XI)
01;
~,
XI) with respect to 1 and
~
exist. They are:
x; eXP[8;(I)(-y: +wI ~D XI]
={ 8;(1) x,
[1 +
t
exp{ 1:+
i=1
wI~:)XI
}
(3.1.8)
and
Olog f( 1,
~,
o~;
xd
={ 8;(1) w,xI
WIX: ex:[6 ,(I) (,: +w, a:) XI] }
[1 +.E eXPh:+ wI~DxI ]
1=1
The second derivatives of log f( 1,
and are:
~,
XI) with respect to 1 and
~
exist
(3.1.9)
88
(2)og
fb,
~, XI)
OriOr;
(3.1.10)
(2)og fb,~,
XI)
Or i orj
X;XI exp[b; +WI~;) XI] exp[bj +W I ~j ) XI]
[
1
+.t
exp{ r;+ wl~; )XI J2
1=1
for i:¢;j, i, j = 1, 2
(3.1.11)
(3.1.12)
(3.1.13)
(3.1.14)
o21 og £("V, ~, XI)
~......::;,.~'.......":"'-~
8~J}~j
=
2 o21 og
wI
fb,J::I,~, XI)
J::I
VriVrj
. ~ • ..
1 2
lor I r J, I, J = , ,
r
(3.1.15)
e·
89
2. Now consider the f(ll; xl) as given in (3.1.6). If we write f({3; x,)=a,x b, with
Of({3;x/)
o;3;r
obi
oa,
..
=a'o ;3;r + b l ;3;r' and for a positive K,
IOf({3;XI)!
o;3;r
I II
obi
oa'l
'
~ a1o;3,r + b 1;3;r =K<oo,
which happens for all i=l, 2 and r=l, 2, ... , p. In the same way, we can show that
1, 2, r, s = 1, 2, ... , p.
Thus, assumption A2 holds.
3.
Finally, consider the first derivatives in (3.1.7). Then, for a positive number
1]
and K, we have
since -y> 0,
~
>
0, exp
(-y~ +w, ~~)
x,
>
0. Then Assumption A3 holds. Consider the
vector of efficient scores
where U-y=(U-Yl U-Y2) and U ~ =(U ~l U ~2) are 2(p+1) vectors such that
90
x; exp( 1:
1
t
+
Xl)
exp(1:
}
(3.1.16)
Xl)
i=l
WlX~ exp( 1:
_
1
+L
Xl)}
eXPh:
(3.1.17)
Xl)
i=l
The sample efficient Fisher information matrix is given by
where
each
element
(vector)
of
the
matrix
above
can
be
calculated
using
(3.1.10 )-( 3.1.15):
o2Iogf(1,~,X/)
fhi01i
I
~= 0,1=1
x;Xlexp(1: X/)2 } r ._ 1 2
2
lor 1- , ,
1
exp{ 1: Xl)2
i=l
(3.1.18)
+L
X;Xl exp(
1
I:2
+L
i=l
xl) exp( 1j Xl) }
exp{li
....
for I#J, I,J = 1,2
Xl)2
(3.1.19)
91
=
-{Ii1 1=1
En {
Xl)
}.rlor I.,....J,
' -J- '
'.
I,
J = 1, 2
(3,1.21)
.r
} lor
C)
I.,....J, I, J = 1 ,-
'-J-'"
(3,1.23)
The maximum likelihood estimators of "'( under H o , say
1, are found as solutions of the
likelihood equations
~
L..
1=1
Olog f( 1, ~,x,) _ 0
[}
-
"'(i
where the derivatives are given
ID
an
d ~ Olog f(-y, ~, x,)
L..
1=1
[}~
= 0,
i
(3.1.8) and (3.1.9), respectively.
conditions, consistency and asymptotic normality of 1 hold:
Under regularity
92
r such
that under H Ol
r
1.
There exists
converges in probability to ,.
2.
{Ii (r -,) has a limiting multivariate normal distribution with mean 0 and
covariance matrix I~,.
3.3.2. The Onion-Intersection Test
In this section we derive the union-intersection test for the hypothesis H o :
against the alternative H a :
that n=nOun l
.
~
> O. Define
no={~: ~=O}
and n 1 ={~ :
Define also a'=(a l a2)' a;=(a;o, a;!' ... , a;p), such that
that n is positively homogeneous and for each ~En let ~=
.fil.
~
a~
~=o
> O} such
O. Assume
For a fixed aER+
3
construct
(3.2.1)
and consider the alternative hypothesis
H * a·.
A
~
_
-
a
(3.2.2)
{Ii.
If we define the set A such that
A
{ a: a~ O}
(3.2.3)
we have
ncu n(a)
aEA
since n is positively homogeneous. Thus, according to the VI principle
(3.2.4)
1*= sup log I
aEA
where
(3.2.5)
where
l' is the maXImum likelihood estimator of , given
likelihood estimator of, given
~=O
~=:h'
1 is the maximum
and L(.,.) is defined in (3.1.5). Then,
93
log I =
f: { i=1
t
0i(I)(
1=1
1: +~
aD x, - log [1 + .\L=2 exp(
1
1: +
w,a: )x I
]
XI]
+ {w,i ia''
2
1+ L exp( 1:+ w ,aDx , }
- log
i=1 2
1+ L exp( x,)
i=1
(3.2.6)
1:
Now, consider a 2(p+1)
X
(p+1) matrix C i , which is partitioned into two submatrices
(p+l) x (p+l), Cii' j=l, 2 in such way that whenever i=j, the submatrix C ij is
identity matrix and whenever i:;f j, the submatrix is zero. Then, we can write
(3.2.7)
Also, we have the relation between
1
and
1:
-
(B '1
(3.2.8)
B'2)
Then, using (3.2.7) and (3.2.8) we have
- B'C·
I
Then,the log of 15 in (3.2.6) becomes
(3.2.9)
94
1
-log
+ . E2 1 exp( 1;, 1=
1
+
2
E
B'C·
_r.:
~n
WI'
1+ ~n
_r.: a;
,
(3.2.10)
exp(l; x,)
i=l
Expanding exp{x} in a Taylor series up to the second order, we have
Using this expansion in the log of Is given in (3.2.10), we have
e-
95
Expanding log(l+x) in a Taylor series, we have
2
1+
L:
,
exp(ri x,)
i=l
Developing the last term and rearranging the terms, we end up with the following
expression:
log I
~ {a.' ~
[0 i(I){Ii
wI
LJ
.LJ
1=1
1=1
XI)WIXll
x, _ {Ii1 1+exp(r:
L: exp(-ri x,)
2
_,
i=l
B' C i
- {Ii
~
[0 i(l)
LJ
1=1
xI' -
l
I
exp(r:
x/)x
2
,
1+ L: exp(ri XI)
i=l
96
+ {iiI
1 a'
(3.2.12)
Therefore, using (3.1.16)- (3.1.23), we have
log 1=
L2a,i
i=1
UA •
L2B, CiU,.
i=1
•
1 2 ,
l'
--2 La. IA.Aai --2a1 I A A a l
i=I'
.U.
U2
1
97
(3.2.13)
The derivation of the VI test statistic 1* as defined in (3.2.5) with log 1 as above in
(3.2.13), can be proceed in the same lines as in Chapter II, by utilization of the KuhnTucker-Lagrange minimization technique.
The test statistic for H o:
~=O
against the
orthant alternative defined in (3.2.2) can be expressed by (2.2.20), in Chapter II.
CHAPTER IV
ORDERED (ORTHANT) ALTERNATIVE TESTS IN A PROPORTIONAL
HAZARD MODEL WITH COMPETING RISK DATA
4.1. Introduction
In this Chapter we study the multisample situation, aiming to compare survival
distributions in a competing risk situation with k causes of failure. Ordered alternatives
are considered and union-intersection test statistic based on partial maximum likelihood
estimators is derived.
The asymptotic distribution of the statistic under the null
hypothesis is also studied, as well as the efficiency of the test when compared to one
with an unrestricted alternative.
Section 4.2 present the problem and the hypothesis.
explained in this section.
Some notations are also
Section 4.3 concerns with the union-intersection test for the
restricted orthant (ordered) alternative.
Section 4.4 studies the union-intersection test
statistic for the ordered alternative.
4.2. Problem and Notation
Suppose we have the following situation: r+l groups of subjects from r+1
subpopulations are followed from a starting point and their survival times (T) are
observed as well as their causes of death C j , i=1,2, ... , k. A vector of covariates z;=(zl'
Z2' ... , Zt) is also observed for the loth individual.
The cause-specific hazard function for
the loth subject having survival time T, failing from cause C j is
(2.1 )
e-
99
where "'Oi(t), the nUisance underlying hazard rate, an unknown, arbitrary nonnegative
function, P'=(-y' 8') is the prx k matrix of unknown parameters, P~=(-y~ 8D is the pr
vector of the parameters for the cause C i , i=l, 2, ... , k, 1~=(1il' 1i2' ... ,1.iP) and
8;=(8;1 8: 2 ... 8~r) where 8: j =(8 ijl , 8 ij2 ,
... , 8ijp )'
The parameter 1 applies to the
control subgroup and 8 j is the differential effect of the j-th subgroup over the control,
j=l, 2, ... , r,
(2.2)
IS
an rp x p matrix such that cj=I p if the l-th subject
IS
from the j-th subpopulation
and c j =0 otherwise.
Suppose we are interested in the null hypothesis of no treatment effect, that is,
(2.3)
against the alternative that none of the treatments is inferior to the control, that is,
1, 2, ... , r
with at least one strict inequality.
studied
In
Chapter
subpopulations.
II
for
the
(2.4)
This situation is a generalization of the problem
two
sample
case
with
control
and
treatment
A second hypothesis of interest may be the ordering of the treatment
effects when H o may not hold, that is,
(2.5)
with at least one strict inequality; this is termed the ordered orthant alternative.
100
Finally, the ordered alternative
(2.6)
may be of interest.
For the hypothesis H 0 and the alternative HI' the cause-specific hazard function
was defined in (2.1). For the alternative hypothesis H 2 and H 3 there is a need to make
a transformation which allow us to write these alternatives as the orthant alternative
given in (2.4).
Let us consider
().) - ()(.)-1 )
j
1,2, ... , r,
°= a
0
(2.7)
so that
t1'.)
(2.8)
is a p x k matrix of the parameters for thej-th subpopu!ation,j=l, 2, ... , r,
t1: j =(6. ijl , t1 ij2 ,
... ,
t1 ijp ) is p-vector of the parameters for each cause and each
subpopulation. The entire matrix of parameters is an rp x k matrix t1'=(t1~ A~ ... t1~).
() i I
Ail
()i2
t1 il +t1 i2
(2.9)
=
°i
rprl
() ir
Ail +A i2 +
... +A ir
100
0
Ail
110
0
A i2
111
1
Air
D
Ai
101
The hypothesis H o: 8=0 and the alternative H 2 defined in (2.5) can be written for
an orthant test on
a:
H o:
a =
0
against H 2 :
a > o.
(2.10)
The cause-specific hazard function (2.1) can be expressed as
(2.11)
where
(2.12)
(2.13)
dj(l)
rpXp
with
1, 2, ..., r, I = 1, 2, ... , n
with
cj(l)
(2.14)
defined in (2.2)
For the hypothesis H o defined in (2.3) and the alternative H 3 in (2.6), we can also
utilize the transformation defined in (2.9).
Ho:~ =0;
leaving
The null hypothesis can be stated as
however, the alternative hypothesis consider the restriction of
~~
(a~ a~
.,.
~~)
unspecified. Then, if we define the (r-l)p x pr matrix
A=[O I]
(2.15)
with 0 a (r-l)p x p matrix of zeros and I a (r-l)p x (r-l)p matrix of ones, we can
writte the alternative H 3 as
(2.16)
102
with A as defined in (2.15).
4.3. Union-Intersection Test for Orthant (Ordered) Alternative
In this section we study the union-intersection test statistic based on the p.m.l.e.
for the hypothesis Ho: 0=0 against the orthant alternative HI: 0>0 and against the
orthant ordered alternative H 2 : 0
~
01
~02~
...
~Or
Let us first consider the hypothesis H o: 0=0
with at least one strict inequality.
and the alternative H1:0>0.
The
cause-specific hazard function is defined in (2.1) and the likelihood function can be
written as
L(P)
k
Il
(3.1 )
i=1
Now, assume that the z 's are random vectors independent and identically
distributed with some mean p. and r=E(z-p.)(z-p.)' positive definite and deter)
Under the conditions studied in section (2.2.1), in Chapter II,
r, the p.m.l.e. of I,
<
00.
under
H o, has the properties of consistency and asymptotic normality, as we saw in Chapter II
in (2.1.16). Also, under a sequence of local alternative H 1n : O=..fn' '/3=(r, a) also has a
consistency property and by contiguity it is asymptotically normality.
Therefore, the
score function
=l l
o
U
81og L(P)I
{"ii
{)O
0-0
-_
- ,1-1
(3.2)
is asymptotically N(O, i AA: ) under H o, and
/
u =1
()
IS
asymptotically
N(a
1 mog
Vi
L(fJ)1
{)()
()=~,r=1
(3.3)
i AA: / ' i AA: / ) under H 1n • U0 and U() are kpr vectors, and
e-
103
I AA :, is a kprx kpr submatrix from
-r;,:oI,orio
1
rAA:,
]
(3.4)
]
J(Pl=[
with dimensions kp x kp, kp x kpr, kpr x kp and kpr x kpr respectively.
The derivation of the UI test statistic follows the same steps taken
for the two sample problem.
III
Chapter II
The test statistic resulting is the same as in (2.2.20),
Chapter II, differing only in the dimensions of the vectors and matrices involved. Then
1*2
(3.5)
where P={l, 2, ... , kpr} and J is any subset of P (there are 2
kpr
sets).
Likewise in
(2.3.1) in Chapter II, the statistic I~ is distributed under H a as a chi-bar, that is,
P{I~ ~cl Ha}
kpr
=E
s=o
w, P{x; ~ x}
kpr
where w, are nonnegative weights with
E w, = 1, w,
s=o
are given by
(3.6)
104
(3.7)
x; represents the chi-squared random variable with s d.f.
The asymptotic distribution of l~ is similar to that in (2.3.4) and the power study also
follows the same results as obtained in section (2.4), Chapter II.
Now, let us consider the hypothesis H o: 0=0 against the ordered orthant
alternative H 2
:
O~01~02~
... ~O,..
The transformation defined in (2.9) allow us to
~,
write the hypothesis H o versus H 2 in terms of
that is, H o:
~=O
against H 2 :
~
> O.
Therefore, the cause-specific hazard function could be written as in (2.16) and the
likelihood function is similar to that in (3.1) with
defined
H2 :
III
(2.18).
Then,
0~01 ~02 ~ ... ~O,.
the
UI
0i
test
changed to
statistic
~i
for
and
wi(l)
by di(l) as
~=O
H o:
against
is the same as given in (3.6).
4.4. Union-Intersection Test for Ordered Alternative
In this section we study the UI test for H o: 0=0 against H 3 :
defined in (2.15) and
~
that defined in (2.11).
A~
> 0, with A as
as defined in (2.9). Thus, the cause-specific hazard function is
The likelihood function, the vector of scores functions and the
sample variance-covariance matrix are as defined in (3.1) to (3.5) with
and
di(l)
instead of
~
instead of 0
ci(l)'
In the subsequent subsections we present the derivation of the UI test statistic (in
subsection 4.4.1), the study of the distribution under H o and H 3 (in subsection 4.4.2)
and the power comparison with the unrestricted test (in subsection 4.4.3).
4.4.1. Union-Intersection Test Statistic Derivation
Define the parameter spaces
no = {~:
such that
~
= O}
(4.1.1)
105
(4.1.2)
. (4.1.3)
where A is defined as in (2.15), A=[ 0 I] such that 0=0 0
U 0 1,
Define similarly,
(4.1.4)
a rp x k matrix with a~ = (a~l a~2
partition a in the same way with Aa ~ O. Let us call a=(ao aD), where aO=Aa~ 0 is a
(r-1)p x k matrix. Assume 0 is positively homogeneous and for each AEO, let A =
-In.
°
For a fixe d a, a- E R +krp construct
{A : AA ~O, A =I 0, A =
O(a)
-In}
(4.1.5)
and consider the alternative hypothesis
°
(4.1.6)
B= {a: Aa~O, aO unspecified}
(4.1.7)
•
.
A
Aa
a
{Ii -- {Ii'
0
H *3
A
u
-
-
A
-Iu1 T
If we define the set B such that
we have
oc U
O(a)
aEB
since 0 is positively homogeneous. Thus, according to the VI principle
106
I; = sup
aEB
2 log 13
(4.1.8)
*)
where
L(t,
L(.y,O)
where
t
is the p.m.I.e of "/ given AA=
(4.1.9)
in and .y is the p.m.I.e of "/ given A=O.
- 0
Thus, the log 12 resulting is the same as obtained in Chapter II in (2.2.6), that is
(4.1.10)
and now the problem is to find sup {2a' U A
aEB
1*3
- inf
Aa~O
where IAA:,,/ is defined in (3.4) and
~
-
a' I A
A.
~~.,,/
a}. Then,
{ 2a' UA - a'IAA:""a}
(4.1.11)
I
iJ A in (3.2), with ()i exchanged for Ai'
The
function to be minimized is a convex quadratic form in a. The Kuhn- Tucker-Lagrange
minimization technique can be applied here in order to minimize h(a)=2a'U A -a'
IAA:,,/a considering h}(a)=-Aa
<
0 and the equality constraint missing. The solution
for the Lagrangean function
L(a, td = h(a)
+ t~
h}(a)
is (a*, t~) which may satisfy the following assumptions:
a) t~~ 0, t~ is a kp(r-l) vector
107
b) A a
~
0
c) t 1*' A a = 0
d) oLea, t 1 )
oa*
(4.1.12)
Note that the vector of score functions Dil is partitioned as
(4.1.13)
where Dills a kpr vector, U~ is a kp vector and D~ is a kp(r-1) vector.
variance-covariance matrix
The
(kpr x kpr) Iilil:, is also partitioned in such way that the
last kp(r-l) columns and rows is denoted by
l~ll:" that is,
1(00)
1(01)
ilil:,
ilil:,
(4.1.14)
I ilil:,
1(10)
1(11)
ilil:,
ilil:,
(4.1.15)
where
E
12
From d) in (4.1.12) we have
=
-
E
11
1(01)
ilil:,
(1(11)
ilil:,
)-1 -
-
E'
21
108
U~ -
A'ti' = 2
a*'A ti' = 2a*'
a*1 I
U~ -
~~:"Y
2 I~~:"Y a*
2 a*1 I~~:"Ya* = 0
a* = a*1
U~
A
Then, we can write h(a) as
h (a) = - a * I I
~~:"Y
a*
(4.1.16)
From d) in (4.12) we can have also
2 I ~~:"Y a
a
*
1
-
*.....,
= 2 U~
= r~~:"Y U ~
+ Vi1
+ A' t *1
*
1
r~~:"Y A' t 1
(4.1.17)
•
1
-
-
}
If we wnte r~~:"Y U ~ =w ~ and A r~~:"Y A'=r we have
-~
A a* = A W
a. o = W- 0~
- ~ =(W 0~
where W
+ 21 r t}*
+ 21 r t}*
-~).
0·
W
SInce A a * ~ 0 and t}*
a. 0 = A a * and ti can be partitioned as
(4.1.18)
> 0 but the inner product is zero,
109
- 0
8,
=
(- 0
8,j
- 0)
8,:7
in such way that i~=O and a;=O, where j is any subset of P=l, 2, ... , kp(r-1) and
3
is the complementary subset (0<; J <; P). Further partitions are necessary:
(4.1.19)
(4.1.20)
Then, (4.1.15) can be written as
Since
t~ =0 and a; =0, we have
(0 t~)'
}
From (4.1.21) we have
0=
(4.1.21)
110
-0
-1
- 0
t;= - 2 r;; W AO) > 0
(4.1.22)
and
- 0
W A(j)
1
-0
+ 2 r j ; t;
- 0*
W A(j) ~ 0
(4.1.23)
Then,
ti
= [
(4.2.24)
-2
and
(4.2.25)
From (4.1.16), we have
1AA:i a * --
U.u
A
+-21 A' t*1
and using (4.1.14),
1(00)
AA:i
1(10)
AA:i
1(01)
AA:i
1(11)
AA:i
aO
aO
-
UO
A
U~
0
+12
t*1
1
(4.1.26)
e·
111
Therefore, from (4.1.26) we have
1
(00)
AA: r
+
aO
1(01)
UO
A
- 0 _
AA: r a
-
(4.1.27)
(
1(00)
AA: r
)-1 1(01)
AA: r
]
a* =
(1(00)
AA: r
UO
)-1
A
Now, we have from (4.1.16) that h(a)=- a*' IAA:ra* which can be written as
1(00)
h(a) = - [ a O
A~:r
aO]
{10)
AA: r
I( 01)
AA: r
aO
1(11)
a-0
AA: r
and using (4.1.27)
h(a) = _
[U A
O' (1(00)
AA: r
)-1 _
a0
1(10)
AA: r
(1(00)
AA: r
)-1
{OO)
AA:i'
1(10)
AA: r
- a0
aO
1(10)
. (1(00)
AA:i'
AA:i'
)-1
J
112
(4.1.28)
Then,
oJ
[f_ii
f~~ ]_1 [W~~j)])
0
f .. f.
JJ
(4.1.29)
JJ
and using
f ii
(fl. cr l
f .-.
JJ:J
JJ
-f;]f) / f ii:} r
fc.
JJ
JJ:J
(4.1.30)
f--
(f ii:) r
JJ
where (fl. -.rl=(f J. J.
-
1f
}if;]
(f~. cr l
JJ:J
f.c f:1c fc.r l , (4.29) can be expressed as
JJ
l
JJ
JJ
113
(4.1.31 )
Since
(4.1.32)
and
we can write (4.31) as
(4.1.33)
and the union-intersection test statistic can be written as
x I{ f
n W- 0~O) < 0 }
-1
4.4.2. Distribution of the Test Statistic
- 0*
x I{ W
~(j)
>0
(4.1.34)
}
n
In this section we study the distribution of the statistic l~, given in (4.1.34), under
the null hypothesis H o :~=O and under a sequence of local alternatives H 3n : A~~,
Aa>O as defined in (4.1.6). The distribution of l~ can be written as
P{13* ~c} =
2:
0~J ~P
P
{[ W 0*'
~
- 0*
E OO : 1 W 0*
~
-1
- 0
+
- 0*'
- 0*
W
~( .) (f 1 .. ,) -1 W
~( .)
W ~(j) > 0, flJ W ~O) < 0
J
}
.
JJ.)
J
]
~c,
(4.2.2)
Now, we want to derive the distribution of 1; in (4.2.2) under the null hypothesis
114
R o: A=O. \Ve want to show that, under R o, the distribution in (4.2.2) is
(4.2.3)
where
(4.2.4)
kp(r-1)
I: w,=1 and X;+kP' 0 ~ j~ kp(r-1) represents a
s=O
central chi-squared random variable with s+kp degrees of freedom.
(that is, s is the cardinality of J),
First, we need to demonstrate that for each J, 0~J ~ P, the pair I(W~~j»O) and
I(f
W~()<O)
1
3:;
are asymptotically independent regardless of R o.
Let consider a
kp(r-1) variate W~ with mean /1 and dispersion matrix f and partition W~according
to sets J and
J.
Let g(W~, /1, f) represent the normal density function of W~.
It
follows that g(W~, /1, f) factors into
(4.2.6)
Then
P
- aA()
A(j) > 0 }·n {-1
fj) W
{{ W- 0*
- 0*
- aA() < 0 } .
< 0 }} = P { W
AU) > 0 } P {-1
fj) W
(4.2.7)
We need to show that W~* and W~()) are independent. In order to prove that,
we need to show that We; is independent of W~. Using the same argument as in first
part, we consider a kpr normal variate W A with mean /1 and dispersion matrix IAA:"Y·
W A and rAA:"Y are partitioned according to the partition of restricted and unrestricted
parameters, that is, according to W~ =(W~ W~) and rAA:"Y as defined in (4.1.15).
Then,
115
Since
f]l:; W~O)
is function of W~, we conclude for the independence between W~*
- 0
and W AO)' Then we have that
and
I{ f]] W~()) < o} are independent.
0*
Using the same argument as above, we can show that W A
- 0*
Aen
and W
are
independent. Therefore,
{ - 0*
and I W t:..U)
>
0
} are mutually independents.
x P{ f
slllce
{
W 0*'
~
EOO:l
-I
n
- 0
W
~())
0*
~
}
- 0*
P{ W
~(j)
2
W O*}
A is distributed as Xpq
and
under R o, s i is the cardinality of J.
- AU)
an d W
<0
Then,
N (0 , f ii:1'
)
}
>0
{ - 0*'
W
Aen
1
1 (fii::;f
W
O*}
Aen
2
as X'i
This follows from the fact that W~* ~ N (0, E OO : 1 )
p
R egrouplllg
.
t he 2l: (r-l) terms
(1< s~kp(r-1)), we have the distribution of
1;
.
III
terms
0
f t he car d'Illal'It)' s
under R o given as in (4.2.3) with w.
given in (4.2.4).
Finally, we want to derive the distribution of
alternative
1;
III
(4.2.2) under the local
116
{ H3n: A
~
=A
_~,
'4
= [(r-1)x
0
pr
A
n (r-1)p x pr
I
l,
(r-1) x (r-1)J
~I #
o}.
Then, under H 3n there exists c> 0 such that
(4.2.10)
where
C 2 (j) =
{
C 3 ()) =
{
Proof: The events
whether or not
{W~~j) >
of
I; given
o}
- 0*
W
~(j)
-)
- 0
r)) W ~()
< 0} ,
o}
{W~~j) >
E(W~)=O.
}
>0
and
{r;l;
W~O) <
However, the events
are not independent when
o}
are independent regardless of
{WZ:) (r~i:Jrl W~~j) ~ c} and
E(W~) is nonnull. Therefore, the distribution
in (4.2.2), under the alternative H3n can be written as
P{1 *3 ~c} =
L
0S;;;J S;;;P
- 0*
W
~(j)
J
- 0*'
- 0*
P {[ W 0*'
~ E oo :1 W 0*
~ +W
~( ') (r 1"'~)-I W
~(')
> 0} P {
1
-I
r))
11.1
)
~ c,
- 0
W
~O) < 0 } .
(4.2.11)
The probabilities in (4.2.11), under H3n, are given below:
117
(4.2.12)
where
*
N( a 0* ,LoOO:l'
~
)
h
~ Loll
~-l a- 0
WO
~,.,.,
were
a 0* = a 0 - LoOl
and finally
is the joint normal probability density function of two independent multinormal variates
0*
- 0*
W ~ and W ~(j)"
(4.1.14)
where
C 3 (j) =
This follows from the fact that
H/Jo
{
r -In
- 0
}
W ~(}) < 0 ,
W~ =rl~:T iJ ~
and
iJ ~ ,.,.,
N(a I~~:T; I~~:T) under
118
4.4.3. Local Asymptotic Power and Efficiency
In this section we present the study of local asymptotic power of the unlonintersection test for the ordered alternative given in (2.16).
From the distribution of the union-intersection test statistic under local alternatives
(4.36), the power of the test can be expressed as
=
0*'
E
0~J~P
x P{ f
-I
0*
- 0*'
1
-I - 0* ]
- 0*
P {[W ~ Eoo:1W ~ +W ~(.)(f. '.") W ~(.) ~c,W ~(.»O
J
- 0
n W ~O)
JJ.)
J
< 0}
}
J
(4.3.1)
Expanding (31 (a) in a Taylor series around a=O, we have
(4.3.2)
where
The two derivatives on the right hand side of (4.3.3) are calculated as follows.
119
-o
0
oa
PH
3
{-1
- 0A(~)
r~~ W
1J
U
1
>
0 }] (4.3.4)
-
The first derivative is zero while the second can be partitioned into two parts since
a]).
o
o~~ P H3 {r1] W~() ~
1
Further,
{
-1
- 0
} =
1lPH r-,w~(~)<O
va
3
1J)
where
o~ 0 P H3 {r1] W~() ~
-
J
C
[
0
}=o.
-o
0
oa
0
P x k
Therefore,
{-1 - 0
PH 3 )
r~~
WU
A(') > 0
)
)-
}J
o} has k(J) elements zeros and k(J) elements
-1
- 0
-0
- 0
-0
r,-.
(W U
A(~)- a,) d N(W A(~); a~ ;
))
))
U))
r~~)
1J
2 (j)
(4.3.5)
=( 0:0 J
g(W
6*, W6~j)
d We: d
W6~j)
C 1(j) n C 2 (j)
J
C 1(j)nC 2 (j)
o*
W- 0*
g (W ~'~(j)
*
)d WO
~
d W- 0*
~(j)
).
(4.3.6)
120
The first derivative can be calculated as
=
whereas the second derivative has kG) elements zeros and k(j) elements calculated as
O.8a~a) at
In order to evaluate each term of
0.81(a)1
oa
a=O it is expressed as
_
(4.3.8)
a=o-
where
A 2 has J elements zeros and the others j are given by
An
= o~~
J
from (4.3.5),
PH 3 {rjl
W6(1)~0~a=0 =-rjl
J W6
(1)
C 2(j)
d
N(W 6 (1)j OJ r,,) ,
121
A 5 also has j elements zeros and J elements calculated by
J-
1 :;
= f ii
- 0*
W 0*
~()) d N(W
~(i)j 0
j
1 :;) from (
)
f ii
4.3.6.
C1U )
- 0*
Since W ~U)
> 0,
1
(fii)r
In a particular case of J
=P
1
-1 - 0
8f31(a)!
is positive definite, f;; W ~cJ) ~ 0 then ---rJa a=o ~O.
and j
=0,
where
at a=O. The first derivative can be calculated using
122
=
J
C
O
EOO1:1 (W ~*
O
~-1 )
a 0*) dN(W ~* ,a0* ''''''00:1'
-
4
whereas the second derivative is zero. The set C 4 is defined as
Then, A 6 =r
A 6 ~ 0 since
-0
JW ~
C4
EoL
- 0
d N(W~; 0; f).
We have Al ~O since is a density function and
is positive definite and
0*
W~
~O.
Therefore,
8 fJ 1( a)1
.
aa
a=o
IS
nonnegative.
We wish to compare the test statistic
that is, when H a :
~
i=
I; with
the test with unrestricted alternative,
O. \Ve know that the test statistic is given by
and the power function is a noncentral chi-squared random variable with kpr degrees of
freedom and noncentrality parameter
(J
=a'
I~~:la.
Following the same lines as in
Chapter II, section 3.4, we can express the power function of this test as
Then, using the same argument as in Chapter II, section 3.4, we conclude that
locally more powerful than R.
I;
IS
CHAPTER V
NUMERICAL ILLUSTRATION AND FURTHER RESEARCH
5.1. Introduction
In this chapter we consider a numerical applications of the UI test in a competing
risk and a two-sample situation, with a proportional hazard model as studied in Chapter
II.
Also, discussion of the results and the conclusions of this study are presented here.
In Section 5.2 we present the data
and we discuss and list some the basic estimates
needed for this numerical analysis. In Section 5.3 we present basically the results of the
analysis using a proportional hazard model.
In
Section 5.4 we present some ideas for
future research.
5.2. Numerical illustration
The following data, extracted from Lagakos (1977), are from a lung cancer clinical
trial being conducted by the Eastern Cooperative Oncology Group at that time. Table
1 summarizes the results of 194 patients with squamous cell carcinoma.
Eighty-three
patients died with local spread of disease (cause 1: C=l), 44 died with metastatic spread
of disease (cause 2: C=2) and 67 were censored (C=O).
Two covariates were
considered: Zl = performance status (ambulatory=O, non-ambulatory = 1), Z2 = age in
years.
A treatment was also considered: W =treatment (W =0, W = 1). The failure
times are indicated by T. Lagakos considered an underlying exponential hazard function
for each cause of failure as follows:
124
and
and the estimated the parameters using the full likelihood function.
Among other
hypotheses, he considered the test of overall effect of the treatments on failure time
expressed by
Ho:.B~ =.B~ =0.
He utilized was a log-likelihood test and for the data and
the particular hypothesis of no effect of treatment on failure times, the result was
x2 =2.23
with 2 degree of freedom; thus not significant at 0'=5%.
Table 1: Outcome of 194 squamous cell patients
Zl W Z2 T
C
Zl W Z2 T C
Zl W Z2 T
1
1
1
1
2
2
2
1
2
0
1
0
1
2
1
1
0
1
2
1
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
1
1
0
1
1
0
0
0
0
0
0
1
0
0
0
1
1
0
1
0
1
1
1
0
0
1
1
0
1
0
0
1
0
1
1
1
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
o
0
0
0
0
1
0
0
1
1
1
1
0
0
0
1
1
0
1
0
0
1
0
1
1
1
0
1
0
1
1
0
0
1
1
0
1
1
1
1
1
1
1
1
1
0
1
0
1
1
0
1
1
1
0
61
52
49
58
43
65
68
47
67
44
73
49
46
60
70
60
67
57
68
67
68
50
60
57
54
65
70
62
55
11
11
5
24
7
17
37
59
22
13
41
11
8
1
21
17
40
56
13
4
27
4
20
19
5
14
17
30
1
1
1
1
1
1
1
1
1
0
0
1
1
1
0
0
1
0
1
0
0
1
1
1
0
1
1
1
1
1
69
60
52
55
68
64
66
48
57
48
67
63
52
70
60
54
60
58
68
58
53
58
64
66
74
62
63
49
57
34 2
22 1
39 1
3 2
12 2
10 0
61 0
11 1
30 0
10 2
24 0
10 0
12 1
26 0
12 2
18 0
31 0
4 0
15 2
13 1
5 1
34 2
6 1
39 1
20 2
19 1
12 0
1010
11 2
1
0
1
1
0
1
0
1
1
1
1
1
0
1
1
1
1
1
1
0
0
1
1
0
1
1
1
0
1
60
76
54
59
50
51
65
69
63
57
51
43
69
70
71
55
62
49
62
64
48
48
54
74
39
59
60
52
59
13
1
18
12
30
6
88
14
4
2
21
19
12
14
11
10
29
8
7
15
27
7
19
11
27
7
4
21
9
C
Zl W Z2 T
2
0
0
0
1
2
1
0
0
0
0
1
2
2
2
1
1
1
1
0
1
1
1
1
1
1
1
1
2
0
1
0
0
0
1
1
1
1
1
1
0
0
0
0
0
1
1
1
0
0
1
1
0
0
0
1
1
1
1
1
0
1
1
1
0
1
1
1
1
1
1
1
1
1
0
1
1
1
1
0
0
1
1
1
1
1
1
74
76
52
39
71
61
42
66
66
68
71
61
74
75
71
50
63
57
39
61
53
49
38
48
55
62
57
61
63
25
7
12
18
35
2
12
3
11
31
55
14
13
17
63
60
13
11
1
13
7
8
19
53
15
38
2
3
26
C
1
0
0
1
1
1
1
2
0
0
2
1
2
1
2
0
2
0
1
1
1
2
0
1
2
0
0
1
e-
125
Table 1: Outcome of 194 squamous cell patients (Continued)
Zl W Z2 T
C
Zl W Z2 T
1
1
0
1
0
0
1
1
0
0
0
0
1
1
0
1
1
1
1
0
2
1
1
0
2
1
2
0
2
0
2
0
2
2
0
2
1
0
1
1
1
1
0
0
0
0
1
1
1
0
0
1
1
1
1
1
1
0
1
0
1
1
1
0
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
0
62
48
53
60
65
44
64
43
65
60
55
42
46
67
59
62
66
50
56
44
21
8
5
14
11
11
8
1
9
14
24
33
84
16
28
29
22
15
9
38
1
1
1
1
0
0
1
1
1
1
1
1
1
1
0
1
0
1
0
0
52
64
79
58
57
48
73
60
43
64
59
70
59
60
71
44
48
59
76
60
4
17
36
25
34
12
21
5
9
62
9
1
2
7
21
28
14
41
34
20
C
Zl W Z2 T
1
2
1
2
0
0
1
1
1
1
1
1
0
1
1
0
0
0
1
1
0
1
0
0
0
0
0
1
1
0
0
0
0
1
0
0
2
1
1
1
1
1
1
1
0
0
1
1
1
0
1
1
1
1
1
1
0
0
52
66
59
65
41
71
62
54
57
72
63
74
58
58
49
58
59
73
61
40
6
27
8
8
34
12
78
11
41
6
14
2
3
21
45
9
5
4
C
Zl W Z2 T
C
1
0
0
1
1
1
1
1
1
0
0
1
2
1
1
0
1
1
0
0
1
0
1
1
0
0
1
0
0
0
0
0
0
1
0
1
1
0
1
2
1
0
1
0
0
0
0
0
0
1
2
0
1
1
0
2
2
1
1
1
0
1
1
1
1
1
0
0
1
0
1
1
1
1
1
1
59
46
64
67
71
57
52
75
44
49
38
69
71
64
68
64
60
70
39
15
11
10
4
5
11
20
18
10
11
7
15
32
34
22
18
6
5
22
5.3 Results
For the proportional hazard model (2.3) defined in Chapter II, we define for the
data above n=194 and k=2. The parameters are then fj~=(1'~, ~~), 1'~ =(-YiO' lil' I;:?)
and ~~ =(~;o, ~il' ~;2)' and the hypotheses are Ho: ~=O against Ha : ~>O.
Since the data present tied failure times, Breslow's (1974) likelihood is used instead
of the one defined in (1.2), Chapter II. The likelihood is
L(fj) =
TIr TIk
exp«()~ SO(
{
j=1 i=1 (
L:
• • J0»
exp«()~
}
(5.1 )
zu) )d;
bER{t;W}
where si(j) indicates the sum of covariates corresponding to the ith cause of failure and
jth failure time, that is
Si(j)=~
Zi(j)' d; is the number of ties in the jth failure time.
J
For the study we reparameterize the model above considering
126
where 1'~=(-'YiO 1'iI 1'i2)' with 1'iO=O are the nuisance parameters related to performance
status and age and ~:=(~iO ~il ~;2)' are the parameters related to the difference of
the control and
treatment groups and z'=(l
performance status and z2 is age.
ZI
Z2)', where ZI
IS the variable
Since the likelihood can be factored into different
causes and there are no common parameters between the two causes, and there is no
commom paramaters between the two causes, two separate analyses can be performed
analyzing each cause of failure separately and treating other causes as censoring at that
time.
The maXImum likelihood estimates of l' under the null hypothesis H o:
~=O
are
calculated solving the system of equations defined by the derivatives of the log of the
likelihood above, similar to those defined in (2.1.1), in Chapter II,
iterative method of Newton-Raphson.
with
~=O
by the
e-
SAS Proc PHGLM was utilized to calculated
these estimators. They are:
-,
l'
-'
-')
( 1'1 1'2
(5.2)
where
Table 2
Performance status
Age
(1)
111 = 0.5783
112=-0.0198
Spread (2)
121 = 0.3282
122= 0.0161
Cause
Local
The scores functions
U~
a.nd the information matrix I(fJ) are calcula.ted using the
•
127
likelihood above and are similar to those defined in (2.1.22) and (2.1.23), Chapter II.
They are calculated by a program written in FORTRAN ( Appendix), resulting in:
-,
U ~ = (5.0545 4.1740 265.6550 2.6408 3.4305
IC/I)
lOX 10
=[
I"
I,~
I~,
I~~
170.3039)'
(5.3)
(5.4 )
J
where the submatrices are given below:
19.3717
41.9168
o
o
41.9168
7924.4219
o
o
o
o
9.4544
25.8006
o
o
25.8006
3678.1194
(5.5 )
I ~~ is a block diagonal matrix with the following submatrices in the main diagonal
(the superscript indicates cause of failure):
(1)
1~~
(2)
1~~
Also
I'Y~
_
-
_
-
15.97
5.42
930.83
5.42
14.96
360.99
930.83
360.99
59988.43
8.00
2.32
487.66
2.32
6.90
162.71
487.66
162.71
32365.12
is a block diagonal matrix with the following matrices in the diagonal:
(5.6)
(5.7)
I~~
= [
I~~=
[
-3.22
12.22
-141.36
12.22
47.85
6434.86
-1.79
5.84
-84.69
-2.17
20.83
2515.83
128
]
]
(5.8)
(5.9)
and finally l Do ')' is a 6x 4 submatrix transpose of I')'Do'
The next objective is to calculate the union-intersection statistic given in (2.2.20),
Chapter II. First, the matrix I DoDo:')' is calculated using the su bmatrices above as
which is a block diagonal matrix with the following submatrices in the main diagonal.
1(1)
15.38
7.40
890.79
7.40
7.19
431.77
890.79
431.77
53156.6
7.66
3.42
469.53
3.42
3.29
211.33
469.53
211.33
29514.9
_
DoDo:')'-
--
and
1(2)
_
DoDo:')' -
The scores
W Do
are calculated as defined in (2.2.11) in Chapter II
and result
III
the
following vector:
- , = (1.13
W Do
and
0.49
-0.018
-0.74
1.27
0.0085)
,
p
129
Both
W ~ and r are partitioned according to
0~j ~P, where P={1,2, ..., p}, p=k x
- *~(j) is calculated as
(r+ 1)=6, the former as defined in (2.2.12). The vector of scores W
defined in (2.2.16) in Chapter II for each j, 0~j ~P. Observe that there exist 2 6 =64
such partitions and the proposed union-intersection test is calculated only on the term
for which both I(W~(j»O) and I(r;;
W ~c.1)~0)
are one. The calculations of these two
indicators are shown below:
Table 3
j
{I}
{2}
j
{2,3,4,5,6}
{1,3,4,5,6}
- *~(j)
W
[0.33]
[0.058]
J
1.74
-26.95
2.64
3.43
170.3
{2,3,5}
0.76
15.15
2.64
3.43
170.30
{2,3,6}
r
{3}
{1,2,4,5,6}
[0.0050]
{1,4,6}
1.04
{1,4,5}
{2,4,5}
{1,3,6}
170.3
-57.35
{4}
{1,2,3,5,6}
[4.80]
0.00
0.00
0.00
-0.174
-0.082
{1,3,5}
2.21
[ -0.23
0.58] [0.76]
15.15
1.28
{2,4,6}
-50.27
[055]
0.0006 [051
-0.068 ]
0.006
0.60
l2.02
3.43
[0.0006
055 ][ -0.92
0.51 ]
6.29
[058]
[0.76]
1.24
15.15
-0.003
-0.14
130
Table 3 (Continued)
j
{5}
{6}
{1,2}
{1,3}
{1,2,3,4,6}
{1,2,3,4,5}
{3,4,5,6}
{2,4,5,6}
[1.04]
[0.0057]
[ 0.09 ]
0.48
[ 1.31 ]
-0.017
5.04
4.17
265.66
-0.93
-50.27
{2,5,6}
5.05
4.18
265.66
-0.068
2.21
{3,4,5}
[-28.56]
2.64
3.43
170.3
[ 2.64
1.79
]
3.43
{1,3,4}
{2,4,5,6}
[ 1.31 ]
-0.017
[ 2.64
1.79
]
3.43
{1,2,6}
{3,4,6}
{1,2,5}
{1,2,4}
{1,2,4}
{2,3,5,6}
[ 0.32 ]
0.34
-26.95
J
2.25
{4,5,6}
{1,2,3}
8.47
{1,5}
{2,3,4,6}
[ 0.32 ]
1.05
[-26.95
1.74 ]
-0.93
-50.27
{1,2,3,4}
{5,6}
2.23
[0005]
[0.60]
1.25
2.02
-0.14
[0.005]
[060]
1.25
2.02
-0.003
[ l.74 ;
6.29
[0005]
[060]
-0.36
2.02
-0.003
{3,5,6}
-0.14
[0005]
[060]
-0.23
2.02
0.011
{3,5,6}
A
~())
[ 0.58]
[015.15
76 ]
1.24
1.28
170.3
{1,4}
)J
-0.003
170.3
{1,3}
r-- W
j
j
-0.14
[-0.74
] [ 4.17
505 ]
1.27
0.008
265.7
0.50 ]
[ 1.13
[ 2.25 ]
8.47
-0.018
0.35
e-
131
Table 3 (Continued)
j
j
{1,6}
{2,3,4,5}
{2,3}
{1,4,5,6}
[ 0.55 ]
0.0006
{2,4}
{1,3,5,6}
[ 0.58 ]
0.34
{2,5}
{1,3,4,6}
{2,6}
{1,3,4,5}
[ 0.58 ]
0.058
{3,4}
{1,2,5,6}
[0.005 ]
0.34
{3,5}
{1,2,4,6}
j
1.74 ]
[ 0.32 ] [ -26.95
0.006
-0.07
2.21
*
w- ~(j)
[ 0.50
1.13 ] [ -0.93 ]
-0.018
-50.27
1.04
{1,2,3,5}
{4,6}
[ 0.51
2.64 ]
3.43
170.3
{1,2,3,6}
{4,5}
[ 15.15
0.76 ]
2.25
8.47
{1,2,4,5}
{3,6}
[0.098
0.48 ] [28.56 ]
-0.23
6.29
1.28
{1,2,4,6}
{3,5}
[0.098
0.48 ] [-28.56]
-0.35
2.23
0.011
[ 15.15
0.76 ]
-0.07
2.21
{1,2,5,6}
{3,4}
[ 0.60
2.02 ]
2.25
8.47
{1,3,4,5}
{2,6}
[ 0.005 ] [ 0.60
2.02 ]
1.04
-0.93
-50.27
{1,3,4,6}
{2,5}
15.15 ]
[ 0.58 ] [076
1.04
-0.93
-50.27
0.50
]
[ -0.018
113 ] [ -0.07
2.21
0.006
0.48 ] [-28.55J
[0.098
-0.14
1.25
-0.003
[ -0.016
1.31 ] [ 1.80 ]
-0.23
6.29
1.28
[ -0.016
1.31 ] [ 1.80 ]
2.23
-0.36
0.011
132
Table 3 (Continued)
j
J
j
{3,6}
{1,2,4,5}
[0.005 ]
0.006
[ 0.60
2.02 ]
-0.07
2.21
{1,3,5,6}
{2,4}
{4,5}
{1,2,3,6}
4.17 ]
[ -0.23 ] [ 5.05
1.28
265.7
6.29
{1,4,5,6}
{2,3}
[ -0.74
1.36 ] [ 1.74 ]
1.27
-26.95
0.008
{4,6}
{1,2,3,5}
[ -0.36 ]
0.01
[ 5.05
4.17 ]
265.7
2.23
{2,3,4,5}
{1,6}
[ 0.0006
0.55 ]
[ 0.51 ]
-0.23
6.29
1.28
{5,6}
{1,2,3,4}
{2,3,4,6}
{1,5}
[ 0.0006
0.55 ]
[ 0.51 ]
-0.36
2.23
0.011
505 ]
[ 1.25 ] [ 4.17
-0.003
265.7
-0.14
[ -0.016
1.31 ] [ 1.80 ]
1.25
-0.14
-0.003
{1,2,3} {4,5,6}
[ 5.24 ] [ 2.64 ]
-0.17
3.43
-0.08
170.3
{2,3,5,6}
{1,4}
[ 0.0006
0.55 ]
1.25
-0.003
{1,2,4} {3,5,6}
[ 0.09]
["28.56]
0.50
2.25
0.34
8.47
{2,4,5,6}
{1,3}
[ -0.74
0.58 ]
0.76
1.27
[ 15.15 ]
0.008
[ 0.51 ]
-0.14
e-
133
Table 3 (Continued)
j
j
{1,2,5} {3,4,6}
j
[0.097]
[-28.56]
0.48
-0.93
1.04
{1,2,6} {3,4,5}
{1,3,4} {2,5,6}
{1,3,5} {2,4,6}
1.05
{1,3,6} {2,4,5}
0.006
{2,3,4} {1,5,6}
4.80
{1,2,3,4,6} {5}
1.13
0.50
-0.018
-0.36
0.011
[2.23]
{1,2,3,5,6} {4}
1.13
0.50
-0.018
1.25
-0.003
[-0.14]
{1,2,4,5,6} {3}
0.097
0.47
-0.74
1.27
0.008
[1.80]
{1,3,4,5,6} {2}
1.13
-0.017
-0.74
1.27
0.008
[1.80]
2.21
[0061]
[0.17]
0.068
50.83
-50.35
0.60 ]
2.02
[6.29]
-50.27
[-0.017
1.31 ] [1.80]
-0.07
1.27
0.008
{1,2,3,4,5} {6}
8.47
[-0.017
1.31 ][ -0.93
1.80 ]
[0.005]
-0.74
[
1.13
0.50
-0.019
-0.23
1.28
2.21
[-0.017
1.31 ][ 2.25
1.80 ]
0.34
{1,2}
-50.27
[0.097]
[-28.56]
0.48
-0.07
0.005
{3,4,5,6}
-*
W.:l(j)
134
Table 3 (Continued)
j
j
{1,4,5} {2,3,6}
j
[ -0.22
0.32 ] [ -26.95
1.74 ]
1.30
{1,4,6} {2,3,5}
~
[ -0.36
033 ] [ -26.95
1.74
0.011
{1,5,6} {2,3,4}
1.25
[ 033
{2,3,4,5,6} {I}
6.29
2.23
J
74
~
-26.95
[1.
-0.003
-0.14
0
0.55
0.0006
-0.74
1.27
0.008
[0.51]
0.061
0.068
11.93
-0.113
-0.11
[0.17]
{1,2,3,4,5,6}-
{1,2,3,4,5,6! }
e·
The union-intersection test statistic as given in (2.2.20) is calculated using only the
term J ={1,2,5} since the terms I{W~(j»O} and I{r
Thus the test resulted
n
W t.() <OJ are both equal to 1.
Ii =6.07676.
The next step is to calculate the P-value given in (2.3.1), Chapter II. The problem
is to calculate the weights w k , where k is the cardinality of J, in our case, k=l, ... , 6.
For k up to 4, the weights of the chi-bar distribution can
be calculated by formulas.
For instance, Cramer (1946, page 290) give the expression for calculating the probability
of both variates from a bivariate normal distribution being nonnegative (J ={1,2}) as
o
p(W > 0)=
0
JJ
f(x,y) dx dy
= ~ + in arc sinp,
-00 -00
where p is the correlation between the two variates.
The probability of both being
135
negative
p{r
n WJ
~ O} is the same as above by symmetry.
The probability of the
first being positive and second being negative (J={l}) is given by ~ -
in sin-1(p).
For
k=3 and k=4, Barlow et al. (1972, page 134-142) present a summary discussion of the
available methods for calculating the weights of the chi-bar distribution when
k~4.
For k>4 the calculation of the weights has to be performed by utilizing simulation
in a computer.
For W
...... N w (0,
1
r~~:/),
multivariate normal replicates and calculate
with w=6, we generate a few thousand
W~
~
r c eWe
and
)J)
for each normal replicate
in order to estimate the orthant probabilities for these variables.
obtained by calculating the proportions which lie in the orthant.
The estimates are
It was necessary to
generate 5000 replicates to obtain the sum of the weights equal to one. The resulting
weights were:
k
weights w k
0
0.1273
1
0.352209
2
0.346452
3
0.146520
4
0.026784
5
0.001190
6
0.000000
Thus, the P-value is calculated as
P=l-{l x 0.1273
+
+
0.986303 x 0.352209
0.892065 x 0.146520
x 0.001190
+
+
+
0.952088 x 0.346452
0.806511 x 0.026784
0.585353 x 0.00000 = 0.0418346
For the unrestricted alternative the score test resulted
+
0.701179
136
-
-
-
1
-
Q=W I~~:"Y W = U r~~:"Y U = 6.85497
and the P-value was P =0.335.
For the next step, the asymptotic power of the two tests was investigated. Since in
the study of the power comparisons in Chapter II, section 2.4, the expression for the
slope of the power curve of the proposed test is not tractable, we supplement the study
with a numerical study.
Utilizing the first 1000 from the 5000 six-variate normal repli-
cates generated for calculating the weights, we calculate new vector by adding each time
a combination of nonnegative numbers for each vector of the scores defined as W. The
value of the union-intersection test statistic
Ii
and Q and their P-values are calculated
and the proportions of times they exceed their respective critical value are observed.
Table 4: Proportions of rejection,
Q
1*1
= 5%.
Q
(0.1
0.0
0.0
0.1
0.0
0.0)
0.101
0.056
(0.3
0.0
0.0
0.3
0.0
0.0)
0.311
0.142
(0.5
0.0
0.0
0.5
0.0
0.0)
0.801
0.472
(1.0
0.0
0.0
1.0
0.0
0.0)
0.993
0.966
(1.5
0.0
0.0
1.5
0.0
0.0)
1.000
1.000
(0.0
0.1
0.0
0.0
0.1
0.0)
0.077
0.053
(0.0
0..3
0.0
0.0
0.3
0.0)
0.180
0.083
(0.0
0.5
0.0
0.0
0.5
0.0)
0.381
0.173
(0.0
1.0
0.0
0.0
1.0
0.0)
0.883
0.681
(0.0
1.5
0.0
0.0
1.5
0.0)
0.998
0.967
(0.0
2.0
0.0
0.0
2.0
0.0)
1.000
1.000
(0.0
0.0
0.01
0.0
0.0
0.01)
0.808
0.545
(0.0
0.0
0.1
0.0
0.0
0.1)
1.000
1.000
(0.1
0.1
0.0
0.1
0.1
0.0)
0.151
0.062
(0.3
0.3
0.0
0.3
0.3
0.0)
0.632
0.331
(0.5
0.5
0.0
0.5
0.5
0.0)
0.963
0.804
e·
137
Table 4 (Continued)
Ii
Q
(1.0
1.0
0.0
1.0
1.0
0.0)
1.000
1.000
(0.1
0.1
0.0
0.0
0.0
0.0)
0.104
0.055
(0.3
0.3
0.0
0.0
0.0
0.0)
0.430
0.207
(0.5
0.5
0.0
0.0
0.0
0.0)
0.821
0.601
(1.0
1.0
0.0
0.0
0.0
0.0)
1.000
1.000
(0.0
0.005 0.005 0.0
0.005 0.005)
0.317
0.143
(0.0
0.01
0.01
0.0
0.01
0.01)
0.i34
0.552
(0.0
0.03
0.03
0.0
0.03
0.03)
1.000
1.000
The proportions of rejections from l~ are nearly twice than the proportions of
rejections from Q in almost all combinations of il studied. Therefore, for H o: il=O, the
test statistic l~ is more porwerful than the test Q, that is, the test with restricted
alternative is more powerful than the test with unrestricted alternative.
5.4. Sugestions for Further Research
In
this
dissertation, we have considered some comparative studies of two
subpopulations in a competing risk situation using some well known analysis models,
such as semiparametric and parametric model.
have also been considered.
studied.
Alternative models for such problems
Extensions to more than two subpopulations have been
We have shown that a restricted alternative test is more powerfull than the
corresponding unrestricted alternative test.
There is a need for more research in this
area, exploring both the applied and theoretical aspects.
further research we intend to undertake in the future:
Following is a synopsis of
138
1.
Numerical comparisons of two survival functions related to the bivariate
exponential model (Sarkar, 1987) and logistic model in a competing risk setting
with two causes of failure are intended for future work.
Also, numerical
application for case of more than two subpopulations in a competing risk
situation should be performed, using a proportional hazard model.
2.
Derive a union-intersection test statistic for comparisons of survival functions of
two subpopulations in a parametric regression model using full likelihood and
considering a exponential family of distributions for the underlying survival
distributions.
with
Efficiency of proportional hazard regression model as compared
parametric regressions
model
can
be of interest
in
assessing how
semiparametric likelihood model behave compared with full likelihood model.
3.
Derive union-intersection test statistic in the semiparametric Cox model as in
Chapter II, for other hypotheses, in a competing risk situation with two causes
of failure.
4.
The estimation of the parameter in the case of rejection of the null hypothesis
may be of interest. This problem is called "estimation after preliminary tests"
and has been investigated in many studies. Basically, this study consists on the
use of different estimation procedures for the parameter in question for the
cases of significant or not significant parameters.
•
139
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III
APPENDIX
FORTRAN Program for calculating the Scores Functions
U~
and the
Informatiom Matrix I(P) using Proportional Hazard Function as given in (2.2),
Chapter II and using Breslow's Likelihood for Tied Data.
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
CALCULATING
U~
AND I(P)
X(I) PERFORMANCE STATUS
X(2) AGE IN YEARS
W
TREATMENT
T
FAILURE TIME
C
CAUSE OF FAILURE
X(3) TREATMENT
X(4) PERFORMANCE STATUS X TREATMENT
X(5) AGE X TREATMENT
G(I) ESTIMATES OF "'I UNDER H o
DELlINDICATOR FOR CAUSE I
DEL21NDICATOR FOR CAUSE 2
TI
DESCENDING ORDERED FAILURE TIME FOR CAUSE I
IMPLICIT REAL*8 (A-Z)
INTEGER I,J
REAL X(5),G(IO)
DOUBLE PRECISION Qll,QI2,QI3,QI4,QI5,Q111,QI12,QI13,QI14,QI15,
1QI21 ,QI22,QI23,QI24 ,QI25,QI31 ,QI32,QI33,QI34,QI35,QI41 ,QI42,
2QI43,QI44,QI45,QI51 ,QI52,QI53,QI54,QI55,U11, U2, U3, U4, U5,111,112,
3113,114,115,121,122,123,124,125,131 ,132,133,134 ,135,141,142,143,
4144,145,151,152,153,154,155
C
C
C
SET VALUES FOR THE ESTIMATES G(I),1=3,4,5
G(3)=O.O
G(4)=O.O
G(5)=O.O
C
C
C
SET SAMPLE SIZE N
N=194
C
C
C
INITIALIZE HT=Tl
HT=-l.O
C
C
INITIALIZE SUM(lJ), J=I,5, THE SUM OF X EXP{BZ} OVER ALL RISK
SET.
C
SUMll=O.O
SUM12=O.O
SUM13=O.O
SUM14=O.O
•
SUM15=0.0
C
C
C
INITIALIZE SUM2, THE SUM OF EXP{BZ} OVER ALL RISK SET.
SUM2=0.0
C
C
C
INITIALIZE THE SCORES FUNCTIONS
U1=0.0
U2=O.O
U3=0.0
U4=0.0
U5=0.0
C
C
C
INITIALIZE THE ELEMENTS OF THE INFORMATION MATRIX.
111=0.0
112=0.0
113=0.0
114=0.0
115=0.0
121=0.0
122=0.0
123=0.0
124=0.0
125=0.0
131=0.0
132=0.0
133=0.0
134=0.0
135=0.0
141=0.0
142=0.0
143=0.0
144=0.0
145=0.0
151=0.0
152=0.0
153=0.0
154=0.0
155=0.0
C
C
C
C
INITIALIZE SUMILM, L,M=1,5, THE SUM OF X 2 EXP{BZ} OVER ALL
RISK SET.
SUMI11=0.0
SUMI12=0.0
SUMI13=0.0
SUMI14=0.0
SUMI15=0.0
SUMI21=0.0
SUMI22=0.0
c
C
C
SUMI23=0.0
SUMI24=0.0
SUMI25=0.0
SUMI31=0.0
SUMI32=0.0
SUMI33=0.0
SUMI34=0.0
SUMI35=0.0
SUMI41=0.0
SUMI42=0.0
SUMI43=0.0
SUMI44=0.0
SUMI45=0.0
SUMI51=0.0
SUMI52=0.0
SUMI53=0.0
SUMI54=0.0
SUMI55=0.0
READ THE ESTIMATES
r
OUTPUT FROM SAS PHGLM
20 READ (7,200) G(I),G(2)
200 FORMAT(2Fl1.8)
C
C
C
READ DATA AND CALCULATE THE SCORES UI, 1=1,5 AND
THE INFORMATION MATRIX IMN, M,N=1,5.
C
DO 400 1=1, N
10 READ (5,100,END=990) X(1),W,X(2),T,C,X(3),X(4),X(5),
9DELl,DEL2,Tl
100 FORMAT (2F2.0,F3.0,F4.0,3F2.0,F3.0,2F2.0,F6.l)
BZ=O
DO 300 J=I, 5
BZ=BZ+G(J)*X(J)
300 CONTINUE
SUMll=SUMll+X(I)*EXP(BZ)
SUM12=SUMI2+X(2)*EXP(BZ)
SUM13=SUMI3+X(3)*EXP(BZ)
SUM14=SUM14+X(4)*EXP(BZ)
SUM15=SUM15+X(5)*EXP(BZ)
SUMIll=SUMlll+X(l)*X(l)*EXP(BZ)
SUMI12=SUMI12+X(1)*X(2)*EXP(BZ)
SUMI13=SUMI13+X(1)*X(3)*EXP(BZ)
SUMI14=SUMI14+X(1)*X(4)*EXP(BZ)
SUMI15=SU MI15+X( 1)*X(5)*EXP(BZ)
SUMI21=SUMI21+X(2)*X(1)*EXP(BZ)
SUMI22=SUMI22+X(2)*X(2)*EXP(BZ)
SUMI23=SUMI23+X(2)*X(3)*EXP(BZ)
SU MI24=SUMI24+X(2)*X(4)*EXP(BZ)
SUMI25=SUMI25+X(2)*X(5)*EXP(BZ)
SUMI31=SUMI31+X(3)*X(1)*EXP(BZ)
SUMI32=SUMI32+X(3)*X(2)*EXP(BZ)
SUMI33=SUMI33+X(3)*X(3)*EXP(BZ)
SUMI34=SUMI34+X(3)*X(4)*EXP(BZ)
SUMI35=SUMI35+X(3)*X(5)*EXP(BZ)
SUMI41=SUMI41+X(4)*X(1)*EXP(BZ)
SUMI42=SUMI42+X(4)*X(2)*EXP(BZ)
SUMI43=SUMI43+X(4)*X(3)*EXP(BZ)
SUMI44=SUMI44+X(4)*X(4)*EXP(BZ)
SUMI45=SUMI45+X(4)*X(5)*EXP(BZ)
SUMI51=SUMI51+X(5)*X(I)*EXP(BZ)
SUMI52=SUMI52+X(5)*X(2)*EXP(BZ)
SUMI53=SUMI53+X(5)*X(3)*EXP(BZ)
SUMI54=SUMI54+X(5)*X(4)*EXP(BZ)
SUMI55=SUMI55+X(5)*X(5)*EXP(BZ)
SUM2=SUM2+EXP(BZ)
IF (DELI .GE. 1.0) THEN
Qll=SUMll/SUM2
QI2=SUMI2/SUM2
QI3=SUMI3/SUM2
QI4=SUMI4/SUM2
QI5=SUMI5/SUM2
Qlll=SUMlll/SUM2
QI12=SUMI12/SUM2
QI13=SUMI13/SUM2
QI14=SUMI14/SUM2
QI15=SUMI15/SUM2
QI21=SUMI21/SUM2
QI22=SUMI22/SUM2
QI23=SUMI23/SUM2
QI24=SUMI24/SUM2
QI25=SUMI25/SUM2
QI31=SUMI31/SUM2
QI32=SUMI32/SUM2
QI33=SUMI33/SUM2
QI34=SUMI34/SUM2
QI35=SUMI35/SUM2
QI41=SUMI41/SUM2
QI42=SUMI42/SUM2
QI43=SUMI43/SUM2
QI44=SUMI44/SUM2
QI45=SUMI45/SUM2
QI51=SUMI51/SUM2
QI52=SUMI52/SUM2
QI53=SUMI53/SUM2
QI54=SUMI54/SUM2
QI55=SUMI55/SUM2
IF (T1.EQ.HT) THEN
DELl=DEL1+HDELl
ENDIF
UI=Ul+X(l)-DELhQll
U2= U2+X(2)-DELhQ12
U3= U3+X(3)-DELhQ13
U4=U4+X(4)-DELhQ14
U5=U5+X(5)-DELI*QI5
111=I1I+DELh(QI1I-QlhQII)
112=I12+DELh(QI12-QlhQI2)
113=113+DEL1*(QI13-Q11 *Q13)
114=114+DELI *( QI14-Q 11 *Q14)
I15=I15+DEL1*(QI15-Q1hQ15)
121=121+DELh(QI21-QI2*Q11)
122=122+DEL1*(QI22-Q12*Q12)
123=123+DELh(QI23-Q12*Q13)
124=124+DEL1*(QI24-Q12*Q14)
125=125+ DELI *( QI25-Q12* Q15)
131 =131+DELh(QI31-Q13*Q11)
132=132+DEL1*(QI32-Q13*Q"12)
133=133+DELI*(QI33-Q13*Q13)
134=I34+DEL1*(QI34-Q13*Q14)
135=I35+DEL1*(QI35-Q13*Q15)
141=I41+DELI*(QI41-Q14*Q11)
142=I42+DELh(QI42-Q14*Q12)
143=143+DELh(QI43-Q14*Q13)
144=I44+DELh(QI44-Q14*Q14)
I45=I45+DEL1*(QI45-Q14*Q15)
151 =151+DELh(QI51-Q15*Q11)
152=152+DEL1*(QI52-Q15*Q12)
153=153+DEL1*(QI53-Q15*Q13)
154=154+ DELh( Q154-Q 15*Q 14)
155=155+DELh(QI55-Q15*Q15)
IF (Tl.EQ.HT) THEN
U1=U1+HDELhHQll
U2=U2+HDELhHQ12
U3=U3+HDELhHQ13
U4=U4+HDELhHQ14
U5=U5+HDELl*HQ15
I11=111-HDELl*(HQI11-H Qll*HQll)
112=112-H DELI *(H Q112-H Ql1*H Q12)
113=113-HDELI*(HQI13-HQ11*HQ13)
114=114- HDELI *(H QI 14- HQ 11 *HQ 14)
115=115-HDELI *( HQI15-H Q11*HQ15)
121=121-HDELI *(H QI21-H Q12*H Qll)
122=122-HDEL1*(HQI22-HQ12*HQ12)
123=I23-HDELl*(HQI23-HQ12*HQ13)
124=124-HDELl*(HQI24-HQ12*HQ14)
125=I25-HDEL1*(HQI25-HQ12*HQ15)
131=I31-HDELh(HQI31-HQ13*HQ11)
132=I32-HDELI*(HQI32-HQ13*HQ12)
133=133-HDELl*(HQI33-HQ13*HQ13)
134=134-HDELl*(HQI34-HQ13*HQ14)
135=135-HDELh(HQI35-HQ13*HQ15)
141=I41-HDELh(HQI41-HQ14*HQ11)
142=142-HDEL1*(HQI42-HQI4*HQ12)
143=143-HDELI*(HQI43-HQ14*HQ13)
144=144-HDEL1*(HQI44-HQ14*HQ14)
145=I45-HDEL1*(HQI45-HQ14*HQI5)
151=151-HDELI*(HQI51-HQ15*HQll)
152=152-HDELh(HQI52-HQ15*HQ12)
153=I53-HDEL1*(HQI53-HQ15*HQ13)
154=I54-HDEL1*(HQI54-HQ15*HQ14)
155=I55-HDELl*(HQI55-HQ15*HQ15)
c
C
C
ENDIF
HOLD THE RATIOS FOR THE NEXT ITERATION
HT=T1
HQll=Qll
HQ12=Q12
HQ13=Q13
HQ14=Q14
HQ15=Q15
HQI11=QI11
HQI12=QI12
HQI13=QI13
HQI14=QI14
HQI15=QI15
HQ121=Q121
HQ122=Q122
HQ123=Q123
HQ124=Q124
HQ125=Q125
HQ131=Q131
HQ132=Q132
HQ133=Q133
HQ134=Q134
HQ135=Q135
HQ141=Q141
HQ142=Q142
HQ143=Q143
HQ144=Q144
HQ145=Q145
HQ151=Q151
HQ152=Q152
HQ153=Q153
HQ154=Q154
HQ155=Q155
HDEL1=DEL1
ENDIF
400 CONTINUE
C
C
C
C
OUTPUT THE SCORES AND THE ELEMENTS OF INFORMATION
MATRIX
WRITE(15,*) U1,U2,U3,U4,U5
WRITE(16,*) 111,112,113,114,115,121,122,123,124,125,131,132,133,
1134,135,141,142,143,144,145,151,152,153,154,155
990 STOP
END