•
•
THE COX REGRESSION MODEL, INVARIANCE PRINCIPLES FOR SOME INDUCED
QUANTILE PROCESSES AND SOME REPEATED SIGNIFICANCE TESTS
by
PRANAB KUMAR SEN
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1208
.
,.
r
January 1979
THE COX REGRESSION MODEL, INVARIANCE PRINCIPLES FOR SOME INDUCED
QUANTILE PROCESSES AND SOME REPEATED SIGNIFICANCE TESTS*
PRANAB KUMAR SEN
University of North Carolina, Chapel Hill, NC
•
ABSTRACT
For the Cox regression model, the partial likelihood functions
involve linear combinations of induced order statistics.
Some in-
variance principles pertaining to such linear combinations of induced
order statistics are studied and the theory is incorporated in the
formulation of suitable repeated significance tests (for the hypothesis
-e
of no regression) based on these partial likelihoods.
•
properties of these tests are also studied .
AMS Subject Classification Nos:
Key Words and Phrases:
Asymptotic power
60B10, 60FOS, 62G99.
Analysis of covariance, asymptotic power,
contiguity of probability measures, Cox regression model, hazard rate,
induced order statistics, quantile processes, repeated significance
tests, survival functions, Wiener process.
* Work supported by the National Heart, Lung and Blood Institute,
Contract-NIH-NHLBI ...71-2243 from the National Institutes of Health.
2
1.
INTRODUCTION
I
14
In the Cox (1972) regression model for survival data, it is
assumed that the i.th subject (having sur1)i1JaZ arne
covariates
(given
Z.
~1
for some
and a set of
has the hazaY'd rate
p? 1)
=z.)
~1
(1.1)
hi (t)
where
Y.1
ho(t),
=hO(t) exp (~' .ti) ,
the hazard rate for
z.
~1
i
=0,
=1 ~
.•• , n,
is an unknown, arbitrary
~
B ) parameterizes the
p
regression of survival time on the covariates. We assume that
the
Yi
t
'
~'= ((31' " ' ,
non-negative function and
is continuous in
.
almost everywhere
hO(t)
(a.e.), so that ties among
may be neglected, with probability 1.
Let
~ = (QI' ••. ,~)',
the vector of anti-ranks, be defined by
Y
(1. 2)
Q1.
where
Y
nl
< ... <Y
nn
=Yn1.,
1:::; i :::; n ,
are the order statistics corresponding to
follow-up (censoring), the partial (log-) likelihood function when all
the failures have been observed [c.f. Cox (1972, 1975)] is given by
logLn =L:=l{~'~i -lOg[Lj=iexp{J!'~j}J} .
(1.3)
To incorporate censoring, Cox (1972) considered a more general
case where there are
m(~
1) ordered failures
(a subset of
t , ..• ,t
l
m
such that at time t. - 0, there is a risk set R. of r.
n1.L
J
J
J
subjects who have neither failed nor been censored by that time, for
the
Y
i :::;j
~m
(so that
times for the
(1,4)
n
t. =Y *
Qj
J
.
'
YI , ... , Yn · Then, following Bhattacharya (1974), ~l' ..• , ~n are
termed the induced order statistics. In the event of no loss in the
R c ... c R )
mI'
If
WI' "', Wn be the censoring
subjects, then, we have
and
R. = {i:
J
Y~ =Y. AW. ~t.,
1
1
1
J
l:::;i :::;n},
l:::;j :::;m,
3
Q* = (Qi~ ••••
where
~1'
is asub-vector of
a Ab =min(a, b),
the COX (J972) partial (log-) likelihood function is
Then~
(1. S)
logL*nm
=L~=l{~'~*
•
J - ' < j -log(L11.'R i exp{~'~.})}
1
Yi =Vi>
[Note that if m =nand
•
and
Q
l<i':n,
then
L*
nm
=L.
n
A
d'i[Jere/;c~
time version of (1.5) has also been considered by Cox (1972) and we
shall refer to that later on.]
For testing the hypothesis of no regression viz ••
(1.6)
8
~
=0 vs.
~
Cox (1972) considered the test statistic
L*
(1.7)
nm
= U*, J* - U*
""l1.m ""11m •
"'11JII
where
(1.8)
Unm
*
= (a;:y/ .......,
aS)logLnm
*
I
Q
IJ=
~
and~-
•
0'
~
*
J""11m
*
= (a~2/ar<ar<')logL
~~
nm
stands for a generalized inverse of
~.
I
R 0
f;:,=~
•
Cox (1972; 1975,
Section 5) argued heuristically that under
HO' L*nm has asymptotically
chi-square distribution with p degrees of freedom (DF).
In a variety of situations, relating to clinical trials and life-
testing experimentations. one may be interested in monitoring the
study from the beginning with the objective of an early termination if
H in (1.6) is not tenable. Such a plan is known as a progressively
O
censo~ed scheme (PCS) [viz .• Chatterjee and Sen (1973) and Sen (1976.
1979)].
Thus, in a PCS, instead of making a terminal test at the m-th
failure
t
,..
a
t.
J
(j
~
1)
J
m
one may like to review the process at each failure
and stop experimentation as soon as
(defined as in
nj
leads to the reJection of
L*
and J * )
nj
for some j ~ m; if L* 1 • " ' , L*
n
nm are all insignificant, then
is accepted along with the termination of the study at the preplanned
(1.7) - (1.8). but., based on U*.
~nJ
4
m-th failure.
Hence~
a repeated significance testing (RST) procedure
Since the
is involved in a PCS.
L. are neither independent nor
nJ
have independent increments, more elaborate analysis is needed to
find out the critical values having prescribed overall level of
significance,
•
We may note that by (1,5) and (1.8),
(l.9)
U*k
n
=t-l{~*
-r~lIR.Z.}
J~j
J 1£ J~l '
for
k =1, •.. ,m ,
and hence, these are all linear combinations of induced order
statistics.
For this reason, we first study some invariance principles
relating to induced order statistics and then incorporate these in
the study of the asymptotic properties of the proposed RST,
These
induced quantile processes are introduced in Section 2 and their weak
convergence results are presented there.
Parallel results for the
discrete time case are treated in Section 3.
Section 4 is devoted to
the study of these invariance principles under (local) contiguous
alternatives.
The concluding section incorporates all these results
in the study of the asymptotic properties of the proposed RST procedures
relating to the Cox model.
2.
WEAK CONVERGENCE OF SOME INDUCED QUANTILE PROCESSES
By analogy to (l,3), (1.5) and (1.8), we let for every
(2.1)
(2,2)
logLnk =
L~=l{~'~.J -IOg{L~=jexp{~'~.})},
1
I
!lnk = (~/1~) logLnk ~=Q
=L.Ilkj =1{ ~.
.
- (n - J + 1)
J
(2.3)
:!nk = - c~2/]~' )logLnk
=L~=l(n
-j +1)-ICn
-l\'n
L. i =j
~.
}
1
I~=O
-j)~j'
say,
k:
I
~k ~n,
.
I
5
s
where
"'1ln
and
=0
S . =(n _j)-lI~=.(~ -Z-~)(~ -f'.l')"
(2.4)
"1lJ
1
J
.I
'<j
.
~
-l\n
~j=(n~J+l)
(2.5)
1 -:::j <n -1;
'<:i
l.i=j~.,
l$j~n.
1
Conventionally, we let
•
uo=o,
"'1l
(2,6)
Vn~l.
JO=O,
"'1l
~
~
First, we study suitable invariance principles relating to the triangular
arrays
0 ~k ~n;
{!!nk:
n
~ l}
and
0 ~k ~n; n ~ l},
{:!.ok:
~ll
Throughout this paper, the covariates
... ,~
are assumed to
be stochastic vectors; there are some simplifications when these are
nonstochastic and these will be briefly considered later on.
Also, in
the usual custom of an analysis of covariance model, we assume that
~l'
•..
,~
are independent and identically distributed random vectors
(i.i.d.r.v.) with
l!. = E~, [= E(~ -l.9
(2.7)
(~
-l!.)'
(both exist),
and
(2.8)
is positive definite (p.d.) with det. [<00.
[
S 1 = (n - 1) -1\. 1 (Z. - Z HZ. - Z ) ,
For clarification of ideas, we note that
(where -Z =n -ll:n. lZ')'
"'11
1= ~
while for
2
~
J'
l.1
"1l
~
=
~l
~l
"'1l
n - 1,
l
.)-l{\n
Z Z' S - (
l.' 1 Z• Z'. - dL.' l;::.n;::.n
"'1lJ. - n - J
1= ~1~1
1= ~i ~i
(2.9)
Z d- 1 7 J(\n
Z' d- l Z')} '
(n - J. + l)-l(\n
l.i=l~i - l.i=l~.
Li=l~i - l.i-l~.
1
and
S
"1ln
= O.
Thus, if
Zl' ••. , "1l
Z
~
of the experimentation, then
&
~k
are all observable at the beginning
depends only on
hence, is observable at the
(k -l)th
~k
and
depends on
Qk+l' ... ,~).
~l'
.•• ,
~
1
failure, for
Ql' ••. ,Qk
Qi' ..• ,Qk-l
2
~k ~n,
and
Also,
(put not individually on
We are primarily interested in the asymptotic behavior
"'1l
6
of the two partial sequences
Let
0::; k ::; n}
Zl"'"
~
.
for
Z
~T1
is nondecreasing in
B
nk
introduce a sequence
{:lnk ~
and
be the
•.• , Z ,
"11
and
Then
{llnk'
l<,k<n,
t (E
a-field generated
while
Hn 0
=B(Zl'
~,
•.• , Z ).
~11
.
For every n (~ 1) 1 we
k (<::; n) •
{k (t) ~
n
0::; k ::; n} ,
= [0,
lJ}
of nondecreasing and
right-continuous non-negative integers by letting
(2.10)
l J k)::; pt}, t (E.
Trace (J"'11n .....n
kn (t) =max{k:
Let then
r
~n
= {t:~n (t),
t (E}
be defined by
(2.11)
Note that in (2.10) and (2.11), we have assumed that
~
J- 2 = B
""nn "'Il
is defined by
B J B'
"11"11n"11
=~
I
,
.J
"11n
the identify matrix of order
Later on, we shall see that by virture of (2.8),
J
"11n
probability, and it is also possible to replace
k (t)
(where
denotes the largest integer
[sJ
n -~ -~!![ntJ'
r
t
£
E.
is p.d. and
is p.d., in
n
by
t: (t)
::; s) and
p.
~n
[ntJ
by
Then, the main theorem of this section is the
following
Under (2.7), (2.8) and
Theorem 2.Z,
to
and
$.
={$.(t) ,
t
£
E}
t: (j) = {~( j) (t),
f (t ) = (t: ( 1) (t),
where
t
£
j =1, ..• ,p
E},
standard Wiener process on
~ = Q,
H :
O
~ weakl:y.oonveztgea
... , ~ (p) (t)) "
t
£
E
are independent copies of a
E.
Before we present the proof of the theorem, we consider the
following lemmas.
Lerrorza 2.2.
Under C2,7) and HO :
martingaZe (for every
n ~ 1) •
~
=Q,
{!!ok' B
nk
,
O::;k ::;n}
is a
,
,
7
Note that by
Proof:
B _ ,
nk 1
(n-k+l) -1 ,
V 1
Ho'
under
(1, .•• ,n)\(Ql' .•• ,Qk.,.J)
E(~
so that
+l)-lL~=k~..
(n -k
k
~
1t
-1,n
~
Now, given
•
for every
U k ~,U k 1 = ~ - Cn - k + 1)
~ -,
~k
(2.12)
set
(~, 2),
~k
Qk
l.' -k~
~-
•
~!
can take on anyone value in the
with the equal conditional probability
IBkl)=(n-k+l) -l{,n
,k-l} =
L'-lZ'-L'-l~
n 11
1~i
Thus, under
J
HO'
E(l!nk -l!nk_lIBnk_l) =0 a.e.,
Q.E.D.
~k ~n.
H '
O
k
E{!lnk-l!nk-1)(l!nk-!lnk-l)'IBnk - 1 } = nn_i +1 ~k' V
By the same arguments, it follows that under
(2.13)
~k
where the
are defined by (2.4).
!1 (~, J?)
(2.14)
=
l~k~n,
Note that if we let
}(~ - J?) (~ - J?) ,, V~, £ £ RP,
then we have
(2.15)
S
"1lk
= (n
,
<P(Z, Z )
k<'
~.
<
_
-l-J_n ~.1 ~.'
J
- k + 1))
2
1 ~k ~n - 1 •
V
As such, by the same arguments as in the proof of Lemma 2.2, we have
H :
O
under
8 = ,..,0,
,..,
(2.16)
= S ,
E(S
"1lklBnq )
~q
Also, note that
S
~n1
Lerruna 2. J.
ES
"1l 1 =r.
,..,
Under (2.7) and
{Sk"1l
,..,r , Bn k ; l~k::;;n-l}
For a
H :
O
~
=Q,
Let
{k}
n
(2,18)
{11~k .... 1:
II,
but
B ; 1 ~ k ::;; n - 1}
nk
n
which
is a martingale.
-J 2
k -+0
n
as
II~II =max{laijl:
n(~ 2), under
l"i,
H : ~ =Q '
O
a non-negative submartinga1e.
be any sequence of positive numbers
k -+00
n
~
n (~ 2) ,
for every
~=((aij))' we let
pxp matrix
,..,1
This leads us to the following
Then, by Lemma 2.3, we have for every
(2.17)
2 , .•. , 2 ,
is aU-statistic (based on
are LLd.r.v.), so that
.
l~q~k~n'-l
'V
(k
n
~n),
such that
n -+00 •
Then, by (2.17), (2.18) and the generalized Ko1mogorov-inequality for
submartinga1es, under
H :
O
~
= Q and (2.7),
j "p},
8
(2.19)
(2.20)
~
Note that under
(1, ... ,n)
every
k:
~ =
H :
O
E -IE
fl,
II S
k
"7ln-
(Ql"
~ r II ~
n
•. ,~)
~n-k =g(~
E
•
>0 •
takes on each permutation of
(n!)~l. Also, by (2.15), for
with the common probability
1 ~k ~n -1,
V
~
, ... ~ ZQ )
Tl .. k
is aU-statistic
n
[Hoeffding (1948)], so that
(2.21)
E II ~m _k - I" = E II g (~
~ ... , ~ ) - III
n-k
n
=E{E[llg(~
, ... , ~ ) -III IBnOD
n-k
n
=E{(k;lfl.
l~ll<"
I.
'<lk+l~n
11~(~i'1 ""~ik+l ) -rll}
=Ellg(~l' ''''~k+l) -III.
Thus,
EII~n_l -III =EII1!(~l' ~2) -III
•
by (2.7) and
<00
(2.22)
On the other hand,
m ~ 2}
{U (Zl' ... , "1ll
Z ),
~~
is a reverse martingale
sequence, so that by the reverse sub-martingale convergence theorem,
Ilg(~l' ... ,~) - III
converges in the first mean to
Hence, (2.22) converges to
0
as
righthand side of (2.19) is
n
-+00,
(7,23)
l~k~n
II.!. J
=p{ max
l~k~
~p{
max
n "7l
k -
(or
k)
n
-+00.
Consequently, the
0
as
E > 0,
~ r II
n
>d
IIIn £1=
\~ l{(n -i)(n -i +l)-l(S
.
.
"7l1
l~k~n-l
m -+00.
and of (2.20) converges to
Note that, for every
p{ max
n
0 as
-0
+ (n -i +l)-lnll
>d
~
,~
1
Iln-lt=l(S k -I) II >E/2} +p{IIIII >2 En},
1
"7l
9
where the second term on the righthand side of (2.23) converges
to
0
On the other hand, using the inequality that
(2.24)
Iln-lI~_l (5
max
l~k~n-1
..
~
•
. -
1--TI1
D II
k
II~k - r: II} + :
{max
l~k~n-k
max
115 k n-k <k~n-l ~
{
n
r: II}
n
we obtain on using (2.18), (2.19). and the convergence results following
(2.22) that the first term on the righthand side of (2.23) converges to
o as n ....
00.
Lemma 2.4.
This leads us to the following.
Under (2.7) and H :
O
(2.25)
we
max
l~k~n
II n- ~ (J
~
~ = Q.
k - k r)
,.."
II L
0, as n .... 00.
By (2.8) and (2,25),\\Te conclude that under HO: ~=Q,
1
(2.26)
n- :lnn L
and is p.d.~ in probability.
r
Note that by (2.2), with probability I,
max Ilu k -U k-1 11 = max II~
-TI
l<k<
~k
_ _n -TI
_ _n
l <k<
(2.27)
::; 2{ max II~.II} =2Z*,
n
_l_n . 1
1 <'<
where, by (2.7). for every
say
E: >0,
p{Z* >
n
(2.28)
,. (n -k + l)-IL~=l~
II
1 ~.1
E:Tn'l .... 0. as
n .... oo
•
[The proof of (2.28) follows from well known results on the maximum
of n
LLd.r.v. 's and hence is omitted.]
lin
Then, for every
E: >0,
-1 n
I E{(U-TIl. -u-TI1.. I)(U-TIl. -u-TIl. 1)'1(lI u
. 111 >E:rn)IBn1.. III
i=1
-TIl. -u~1-
1 n
=lln- . 1 n
1=
I
_~1.
n
+
1
L (~Z . -1~)(~Z
-f~)'I(II~Z
-1~11
1
j
1
j
1.
.•
J=1.
J
.
::; (n -1 ~l. Trace
[Cn - i) (n - i
.1= 1
+ 1)
1 r
5.] ) 1CZ * >-2
E:rn)
NTIl
n
'
-1
1 E:rn)
r
= (Trace [n -1 J-TIn ]) 1(Z n* >-2
...£. 0, as n .... oo [by (2,6) and (2,28)].
>E:rn)11
.
I
10
Let us now return to the proof of Theorem 2,1,
By virtue of
(2,10), (2,11), (2.13), (2,25), and Lemma 2,2, for every
0
<t·, I,
<:5
(2.30)
I
-~
=J
"'11n
-~
=:lnn
{k
n (s)
I I =l
.
} -"2
1
n - 1.
.'. J
n-
1 +
-~
:lnkn (s) :lnn
1
~I
"'11n
•
.E.~ s1 = (s At)1
7
p
p
=E{s(t) [s(s)] I} ,
"'"
"'"
This, intuitively suggests the process
f
in Theorem 2.1.
To prove
the desired weak convergence result, we make use of the martingale
property in Lemma 2.2 and appeal to functional central limit theorems
for (a triangular array of)
martingales [viz., Scott (1973)]; we only
need to extend these to the multivariate case, but that poses no
problem [the usual Cramer-Wold type arguments filter through without
demanding any extra condition].
Our Lemma 2.4 and (2,29) insure that
both the conditions of Scott (1973) hold in this multivariate case and
the proof of the theorem is
the~fore
complete.
and (2.25), it follows that we may also take
sn (t)
1
By (2.10), (2.11),
k (t) = [nt] ,
n
t
£
E
and
1
=n-~r-~[ t]'
"'"
"'" n
t £E.
Note that by virtue of the Courant theorem (on the ratio of two
quadratic forms) and by (2.25),
(2.31)
max
-1
U k -1 I
IU'kJ -1 U k/u'k(nr)
"'11 "'" "'11
1 s:k ~ "'11 "'11n"'11
l
s:max{IChl(nrJ- )
""""11n
p 0 " .as
~
(where
roots) ,
chI
and
ch p
n
-1/,
1
Ich (nrJ- ) -II}
p ""''''11n
-+00,
stand for the largest and smallest characteristic
on the other hand, by Lemma 2,2, under Ho: f? = Q,
I.
I
11
(2,32)
Also, note that
E{n
-1 U'kr ~l U k } = E{-l
-1 kU1·k] }
n Trace [r U
"'11 . "'11
"'11 "'11
~
= E{n
-1
~
Trace [r
~
=pn-1I~=1
~pk/n,
-1·}
J"'11.
k]
=n
-l\'k
n-i
-1
L' 1 - - .-1 E (Trace [r s. ] )
1= n - 1 +
~
"'111
[as ES
"'111. = r. V 1
(n - i)/(n - i + 1)
V 1
~.
~
i
~
n - 1]
~k ~n.
Therefore, by (2.32), (2.33) and Theorem 2.1 of Birnbaum and Marshall
(1961) for any
(2.34)
"-
P
~
a
n2
~
. .. a
nn
}, 0 < 0
I(
1
'
e: > 0 •
I e: }
max a k In -1 U'k I -1 Uk>
1sks[no] n
"'11
"'11
{
~ pn
Thus, i f
{a
nl
-1
(anl +···+ an[no])/e:
q={q(t), tEE}
continuous function of
(2.35)
r
t
be a non-negative, and non-decreasing and
such that
[q(t)]-2dt <00,
0
-1 2
2
(-2
then n (q (lin) + ... +q ([no]/n)) S J [q(t)] dt, V 0 <0 sl, From
O
(2.35), it follows that by choosing 0(>0) sufficiently small, the
o
righthand side of (2.34) can be made small when
On the other hand, for
o
~
t
~ 1,
a. = q-2 (i/n),
n1
t >0(> 0), q -2 (t) Sq -2 (0) <00,
the weak convergence of
~
i
~ 1.
so that for
in the uniform (or Skorokhod
metric) insures the same under the "sup norm" metric
(2.36)
j
P (x, y)= sup{lx(t) -y(t)l/q(tl,
q
te:E},
Thus, from Theorem 2,1, (2.31), (2,34), and (2.35), we arrive at the
following.
12
The weak convergence in Theorem 2.t holds in the sup
Theorem 2.6.
nom metric
when (2.35) ho Zds •
p
q
Both of these theorems are useful for the proposed RST procedures.
3,
WEAK CONVERGENCE IN THE DISCRETE TIME MODEL
•
As in Section 1, we conceive of
where
R
k
o = to < t
has
r
k
(3.1)
(3.2)
s.
J
l
2
~
~
, k
k
R
m
1,
j =0, I, ••• , m.
failures in the interval
D. (j
J
n~. =Lk>.n. , for
n .. =I(Y. £0.),
1J
1
J
t
::> ••• ::>
and we denote by
D. = [t .• t. 1)'
J
J' J+
Let there be
R ::- R
subjects whose survival times are
< ••• < t . < t
= 00
m m+l
l
m risk sets
1J
-J 1k
~O).
Also, let
O~j ~m,
i=l, ... ,n.
Then, proceeding as in Section 6 of Cox (1972), we obtain the derivatives
of the partial log-likelihood functions (evaluated at
U*k=li'~
1{1i'~
l(n1J
..
"11
l.J=
l.1=
(3.3)
*
:lnk
(3.4)
Ii' k
= l.j =1
-n~.s./r.)z.},
k=l,
1J J J ~1
s.(r. -s.) n
J J J Ii'
r. (r. - 1) . ~ (.tinij
J J
1-1
=L~J= 1 (s.J (r.J
.@,
- s -llr.) S* . ,
J
J "11J
= Q)
as
e"
.. .,m,
...'
-*
*-*
-.tj H.tin ij - kj) ,
say,
where
-=*
z.
(3.5)
~J
In
* )/r.,
= ( . 1z.n..
1= "'1 1J
J
[The close relationship between
need not be overemphasized.]
sl'
• , 'J
s.
J
m
B~O = B(~l'
r , ... , r ,
m
l
•• •, Z ,
"11
sl'
in (2.4) and
In this case, we let
n .. , j
1J
•, •, S
~k
J' =1, ... ,m.
,
m
~k,
1 :s: i :S:n)
r , .•• , r ) .
l
m
for
S*k
"'n
in (3.4)
B*k
.•. , "11
Z ,
n = B(Zl'
~
k = 1,
I
••
,
rn,
while
Then, by arguments
very similar to those in Lemma 2,2, we arrive at the following
Lemma 3, l.
Under (2. 7J and HO:
~ = Q.,
for every n,
{~k' B~k' k :s:m}
is a martingaZe.
We may also note that by arguments very similar to those in
I
13
Section 2, under (2.7) and
~
HO:
= Q,
'\k
n -1 (J * k -/"_lSj
(r.
II
l~k~m
"1l
J~
J
max
(3,6)
- s.)r.~ 1[)
J
II
P
.--+ 01
J
as
n
-+0",
Now, to study the desired weak convergence results, we consider two
•
different cases:
(I)
is fixed, so that as .n
m
for every
J
both increase
j (= 1, •. " m), This situation arises when we have a given
number of ordered categories, while
(II)
and
s.
-+00,
m =m(n)
-+00
but
max
n
s./n
l~j ~m(n) J
arises when the width of D. (j
~
J
is large.
-+0,
as
n
This situation
-+00.
is small, so that there are a
1)
large number of cells and the possibility of ties is no longer
negligible.
In case (I), by virtue of
uncorrelated.
(3.7)
n
B~k_l'
Moreover, given
-~
*
*
(U
k - "'l1
U k - 1) =n
"'l1
is generated by the
(over the set
Lemma 3.1,
rkl
* - ~k-l'
*
~k
k
~
the conditional distribution of
-~.n
-1
l'_lZ'
(n'1 k - skrk
1- ......1
m1 k )
,
equally likely realizations of the
Rk = {i: nik = I})
are all
1
n
ik
and by an appeal to the classical
permutational central limit theorem [c.f. Hajek (1961)], we conclude
that this conditional distribution is asymptotically (in probability)
multinormal with null mean vector and dispersion matrix
(3.8)
. (k
Thus, using a chain of conditioning
some routine steps that given
t
of
* - ~k-l) ,
{n -~ (!!nk
B~o,
1 ~ k ~m}
=1,
" ' , m) ,
(k =m -1, ••• ,1),
i f follows by
the joint conditional distribution
is asymptotically (in probability)
multinormal with null mean vector and dispersion matrix which is
block-diagonal with the matrices
b(k)'
1 ~k ~m
in (3.8),
This, in
14
(2.29) that for every
Iln~1
(3.9)
men)
iL
£
>0
E{~k -~k-l)(~k -~k_I)'I(II!:!nk -!:!nk.l"
>£Iil)IBnk _l }
Lo,
•
I
so that by Lemma 3.1, (3.. 6) and (3.9L the invariance principle for
martingales, developed in Scott (1973), holds for the multivariate case
treated here.
The result is parallel to Theorem 2.1, where we replace
~k' ~k and:Ink
4.
by
~k' ~k and ~k' respectively.
INVARIANCE PRINCIPLES UNDER LOCAL ALTERNATIVES
With a view to study the asymptotic power properties of the RST
(to be considered in Section 5), we proceed now to extend the results
HO: g =Q may not hold, We conceive of a
of local (Pitman-type) alternative hypotheses, where
of Sections 2 and 3, when
sequence
{K }
n
g = B~(n) .= n -~~, for some A £RP
n
'"
'
and desire to study the weak convergence results under
(4.1)
K :
This
n
task is greatly simplified by employing the concept of contiguity of
probability measures
Pn
v
[as in Chapter 6 of Hajek and Sidak (1967)].
be the joint distribution of
~
{K }.
(Y ", Z.),
1
~
i
=Q and P*n be the same under kn . As
Let
=1, "" n under
a first step, we like to
show that under (1.1), (2,7), and (4.1),
{p~} is contiguous to
(4.2)
Let
which
HO(X)
n
be a non-decreasing and non.negative function for
(d/dx)HO(x)
=hO(~) , defined in (1,1), and let g (!) be the
joint density function of
(Y,~)
{p } ,
is given by
~.
Then, by (1,1), the joint density of
*
15
fey, ~; ~) =g(~)ho(y)exp{~'~-e~·~L
"CIO(y)}
(4,3)
13'z
=f(y, ~; Q).exp{~'!+HO(Y)C1-e~~)L
Thus, for testing the simple null hypothesis
~
H :
O
= Q vs. K :
n
t:l.
~
=n
.
~
>..
",'
the (log-) likelihood ratio statistic is
n
(4.4)
10gLn =\.
11og{f(Y.Z.;
n
1.. =
1~1
1
-~.
>")/f(Y.,
Zi;
'"
1
~
0)
~
~ n
~
= n - 2\. 1{>.. ' Z. + H (Y . ) n 2(l - e
/.1= '" "'1
o 1
n- 2>..Z
~i
)}
where we note that
(4.5)
~
n
n 2(1-e
-~>"'Z.
~ ~J..)
1 1
2
=->"'Z. --n-~(A'Z.)
""1
2
"" ""1
t"oJ
1 -~
2
-~
+ -2 n 2 (>.. ' Z.) [1 - exp (en 2>.., Z. ) ]
"" ""1
-..
\
and by (2.28), the last term on the rhs of (4.5) is
uniformly in
(4.7)
i:
10gL
n
~i ~n.
1
(0 < e < 1)
"" ""1
0
-~
P
2
(n 2)(>..,Z.) ,
~
"'1
Thus, by (4.5) and (4,6),
=n-~t1= l(>,,'Z.)[l-HO(Y')]
'" "'1
1
l t 1(>..'Z.)2 + on).
--2
n 1=
~
~
"'1
Now, by the Kintchine law of large numbers, under (2,7),
(4.8)
Al so, under
H :
O
= Q,
~
Y and
~
are independent with
f
(4.9)
E[Ho(y)]r = [Ho(y)]rexp(-Ho(y))dHo(y) =r!,
Thus,
E(~'~i)[l -HO(Y i )]
=
(~'lA)
0 =0
and
V r = 1,2, •. ,
E[(~'~i)(l -HO(y i ))2] =~'I.6.
Hence, from (4.7), (4.8), (4.9), and the above, we conclude that under
H :
O
~ = Q,
variance
10gL
n
.6'I.6.
is asymptotically normal with mean
-
~.6' 11
and
By virtue of LeCam's third lemma [c,f. H~jek and Sidak
(1967, p. 208)], this ensures that
{P*}
n
is contiguous to
{p}.
n
By using (4,3), we obtain by some simple steps that the conditional
density of
~
given
Y = y,
under
K,
n
is given by
16
!-<
!-<
g(~ly) =f(y~ f; ~=n-2~)lff(y~ ~; ~=n"2~)df
(4.10)
= g C.~)
so that under
K
I1
+ n-
!-<
2 (1
- H (y)) (~ O
1\)
'~ + 0 (n '"'
~
E(Z~1. I Y.1 = y) = N11 + n- 2 (I - H (y) ) ~~
fA + 0 (nO
Also, i f
FO(x)
I
~
2) •
be the marginal density of Y (same in both the null
and non-null cases), then
I-FO(y) =exp(-HO(y)),
has the simple exponential distribution,
(4.12)
2) ]
~
n
(4.11)
!-<
[1 -HO(y)] -
so that
Ho(y).
Hence, for any y,
U-FO(y)}-lflO[1-HO(~)dFO(~)
y
oo
={l-FO(y)}-l
fY
(00 -H
[1 -FO(~)dHO(~) =U-FO(y)}-lj, e
(y)
0
Y
dHO(~) =1.
From (2.2), (4.11), and (4.12), we obtain by some routine steps that
under
in (4.1),
{K}
n
(4.13)
n
Thus, if we define
(4.14 )
lim
n-+oo
-1
~n(t)
E{~
(t)
n
k
~a(O
< a < 1)
~
as in (2.11), then
1/
IKn } ~tr'2A
~
~
, V t cE.
Now, under the null hypothesis
independent of Y,
HO: ~ = Q and (1.1), Z is
so that by using Lemma 1 of Bhattacharya (1974),
it follows [as the conditional df of
~
given
y does not depend on
HO' ~l' ""~n are i.i.d.r.v. with the density q(f)·
Hence, by an appeal to Theorem 2,1, the contiguity of {p*} to {p}
y] that under
n
n
and a theorem of Neuhaus and Behnen (1975), we conclude that for any
k:
kin
~a:
with mean
o <a :51, n -~~k is asymptotically normal (under
aI~
and dispersion matrix
employed to show that for finitely many
aI.
(m)
{K })
n
The same technique can be
Points,
!-<
n- 2(U
U)
'""11k l' ••• , '""I1k
m
17
(where
k .In
J
-+ a.
J
has asymptotically (under
:
variate normal distribution with mean vector
dispersion matrix
{K })
a pm
n
and
(a ••••••. am) 0 ...,....,
fA
l
This establishes the convergence of
«(arAa.))
0r,
J
~
in (4.1).
n
As in the proof of Theorem 2 of Sen (1976) , we conclude that the
finite~dimensional
{p*}
n
contiguity of
distributions of
{p }
to
{F,;n}
under
and the tightness of
n
(insured by Theorem 2.1) imply the tightness of
well.
{K }
{F,;n}
{F,;n} under
under
H
O
{K }
n
as
This leads us to the following
Theorem 4.l.
~ +
defined by (2.11) converges weakly to
.
Theorem2.land
~
{Kn } in (4.1) [relating to (1.1)],
Under (2.7) and
£,
where
f
is defined in
!.:
F,;={t:(t)=tr 2 A,
~
'"
,...",
tEE}.
I'J
We conclude this section with the remark that similar weak
convergence results hold for the discrete time case treated in Section 3.
5.
RST PROCEDURES RELATED TO THE COX REGRESSION MODEL
We start with the remark that if one works with U
""11n
in (2.2)
and constructs
r U
nn =U'
""11n""11n""11n'
L
(5.1)
is defined by (2.3), then, under HO: ~ = Q and (2.7) - (2.8) ,
Lnn has asymptotically chi-square distribution with p degrees of
freedom (DF) (by Theorem 2.1) and by Theorem 4.1, under {Kn } in (4.1),
where :Inn
has asymptotically a non-central chi-square distribution with p DF
nn
and the non-centrality parameter
L
~L =~'£~
(5.2)
Now 1 suppose we work with
e.
a sub-vector of
g,
L~,
.
defined by 0.7).
Since
~t
the martingale property in Lemmas 2,2 and 2.3
remain true for these subsequences
{~k} and {~k}
and also
is
18
{J*k
",n L
Lemma 2,4 extends to
{L* }
Thus, the weak convergence of
run
follows on the same line as in the proof of Theorem 2.1, provided
m(= m(n))
-+00
as
n
-+00,
Further, the contiguity being assured by
(4.2), Theorem 4,2 also holds for this subsequence,
claim that under
(2.7)~(2,8)
and
H :
O
~=QJ
chi-square distribution with P DF , provided
As a result, we
L*
has asymptotically
men)
-+00
nm
as
n
-+00.
Also,
if, in addition,
1 .
(5.3)
lim n- men) = '[:
n-+oo
then under
0::; '[ ::; 1 ,
L*tun has asymptotically a non-central
{Kn } in (4.1);
chi-square distribution with p DF
and non-centrality parameter
!.l L* = '[2 (~' r~) = '[2 .!.l L '
(5.4)
Thus, the asymptotic relative efficiency (A.R,E,) of
to
L*nm with respect
Lnn is equal to
(5.5)
In particular, if
'[
is close to
this sense, i f m(n)/n
regression based on
-+1
as
n
1,
-+00,
then (5,5) is also so and in
then the Cox (1972) test for no
L*nm in (1,7) is asymptotically fully efficient -
this result has been obtained for the discrete time model by Efron (1977)
from a somewhat different consideration.
for
'[<1,
However, we may note that
the A.R.E, in (5.5) is less than 1 indicating that too
much of censoring takes away some information,
A common feature of the two tests based on
they are based on the partial likelihood
Lnn and L*nm is that
Lnn and L*run'
respectively,
and, thereby, demand that the experimentation be continued until all
or m of the failures occur,
In the proposed RST procedure, we like
to update the picture as successive failures occur and consider the
following time-sequential procedures,
These procedures are based on
..
19
the partial sequence
{L
nk
~n}
k
,
of partial likelihood ratio statistics,
Note that by (2,10)-(2,11)1
.
~
-1
~
,Lnk (t) = [~(t)]' [:!nn:!nk (t):!nn] [~(t)], V t cE ,
n
n .
(5.6)
where by Lemma 2,4, under (2,7)1 (2,8), and
~
(5.7)
P
-1
~ t
I,
"'P
~
J""1ln"'1l
J k (t)J"'1ln
n
Thus,
L
nk
hand, for
(t)
n 2
behaves like
q (t) =t
-1
,
.§=Q,
H :
O
V teE.
t -1 [~(t)] , [~(t)]
as
(2.35) does not hold and
not behave very smoothly when
t+O;
t-
n
1
-HXl.
On the other
[~(t)]'[~(t)]
does
in fact, by the law of iterated
logarithm for Brownian notions,
lim sup t
t-+O
(5.8)
-1
.
[~(t)]'
[.s(t)] =00,
with probability 1.
Hence, to make statistically informative use of the partial sequence
either we use some appropriate weight function or we
\
start monitoring of the experimentation only when a given number of
failures have already taken place,
This leads us to the following
two types of RST procedures.
(i) Type A RST procedures.
o
k = [fiE:].
n
N =
{
c n,
(5.9)
"
c*
n
and let
c:
Define then the stopping number
min{k (k
where
We choose some
if
0'
n
~ k ~ n)
:
L k > c*},
n
n
Lk~c*,
VkO~k:<::;n.
n
n
n
is a suitable constant, to be defined later on.
operationally, starting at thekO-th
n
failure
o
Y 0'
nk
Then,
we monitor experi-
n
mentation and at each failure Y (k ;::: k ), we compute L : if L
is >
nle
nk
n
nk
c~,
we stop experimentation at that time along with the rejection of
O:
H
~
=Q and if Lnk
~
c*,
n
we continue experimentation until the next
failure occurs and reinspect the process at that failure,
If, we do
20
not stop on or before the n-,th failure, experimentation terminates
along with the acceptance of
allowing
k to range over
and, in that case,
N
E:
HO' In (S,g), we may modify N( by
k0 ~ k ~ m for some predetermined m(~ n)
n
with probability
~m
In theory, it does
I,
not make any difference, but, in most practical problems, we may be
inclined to choose an
m well below
n,
depending on the time and
cost of experimentation (at our disposal).
We need to choose
c*
n
in
such a way that
PUnk >c~
(5.10)
where
a(O < a < 1)
for some
k~ ~k ~nIHo: f? =Q} = a,
k:
is the desired level of significance of the RST.
(ii) TYpe B RST procedupe.
Here, we do not use a constant
{c~k' 1 ~k ~n}
boundary and choose a sequence
of positive numbers
and denote the stopping number by
(5.11)
N* =
min k(l ~k ~n):
{n,
L
>c~k'
nk
L
~c~k' V k ~n .
nk
if
Then, we proceed as in the Type A procedure, but using the variable
boundary instead of the constant
c* .
n
Here, we need to have
(5.12)
[We do not need to wait until the occurrence of y
c*n
PUnk >A,
-I-
n
E
for some
>0
and
say,
under HO:!2, = Q,
A > 0,
k: k~ ~k :s;nl~o}
p{t-l[~(t)]'[~(t)] >A for some
= P (A; E),
n
in (5,9)-(5,10), we now appeal to Theorem 2.1
and conclude that for every
(5.13)
and the process
kO_th failure.]
may terminate even prior to the
To determine
nk
0
E
~t:s;l}
21
TABLE 1
Table for the simulated values of
-
p =1
a=,Ol
.
--
A
a
P =3
P =2
,OS ,10
,01
,OS
(a,
for typical
£
,01
.10
£)
-
,OS
=
P =4
,10
.01
.05
.10
.005
12.11 9,24 7.78
15.42 12,46 10.68
18,19 15.14 13.10
20.97 16.83 14.99
.010
11.97 9,06 7.62
15.24 12.23 10.52
18,10 14.52 12.85
20,44 16.76 14.86
.050
11.49 8,35 6.86
14,85 11,90
9.88
16.99 14,19 12,37
19.72 16,18 13.94
.100
11. 09 7,79 6.35
14,69 11.48
9,39
16,92 13.98 11,87
19,58 15.72 13.46
.150
10.63 7.51 6.10
13.94 10.99
8.97
16.75 13.60 11.47
18.62 15.24 13.18
.200
10.48 7.34 5.90
13.92 10.55
8.87
16.50 13.02 11.21
18.10 14.96 12.96
Aa£ be the solution of peA;
Thus, if
{c*}
n
£) =a,
any sequence of positive numbers for which
then we may choose
c* +A
n
a£
Aa£ are not yet available.
Analytical solutions for
as
n +00.
However, certain
simulation studies are made by Majumdar and Sen (1977) and Majundar (1978).
We report above, in Table 1, some of these simulated values for
c~k
For (5.11) - (5 .12), we take
= (n/k) c~,
1
~ k ~ n,
(5.6), (5.7), (5.12), and Theorem 2.1, we have for
P{Lnk > c~k'
(5.14)
+
Thus, if
A~
{*Lnk , 1 ~k ~n}]
p >1
k ~n IH O}
p{ [~(t)] , [~(t)] > A,
for some
be the solution for
[For this choice of
for
for some
(for
p
c~k'
Pt =a,
4.
'
=Pt '
say.
then we have choose
c* = A* .
n
a
we really work with the partial sequence
Again, analytical solutions for
=1,
tEE}
~
so that by
c* +A
n
p
A~
are not known
this is, however, known), and, we quote the
following simulated values obtained by Majumdar and Sen (1978).
By virtue of Theorem 4,1, we are also able to express the
asymptotic power of these RST procedures interms of the boundary crossing
probabilities of
{~(t),
t EEl.
From (5.9), (5.10), and Theorem 4.1,
22
TABLE 2
Table for the simulated
values of A* for some typical
(a, p)
a
A*
a
a
P
la
,01
,05
,10
7.90
5,02
3.84
2
10.37
7.29
5.52
3
13.76
9.30
7.73
4
15.13-
10,96
9.24
a)Exact values.
we have that under
{K}
in (4.1), the asymptotic power of the Type A
n
RST procedure is equal to
(5.15)
Similarly, for the Type B RST procedure, the asymptotic power under
{Kn } in (4.1) (when we choose
c~k
!-.
= (n/k) c~, k $ n) is given by
!-.
p{(f,;(t) +tr 2 A)'(f,;(t) +tr 2 A) >A*ex.
(5.16)
,.....,
I"'OV
f'Jf'OJ
for some
tEE},
,....,,....,
A comparison of (5.15) and (5.16) demands extensive numerical integra-
tion and simulation studies.
So far, we have considered RST procedures based on
It
{L~k'
is also possible to use
k ~m}
{L
nk
when m-+00 with
, k ~n}.
n -+00.
The stopping variables are similar to those in (5.9) and (5.11) and
when
min -+ 1,
min -+T,
(5.13) and (5.14) also stand valid.
for some
the range of
T < 1,
t(E~t~l
However, if
then in (5.13) or (5,14), we have to replace
or
tEE)
by
E~t~T
similar changes are needed in (5.15)-(5.16),
(or
tEE[O, TD
For the discrete time
parameter case, treated in Section 3, when m =m(n) -+00 with
(i.e"
case II), the same results apply.
and
In case I (i.e.,
n-+ oo
m fixed),
23
~n(t)~
instead of the process
~n(tj)'
t j EE,
j =1, ""m,
t EEl
we have only finitely many
As such, t;he useof\).E or
A~
(5.13) or (5,14) will result in a more conservative test i,e"
actual level of significance may be much below the specified
in
the
a.
However, as in Majumdar and Sen (1977), this will continue to be a nice
test.
Finally, throughout the paper, in the usual fashion of an
analysis of covariance model, we have taken
the other hand, the
Z.
~1
Z.
~1
as stochastic.
If, on
are non-stochastic, the problem becomes a
little more simple and a direct multivariate extension of Sen (1975)
yields the desired results.
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t