~.l-\RAMffi'RIC
TESTING AGAlNST RESTRICI'EJ) ALTERNATIVES
UNDER Pl{)GRESSIVE CENSORING
by
Pranab Kumar sen
Depart:;rrEnt of Biostatistics
University of North Carolina at Chapel' Hill
Institute of Statis.ti,c$ Mi:mao ::;eries No. 1473
December 1984
NCNPARAMI::TRIC TESI'ING AGArnST RESTRICI'ED ALTERNATIVES
UNDER PRX;RESSIVE CENSORING
Pranab Kumar
sen
Depar1::Irent of Biostatistics
University of North carolina.
Chapel Hill, NC 27514, USA
Key Words and Phrases: Bessel-bar process; invariance principles;
Kuhn-Tucker-Lagrange fo~la; linear rank statistics; progressive
censoring schemes; restricted alternatives; simple regression model; time-sequential tests; union-intersection principle.
Nonparametric testing under progressive censoring scherres are
often adapted in clinical trials and life testirig experimentations.
In a variety of situations, a null hypothesis (of the hcm:>geneity
of distributions ) may be tested against a restricted ( viz., orthant or ordered) alternative. Though such tests have been considered in a CCItPlete or a single-point censored (or tnmcated ) experinental setup, there has not been much progress with these tests in
a progressively censoring setup. Genuinely distribution-free tests
based on sirrple linear rank statistics and the classical union-in-
tersect ion principle are considered. The theory of such tine-sequential tests rests heavily on sc:.m= basic invariance principles,
and these are studied as well. The case of proportional hazard
nroels is also treated briefly. The main errphasis is laid on the
developrent of the relevant asymptotic theory, and, in this context,
sc:.m= (sub-) martingale characterizations of progressively censorerl
rank statistics play a vital role.
1. lNl"roIXJCI'ICN
V«= consider here a sinple regression m:x:1el. let Xl' ••• 'X
be n
n
iIrleperrlent randan variables (r. v.) with continuous distribution
functions (d.f.) F , ... , F ' respectively, all defined on the
n
l
real line E = (-00,001. It is assurred that
F. (x) = F(x - B'c. ) ,
1
~ ~1
X
E: E ,
i =l, ... ,n,
,
are q-vectors (for SCIre q >
~1
1l
1q
known regression constants , not all equal, B = ( B , ••• ,
l
a q-vector of unknown (regression) parameters, and the d. f.
where the c. = (c. ' ••• , c.)
(1.1)
1) of
,
B)
is
q
F is
also of unspecifiErl fonn. [ In the context of a clinical trial or a
life testing experinentation, the failure ti..mes are nonnegative r. v.
There the Xi may be taken as the log-failure ti.n'es, so that (1.1)
will relate to a regression in the scale pararneter m:x:1el. J
In a nonpararnetric testing problem, we consider the null hypo-
thesis ( H
O
) of the harogeneity of the F i ' i. e. ,
H : F = ... = F n = F (unknown) or
O
l
0 in (1.1),
...B = _
(1.2)
against suitable alternatives vitiating the harogeneity of the F .•
1
For the regression m:x:1el in (1.1), a global alternative hypothesis
is of the fonn
H*
...B ~
-
O.
(1.3)
In many problans of practical interest, in testing for H
in (1.2),
O
one may be interestErl in llDre restrictErl alternatives having nore
relevance to the given experinental setup. For exarrple, we may let
n = nO + n
the n. are all positive integers),
J
,
and take c = ••• = c
=
0
,
c
+1
=
•••
=
c
+
=
(1,0,
•••
,0)
... l
...n
~
~nO
...n n
O
, O l
c~O+n +1 = ••• = c
+
= (0,1,0, ••• ,0) , •••• , c
+n _ +1
...n +
_nO+•••
q l
l
O nl n2
= ... = ~ = (0, ••• ,0,1)'. Then, (1.1) represents a nulti«q+l)-)
l
+ ••• + n
q
(where
sarrple 1ccation llDdel, with the subscript 0 for the control arrl j
(1, ••• ,q) for the treat:Irents. Thus, B. starrls for the jth treat:Irent
J
effect (over the control ), for j = 1, ••• ,q, and H in (1.2) relates
O
to the hypothesis of no treatJrent effect. In such a case, one may be
interested in an alternative hypothesis that none of the q treat:rrents is \\1Orse than the control, leading to the so called orthant al-
ternative
H+ : l3. > 0, j=l, ... ,q , with at least one strict
J -
inequality
(1.4)
sign being true.
Another possible hypothesis of considerable interest relates to the
ordering of the treat:rrent effects i.e., an ordered alternative
>
H :
>••• > l3 , with at least one ·strict
q - 1
inequality sign being true.
13
A third possibility is to consider an
(1.5)
ordered orthant alternative
H+> : 13 >••• > 13 > 0 , with at least one strict
q- - 1 -
(1.6)
inequality sign being true.
q
Note that whereas in (1.3), ~ may belong to any where in E , in
each of the other three cases, it may belong only to a restricted
q
subspace of E • SUCh an alternative will therefore be tenned a res-
tricted alternative. A rrore general definition will be given later
on.
In testing for H
in (1.1) against a restricted alternative,
O
such as in (1.4), (1.5) or (1.6), the conventional tests (designed
for testing H against H* ) may still remain valid • Nevertheless,
O
they may not be the rrost desirable ones. This is mainly because of
the fact that the local or global optimality considerations underlying the construction of these (global) tests may not retain in
tact when one restricts
interest to a rrore restricted class of al-
ternatives ( for which rrore powerful tests may be constructed by
sare alternative approaches).
In a general class of pararretric nodels, such tests against
appropriate fonns of restricted alternatives have been sttrlied in
detail in BarlCM, Bartholarew, Brera1er and Brunk (1972), where an
extensive bibliography on this topic has also been provided. SCJt'e
developrents on nonparanetric tests against restricted alternatives
are accounted in Chinchilli and Sen (1981 a,b; 1982), Sen (1982) ,
Boyd and
sen (1983,19841
and others ; references to earlier \\1Orks
have also been cited there. All these sb.xlies relate to the conventional ccnplete sarrple case where the tests are based on the set
of all sarrple obsel::vations.
In clinical trials and life testing experimentations, one has
essentially a follow-up study, where, typically, the response (Le.
failure ti.nes ) occurs sequentially over ti.nE. In such a case, because of t:ilre, cost and other limitations, one may have to curtail
the study at an intenredi.ate stage where
salle
of the responses are
already obtained, while the others are censored. 'Ib eliminate the
usual limitations of a signle point censoring or tnmcation scherre,
often, in this setup, a statistical nonitoring is advocated: Fran
the very beginning, the study is nonitored with the objective of a
possible early tennination contingent on the accumulating statistical evidence. This is referred to as a progressive censoring sch-
eme (PCS). For the accumulating data under PCS , the picture can
be viewed as a ti.ne-sequential one , and for this a general class
of nonPararretric testing procedures has been developed by Chatterjee and Sen (1973) with further extension by Majurrrlar and Sen
(1978) and others. A general account of nonPararretric testing under progressive censoring is given in Sen (198la,Ch.ll). Again, all
these tests have been designed primarily for testing H in (1.2)
O
against the global alternatives in (l. 3). '!here has not been rruch
progress with the nonPararretric testing against restricted alternatives under progressive censoring, and the current investigation
is primarily focused on this topic.
Along with the preliminary notions, the proposed test statistics (based on suitable linear rank statistics and the union-intersection (UI-) principle ) are introduced in section 2. A submartingale property of these PCS rank test statistics is considered
in section 3 , and this yields suitable probability inequalities
which are used in later sections for the study of the properties
of the tests.
SCIre
invariance principles relevant to the PCS test
statistics are considered in section 4. Asynptotic properties of
e
the proposed tests are studied in section 5. The last section deals
with the proportional hazard nodel, and PCS tests for restricted
alternatives are presented there •
'lEE PROPOSED TESTS
2.
By virtue of the asSUlreCl continuity of the d. f. 's F1,· .. ,Fn '
we ma.y neglect ties anong the observations, with probability 1.
Let R . be the rank of X. arrong Xl' ••• 'X , for i=l, ••• ,n. Also,let
m
-1.
n
a (l), ••• ,a (n) be a set of scores which will be defined rrore forn
n
ma.lly later on. Then, we nay consider (a vector of ) linear rank
statistics of the form
= r.n1.=1
L
-n
(c. - c- )an(R,...;) ,
-1.
-n
.~
(2.1)
-1 n
where c = n r. lC.' , and the c. are defined as in m:::rlel (1.1).
-n
1.= -1.
-1.
Note that under the null hypothesis in (1.2),
EL
-n
= 0
-
and
EL L' =
-n-n
C
-n
2
.A
(2.2)
n
where
C
-n
•
= ~1.=l(
c.
-1. - C
_n
)( c.
_1.
- ~_n )' ,
(2.3)
a
= (n-ll-lr~ II a (i) J2
(2.4)
1.=
n
n'
t. .
.
*
and -an = n -l~n
L.i=lan (1.. ) • 'C1
~or tes l.ng H l.n U.2) agal.nst H in
O
(1. 3), usually, one considers a test statistic of the form
2
A
n
' - L },
Tn = An-2 {L
(2.5)
-n C
_n -n
where C- is a (reflexive) generalized inverse of C • T is genuin_n
-n
n
ely distribution-free under H ' and for large sample sizes, urrler
O
fairly general regularity conditions, the null distribution of T
n
nay be approximated by the central chi-square distribution with r
degrees of freedan (OF), where r is the rank of C
-n
The asyrrptotic distribution of T when
~
1
= n-'Z
r , for
SCJ1E
fixed
n
r":f 9
)
(in the limit).
(3 is close to 0 (i.e., for
-
-
is ooncentral chi square with
r DF and an appropriate noncentrality pararreter 6 which depends on
r
,the scores and the d. f. F. For a given F, the scores ma.y be
so defined that 6
is naximised , and this leads to the asyrrptotic
(within the class of rank. statistics).
(local} q>tiInality of a T
n
Since F is generally not known, in practice, one usually chooses
the scores in such a manner that T has the asynptotic c:ptimality
n
for certain anticipated d.f. F ,and at the sane ti.rre, it retains
efficiency-robust for other d. f.
I
S
not
IlUlch
different fran the
anticipated one. The Wilcoxon scores or the nonnal scores statistics are the classical exanples of this type.
let us nON consider a restricted alternative. For sare set
q
E
£
,
fram: an alternative hypothesis
\\'e
Hr :
B
£
r,
where
r
is positiveZy homogeneous.
(2.6)
Note that a set A is positively haTogeneous if for every a
M > 0, Ma
r
£
A. It is easy to veify that in (1.4),
r is
£
A and
the positive
q
< a <••• <a land in the
l - 2 - q
other case, it is the cone in the positive orthant, i. e. , {o< a < •
orthant in E , in (l.5) it is the cone { a
1
•• < a } , and , in any case, r is positively honDgeneous. To derive
- q
a suitable test statistic for testing H against
\\'e may make
O
use of }by's (1953) UI-principle (along with further anendnents
Hr '
fran Sen (1982) ), and, for every
k
T (y) = y'T / (ylC y) 2A
r
£
r,
we
may write
•
(2.7)
n ...
- -n - ...nn
Point wise, for testing H against H : B = y , the statistic T (y)
O
Y - n ...
in (2. 7) may have better power than other canbinations of T , and,
...n
under HO ' Tn (X) has mean 01 variance 1 and asyrrptotically a starrlard nonra.l distribution. Since Hr = U r H , in accordance with
y£
Y
the UI- principle, we may consider the overall test statistic
~n = sup{ y'T
Y)~An : Y
(2.8)
......n/(y'C
... ...n...
... £ r } •
r
Note that for H* in (1.3), r = ~, so that T in (2.8) reduces to
n
.
) Hor.Yever, ~
. general , the fo:rm of Tr may
T~ , def~ed
by ( 2
.5.
n
n
depend (in a rather involved way) on the particular fo:rm of the
set r • It also follONS fran (2.8) that like Tn ' for any chosen
r,
the test statistic
under
SCIre
and the set
rr;:
is also distribution-free urrler H ' and
O
fairly general regularity conditions on the scores, F
r ,
the asymptotic null distribution of
~n is given
by a convex mixture of r+1 chi distributions with DF 0,1, ••• , r ; in
the literature I c.f. Bar1C1tl et al (1972)] this is known as the
r in (2.6) is defined
chi-bar distribution. When the set
of sore inequlaity retraints, as is
(1. 6), for the c:xJtl?utation of
r
T
by means
the case in (1.4), (1.S)
am
in (2. 8), one may use sare stan-
n
dard results in non-linear progranrning, and in this context, the
Kulm-Tucker-Lagrange IX>int formula plays a vital role I viz., Chinchilli and
sen
'
(1981 a,b ) ]. Finally, for testing H against Hr
O
though Tn may be a valid test statistic, the asyrrptotic power of
r
n is generally greater than that of Tn I viz., Chatterjee and
T
(1974) and Chinchi11i and
sen (1981
De
b) and others].
Let us now look into the same testing problem under PCS. In
this case, we denote the order statistics correS};X)nding to Xl' ••• '
Xn by Xn: 1 < ••• < Xn:n , reSPeCtively. Then, note that X.~ = XnR: .
n~
for i=l, ... ,n.
Side, by side, we define the vector of anti-ranks
S
-n = (Snl , ••• ,Snn )
that R
nS
sm '
by letting Xn.~,
.' = X
for i=l, ••• ,n. Note
' = SnR . = i, for i=l, ••• ,n. In a PCS, it is nore conve-
nient ton~descr~ the observable r.v. 's in terms of the order statistics and the anti-ranks.
Note that in a tine-sequential setup,
at the kth order statistic X k ' we have the set of observations
n:
(Xn : 1 ,Snl) , ••• , (Xn:k,Snk) along with the partial infonnation on
the canp1enentary set of anti-ranks without the actual realizations
., i > k • Since L
and without the realizations of X
-n
n:~
in (2.1) is
indePendent of the order statistics, but dePends on the X. through
~
their ranks, or alternatively, the anti-ranks, we may take the conditional expectation of L
denote this by
!:nk '
-n
given S (k) = (Snl' ••• ,Snk) , (under H
-n
and Sen (1973) and Majurrrlar and
~~
= -k
r-'--l (c
-S
~-
),
and incorIX>rate this in the formulation of a
test statistic at that stage, for k
L_l,
O
.
m
- c-
-n
sen
~
1. It fo11CMS fran Chatterjee
(1978) that
) { a (i) - a * (k) } ,
n
n
(2.9)
where
a * Ck)
n
(2.10)
Note that under
11>
in 0-.2},
E( ~nk } =
2
and E( !:'nkk ) = Ank.£n '
2
I
(2.11)
where
2
-1
k
2
* 2 -2
A = (n-l) {Li=lan (i) + (n-k)I an (k)J - nan },
(2.12)
nk
222
2
2.
for k= 0,1, ••• ,n. Note that A = 0 , A = A 1 = A , and A 1.S
no
nn
nnn
hk
nondecreasing in k (0<k<n) , for every n ~ 2; \oJe may refer to Chatterjee and
sen
(1973) for sate of these details.
Now at the kth stage, based on the set (X : ,8
8
nk
),
\oJe
n l
may define parallel to T (y) in (2.7),
Tnk
nl
),···, (Xn : k ,
n ...
!-<
t:r}
= (r'!:'nk)/{ Ank (r'~nr) 2} ,
and par(~lel to ~
T~ =
In a PCS, as
in (2.8), we let
sup{ Tnk(r) : r e:
\oJe
(2.13)
r
}
(2.14)
have noted earlier, one usually nonitors the sttrly
frau the beginning with a view to having an early stopping whenever
there is enough evidence that the null hypothesis H is not tenable.
O
For this reason, at each failure point X k' k > 1, one looks at
n·
the test statistic
T~ , defined in (2.14) ,
-
arxi proceeds in a qu-
asi-sequential manner. Thus, oPerationally, the test is basal on
the Partial sequence
{T~ ; k .:: n}, and the basic problem is to de-
fine a stopping rule by which the ti.rre sequential procedure is well
defined, and the stopping rule as well as the test procedure have
sate
desirable properties. we consider here two related stopping
rules based on horizontal and square root boundaries reSPeCtively.
First, consider the m:::xlified version of (2.14), arrl define
Tnk*r = sup{ (yIL~~)/{A
y)~}
...
n (ylC
_ -n.:..
¥~
= (Aru!An )
X e:
r }
1
sup{
r
= (Aru!An).T
nk
Note that
:
A~A~ describes
(r'!:'nk)/{Ank(r'~nX)~}: Xl: r }
' for k =O,l, ... ,n.
(2.15)
the proportion of variability ( in the
scores) when censoring is made at the rank k, and this quantity
IIDnotonically goes to 1 as k increases fram 0 to n. For this
reas:>n (and the fact that in later manipulaticns we shall make use
2
2
of a mapp.ing ( scale transfonnation l: k/n .... A - /A ; k < n ), we
hk'n
-
tenn AnJlA as the 'square root' factor. With this , the 'bJo botmn
daries to be considered may be introduced as follCMS:
(al Horizontal Boundro:y.
The stopping tine K';:) is a p:>sitive
integer valued r. v. defined by
(I) __ {rnin{k: l<k<n and T:' ~ i:~I)} ,
K
n
n , if no such k exists.
(2.16)
i: (I) is a suitable p:>sitive constant to be detennined in
wnere
n
such a way that the Type I error of the test under PCS is equal to
salE
preassigned mIlTber a : 0 < a < 1; a is tenred the level of
significance of the test.
(b) Square-root Boundro:y.
'!he stopping tine K(II) is a p:>si-
n
tive integer valued r.v. defined by
K(II)
n
=
. {
{
*r
nun k:l~<n: Tnk/(AnJlAn }
n , if no such k exists,
where again the p:>sitive oonstant
>
or
L
i:~II} is to
(II)}
'
n
be so detennined
that the test under PCS has the significance level a •
T: '
Note that we have for oonvenience defined the 'bJo boundaries
in tenns of the m:x1ified statistics
of the
~
k
~
n. Defined in tenns
, the square-root boundary in (2.17) would reduce to a
oorizontal one, while the boundary in (2.16) will be a reciprocalsquare-root one. we may also note that it is not necessary to use
n as the maxi.nu.Im p:>ssible value for either of the stopping number.
In either (2.16) or (2.17), we may conceive of a p:>sitive integer
r
= rn
, such that n
-1
r
n
converges to
sate
number p : 0 < p < 1 ,
and adjust the definitions wherein n is replaced by r
range of k, but not for the definition of the T;r ).
-
~
(for the
shall see
in a later section that this adjustment can be made with mi...n.:i.mum
Pain.
From the operational p:>int of view, it may be quite inpJrt-
ant to choose a value of p less than 1 (depending on the expected
duration of a clinical trial or a life testing experi.Irentation we
are concerned withl and to eliminate the possible extended duration
of nonitoring in PCS t when H is likely to be true). Of the 1:\\10
O
testing procedures considered, the second one is m::>re in the spirit
of repeated signifiaanae testiY/B as the rrI'nk are ~. approriate test
statistics for the censored experi.nents (at the various failure p:>ints ),and the first one is akin to a trunaated sequential
testing~
though here one does not have the usual independent am harogeneous
increrrents of the process on which the testing is made.
For either of the testing procedure (under PCS), the basic task
is to detennine the critical level ( T (I) or T (II) ). Toward this,
n
we may note at this stage that under H
O
n
in (1.2), individually, each
T~ is genuinely distribution-free, and the sane conclusion holds
for the entire set{ T~ ; k ~ n} • Thus, the distributions of the
stopping numbers K (I) and K (II) (under H ) do not depend on the
n
n
O
basic d. f. F in U.l), nor the critical levels
T (I)
n
and T (II) den
pend on F. Hence, both the proposed tests are genuinely distributi-
on-free under H • For the actual null hypothesis distributions of
O
maxi T*r : k < n} or maxi Tr : k < n} ( as well as K(I) am K(II)
nk
nk
n
n
and other related statistics), one needs to consider the unifonn
(peDmltation) distribution of the set of ranks ( or anti-ranks) on
the set of peunutations of (1, ••• ,n), and for specific set of the
c. am n, this can be done when n is not very large. In sene other
-1
specific m:::x:iels ( such as the classical multi-sarrple m:rlel referred
to in section 1), the task beoanes even sinpler, as the number of
distinct pennutations reduces to a nuch smaller quantity. However,
in any case, the actual (peDmltation) distribution of any such statistic will depend on the c. , n as well as on the dana.in rover
-1
which the supremum in (2.14) or (2.15) is extended, and as n beco-
nes large, the task of evaluating the exact pennutation distribution becares prohibitively laborious. For this reason, we consider
sene distribution-free probability inequalities on the proposed
test statistics, where sene sub-martingale characterizations of
the
Tl
playa vital role. These results are presented in the next
section. Invariance principles needed for the study of the asymptotic· distribution theory are then considered in Section 4.
tit
3.
TEST STATISTICS: <:XMVI'ATICNAL SCHEMES
AND SGffi proBABILITY INB;:VALITIES
we
r
have noticErl earlier that for
q
= E ,
A~{ ~~~~~nk}\
T~
rErluces to
T~
=
for every k ~ n. For r being sare restricted part
q
of E , the canputation of the ~nk may deperrl very ITUlCh on the actual danain of
r •
For the sake of illustration, we consider here the
othnant alternative in (1.4), and for this the carputational scherre
may be presentErl as follCMs. Note that here r
r'!:'nk
- --
{y: y'y >0, Y > OJ.
""""
obse~that
Thus, looking at (2.13) and (2.14), we
mize
=
--..
-
we need to maxi-
subject to the inequality restraint r ~ ~ arrl,because of
the positive horrogeneity of r , the equality restraint that y'C Y
- -n-
-r'!:'nk '
=
constant. If we let her) =
~ (r) = -r and h 2 (r) = (r'~nr
- 1) , then the problem rErluces to that of minimizing h (r) subject
and h (r) = O. For this non-linear
2
progranrning problen, the Kuhn-Tucker-Lagrange (KTL-) theorem yields
to the constraints :
~l
(r)
~ ~
the follCMing solution { see
sen
let a be any subset of Q
ary subset (
~
c a
~
(1982)
for details ].
= {I, ••• ,q}
and
a be the canplerent-
Q ). For each a , we partition (follCMing poss-
ible rearrangerrent ) L_"
and C
-llh.
_n as
L'
...hk
= ( L
,
,L'
-) ,
:hk(a) ...hk(a)
Also, for each a
£ Q,
C =
...n
(~n (00)
,
~n (aa))
C
...n (aa ) , C
...n (--)
aa
•
(3.1)
let
"
L
- C
- C
-- L
- ,
:nk (a) = L
...11k (a)
...n (oo)
...n (aa)
...11k (a)
(3.2)
~(a) = Sn(aa)
(3.3)
-
~n(aa)~n(aa)~n(aa)
•
Then, T~ in (2.14) , is given by
~
~~aSQ
A~ {(~(a)~(~)~(a))~I(fnk(a» ~).
I (C-(--)L (-) < O)} ,
...naa ...11k a -
(3.4)
where I (A) stands for the indicator function of the set A, and in
q
(3.4), only in one of the 2 tenns , both the indicator :furr=tions
are equal to 1. Thus, essentially, we have a single ( randan) tenn
in (3.4) representErl in this manner to depict the full picture.
r,
we consider now a nore general fonn of
I
where we partition
I
Xas (X (1) , X(2) )
where X(1) is a p-vector arrl
X(2) is a (q-p) -vector, for SCIIE positive p ~ q. A generalized orthant alternative may then be franEd as
the q-vectror
Ha: ~ =
~
I
,
against ~ : ~ e:
r
'I~,
X(1)
In this case, we define first a as (1, ... ,p) and
a as
r = {X:
~~}.
(3.5)
(pt1, ... ,q) ,
and define !:'nk(2:l) and ~n(22:l) as in (3.2) and (3.3). Let then
!:'nk(l) denote the p-vector corresponding to the set (1, ••• ,p) and
let ~n (11) be the corresponding partition of ~n' Now, we consider
the set P of all possible susets of {l, ••• , p} , and for each a : ~
•
A
I
-
0
f: a ~ P , we defllle the !:'nk (la) , !:'nk (la)' ~n (lla) and ~n (llaa) as
in (3.1)-(3.3). Then, Parallel to (3.4), here, the ~nk are given by
-1
AI
0 -
~
A
Ank{(~nk(2:l)~n(22:l)~nk(2:l) + ~nk(la)~n(lla)!:'nk(la») •
I( fnk(la) >
~)I(~~(llaa)!:'nk(la) ~ ~)}.
(3.6)
In particular if p =q, then (3.6) reduces to (3.4).
The corrputational schare for the ordered alteD1ative problan
is very similar to (3.6) when we IIEke an initial transfonnation on
the !:'nk •
Yq
= 13q
we may set here Yl = 132 - 131 , ... , Yq - l = Bq - Bq _ l
'ihen we write
•
= ...,...,
aE
BI L
...,.....,IlJc
where
~=
I
I
(E I ) -IL
-IlJC
lI'V
-1~ ~~10~.:::~
... 0)
(o
••
••
o 0.· -1 1
0· .. 0
= Y- L""",M
* ,
I
andC
say,
* =ECE
- .....n-
to replace
;re !:'nk
and C by C
-n
•
(3.7)
I
(3. B)
lI'Vn
1
Thus, we may proceed as in (3.6) where we have p
-n
and
= q-l
arrl we need
*
( or its partitions) by the corresporrling !:'nk
The cCJTPUtational scheme for the ordererl orthant
alternative problan follows directly as in (3.4) after making use
Yl = 131 ' Y2 = 62 - 131 , ... , Yq = 13q 13q - l and using the a;rresporrling elementary matrix to transfonn the
!:'nk to appropriate!:'nk on which (3.4) applies.
of the reParaITEtrization
With a view to deriving
(distribution-free) probability
sc::Ire
inequalities for the proposed test statistics (under H ) , first,
O
lVe oonsider the fo11cMi..ng sub-martingale characterization. let ~nk
= ~ (S (k» be the sub-sigma field gE"nerated by S (k) , for k= 1, ••• ,
....n
....n
n and let ~nO be the trivial sub-sigma field • Then, for every n
(~
1), (Bnk is noroecreasing in k : 0 .s. k .s. n. Note
that under H '
O
given iBnk ' for every j (=1, ••• ,n-k), Snk+' can assurre each of the
rerraining (n-k) p:>ssib1e values with the c~ probability (n-k)-l
Hence, rewriting L
....n in (2.1) as
that
I
Eol !:n
6?>nk J =
t':-1
(c....
1.S
.
nl.
!:nk'
V
cn
-
)an (i), it fo11o.vs
....
k : 0 .s. k .s. n ,
(3.9)
where the !:nk are defined by (2.9). Thus, under H
(2:.
1) , {!:nk: 0
• Also, note that by (2.14), for every k ( < n),
right by L
r
....n
Tnk
.:s.
' for every n
O
k .s. n} is a martingale sequence, closed on the
-
k
sup{ (r'!:nk)/Ank(r'~nr)2
-1 ~
~l
~
~
= Ank(rrikLnk)/(rnk~nrnk) ,
=
YEr}
(3.10)
~
where
rnk is ~nk""'ireaSUrab1e and it belongs to
r ,
with probabi1~ty
1. Therefore, by (2.15) and (3.10), we obtain that
*r
1'0
-1
.
~
\..Onk J = AnLOI sup{ (r'!:nk+1 J / (r ~nr) :r
EOI Tnk+1
I
E
I
r} CBnk]
2:. A~~O I (r~!:nk+1) / (r~~nrnk) ~ I ~ nk]
-1 ~l
~,
~
k
= An (rnkEOl!:nk+1 \CB nk] )/(rnk~nrnk) 2
-1~'
~,~
= An
(rnk!:nk) / (rnk~nrnk)
= T*r
nk
a.e. ,
~
(3.11)
for every k : O.s. k .s. n-1. Since under H ' ~ = ~ , the region r
O
contains the orogin 0 as a boundary p:>int , and hence, it is easy
* ..
to show that the TJ are all nonnegative r.v.s, so that from the
above inequality, we conclude that under H ' for every n
O
{T;r ; 0.s. k .s. n}
is a nonnegative sul:martinga1e.
~
1,
(3.12)
Consider now a nonnegative, nondecreasing and convex function
h = {h(t), t ~ O}, such that for every p:>sitive integer r (.s. n),
Eolh(T:')] <
Q)
•
Then, by virtue of the sub-martinga1e maximal
inequality, we have for every A > 0 ,
pi
max
*r
0 <k< r Tnk
for every r
~
IH
*r
(3.13)
{EOh(Tnr ) }/h(A)
2
n. In particular, for h (t) = t , we note that
E I(T*r)2J
o nr
> A
O} ~
sup{ (y'L )2/{A2 (y'C y)}:y
- -nr
n - -n..
= EOI(
£
r })J
~ EOI( sup{ (r'~nr)2/{A~(r'~nr)}:r £
= EOI (L'
C- L ) /A2 J
-nr -n-nr
n
q
E })]
2
2
= (A;!A
n
)
~
~
1, for every r
n.
(3.14)
As such, for square integrable score functions, we have for every
r~nandA>O,
p{
O~r T~ ~
A
I HO
}
~ A-2EO(T~)2
,
(3.15)
where in various particular cases of r , using expressions similar
to those in (3.4), (3.6) an:i others, one may evaluate the secorrl
IIDIll9nt of
T~
(urrler H ) an:1 obtain an exact expression for the
O
right hand side of (3.15). In any case, a (very crude) upper rourrl
for the right ha.rrl side of (3.15) is simply
for
A-2. In general, if
> 1), (L'c-L)m has a finite expectation, proceeding as
-n-n-n
*r 2m
in (3.14), we conc1ooe that EOI (T ) ] is finite, for every r < n,
nr
so that by (3.13), for every A > 0,
SCITe m(
max T*r > A I H } < A-~ (T*r)2m
(3.16)
O<k<r nk
0
0 nr
'
which may result in a sharper round ( than in (3.15». In actual
p{
practice, often, the scores a (i) are generated by a score function
n
1
¢ = {¢(U)iO<u<l}, such that for same u > 0 , f exp(u¢(u»du < 00.
O
The Wi1coxn scores, nonnal scores, log-rank scores arrl the other
notable ones all satisfy this conclition. In that case, noting that
EOI exp(UT*r )J < EOI exp(uA-1 (L'C-L )~ J < o:l , we obtain from
nr
n -n.-.n..-n
(3.13) that for every A > 0,
r:: n ,
max *r
p{ O<k<r Tnk
> A
I
inf
HO} ~ u > 0
{-uA
e
*r it
EOIexp(UTnr )]j(3.17)
and this provides a rrore sharp rourrl. Note that the right ha.rrl sides
of (3.13), (3.15), (3.16) and (3.17) can all be evaluated by using
the perrm1tation distribution of the ranks, arrl hence, these are all
distribution-free. Note that for the stopping t.ine in the horizontal case in <.2.161, we have for every r: r .:: n-l ,
.
I H0
K(I) > r
p{
n
}=
p{
max T*r
k < r nk
<
I --U
H_
L CI)
n
}
(3.18)
'
so that the rounds in (3.15), (3.16) and (3.17) also apply to the
distribution of the stopping time •
let us
TY::M
consider the case of the square rr-ot rourrlary in
(2.17). By virtue of (2.15), for this stu:1y, we need to consider
max{ ~nk : k.:: r}, when H
O
in (1.2) holds. Since for every n , AnA~ is nonincreasing in k :0
suitable probability inequalities for
~ k ~ n , by using (2.15), (3.12) and the H£jek-Re"nyi-chow inequal-
i ty for sul::rnartingales, we have for every r
max
r
p{ k < r Tnk ~
I
).
HO }
= p{
~
). >
n,
max
-l_*r
k < r AnAnk~nk
>).
0,
I
HO }
-2
*r 2
*r 2
2 -2
An).
ILk < r Ank E { (Tnk ) - (Tnk- l ) }].
(3.19)
O
In a similar way, the rounds in (3.16) and (3.17) can be m:xlified
<
to suit this case too. There is , hoNever, a basic concern here.
we
may note that in (3.19) (or in the other p::>ssible formulae), we may
have to limit k to k
< k < r , where k
0-
-
0
= min{
k :
A~l'
JIh
> 0
LEven
so, for any fixed k ' as n becates large, the right han:l side of
o
(3.19) may not converge to a limit less than one • Rather, it may
increase with n , typically, at the rate of log n • This feature is
naturally discouraging, and is also shared by the PCS tests for the
global alternative case.
'Ib eliminate this drawback, as is usually
done with the square root boundary, we conceive of a postive number
£ : 0 < £ < 1 (usually
£ is quite small), and, for every n , we
define a p::>sitive integer k (£) by
n
2
2
k (£) = min{ k : A _ /A > £
n
hk'n-
, k _> I},
(3.20)
and the PCS testing scharE starts only after the k (£)th failure
n
has occurred. With this adjustment, in (3.19), the range of k is
restricted to
k (£) < k < r ( <n),
n
-
-
-
am
within this range, the rig-
ht hand side of (3.19) converges to a limit (depending on £ and r/n)
as n ....
Similar results hold for distribution of the stopping variable K(ll) , defined in (2.17).
(X).
n
4.
INVARIANCE PRINCIPLES FOR PCS TFSI'-STATISTICS
Invariance principles for the Lnki k
~
n, playa vital role in
the asynptotic theory of PCS rank tests for global altE!matives i
these are discusse:l. in detail in Sen (l98la, Chapter 11). we extend
these results to the case of restricted altematives, arrl in section
5, we incorp;:>rate these findings in the study of the asyrrptotic theory of the prq;x:>se:l. tests.
For every n , let {k (t) i 0 < t < I} be a sequence of integern
--
valued, nondecreasing and right-continuous function ( of t) defined
on the unit. interval 10,1] in the following nanner :
2
k (t) = max{ k : A
< tA2 }, 0 <t<li k (0) = 0 •
hk n
n
-n
Define then
B = {B (t) i 0 < t <l}
n
n
--
-1'
by letting
-
Bn (t) = An (~nk (t) ~n~nk (t»
n
n
(4.1)
Also, for every £ : 0 < £ < 1, define
~
, 0 < t < 1.
*
B
=
n£
(4.2)
*
{B (t)i£
n
< t < I}
by
letting
*
Bn(t)
=
-1
Ank
(t)AnBn(t) , £ ~ t
n
< 1 •
(4.3)
,
let then
W = ( WI , ••• , H) be q independent copies of a standard
...
q
BrCMIlian notion on 10,1], i.e., each W. = {W. (t), 0 < t <I} is a
J
J
--
standard BrCMnian notion an 10,1]. A q-paraneter Bessel process B =
{B (t) i 0 ~ t ~ I} is then defined by
,
-
l.:
B(t) ={f W(t)] IW(t)]} 2,0 < t < 1 •
--
,..,
(4.4)
Similarly, a staI'rlardized Beseel process is defined by B£* ~ {B* (t)=
t
-l.:
~ (t)
, £ :5.. t :5.. I} , defined over [£,1], for every £ : 0 < £ <1.
If we assurre that as n
n-lc
-n
-+
C
...0
-+
00
,
(positive definite)
(4.5)
and the scores are generated by a square integrable score function
( of bounded variation an Ic,d] for every 0 < c < d < 1 ), then under H in (1.2), as n
O
B
n
3:1
-+
00
,
B, in the Skorokhod Jl-topology on 010,1],
B* 3:5-+ B* , in the Skorokhod Jl-topology on 01£,1],
n£
£
(4.6)
(4.7)
for every E : 0 < E < 1. These results are discussed in detail in
sen
(198la,Cbapter II) •
For the restricted alternative test statistics,
= { Br(t}i
n
0 < t < I}
--
r
n
B (t)
=
\\'e
define B
r
n
by letting
*r
T
(t) ,
nk
n
t E 10,1] ,
(4.8)
where k (t) and the T~ are defined as in before. Similarly, for
n
*r
*r
-k r
r
every E : 0 < E < 1, \\'e define B = { B (t) = t ~n (t) = T
(t) i
n
n
nkn
E < t < l} • Note that in (4.5), C has been assurrai to be p:::>sitive
-
-
-0
,
definite (p.d.). Thus, there exists a p.d. 0r\ , such that C
---
We define then a stochastic process Br
= { Bf (t) it
-0
= D_0-0
D •
E IO,l]} by let-
ting
Bf (t)
= sup!
-
k
(y'D W(t»/(y'D D' y)2
: y
-
~ ~~
for every tEla, 1]. Further,
*r
BE
= {B*r (t) , E ~ t ~ I}
~-
E r},
(4.9)
define, for every E : 0 < E < 1,
*r
-k r
by letting B (t) = t ~ (t) , for
\\'e
every t E IE,lJ. Now, proceeding as in (3.4) or (3.6), it follows
r
that for a general class of r , for each t, B (t) can be expressed
as a convex mixture of quadratic fonns in
~(tl
of ranks 0,1, ••• ,q,
where the mixing coefficients are the indicator functions of various
linear transfonns of Wet) (and are therefore stochastic in nature).
r
In such a case, for ~ry fixed t E IO,lJ, the distri.b.1tion of B (t)
can be expressed in tenns of a convex mixture of
SCIre
chi distri.b.1-
tions with DFs O,l, ••• ,q , and in the literature I viz., BarlCM et.
al (1972)J, this is known as the chi-bar distribution. In view of
r
this, we shall tenn B as a Bessel-bar process on [0,1]. Similarly,
B;r will be tel:Irl:d a standardized Bessel-bar process on [E,l]. The
main theorem of this section is the follCMing.
Theorem 4.1. Urrler the assurrai regularity corrlitions, ~ H in
O
(1.2) holds,
B~ ~--
~
n increases,
Br , in the Skorokhod Jl-topology on Dl0,lJ,
(4.10)
and, for every E : 0 < E < 1,
*r ~
*r
B
~ BE' in the Skorokhod J l-topolm on DIE, 1] •
nE
(4.11)
Proof. VE consider first the case of (4.10). we need to establish
(i) the convergence of the finite dim.:msional distributions (f .d.d. )
of
r,
B~
to B
and (ii) the tightness
{B~}
• Note that under H
O
and the asst1l1B'l regularity conditions, for every m( ~ 1) and arbit-
rary 0 < t <••• < t
1
-
-1 -1
of
< 1,
m-
1?o An {~nk (t ) , ••• , ~nk (t ) }
n
n
1
~ {!(~)
m
, ••• , !(t }} ;
m
(4.12)
we may again refer to sen (198Ja, Chapter 11) for the details. Then,
by virtue of (2.14), (2.15) and (4.9), (4.12) ensures that for every (fixed) m ( > 1) and t , ... ,t (belonging to {O,l]), as n -+ 00,
1
m
r
r
~
r
r
{B (t ), ••• ,B (t)}
~ {B (tl), ... ,B (t) },
(4.13)
n 1
n m
m
r
which proves the convergence of the fed.d.' s of {B } to those of
n
r
B • By virtue of (4.8) and the sub-martinga1e propoerty in (3.12),
we use the (Brown) maximal sub-martingale inequality and conclude
r
r
that the convergence of fed.d.'s of {B } (to those of B ) also
n
i.rrplies the tightness of the sarre. For the weak convergence result
in (4.11), we note that for every e: > 0 , t
~
e:
, t
-k
2
< e:
so that (4.13) ensures that for every (fixed) m (> 1) and
-k
2
<
00
,
t , ... ,
1
t , all belonging to Ie:, lJ ,
m
*r
*r
~
*r
*r
{B (t ), ... ,B (t)}
.~ {B
(tl), ••• ,B (t)}.
n
n m
m
1
(4.14)
Finally, the continuity and boundedness of t - \ for t e: {e:,l], and
r
the tightness of {B } ensure the tightness of {B*~} • Q.E.D.
n
n~
So far, we have confined ourselves to the case where II
in
O
0 , under the assurral regularity
- :f -
(1.2) holds. For any (fixed) B
conditions, as n
{n-~nkn(t}
where
-+
00
,
p
- An(t) , t e: LO,lJ}
A (t) does not vanish (as n
n
VE may refer to Chatterjee and
-+00
-+
)
sen (1973)
ssed in detail for the sinple case of q
holds for the general case of q
~
0,
(4.15)
on the entire interval (O,l).
where this has been discu-
= 1,
and the sarrw= picture
1. As such, for a fixed
~:f ~
,
e
Bnr (or Bn*r ) may not converge weakly to a drifted Besselbar process. In fact, they may not have nondegenerate asyrrptotic
6 E
_
r,
f.d.d. IS. 'Ib avoid this limiting degeneracy arrl to obtain sane meaningful aSyIl1ptotic results, we confine ourselves to sane local alternatives {K } , where referred to the nodel (1.1),
n
-1:
(4.16)
Kn : 6 = -6 (n) = n 2 A
and
A £
r •
_
Note that the positive h:::m::>geneity of
~
ensures that tmder K '
n
and that for any fixerl ~ E r , the alternative
r ,
(n) belongs to
r
approaches the null point as n
-+
00
•
we also assurre that the d. f. F
has an absolutely continuous p.d.f. f with a finite Fisher infonnation I (f)
1/J (ul
•
=
! (f' If) 2dF
= -f' cr-l
_
¢t(u}
=
n(t)
=
«
(0).
let us then define
(u»/f (F-l (u»
,
a
(4.17)
< u < 1 ,
¢ (u) , a < u :- t ,
-1 1
(l-t)
f ¢(u)du
t < u < 1
t
fa1 ¢t(u}
1/J(u)du , for every t £ 10,1].
{
(4.18)
(4.19)
Recall that we have set (without any loss of generality) that ¢ =
1
1 -2
1 2
.
fO¢(u)du = O. Also, note that n*(t) = fa ¢t(u)du If a ¢ (u)du ~s a
nondecreasing function of t on 10,lJ, with
1. we define then
set)
= inf{
s
=
u
n*(u)
2: t } , for t
not be linear in t • Defining the
r
~(tl
£
£
n* (1)
=
(4.20)
10,1].
10,lJ, though, it need
as in before, we set
= sup! (x' I~~(t)~~n(s(t»J)/(r'~r)
-
= a arrl
{s (t), t E 10,1]} by letting
l'bte that set) is also nondecreasing in t
B). (t)
n* (0)
1:
2: r £
r },
(4.21)
r = IBr (t), t E 10,1]} and,
A
*r
='1: r
for every E:O < E < 1, let B = {t~). (t); t E I£,l]}. l'bte that
AE
for every t £ 10,lJ. Finally, let
B).
the contiguity of the probability measures tmder {K}
to those un-
n
follo.vs as in (11.4.34) in Sen (198:lP, Chapter 11), arrl his
der H
O
Theorem 11.4 also leads to the convergence of the f.d.d. 's of our
"""-'-.. these CULl
--~ .1..0
e 1 lCMlllg
.
.
Br and B*r • UAl.UJmmg
sane routme
steps, we
n
n
arrive at the follOlNing :
Under {K } arrl the asst:nTed regularity coooitions, as n
n
-+
00,
B~ ~ B~
, in the Skorokhod Jl-topology on 010,lJ,
and, for everi"£ : 0 <
*r
Bn £
(4.22)
£ < 1,
*r
~
~
l\£ ' in the Skorokh::xl Jl-topology on OI£,lJ. (4.23)
q
In particular, for unrestrictErl alternatives, L e., for r = E ,
and
B~r reduce to a drifted
I{
and drifted standardized Bessel process,
respeCtively.
s.
ASYMPTOI'IC PROPERrIFS OF THEPCSTESTS
For the proposed PCS tests, the stowing times K(I) and K(ll) ,
and the critical levels
n
n
T(II) are defined as in (2.16)
T(I) and
n
n
and (2.17). Asynptotic solutions to the critical levels are provided
by the basic invariance principles in Theorem 4.1.
Also, by virtue
of (3.18) and a similar identity for K~II} , the asynptotic distribution theoxy of the stopping times ( under H ) can also be formuO
lated in tenns of the first exit time distributions related to the
two bessel-bar processes Br and B*r • For local alternatives, such
£
as in (4.161, the asyrrptotic power function of the proposed PCS tests as well as the asynptotic distribution of the stopping t:irres
can similarly be fontllalted in terms of the two processes
B~~ • For
Bi
•
and
the classical Bessel process and the starrlardizerl Bessel
process, critical levels have been canputed by various workers; we
may refer to DelDng (1980,1981) where other references are also
cited. Using the same inequality as in the first step in (3.14), it
q
is easy to verify that for any restricted subset r £ E , the critical levels for the Br
and
B*r are daninated by those of B and B* ,
£
£
reSPeCtively. On the other han:1, looking at (4.21), we observe that
r
~
B). (t) ~ (~'~(t» (~'So~r + 7T (s (t»
wheli
0 'So~) ~2,
t £ 10,1] , (5.1)
for every ~ ~ ~ ,
1
{(A'O W(t»/(A'C ).)~ ;t £ 10,1]}
.., ...,0.....
-
~-
T1u.ls, noving A may fran
~
a
~
~
= {W(t);t
-
(Le., making Atc A
~-
£
IO,lJ } •
ts.21
adequately large),
and using the boundary crossing probabilities for an ordinary (dri-
£ted ) Brownian notion process, it follows that the power of the
PCS test can asyrrptotically be made arbitrarily close to one. In
fact, (5.1) and l5.2} provide a sinple lower bound to the asyrrptcr
tic power functions for the prqx>sed PCS tests (against restricted
alternatives). Consider the two bessel-bar processes Br and B*r,
•
and let
L~I}
and
L~II)
distributions of sup{
be respectively the
Br (t) : O;.t<l}
u~r 1000.%
pointEof the
and sup{B*r (t) : E;.t<l}. 'lben, un-
der {K } and the assurred regularity conditions, a lONer round for
n
the asyrrptotic power function of the test based on the stoWing rule
in (2.16) is given by
p{ W(t) + n(s(t»
A)~
T(I), for same t E lO,I]},
a
and the parallel result for the stoWing rule in (2.17) is
(AIC
-
>
(5.3)
.;.0-
p{ wet) + n(s(t}) (AtC A)~ > t~(II) ,for SCIre t E IE,l]}; (5.4)
- -0a
for roth the fo:rmulae, s (t) and 1T(u) are defined as in (4.19) and
(4.20), and, without any loss of generality, in (5.11- (5.4),
\.IllE that the score function
= f~
cj>2 (u)du
= 1.
cj> is so nonnalized that
Further, for the stopping rule in (2.17),
O
O
restricted ourselves to the case of k > k where k
2
- n
n
> EA }, corre5fOn::1ing to the chosen E: 0 < E <1.
-
~ =0
= min{
~
and
ass-
A~
~ have
2
k : A
hk
n
In the conventional tenninal test (based on all the observat-
ions), the FQWer' superiority of the tests for restricted alternatives over their counterparts for the global alternatives has been
studied by a host of v.urkers. we may refer to Chatterjee and De
(1974) and Chinchilli (1979) where other references are also cited.
While theoretical results have only been obtained in sane very simple cases (and un::1er rather stringent regularity con::1itions), simulation and nurrerical sttrlies reveal that in::1eed there is gain in the
power for the restricted alternative tests. A fully general result
..
in this direction for a general class of restricted alternatives has
yet to be fo:rmulated, and an extension of this result to the setup
of PCS ranains as a challenging mathanatical problem. HcMever, the
prospect of numerical and simulation studies in this direction looks
very bright. First, ~ Catmmt on the n\.IllErical carputations of the
critical levels T (I) and T(II). For the special case of r = Eq ,
a
a
these may directly be obtained fran DelDng 0-980 , 1981}. In sane
r
other specific cases (e.g.,
q
as a linear subspace of E ), it
nay be possible to errploy DeIDng' s nurrerical integraticn technique
to evaluate the critical levels. lbwever, in genral, the nurrerical
evaluation of the Fburier coefficients , needErl in this context,
nay becare so involved that his technique may not work out well.
The prospects for a !vbnte carlo study is good. For
SCIre
large
positive integer N, let N* = Nq and generate N* irrlependent standa, ... , Y. ) ", i=l, ••• ,N.
il
1q
r'<k Y. , k= 1, ... , N and let W_(o) = o.
rd nonnal variables , blocked as Y. = (Y
_1
Define then
W_(k/N) = N "2
~~
-1
1
~N
-1
-
Parallel to that in (4.9), let
r
~
r
B(N) (k/N) = sup{ (r'~~(k/N))/(r'~)2 :
£ r }
(5.5)
for k = 1, ••• ,N. For the CCIlputations in {5.5}, the algorithrnE in
section 3 may be used with advantage. Ccrnpute then
max
r
k ~ N B (N) (k/N)
or
max!.:r
N£<k <N (N/k) ~ (N) (k/N)
(5.6)
Repeat this procedure a large number of tines, and plot the enpirical d.f. 's for the statistics in (5.6) ; obtain the critical levels
fran these empirical plots. By choosing N and the number of replications adequately large, this simulation rret:h:rl can be used to obtain quite reliable approximations to the true critical values. In
a different context ( arising in PCS ), Sinha and Sen (1982) have
used a similar simulation technique which is quite econanic and
accurate too. For general
r ,
this may be recClllTeIrled without ITUlch
reservation. The other possibility is to use the peIllUltation distribution as has been rrentioned in Section 3. Note that the.f
nk
(or
T: ) are rank statistics , so that their null distributions are
generated by the n! equally likely permutations of the vector of
ranks ( over the set of all n! perm.1tatians of (1, ••• ,n) }. Thus,
it may be siltpler to use a subset of N randan peJ:JTU.ltations of the
set of all possible perm.1tations of (1, ••• ,n), and for each such
permutation, the actual test statistics can be COllp.lted, a"ld the
erpirical distri..l:Jution of tlan may be used to detennine a:!pirical
cr'itical levels which are good approxinations , when N is large. It
is quit-..e clear that bJotstrap net:lx::ds can also be used in this
context.
~er, ~e ~
are dealing with rank statistics, there
may not be any extra advantage of using the lx:x:>tstrap nethod over
the pennutation nethod , nentioned earlier.
...
The nunerical arrl sinulation nethods considered aOOve are also
applicable to the stooy of the distri.b.1tion of the stopping ti.lres
in (2.16) or (2.17) , when H ooIds. Also, (5.1) arrl (5.2) , in such
O
a problem, provide suitable rounds for the asymptotic distri.b.1tions
of the stopping ti.lres when H may not mId.
O
The t-bnte Carlo sttrly schaTe, discussed earlier, may also be
IIDdified to study the asymptotic non-nu1l distribution theory. In
such a case, we note that s(t), defined in (4.20), dePending on the
chosen score function ¢ , is a known function of t on 10,1]. Thus,
for given A, C and D ,we may define
...
-0
-0
r A J <
B(Nl(k/Nl = sup{ (r'I~~(k/N)~~7T(S(k/N»J/(r'~x)2:r£ r }
for k=l, ••• ,N, where the
•
~(k/N)
are defined as in before (5.5).we
may then consider the maximal statistics as in (5.6), and , then by
replicating the experilrent a large m.nnber of tines ,
te the asymptotic
p:JWer
~
may estima-
curves by the corresponding empirical plots
( counting the prop;::>rtion of ti.lres a rejection is made ), varying A
over a range in
r.
...
In this context, note that ¢(u), u £ (0,1) act-
ually depends on the score function ¢ and the underlying p.d.f. f
(through the score function '1/1), and hence, a specific fonn of 7T (u)
needs to be used in si.1.nul1ating the test statistics. Or, in other
~rds,
in the Monte carlo sttrly of the
p:JWer
properties, the inpact
of the underlying d.f. F is through the function 7T(U),U
£
(0,1), so
for actual numerical studies, specific fonn of 7T (u) has to be used.
In the so called, locally IIOst powerful rank statistic case, we ha-
.
ve ¢ =
'1/1 ,
yielding 7T (u) -= u: O<u<l, so that working with 7T (u) = u
provides the picture for this locally optimal score function. In
practice, however·, F may not be known, and hence, for a chosen score
function, the
p:JWer
can only be studied by reference to appropriate
d.f. F , for which the corresponding 7T(U) is used in the simulation.
A very similar case holds with the simulation sttrly of the asympt0tic distribution of the stopping ti.lre under local alternatives.
5.
RESTRICTED ALTERNATIVE PCS TESTS FOR THE COX MJDEL
Cox (19721 considered a semi-nonpararretric llDdel which has re-
ceived considerable attention in survival analysis. In his setup,
the hazard function is of arbitrary fonn, though the effect of the
covariates is of
SCIre
asSl.lllEd structural fonn. '!he nice thing aOOut
the partial likelihood approach of Cbx, is its flexibility to vari-
ous types of censoring which may otherwise cause serious concern to
a conventional full likelihood awroach. Invariance principles for
the usual scores statistics relating to the Partial likelihood have
been studied by a host of w:::>rkers. Sen (l98lb) has errployed these in-
variance principles for constructing tests for the null hYPJthesis
of harogeneity in a PCS. Later on, Sen(l984)
has also considererl
tests against restricted altematives based on the Cox Partial likelihood ratio statistics • These tests are corrparable to the conventional tenninal tests, though, they incorporate various types of
censoring in a simple marmer. The asymptotic distribution of such
a test statistic (under H ) e<:ITeS out in the fonn of a chi-bar disO
trihltion, and , thereby, is very I'll.1ch canparable to the conventional case of other likelihood ratio tests, treated in BarlCM et. al
(1972) •
we may refer to Section 2 of Sen U984) for sare of the details
of the Dr-tests for restricted altematives for the Cox regression
nodel. Though specifically the tenninal test statistic was constructed there, at each observable failure point, a very similar version
of this test statistic can be carputed. By virtue of the martingale
properties of the Partial likelihood ratio scores statistics, studied in detail in Sen (l98lb) , we have a sub-martingale propoerty for
the PCS Dr-test statistics related to the Cox nodel ( under H
),
O
and as such, the theory developed in Sections 3 and 4 here, ranains
applicable to the Cox llDdel too. Thus, we have similar Bessel-bar
process approximations for the Dr-test statistics in the Cbx llDdel.
The discussions in our Section 5 therefore pertain to the Cox nodel
as well; however, unlike the rank tests, the other tests may fail
to be genuinely distribution-free under
Ha.
•
'!his work was supported by the National Institutes of Health,
..
Contract: NIH-NHIBI-71-2243-L fran the National Institutes of Heath. The rrain therce of this pa.per was presented in a contri..b.1ted
pa.per session at the XII International Biaretric Society Meeting in
'Ibkyo (sepet:mber, 1984), and an informative abstract appeared in the
proceedings of this conference.
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l2~
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•
..