"
RANK ANALYSIS OF COVARIANCE UNDER
PROGRESSIVE CENSORING
by
Pranab Kumar Sen
Department of Biostatistics
University of North Carolina at Chapel Hil
•
Institute of Statistics Mimeo Series No. 1118
May 1977
RANK ANALYSIS OF COVARIANCE UNDER
PROGRESSIVE CENSORING*
by
•
Pranab Kumar Sen
University of North Carolina, Chapel Hill
ABSTRACT
In the context of survival analysis under a progressive censoring
scheme, a class of analysis of covariance tests based on suitable linear
rank statistics is proposed and studied.
Some invariance principles for
certain (multivariate) progressively censored rank order processes are
~
established and incorporated in the study of the asymptotic properties
of the proposed tests.
AMS 1970 Classification Nos:
•
60FOS, 62GlO, 62L99.
Key Words & Phrases: Analysis of covariance, asymptotic power; asymptotic
relative efficiency; Brownian motions, censoring, invariance principles;
linear rank statistics; permutational invariance; progressive censoring.
*Work supported partially by the Air Force Office of Scientific Research,
U.S.A.F., A.F.S.C., Grant No. AFOSR-74-2736 and partially by the (U.S.)
National Heart, Lung, and Blood Institute, Contract NIH-NHLI-7l-2243 from
the National Institutes of Health.
-2-
1.
INTRODUCTION
In clinical trials and life testing problems, due to practical limitations, often, statistical tests are based on censored or truncated data.
In this context, a progressive censoring scheme (PCS) incorporates a continuous monitoring of experimentation from the beginning with the objective
of an early termination depending on the accumulated statistical evidence.
In this sense, a PCS involves a time-sequential test.
For censored or
truncated tests based on ranks, we may refer to Breslow (1970) and
Halperin and Ware (1974) where other references are cited.
For nonpara-
metric testing under PCS, Chatterjee and Sen (1973) have formulated a
general class of tests based on linear rank statistics, censored at successive failures.
Further works in this direction are due to Sen (1976a,b),
Majumdar and Sen (1977) and Davis (1977), among others.
In all these stu-
dies, PCS tests have been developed only for the simple analysis of variance problem.
In application to clinical trials, besides the primary variate
[viz., failure time due to a heart-attack], there may be other concomitant
variates [viz., initial blood pressure, cholesterol level etc.] which possibly
acco~nts
for some assignable variations in the observed responses.
Thus,
in this setup, an analysis of covariance (ANOCA) in the context of survival
analysis is deemed appropriate.
In the present paper, we desire to develop
such rank based ANOCA tests under PCS.
A variety of rank ANOCA tests, considered by Quade (1967), Puri and
Sen (1969a) and Sen and Puri (1970), being based on the complete sample
-3-
observations, is not directly usable in a PCS,
Basically, the repeated
significance testing involved in a PCS (relating to an increasing
dimen~
sion of dependent data) introduces extra complications in a valid statistical analysis and calls for more refinements.
The theory developed
by Chatterjee and Sen (1973) is extended here to the general ANOCA model .
.
The proposed tests are based on progressively censored linear rank statistics in a multivariate setup and rest on a permutational-invariance
principle developed earlier by Chatterjee and Sen (1964) [and incorporated in the ANOCA problem by Puri and Sen (1969a)],
Along with the preliminary notions, the basic problem is formulated
in Section 2.
The proposed PCS tests are developed in Section 3,
totic distribution
~
Asymp-
theory of the test statistics, under the null and
local alternative hypotheses, are developed in Sections 4 and 5.
The
concluding section deals with some general remarks,
2.
Let
{X~
-1
PRELIMINARY NOTIONS
= (XO"X!)';
X! = (Xl.,
•.. ,Xp1.);
1 -1
-1
1
independent random vectors (rv), where
are the primary variables and the
It is assumed that
tion (df)
F~(x),
1
X~
-1
p
X.
-1
has a continuous
i~l}
be a sequence of
is a positive integer, the
are the concomitant variates.
(p+l)-variate distribution func-
and we denote by
-
F. (x)
-
~1
= F~(oo,
x) = p{X.
~ x} ,
1
-1P{X o 1' ~ xolx.
=x} ,
-1-
(2.1)
(2,2)
-4-
F.1 0 (x)
=
p{X 1
= F. (x , 00) ,
. ~ x} 1
O
'rJ i ~ 1 .
E,
X E
(2.3)
As is usually the case with ANOCA models [with stochastic covariates; cf.
Scheffe (1959, Ch, 6)], we asstune that the df
on
i(
F.
in (2.1) does not depend
1
so that
~l),
F.1 (x)
= F (x) ,
,...",..."
~
V i
~.
1 ,
.-
E
EP •
(2.4)
Our basic problem is to test the null hypothesis
(2.5)
Note that in view of (2,4),
F~ (x)
1
-
H implies that
O
= F* (x)
,
V i
~
1 ,
-
~E
EP+l
(2,6)
•
Keeping in mind a simple (ANOCA) linear model, we set
. > 1
1 -,
are known constants (not all equal) and
where the
parameter.
df's
E
EP + 1
,
(2.7)
B is an unknown
For example, for the so called two-sample problem, we may set
n -- n 1 + n 2'
so that
(x 0'X t)'
B
n
~
I,
n
~
2
1,
c
1
= ,.. =
c
°
=
and c +
=
= c = 1
1
nl
n '
l
stands for the difference of locations of the two conditional
l
n
o
HO: B
In this setup, (2.5) reduces to
and
= 0,
In the context of a life testing problem, we conceive of the set
{X~;
-1
i
= 1, ... ,n}
and assume that the covariates
Xl, .. "X
-n
-
are all obser-
vable at the start of the experimentation. but the primary variates
. '.'X on
ing to
are not so.
XOl '·· .,X On ;
X '
Ol
be the ordered rv t s correspond-
ZO 1 ~ ... ~ ZO
n,
n,n
by virtue of the assumed continuity of the df's, ties
Let
-5-
th·~
among
(and hence, the
XOi
are neglected, in probability,
Let then
for
~s i '
so that
~
=
and is a (random) permutation of
2,1 =
ZO k'
n,
(1,. ",n).
,1
~S,)
Z
n,l"
1
,
(2.8)
In a life testing situation,
for
i
= 1, .. "
k ,;. k:"> n •
In a fixed-point censoring scheme, for some pre-fixed
zO
n,r
mentation is curtailed when
(2.5) is then based on
XOi
the observable rv's are
°
(S"
= 1, , .. ,n
represents the vector of anti-ranks of the
(Sl"",Sn)'
at the k-th failure
i
Ql'" .,Qr'
r(:"> n) ,
(2.9)
experi-
H in
O
one proceeds to
has been observed and a test for
In contrast, in a
pes,
construct a time-sequential test based on the progressively available (partial) sequence
.2r.
21 ""
prior to observing
ZO
n,r
so that the option of stopping experimentation
is left open.
In this setup, depending on the
accumulated statistical evidence at the various failures, one may stop
when, for some
zO
n,k
k:"> r,
is observed.
To formulate suitable rank-based
or
according as
0
R..
J1
for
i
=
=
u
is
ta= lC(X,.J1 -X.Ja)
l, ... ,n
and
or
~
=
< 0
Rank of
j = 0,1, ... ,p,
pes
tests, let us denote by
=
((R .. )) '-0
J1
= 1
and let
X..
J1
among
X'J
,X.In
O""
'
(2.10)
Thus, for the complete sampling
scheme, we obtain the rank-aoZZeation matrix (of order
R
-n
c(u)
J- ," .,p,' 1
- ,•••1
,n '
(p+l) xn);
(2,11)
-6-
Consider now a permutation of the columns of
in the natural order [viz.
l •.•.• n].
R
41
so that the top row is
and denote the resulting matrix
[termed the peduced pank-coZZection matpix] by
R
41
+
R*
""Il
=
n
2
1
so that
R*
R*
R*
11' 12· .. ·• In
(2.12)
~;;:'~;~:::::~;~J
Note that by (2.8). (2.11) and (2.12).
ROS .
=i
R~.
and
J1
1
= R·J s .
for
j = 0.1 •...• p
and
(2.13)
i = l •... ,n .
1
Thus, corresponding to the partial collection
(21'" .• 2k)'
we have the
paptiaZ peduced pank-coZZection matpix
R*
""Il,k
*
((R J1
.. )) J. -0 , 1 ,.··,P.1.. -1 , ...• k
=
for
k
=
1, .•. ,n
(2.14)
S
""Il.n
(2.15)
We also denote by
S
""Il,k
=
(Sl' ... ,Sk) ,
l::;k::;n •
so thCl.t
S
""Il
=
Since we are interested in developing a rank test under PCS, we confront the problem of constructing a (sequential) test based on the partial
{R* k; k::; r} (where, we may even let r =n) . [Note that as the
""Il •
covariates are all observable at the beginning of the experiment, we have
sequence
S k and R*
at the k-th failure Z k' k::;n.] As
""Il,k
41,
""Il •
or the corresponding (R. S ' 0::; j ::; p) are not independent for
the knowledge of
the
S.
1
different
J i
i
[and the distribution of
on the underlying
F*
(even under
HO)'
R
""Il
R*k·k::;n) depends
""Il •
unless the coordinates of X~
(or any
~1
e
-7-
are mutually independent], in a PCS, the repeated significance tests involve
an increasing sequence of dependent data, and thereby, poses additional complications.
Our proposed tests are based on linear rank statistics.
In view of
(2.7), we consider the statistics
•
L = T(O),
... ,T(P))' = t l(c.-c)(a O(RO.), ... ,a
(R .))t
n
n
1=
1 n
n,
1
n,p p1
(2.16)
,
u
where the
c
are known (regression) constants, not all equal,
i
-l\,n
c n = n Li=lc i ,
the
R.. are defined by (2.10), and for each j(=O, .. "p), a .(l), .. "a .(n)
1J
nJJ
n,J
are (real valued) scores (not all equal) which we shall define more formally
later on.
based on
When all the
T
X.*
[viz., Puri and Sen (1969a) for motivations].
-0
o
setup, at the k-th failure
S
-0,
are observable, the usual rank-ANOCA tests are
-1
we are provided only with
Zn J k'
and hence, we need some modifications.
k (k ~ ri),
.
T
-0
In the current
R*
-o,k
We rewrite
\,n
* ... ,a (R * .))'
= L.
l(c -c- )(a O(RO.),
Si n n ,
1=
1
n,p p1
and
T
-n
as
(2.17)
.
* ... ,X * are independent and identically
H in (2.5), Xl,
O
-0
distributed (i.i.d.) rv, and hence, their joint distribution remains invari-
Note that under
ant under any permutation of the
the set of n!
n
vectors among themselves.
matrices obtained from
of the columns of the latter.
R*
-0
Let
R*
n
be
by all pos:ible permutations
Then, as in Chatterjee and Sen (1964) [or
Puri and Sen (1969b)], under (2.4)-(2.5), the conditional distribution of
R
-0
over
R*
is uniform [with each element of
n
ditional probability
measure by
P .
n
(n!)
-1
].
R
n
having the common
con~
Let use denote this conditional probability
In passing, we may remark that the unconditional df of
-8-
T
"1l
generally depends on the unknown
H '
distribution-free under
HO'
tion-free under
O
*
F.
and hence.
T
"1l
is not genuinely
Nevertheless. it is conditionally distribu-
Then, motivated by the line of attack of Chatterjee
and Sen (1973), we define
T
= (T(Ok) •... ,T(Pk)), = Ep (T Is k)
""Il.k
n,
n,
n "1l "1l,
=
I~-l(CS
-c )(a o(i).an, l(R lS i ), ...• a n,p (Rp Si ))1
1inn,
+
{I~-k
1- + l(c S . -cn )}(a*n, O(k) •... ,a*n,p (k))'
(2.18)
1
=
I~=l(Cs
1
. -cn )(an, o(i)-a*n, O(k),a n, l(R ls . )
1
1
-a* l(k), ... ,a
(R S )-a*
(k))
n,
n,p p.
n,p
1
1
where for
l~k~n-l,
a* .(k)
n,J
= (n-k) -l\n
[.·-k
lan,J.CR·] s i )
1- +
=
(n-k)-l{L~1= lan,J.(R~.)
J1
= (n-k)-l{ni . n,J
L~1= lan,J.(R~.)}
J1
-
I~_la
.(R~.)}·,
1- n,J J1
(2.19)
j = O,l,., .• p.
and
a
. = n
n,J
-l\n
.
[.. la . (1)
1= n,J
Conventionally, we let
we let
T 0 = O.
""Il,
a* . (n) = 0, 0
n.J
Thus,
a = (i o, ... ,i
)'
""Il
n,
n,p
j = O,l, ...• p
T k
""Il ,
~
j
~
P.
so that
is defined for every
T
= T
""Il,n
~n'
k: 0
(2.20)
and
~k ~n.
Our task is to consider a suitable sequence of covariate_adjusted
statistics, viz.,
L
n,k
= Ln (T"1l, k)
,
k=O,l, ...
,r(~n)
,
(2.21)
e
-9-
and continue experimentation as long as
value
Cl: 0 < Cl < 1
(where
L
n,k
lies below a critical
is the desired level of significance
of the PCS test); if, for the first time, for some
~
k = N(S r) , L
n,N
along
failure ZO
n,N
exists~ experimenta ..
len)
the experimentation is curtailed at the N~th
Cl '
with the rejection of H ' and if no such k(s r)
O
tion stops at the preplanned r-th failure zO
along with the acceptn,r
ance of
H '
O
Thus,
is the stopping number and
N(S r)
is the
stopping time of the time-sequential procedure in the prescribed PCS.
Finally,
P{L
~l(n)
n,k
for some
Our basic problem is to construct
l~n)
ksrIHo}SCl.
{Ln, k;
, k S reS n)}
(2.22)
and to choose
such that (2.22) holds.
3.
THE PROPOSED PCS TESTS
Let us define for every
k: 1 S k S n,
*
.(R *.. )a o(Ro·)+(n
..k)a * .(k)a * o(k) } .. a .a (),
n,) n,-t.
i=l n,) )1 n,-L -L1
n,)
n,-t.
_ 1 { kLa
v (k).()--
n,)-L
for
n
j, l = O,l, ... ,p,
V
'""Jl,k
and set
= ((v(k~ )).
n,)l
(3.1)
lskSn
),l=o, . .. ,p
Also, let
2
Cn
,n l(c.-c)
- 2
= /...
1=
1
n
V
'""Jl,0
=0
~
.
Then, following the line of attack of Chatterjee and Sen (1965) and Puri
and Sen (1969b), we have by (2.18) and (3.1),
(3.2)
(3.3)
-10-
= Ep
(In)
= 0,
"J k
(3.4)
n
$
n
2
{T
} = C •V
n ""Tl,k
p ""Tl,k
V
for
k = O,., •.. ,n .
(3.5)
n
To utilize the information contained in the concomitant variates, we fit
a linear regression of
duals.
T(O)
n,k
on
(1)
(p)
(Tn, k'· .• ' Tn , k)
and work with the resiC-1T
n ""Tl,k
[In view of the asymptotic multinormality of
(to be proved
in Section 4), the fitting of linear regression seems justifiable.]
Let
then
T(O)
T*
n,k = n,k
=
L~=o
T(O)
n,k
(fitted value of
(1)
(p)
(Tn, k"" ITn, k))
on
(v pj IvO O )T(j) ,
n,k n,k n,k
(3.6)
where
jt
-1
((v
)).
V
""Tl,k =
n,k Jlt=O' ... ,p
and, for the time-being, we assume that
k
~
1 ,
e
(3.7)
is positive-definite (p.d.).
V
""Tl,k
Also, note that
k
~
1 .
(3.8)
Let us introduce the standardize variates
_
E;;n , k
{
C_l(VOOk)~~*
k'
n
n,
n,
0,
if
otherwise, for
V
""Tl,k
0
$
k
$
is
p.d. ,
(3,9)
n .
Then, depending on the one or two-sided alternatives, we may have in mind
[viz., (2.7):
S > 0 or S ~ 0], we set
Ii.:n , k I ,
k ~0 .
(3.10)
e
-11-
Thus, our proposed PCS test is based on the standardized and covariateadjusted linear rank statistics at the successive failures for the primary
variates.
Though, in the above formulation, we have made use of
max
order to enumerate the permutational distribution of
for the determination of
..
zO
n,r
This is in contrary
to our aim of progressively censoring from the very beginning.
00
[required
k~r Ln,k'
len)
in (2.2)], we need the knowledge of R*
"11,r'
a.
so that we have to wait till the r-th failure
v
n,r
Pn , in
Also,
is not known until the r-th failure has occurred, and hence, unlike
in Chatterjee and Sen (1973), in (3.9), at the kth stage, we take
y
00
k but
n,
not yOO (for k < r). For these reasons, we do not use the permutation
n,r
distribution [as in Puri and Sen (1969a) dealing with the complete sampIe case].
Rather; we proceed to develop certain invariance principles
for the partial sequences
{T k ; k ~ n}
n,
and
{l;
n,
k; k ~n}
len)
ate them in the study of the asumptotic form of
a.
and incorpor-
'
•
4.
ASYMPTOTIC DISTRIBUTIONS UNDER
H
O
For the study of some weak convergence results for the partial sequences
{T
"11,
k; k
~n},
{T * k; k
n,
~n}
and
{l;
n,
k; k
~n},
we make the following assump-
tions:
r
=
r(n)
+
00
~
n
+
00
with
lim r(n)/n
n-+oo
(II)
we assume that as
For the maximum censoring point
(I)
The scores
the following way
a
. (i)
n,J
= T:
0
<T~
1
(4,1)
•
are generated by a score function
cp.
J
in
-12-
or
where
U < .. , < U
nl
nn
the rectangular
E<I>. (U .) ,
J n1
j =O,l, •.. ,p
are the ordered rv's of a sample of size
df, and for each
(0,1)
where
<1>. k(u)
J,
grable inside
J,
O<u<l,
J,
(4.3)
(0,1).
c*. = (c.-c )/C
n1
from
is non-decreasing, absolutely continuous and square inte-
Concerning the
(III)
n
j (0 $ j $ p) •
<I>.(u) = <1>. leu) - <1>. 2(u) ,
J
(4.2)
1
n
we let
c. ,
1
1
$
n
i
$
(.~ L,~1=lC*'
n1
n
=
O.
,n (c*)
Li=l
ni
2-- 1)
,
(4.4 )
and assume that
I I} -- 0 .
lim { max
*
n+oo ,l$i$n c ni
(IV)
Let
the marginal df of
X.. ,
variate (if
df of
joint df of
J1
P ~ 2)
sO
0
=_
0< t < 1, F [0] (St) = t
1
+
If
(of$j
= I\.
o
so that
¢.
J
•
andthetri-
all under
H '
O
has a unique solution
Also,
0
(St) ,
Let then
00
O$t<l ,
4'Ot = (l-t)- t <1>0 (u)du •
<P jt = (l-t) -1
be respectively
(X .. ,X o .) (O$j~£.$p)
J1
.(..1
(XO.,X .. ,X o .), l$j ~l$p,
1 J 1 .(..1
sO =
00 ,
F[O,j,£.](x,y,z)
o
we assume that for every
and we let
and
F[j](X)' F[j,l](x,y)
(4.5)
¢OO
= ~o
O$t<l ,
(F [j 1(y))dF [0 ,j J (x .y)
¢Ol = 0
j=l, ... ,p ,
(4.6)
(4.7)
t-ClO
J
(u)du
VOO(t) =
= 4'jO and let ¢jl = 0 , Vl$j$p.
r
o<PO2 (u)du
+
(l-t)¢~t
-2
<1>0
O$t$l ,
Also, let
(4.8)
e
-13-
_00
lP
poCO
+ (l-t)4)Ot jt -
=
..
lPOlPj
,
j
... CD
j,l
~(t) =
~(O)
((V.o(t))). 0-0
J'
=].
J,'-
! •••
= 1, ... , p,
for
,p
0
~
t
~
0 ~t ~ 1
(4.10)
(4.11)
1 .
We assume that
~(t)
~
(4.9)
f'~[" ["$j(F[ij(Y))$i(F[ij (z))dF[O,i,ij (x,Y,z)
_00 - 00
Note that
0~t ~1 ;
= 1, .. " p ,
is p.d. for every
0 <t
~
(4.12)
1 ,
and denote the reciprocal matrix by
~
Theorem 4.1.
-1
=
(t)
((v
jl
(t))). 0=01
J,'
, , ... ,p
for
O<t~l
Under (2.6), (4.2), (4.3) and for continuous
(4.13)
.
*
F,
(4.14 )
Outline of the proof.
l ~ p,
(k)
max
k< v . 0
_n n,J'
I
1 ~ j. l ~ p;
-
V.
It sufficies to show that for every
(k] P
°- I
J' n
-+
0
as
the case of either of
parative1y simpler manipulations.
F [.](x)
n J
= n -ltL.n1= 1c (x-X J1
.. )
,
n
-+
j
and we prove this for some
00,
or
(j ,l): 0 ~ j ,
l
being
0
follows by com-
Let us define
F [. 0] (x, y)
n J, ,
= n -ltLn1. =1c (x -X.J 1. ) c (y.,. X'0.)
1
,
(4.15)
-14-
for
~
j
i = 0,1, ... ,p,
and also for
Fn[O,j ,il (x,y,z) = n
j
~
i = 1, ... ,p,
we let
-l,n
[.. lC(x~xO')c(Y~X .. )c(z~xo.) .
1=
1
J1
-(..1
(4.16)
Further, we let
~~n)(u)
=
J
a
.(i)
n,J
for
i-I
n
-<
u::S;
i
n
j=O,l"."p
Then, by (3.1), (4.15), (4,16) and (4,17), we have for
(4,17)
k <n,
(4.18)
+ :k
~z~):$t)
-~:$t)
and for
k = n,
(Fn[j]
(y)
)dFn[O ,j] (X,y]
(Fn[j J(x) ldFn[j J(x]
~Z~'k[$t)
(Fn[!] (,)dFn[O ,!] (X,']
~:$Y) (Fn[!] (y))dFn[!] (Y1J
'
•
the second term on the right hand side of (4.18) droppsout.
Now, under (4.2) and (4.3), we may use the Hajek (1968) polynomial
approximation for the
~.
J
(and hence, the
a
. (i)),
n ,J
and proceeding as in
the first part of the proof of Theorem 3.1 of Puri and Sen (l969b), for
into a polynomial part (say.
J
n (k)
~.) and a residual part (say, "~.), such that, definining the v .t as
n, J
J
J
in (4.18) with the ~. being replaced by cjI1. , O::s; j : s; p,
J
J
every
n > 0,
we have a decomposition of
~.
j,t
Similarly, if in (4.8)-(4,10), we replace the
resulting quantities by
v.n (t)
J1
,
= 0,1, •.. ,p
¢.
J
by
~.
J
.
and denote the
then we have
(4.19)
.
-15-
Ivj i
(t) .. v ~li (t)
I
< n,
v 0S t
s1
and
v~~~i
Hence, it suffices to show that (4.14) holds with
replaced by
vn(~l
n,J~
and
V~D(~)'
n
J~
i sp .
0Sj ,
respectively.
and
(4.20)
vji(*J
~~J
Note that the
being
are all
polynomial (and hence, are bounded and continuously differentiable) and the
Glivenko-Cantelli Lemma insures the almost sure (a.s.) convergnece of
(to
0),
V 0 s j S p;
similar a,s. convergence
results also hold for the bivariate and trivariate df's in (4,15)-(4.16).
max IF
(ZO)
0
kSn n[O] n,k - F[O] (Zn,k)
.
F1nally,
as
n
~
I
max I
(
0
= kSn F[O] Zn,k) - kin
I~
0
Hence, by some standard steps, it can be shown that as
00.
a.s"
n
~
00 ,
(4.21)
and the desired result follows from (4.19)-(4.21).
F*
Note that by the continuity of
a continuous function of
"
v ji =
t
~
[0,1]
and
and the score functions,
~(l) = ~ = ((V
r rXl~j(F[j](X)Hi(F[i](Y))dF[j,i](X,y)
_00
Q.E,D.
-
ji
~j~i'
)),
~(t)
is
where
OSj,
isp.
(4.22)
_00
It follows therefore that
(4.23)
Further, note that by virtue of (4.12) and Theorem 4.1, for every
0<£<1,
the matrices in the partial sequence
positive definite, in probability, as
Let now
for
(3.4) ,
n(~
~
~n,
k; n£
S
k ::; n}
are all
00 ,
Bn, k be the sigma-field generated by "'1l,
S k (under P),
n
k = l, ... ,n
every
n
{v
£:
and let
1), B k
n,
B 0 be the trivial sigma-field. Then, for
n,
is non-decreasing in k(S n). Note that by (2.18) and
-16-
(4.24 )
and hence, for every
n(~
1),
P, {T k' B k; 0 s k S n}
n
"7l,
n,
under
tinga1e, closed to the right by
T .
"7l
v
For every
Lemma 4.2.
is a mar-
- "7l,k
V
"7l,k+1
is positive
sem~-
definite (p.s.d.).
Proof.
By (3.5) and (4.24), for every
= C-n 2
=
• V (T
- T
)
..... p "7l k+1
"7l,k
n
-2
Cn Ep
n
m(~ 1), (0 S)t
-1
k
(T
T , k)(T
T , k)'}
"7l, k + 1 - "7l
"7l, k + 1 - "7l
Under Assumptions (II), (III) and (IV), for every (fixed)
Theorem 4.3.
n-+oo n
{
(4.25)
'
Q.E.D.
and hence, is p.s.d.
1im
Osksn-1,
1
tm(S 1)
and every
r = 1, ... ,m,
when
< ••• <
r = tr,
L (C
-1
n
(T
k"'" T
{k ,··. ,km}
1
H
O
k))
"7l, m
"7l ' 1
..
satisfying
holds ,
-+
N( 0 , r
""1Il
)
(4,26)
where
L...
= (( Y , J' , • rr ' =
J
Outline of proof,
,
v JJ
.. , (t r A tr ' ) )),J,J., -- 0 , .. "p,r,r
. ,-1
- , .. "m .
For simplicity, we take
lengthy proof holds for any
~ =
(Ao, ... ,A )' (;t 0)
p
and
m(~
1),
m = 1;
(4,27
a similar but somewhat
Note that by (2.18) and (4.4) for any
k(lSkSn),
-17~
y
n,k
= C~lA'(T
n ~ "1l,k
=
Yn,O
and
- T
)
"1l,k-l
r* -
= O.
nSk
n-k+l
1
tJ-k- c*nS.J~PL. OA.{a
)
)=)
~
.(R.* ) ~ a* .(k)}
Jk
n,)
n,)
(4.28)
Then. by (4.24) and (4.25),
(4.29)
E (y2
p
n,k
n
)
=
V k) A(~ 0) , IV 1 S k S n .
-N (V
"1l, k +1 - "1l,-
(4.30)
Therefore, by Theorem 4.1 and (4.30),
lim kin
n+oo
=t
=> Vp
[2~1= lYn
n
e
First, we establish (4.26) under
.J
1
+
A'V(t)A
(> 0, V O<tSl) .
~
~
~
(4.31 )
•
p .
n
By virtue of (4.29), (4.31) and the
main theorem of DWl'etzky (1972). it suffices to show that for
as
n
+
<Xl
•
k/n+t e (0,1],
<Xl
(4.32)
..
and, for every
E > 0,
~k
2
2
L'_lE
pn {y n,1.I(Yn,1." > E)}
1-
where
I(A)
£ 0,
as
n
+
stands for the indicator function of a set
Now, under assumption (II), denoting
Tnn, k'
(4.33)
A.
the vector of rank sta-
tistics when the scores are generated by the polynomial score functions
<P
n , osj sp,
j
we have
(4.34)
an~
hence, by (4.19), (4.34) can be made arbitrarily small, by choosing
-18 ..
n(> 0)
arbitrarily small.
go ~
"" Tn)
C-IAl(T
n"" -n,k
-n,k
Hence,
under
and to prove (4.32)",,(4.33), it suffices to take the score functions
o :s; m:S; p all polynomials inside (0,1)
continuous insiqe
(0,1)].
P,
n
¢m'
[which are therefore boundedly
As such, for polynomial score functions, it
follows by some standard setps that with probability 1,
I
max {max
(R *)
* (k)
O:S;j:S;p l:s;k:S;n an,j jk - an,j
I} =
so that by (4.5), (4.28) and (4.35) , for every
positive integer
n £'
such that for
max {y 2 . }
< £ ,
l:s;i:S;n
n,l
and (4.36) insures (4.33),
2
Ep (Y .
n,l
n
n~n
> 0,
there exists a
e;'
with probability
1 ,
(4.36)
Also, by (4.28), we have
p
*
.(R .. )
lB.
n,1- 1) = L.J= OA.{a
J n,J Jl
~
.
£
(4.35)
0(1) ,
- a
~2
*
(4.37)
.(i)},
n,J
~
.
Thus, for every
k: l:s; k :s; n,
,k
2 . IB . 1)
t·1= lE pn (Y n,l
n,1where
gn , k(·)
=
(4.38)
g k(c*s , ... ,c*S )
n,
n 1
n n
is a quadratic form in its
n
arguments.
Note that
(4.39)
Ep g k(c*S , ... ,c*S )
n n,
n 1
n n
while by using (4.5), (4.14), (4.37) and following some standard steps,
it follows that (for polynomial score functions)
Ep g2 k(c*S , ... ,c*S )
n n,
n 1
n n
=
(A IV
A) 2 + 0 (1)
- -n,k-
,
as
n
-+
00
•
(4.40)
e
-19-
Hence, (4.32) follows from
~ ~
-+ t
-
n
Since,
,,'">
L
P
.
P{C-1T
n "'Il, ksxlHO}
~
(4.39) and (4.40),
(4.38)~
( C-1 T
n "'Il,k
Thus,
)
(4.41)
n'
= E[P{C-1T
n "'Il, kSx,
~
P }], V ~,
under
n
the desired
Q.E.D.
result follows from (4.41) and Theorem 4.1.
Now, by virtue of Lemma 4,2, whenever properly defined
v 00k
n,
00
v
n,k
and we take
whenever
{W *
n,r
V
"'Il,r
is
'\..
~ in
to be equal to
+
k
for
1s k
00
whenever
~
(4.42)
n ,
V
"'Il,k
is not p.d.
is p.d. , we consider a stochastic process
(t); £ s t s l}
(where
W*
n,r
(t)
0 < £ < 1),
-1
= Cn
W*
£ n,r
Then,
=
by letting
00
k *
(vn k
(t))"n k
(t)
, n,r
' n,r
(4.43)
= ~n , k n,r ( t ) '
"
where
kn ret)
,
Then, whenever,
cess
W*
00
n,r
00
00
vn,r S tv n, k}'
is
and finite, for every
> 0
belongs to the space
£ n,r
topology.
v
= max{k:
Let
and for every
W=
0[£,1],
{Wet), ostsll
0 < £ < 1,
£StSl
(4.44)
0<£<1,
the pro-
endowed with the Skorohod J 1
be a standard Wiener process on
[0,1],
we define
(4.45 )
Our basic contention is to show that
£
W*
n,r
weakly converge to
£
W*
-20-
By virtue of Theorems 4,1 and 4,3 and the definitions in (4,43) and
(4.44), we arrive at the following theorem by some routine steps.
Theorem 4.4.
H and the Asswnptions (I)
O
Under
for every (fixed)
m(~
1), £(0
<
£
1)
<
and
(II), (I II) and (IV),
I
(£:5) t
l
< '"
<
t (:5 1),
m
{W * (t )" .. ,IV * (t)} V
~ {W * (tl), .. .,W * (t )} .
n,r l
n,r m
m
Let us now consider
(p+l)
O:5t:5ll, j =O,l, ... ,p,
stochastic processes
(4.46)
W(j) = {W(j) (t) .
n,r
n,r
'
by letting
(4.47)
= max{k: v(k~. :5 tv(r~.}
n,))
O:5t:5l,
n,))
Then, each of these processes belongs to the
(4.48)
j =O,l" .. ,p
D[O,l]
space.
By virtue
of (4.23), the martingale property (4,24) and Theorems 4.1 and 4.3, we
may proceed as in the proof of Theorem 4.2 of Chatterjee and Sen (1973)
and show that on defining (for
0 < IS < 1)
wo(x) = sup{lx(t) .. xes)
£>0
that for every
tive number
nO'
and
n > 0,
such that for
I;
0:5
S <
t:5
there exist a
n
~nO'
p{w (W U )) >d < (p+l) -1 n
o n,r
,
under
for
S
+ O:5ll
(4.49)
I
0: 0<0<1
and a posi-
HO'
j=O,l"",p ,
The convergence of finite-dimensional distributions (f.d,d.)
. of
those of
W,
•
(4.50)
w(j)
n,r
to
the standard Wiener process, follows readily from Theorem
4.3, and hence, by (4.50), we claim that under the hypothesis of Theorem 4.4,
-21~
(4.51)
and note that (4.51) insures that
I
max {SUp
(j)
O~j~p O~t~l Wn,r(t)
I}-- 0p(l)
(4.52)
.
Now, by virtue of (4.12) and (4,23), we claim that for every
max
I OJ (t) - v OJ (s) I ; £
< "< { Sup [v
0-J-P
~
S<t
~
s + 0 ~ 1] }
-+
0
as
0
~
£: 0<£<1,
0 ,
(4.53)
and by (4.12) and Theorem 4.1,
max
{ max
I OJ
as
<"<
< < vn,k
O-J-P
n£_k_n
As such, for
k
~
q
~
rand
v
00 <
n,k
00
(~>
v
n -+
00
(4.54)
•
00 < 00)
n,q'
•
(4.55)
Hence, by using the definitions in (4.43), (4.44), (4.47), (4.48) and (4.49)
along with (4.50), (4.52) and (4.53)-(4.54), it follows from (4,55) that
p
Sup{lw * (t)-W * (s)/: £~s~t:5s+o~l} ~
0
n,r
n,r
as c~O,
'If
0<£<1.
(4,56)
From Theorem 4.4 and (4.56 L we obtain the following:
Theorem 4.5.
Undel' the hypothesis of TheOl'em 4.4, fop evel'y
w*
£ n,r
£ £ w*
in the J -topoloty on
1
D[£,l] .
£: 0 < E < 1 ,
(4.57)
-22~
For our proposed pes tests, (4.57) provides the key result.
procedure sketched in (2.21)-(2.22) with the
we conceive of an initial failure number
implemented only when the failure
L
n,k
kO(~
defined by
For the
C3.9)~(3.l0),
such that the pes is
2),
ZO
n,k
has occurred; the basic idea is
O
not to reject H until at least a few observations are at hand. In order
O
to choose k
properly, we note that though v (r)
is known in advance,
O
n,OO
00
v
is not known until the r-th failure has taken place. But, we have
n,r
the inequality that v (r)00 >- l/v 00 , so that for every k; ne: ~ k ~ r,
n,
n,r
l/(v(r) v OO )
n,OO n,k
(4.58)
Hence,
where the right hand side is observable at the k-th failure
if we define for any given
k
O
=
e:: 0 < e: < 1,
min{k; (v(r) vOO ) ~ e:- l } ,
n,OO n,k
(4.59)
then we have
•
and a similar inequality holds for the two-sided case.
hypothesis of Theorem 4.5, for every real
p{
Thus, under the
A,
W* (t) > A} ,
sup
£~t~l
(4.61)
(4.62)
Thus, if
A+(e:)
ex
and
A (£)
ex
be the values of
sides of (4.61) and (4.62) are both equal to
A for which the right hand
a,
for
then in the asymptotic case, in (2.22), we may take
O<ex<l, O<e:<l,
.e(n) = A+ (e:)
a
ex
or
A (e:).
ex
e
Analytical solutions for
and
A-+(E:)
ex.
A- (E)
ex.
are difficult to workout.
Majumdar and Sen (1977) have made some simulation studies and we report
some of their values here.
TABLE 1
and
Simulated values of
for some typical
A- (E)
ex.
+
A- (E)
A- (E)
ex.
E
ex.
= .01
ex.
(E,ex.)
ex.
= .05
ex.
= .10
ex.
= .01
ex.
= .05
ex.
= .10
0.005
3.32
2.76
2.46
3.48
3.04
2.79
0.01
3.28
2.72
2.42
3.46
3.01
2.76
0.05
3.19
2.60
2.29
3.39
2.89
2.62
f).lO
3.05
2.50
2.19
3.26
2.79
2.52
•
For larger values of
to
+
E, A- (E)
ex.
and
A- (E)
ex.
are not very sensitive
E.
5.
ASYMPTOTIC DISTRIBUTION THEORY UNDER LOCAL ALTERNATIVES
We shall now consider the non-null
(2.7) holds for some
8> 0
(or
;t
0).
c~se
,.'here (2.5) does not hold, but
For apy fixed non-null
8,
the
consistency of the proposed test can be proved along the same line as in
Chatterjee and Sen (1973).
~
Hence, for the study of the asymptotic power
of the proposed test, we confine ourselves to some local
alternatives
~24~
S to be close to
where we allow
we frame a sequence
(or the c. to be so),
°
of alternative hypotheses, where
{K}
n
Kn 1 F?(xolx)
1
~
e
Specifically,
= F?1,n (xolx)
~
=
with a real
1
pO(X ~ eC~il~)
,
o
(fixed) and the
~ i ~n
(5.1)
,
given by (4.4).
c* .
nl
Under (5.1) [and (2.4)], for
1
e~o,
the basic permutational
invari~
ance structure of Section 2 does not hold, and, as a result, neither (4.24)
nor the martingale-proof of Theorem 4.3 holds.
asymptotic normality of
l
C- (T
n~,
k - ET
~,
k)
A general proof of the
may be worked out by using the
basic projection technique of Hajek (1968) [as extended to the multivariate
case by Puri and Sen (1969b)].
However, in the absence of·the martingale
property (4.24), this approach does not yield the tightness property of
W*
e: n,r
[in (4.56)] when
H may not hold. For this reason, we employ
O
v
here the notion of contiguity [viz., Chapter VI of Hajek and Sidak (1967)]
which provides (in a reasonably simple way) a natural extension of Theorem
4.5 under
{K
n
L
possesses an absolutely continuous probabi-
We assume the the df
1ity density function (pdf)
the pdf when
8=0
ate pdf for the
·0
[in (5.1)].
(O,j)th
Vi;:: 1
Also, let
variates, for
f[0,j](x 1 ,x 2 )
1~ j
f[0,j](x 1 ,x 2 ) = _(a/ax1)10g f[0,j](x 1 ,x 2 )
and let
~
p,
_00
* (~)
be
be the bivari-
and we assume that
exists (a.e.) and
v
-00
f
j = 1" .. ,p .
(5.2)
-25-
Let then
(5.3)
...
for
0::::; t : : ; I
and
j
= I , ... , p ;
(5.4)
where
(5.5)
and
~Ot
is defined by (4.6) for
v OO (t)
that
Defining
T
~O
~O.
being replaced by
is a continuous and non-increasing function of
Further, note
t
€
(0,1] .
as in (4.1), we let
(5.6)
Finally, for every
•
0 <
£:
£ <
I,
we
P
O·
by
define
letting
lJ*(t)
T
=
Theorem 5.1.
e{v
00
1
(n (t))}-~~. OV J(n (t))lJ.(n (t)) ,
T
J=
T
J
(5.7)
T
Undep Assumptions (1)1 (II), (III), (IV) of Section 4,
in (5.1) and (5.2) fop every (fixed)
m(~ 1)
and
(0 <)t
l
{K }
n
< ..• < tm(~ 1),
{W * (t )1""W * (t)} V
-+ {W * (t ) +lJ * (tl),. .. ,W * (t ) +lJ * (t)}
n,r I
n,r m
T
m
T m
I
whepe
*
Wand
Proof.
for any
n,r
W*
ape defined by
Here also, we prove (5.8) for
m ~ I.
Recall that for every
(4.43)~(4.45).
m= 1
only; a similar proof holds
j; 0::::; j : : ; p
and
k(::::; n),
(5.8)
-26,..
~n
* {
(n)
1= lC n.~.
l)
= L'
Define then for every
h* .(t) =
n,)
0
0
}
(F n)
[.](x )1
.. ))c(Z
. n~ k-XO')
1 + a*n,).(k)c(Xo·"'Z
1 n, k-)
0 <t
~
1
°
and
j
~
~
(5.9)
p,
L~1= lc*.{~.(F[.](X
.. ))C(so-xo·)
n1)
)
)1
t
1
¢.) t C(Xo''''so_)}
1 t
+
(5.10)
As in Theorems 4.1 and 4.3, under Assumption II, we use the polynomial decomposi tion of
0
~
kin
+
~,
m
that whenever
m ~ p,
t
E
and then it can be shown by some routine steps
[0,11,
as
n
~
°
00
~
j
~
p,
under
(5.11)
H .
O
We denote the joint distribution of
p
K
n
(and p 0' respectively). Then, from the results of Chapter IV
n,
v
of Hajek and Sidak (1967), it follows that under the hypothesis of Theorem
by
n,n
{p
5.1,
n,n
}
is contigious to
{p
n,
o},
and hence, by (5.11),
under
On the other hand, for any
o ~p
C(XO'-St-)L'
1
-}
OA'~'t
J=)
)
the contiguity of
{K}.
n
(5.12)
°
p OA.h * . = ~.n lC*'{c(s -XO')~'
p
A~O,~.
.. )
~ L)= ) n,)
L 1 = n1
t
1 LJ= OA.~.(F[.](X
))
)
)1
involves a linear combination of independent rv's, and
e~~
OA.]..I. (t)
LJ= ) )
{p
as well .
~
hence, by the central limit theorem, under
normal with mean
{K}
n
}
to
{K },
n
~'~(t)~.
and variance
{p
O},
n,n
n,
As such, by (4.12), for every
is asymptotically
Further, by (4.14) and
we claim that (4.14) also holds under
0 <t
~
1
and
O~j ~p
Hence, by the Slutzky theorem, under
I.pJ=)
OA. h * .
n,)
kin
,
+
under
{K} kin + t=>
n '
t,
{K}
n
(5.13)
+
(5,14)
where the right hand side of (5.14) is asymptotically normal (under
with mean
e{vOO(t)}~~Ir=ovOj(t)~j(t)
and variance
then follows by using the definitions in
,.
Note that (4.56) insures the tightness of
E
The desired result
I,
(4,43)~(4.44)
{K })
n
and (5.6)-(5.7).
*
Wunder
n,r
H '
O
Q,E.D.
and
hence, proceeding as in the proof of Theorem 2 of Sen (l976a), we conclude
that by the contiguity of
remains tight under
as well.
{K }
n
W*
n,r
Defining
Then,
(4.56) and (5.8),
A},
V
-+- W*
in the
E
J
L by (4.1), for every
1
(4.59) and (5.16), we claim that under
and
W*
n,r
0
von /v OO k
n,r n,
°<
E
< 1,
-topology on
1,
< E <
(5.15)
we let
(5.16)
,
to = VOO(L)/VOO(to) ~ 1/[VOO(L)VOO(to)] ~E.
n -1 ko ~ to
E
Hence, we have the following.
. f {t,' Von ( L) V00 ( to ) <
to -_ III
- E -1 }
"
n,
Under the hypothesis of Theorem 5,1, for every
Theorem 5.2.
E
{p
As such, by (4.58),
{K},
n
° ~ to
,
as
n -+-
ex>
,
(5.17)
and hence, by Theorem 5.2 and (4.58)-(4.55), we conclude that
lim
n~
..
p{
max
kO~k~r
F,;
n,k
> A+ (E) IK }
a
n
+
*
= p{W * (t) > Aa(E)
- ~L(t)
,
= p{W(t) > t
!.: +
2[A (E) - ~* (t)]
a
L
for some
t
for some
°~ t ~ 1}
t
°~ t ~ 1}
(5.18)
-28'\"'
IK}
lim p{ max I~
I > Aa. e: n
n-+oo
kO~k~r n, k
= p{W(t)
where
!.:
t
t z[± A (e:) .,.lJ* (t)]
a.
W = {Wet), t
~o}
T
for some
t; t
0
is a standard Wiener-process,
~ t ~ l} ,
(5.19)
...
Thus, in either
case, the asymptotic power function is expressible in terms of the boundary
o
crossing probabilities of a standard Wiener process (on
[t ,1])
where
I
the boundaries are generally non'l"'linear.
SOME CONCLUDING REMARKS
6.
Note that for
1
T=t=l, lJi(l) = e[v
~ j ~ p are equal to
!.:
(l)]zlJO(l),
as all the
lJj(l),
Similarly, for the ANOVA test (based on
O.
. mean of
alone ) , we h ave t h e asymptotIc
equal to
00
!.:
e[VOO(l)]-~O(l).
C.. l[v(n) ]-~(O)
-.
n
n,OO
n
(under
T (0)
n
{K })
n
Hence, for the complete sample (non.. sequential)
case, the asymptotic relative efficiency (A.R.E.) of the ANOCA test with
•
respect to the ANOVA test (based on the same score function) is given by
(6.1)
where the quality sign holds only when
vOj(l) = 0,
context of PCS, we note that for
(even when
t <1
V l~j ~p.
In the
is rather
T, t, <PO"" ,<P p and f [O,j]' 1 ~ j ~ p) . If, we
do not have any concomitant variate, we may develop a parallel PCS ANOVA
a complicated function (of
test, where in (3.9), we shall take
~n , k
=
(v(k) )~C-1T(0)
n,OO
n n,k
{ 0, otherwise
.
(6.2)
-29-
where
v(k)
n,OO
°'
>
V ko:S k:s n.
In that event, we need not use the inequa-
lity (4.58), so that in (4.61]-(4.62), the
sign.
=
.'
sign may be replaced by an
Also, in (5.18),,(5.19), we will have then
to
= £
while for
ll* (t),
'T
given by
(6.3)
(6.4)
In general, neither
and
llO(t)
'T
ne(t)
are equal (for all
are either dominated by the other.
t:S 1) ,
nor
Hence, unlike (6.1),
in general, it is difficult to conclude whether the ANOCA always performs
(asymptotically) better than the ANOVA rank test,
Chatterjee and Sen (1973) have studied the Bahadur-ARE of the PCS
4It
ANOVA rank test with respect to the single-point censoring test and have
shown that under very general conditions, the PCS test performs better.
•
In
the current setup, such a study can be made provided we are able to show
that for (4.61) or (4.62), as
c
>
0;
A~
00
, - log p{ .•.. }
=
2
2
cA + o(A ),
where
this remains an open problem.
Finally, batch-arrival models or staggering entry plans are often more
flexible to suit practical problems.
In such a case, as in Sen (1976a), one
needs a two-dimensional time-parameter stochastic process to formulate the
ANOCA tests.
This is possible - though the formulation needs much more com-
plications, and will not be done here.
..
Some simulation studies for the
asumptotic power of the proposed tests are intended to be made and published
in a separate communication.
-30-
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[1]
Bre$low, N. (1970), A generalized Kruska1-Wa1lis test for comparing
k samples subject to unequal patterns of censorship. Biometrika
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[2]
Breslow, N. (1974). Covariance analysis of censored survival data.
Biometrics ~. 89-99,
[3]
Chatterjee, S.K. and Sen, P.K, (1964). Nonparametric test for the
bivariate two~sample location problem, Calcutta Statist. Assoc,
BuZZ . .!l, 18,.58
[4]
Chatterjee, S,K. and Sen. P.K. (1973). Nonparametric testing under
progressive censoring. Calcutta Statist, Assoc, Bull, 22, 13-50.
[5]
Davis, C.E. (1977),
censored data.
[6]
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[7]
Hajek, J. (1968), Asymptotic normality of simple linear rank sta",
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'.
.
A two~samp1e Wilcoxon test for progressively
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v
Theory of Rank Tests,
[8]
Hajek, J. and Sida'k, Z, (1967),
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Academic
[9]
Halperin, M. and Ware, J, (1974), Early decision in a censored twosan~le test for accumulating survival data.
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[10]
Majumdar, H. and Sen, P.K. (1977), Rank order tests for grouped
data under progressive censoring. Communications in Statistics
§.., in press.
[11]
Puri, M.L. and Sen, P.K, (1969a). Analysis of covariance based on
general rank scores. Ann. Math. Statist, 40. 610_618,
[12]
Puri, M.L. and Sen, P.K, (1969b).
a general linear hypotheses,
[13]
Quade. D. (1967). Rank analysis of covariance,
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[13]
Scheffe, H, (1959).
A class of rank order tests for
Ann, Math, Statist, 40. 1325-1343,
The Analysis of Variance.
Jour, American Statist.
John Wiley, New York.
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•
•
[15]
Sen, P,K, (1976a), A tWQ~imensional functional permutational cen~
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[16]
Sen, P,K. (l976b). Asymptotically optimal rank order tests for progressive censoring. CaZcutta Statist. Assoc. BuZZ. ~, in press .
[17]
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