RANK ANALYSIS OF COVARIANCE UNDER PROGRESSIVE CENSORING, 11*
by
Pranab Kumar Sen
,.
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1275
MARCH 1980
RANK ANALYSIS OF COVARIANCE UNDER PROGRESSIVE CENSORING, 11*
by
PRANAB KUMAR SEN
Department of Biostatistics
University of North Carolina, Chapel Hill, N.C., U.S.A.
SUMMARY
For some general analysis of covariance models, a class of
progressively censored nonparametric tests based on suitable rank order
statistics is considered.
The proposed procedure is a generalization
of an one (for a simpler model) in Sen (1979b).
Along with some in-
variance principles for progressively censored (multivariate) linear
rank statistics, asymptotic properties of the proposed tests are studied.
1.
INTRODUCTION
In clinical trials or life testing problems 3 a progressive
censoring scheme (PCS) incorporates a continuous monitoring of experimentation from the beginning with the objective of an early termination
(contingent on the accumulating statistical evidence) without increasing
the margin of the type I error.
Chatterjee and Sen (1973) have
formulated a general class of PCS nonparametric tests for a simple
regression model (which includes the classical two-sample problem as a
*Work supported by the National Heart, Lung and Blood Institute,
Contract No. NIH-NHLBI-7l-2243 from the National Institutes of Health.
-2special case); their theory rests on some invariance principles for
some PCS linear rank statistics.
Sen (1979a) has developed some
invariance principles for some related quantile processes arising in
PCS; some generalizations of these are due to Sen (1980).
Sen (1979b)
has incorporated concomitant variates in the simple regression model
and studied some PCS analysis of covariance (ANOCOVA) tests; references
to other relevant works are cited in these papers.
PCS nonparametric
tests for multiple regression (containing the several sample problem
as a special case) have been considered by Majumdar and Sen (1978).
The object of the present investigation is to incorporate (stochastic)
concomitant variates in some general mu1tiparameter models and to
formulate appropriate PCS rank tests for such general ANOCOVA models.
Section 2 deals with the basic ANOCOVA model and the preliminary
notions.
The proposed PCS tests are then developed in Section 3.
Section 4 is devoted to the asymptotic distribution theory of the
allied test-statistics, both under the null and some local alternative
hypotheses.
Since our proposed procedure is a natural extension of
Sen (1979b), in the sequel, often, to minimize the technical manipu1ations, we shall omit some details by suitable cross references to the
earlier paper.
Some general remarks are made in the concluding section.
2.
Let
X~
~1
=(X 1"
O
PRELIMINARY NOTIONS
X 1., ... ,Xpl.)' =(X 1"
O
1
X!),,
~1
i =l, ... ,n
independent random vectors (r.v.) with continuous
distribution functions (d.f.)
Ft,
i =1, ... ,n,
primary variates with (marginal) d.f.
and the
X.
~1
F '
Oi
be
(p +l)-variate
where the
defined on
XOi
are the
E(=(_oo, 00))
are the concomitant variates with marginal (joint) d.f.
-3-
defined on
P. ,
1
d.f. of
for some
given
XOi '
X. =x,
"'1
p
p~(ylx)
Let
~l.
1
i=l, ... ,n,
'"
in an ANOCOVA model, we assume that
PI
'"
be the conditional
and, as is usually the case
="'= Pn =P
(unknown).
Our
basic problem is to test for the null hypothesis
H : P~
O
=...= P~ = po
(unknown),
(2.1)
against an alternative that they are not all the same.
pOi(ylx) =po(y; Mc.
-c)
Ix),
f'.Il
"11
f"'o"J
~
matrix
(m :2:1, q
i"Y
where
the
is an
m xq
-c
are specified q-vectors and
c.
"'1
'"
~ EEP):
y EE,
the model in Sen (1979b), we let (for every
Generalizing
i=l, ... ,n,
~l)
(2.2)
of unknown parameters,
=n -lIn1=
. lC"
"'1
The classical
ANOCOVA model relating to the one way layout is a special case of (2.2)
where
V
i
m =1, q ~l
pO(y; ~(c. -c) Ix)
'" "'1
'""1l
= pO(y
Purther, in this special case, we have
~l.
sizes
and
nl, ... ,nk ,
respectively
k
(n =\.
In.),
L = 1
1
-~(c. -c) Ix),
'" "'1
'""1l
k(=q +1)
'"
samples of
c. ="'=c'""1l =0",'
"'1
1
c'""1l +1 = ... = -n
c +n =(1
. , O,···,O)"···,c
",n + .. '+n +1 ="'=c
.....n +.. '+n
l.q
1
k
1
1 2
and Q stands for the vector of treatment-effects.
= (0,···,0,1)'
More general models may be conceived by allowing the location and
scales to vary under alternatives.
Now, under (2.2), we may recast
(2.1) as
(2.3)
In a life testing situation, though the concomitant variates
Xl'
... ,X'""1l
,..,
may be observable at the beginning of the experimentation,
the primary variates are not so.
rv's corresponding to
XOl"",X On
If
ZO
nl
Sl' ... , Sn
ZO. =
nJ
nn
be the ordered
(ties neglected, with probability
1, by virtue of the assumed continuity of
anti-ranks
< .•• < ZO
PO) and if we define the
by
for
j =l, ... ,n,
(2.4 )
-4-
Z~k'
then, at the k-th failure
o
n1
Q. =(S., Z ., X ),
1
~1
the observable rv's are
i =l, ... ,k,
S
~.
for k =l, ... ,n.
(2.5)
1
[Though the complementary sets of concomitant variates are known,
their anti-ranks are not specified, in advance.]
Nonparametric ANOCOVA tests (based on the entire set
{X *, ... , X* })
1
~
~
have been considered by Quade (1967), Puri and Sen (1969a) and Sen
and Puri (1970), among others.
Under PCS, a special case of a simple
regression model, has been studied by Sen (1979b).
general model in (2.2) (for
m 2:1, q 2:1) will be considered.
introduce the following notations.
among
xjl,···,X jn ,
for
In this paper, the
Let
i=l, ... ,n;
R..
J1
We
be the rank of
X..
J1
These yield the
j =0,1, ... ,p.
rank-collection matrix R =((R .. )) (of order (p +1) xn), and
"'11
J1
permuting the columns of R, so that the top row is in the natural
"'11
order, we obtain the reduced rank-collection matrix
R*
~
=((R~.)),
where
J1
by (2.4),
ROS . =ROi =i
1
and
R·J S .
1
=R~.
J1
,
for 1
~j
~p,
i=l, ... ,n,
(2.6)
- ( (1).C)
C)) , , i =1, ... ,n}
. (i) -a
1 , ... , a(m)
.1
n,J
n,J
J
be a set of scores (vectors), which are chosen with the model in (2.2)
For each
j(=0, ... ,p),
in mind.
For example, if we restrict ourselves to the multiple linear
let
{a
"'11,
regression model, we have then
[Wilcoxon scores] or
m =2,
1
~
a
. (i) = i/(n +1)
n, J
i ~ n, where <P
If we have the joint location/scale model,
and besides the above scores (suitable for location
alternatives), we need to choose [for
for scale alternatives.
later on.
and we may choose
[normal scores],
is the standard normal d.f.
then
m =1
a (2~ (i)]
n, J
some scores suitable
These scores will be defined more formally
As in Puri and Sen (1969b), we consider the (multivariate)
-5-
linear rank statistics
=t
T
~1=
~n
By (2.6), we may rewrite
~1
T =I~1= l(c"'.
s
~
-c
~
1
(2.7)
)[a l O(RO1')," .,a'
(Rp1.)]
~,p
~.~,
=n -lIn1=
. lC"
-c
where
-c
l(c.
~1
l
)[a~,
O(i),
T
~
as
a'l(Rl*.),
' (R*.)].
1 ... ,a~,p
p1
(2.8)
~,
Keeping in mind the observable r.v.'s in (2.5), as in Sen (1979b), we
let
Pn be the conditional (permutational) probability measure generated
by the
nl
(conditionally) equally likely column permutations of
R*)
(given
and let
~
In , k
for
S k = (Sl ' . . . , Sp ),
~,
k=l, ... ,n.
Then, we let
=E o (In I~ k)
.<.n
'
k
l
=I1"--1(£'S. -c)[a
o(k), ... ,a"11,p
(k)]
' (R*.)
~
"11, o(i) -a*
~,
p1 -a*I
~,p
(2.9)
1
where
a* I
.
"'n, J
1 :s;k :s;n -1
I~1= la"11,J
I
. (R~ . )} ,
J1
(n -k) -l{ na
I " ~,J
(k) =
{ 0,
(2.10)
k =n,
and
a
l
.
~,J
= n -1 In1=
. la .
"11, J
I
Conventionally, we let
(.)
1
for
,
T 0 =0, V n
"11,
on the partial sequence
'"
{T
"11 ,
j =0, 1, ... ,p.
The proposed PCS tests rest
~l.
k; k :s;n}
(2.11)
and are formulated in the next
section.
3.
THE PROPOSED PCS TESTS
Let us define
t
£n = 1=1 (c. -c Hc. -c ) I
··u
and assume that
~1
~
~1
(3.1)
~
is positive definite (p.d.).
Further, the
statistics in (2.9) are location invariant, and hence, without any loss
of generality, we may set
a"11,J. =0,
~
for
j =0, l, ... ,p.
(3.2)
-6-
For every
~k ~n,
k: 1
and
j, £(=0, l, ... ,p), we define
y(k~o
=n-l{L~1 = la"'l1, J.(R':.)a'
o(Rt.)
"'l1, J.{..
J 1 "'l1,.{.. .(..l
and consider then the
y
"'l1,k
for
m(p +1) xm(p +1)
y(k)
y(k)
y(k)
"'l1,00
"'l1,Op
"'l1,00
=
=
y(k)
y(k)
"11, pO
"'l1, pp
k = 1 , ... , n .
l'
"1l,k
"'l1, J
"'l1.,k
(3.3)
matrices
y(k)
"11,0*
mxm
m xmp
y(k)
y(k)
"11,*0
"11,**
mp xm
mp xmp
Al so, we rewrite
= (1'(0)
+(n -k)a* .(k)a*'o(k)},
l'
"'l1,k
, say,
(3.4 )
in (2.9) as
T(lk)'" ',T(P )) = (1' (Ok) , T(*k)), say.
~n, k
~n,
~n,
(3.5)
"11,k' "'l1,
Then, following the line of attack of Sen (1979b), we have by (2.9),
Ep (1' ) = 0,
"'l1
~
n
yp (1'
n "'l1.,
k) = C @y k'
"'l1
NJ.1 ,
(3.6)
V 1 ~k ~n.
(3.7)
To eliminate the effects of the concomitant variates, we take into
account the asymptotic multinormality of
linearity of regression of
l' (Ok)
"'l1,
on
l' k
"'l1 ,
T(*k))
(insuring the asymptotic
and work with the
"11,
residuals:
TO
"11 , k
= T(Ok) - (fitted value of 1'(0)
"11 ,
"11 ,
= T(Ok) - y(k ) (v(k~ ) - l' (*k) ,
"11 ,
"11 , o* "11, * *
"'l1 ,
where
k
on
(1 ~k ~n),
stands for the generalized inverse of
~
1'(*))
"'l1 , k
~.
(3.8)
Note that
o
Ep l'
I = 0 and
"'l1 (
n
'
~
We let
(3.10)
(3.11)
t: n k =Trace [G kH-k]'
"11
~n
1 ~k ~n.
(3.12)
-7~nk
Then,
is the covariate-adjusted rank test statistic (for testing
HO in (2.1)) based on
r(=r)
n
gl' ... ,gk'
for
k =1, ... ,n.
Suppose now that
rln +T,
is some prefixed positive integer, such that
~1.
0 <T
Then, the proposed PCS can be formulated as follows:
Continue experimentation as long as
for the first time, for some
k
=N(~r),
k
~nO
~nN
is
Z~N
mentation is curtailed at the N-th failure
rejection of
HO.
k(~r)
If no such
BO.
o(a).
lea)
if,
'
then experi-
n
along with the
ZO
nr
along with the acceptance
is the desired ZeveZ of significance of the
a(O <a <1)
Here
>
<
C
"'nk - -l-n
exists, then experimentation is
stopped at the prep1anned r-th failure
of
and
l~a) (and the. initial number nO)
PCS test and the critical value
are
to be chosen in such a way that
e
p{~
Note that
nk
>l (a)
n
k: no~k~rIHo}
for some
N is the stopping number and
0
is the stopping time.
ZnN
Our task is to develop theory leading to the choice of
no)
(3.13)
~a.
l (a)
n
(and
satisfying (3.13) and to study the (asymptotiqproperties of the
proposed tests.
This is done in the next section.
4.
ASYMPTOTIC PROPERTIES OF THE PROPOSED TESTS
Note that if in (2.7) -(2.9), we replace the
where
c.
.....1
by
d. =Dc
.
..........1
.....1
Q is any p.d. matrix and proceed as in (3.11) -(3.12) (with
replaced by
choice of
Q.
DC DI),
then the
"""11.....
remain invariant under any
Hence, without any loss of generality, in (2.7) -(2.9),
and elsewhere, we may replace
n
. c.
I 1=1"'l11
=0
.....'
(c. - c)
\,n
.....1
"'l1
c
c'
Li=l"'l1i"'l1i
by
c .,
"'l11
= C"'l1 =.....I q
1
~i ~n,
where
(4.1)
and we assume that
max (c'.c .)
"'l11"'l11
.
1 ~1~n
--+
Q"
as
n +
00
•
(4.2)
-8-
Secondly, defining
r =r,
n
limn-+oon
Concerning the scores
a
-1
{a
r n = T:
.(i)},
.(i) =E.c£.(U,)
J
U < ... < U
nl
nn
from the rectangular
0 <T
or
n1
(4.3)
~l.
we assume that for each
"11,J
"11,J
where
as in after (3.12), we assume that
.c£.(i/(n +1)),
1
J
j(=O,l, ... ,p),
~i ~n,
(4.4)
are the ordered r.v.'s of a sample of size
n
.c£. =(~l"""~ ,)1 is expressible
(0,1) d.f. and
J
.J
fllJ
as
</> ,(u) =</> . leu) -</> . 2(u),
SJ
sJ,
where the
</> ' k
SJ,
0 <u <1,
sJ,
1
~s ~m,
0
~j ~p,
(4.5)
are all nondescreasing, absolutely continuous and
square integrable inside (0, 1).
F[j]' F[jl] and F[Ojl]
Let now
of
X.. ,
the joint d.f. of
J1
(X Oi ' Xji , Xli)'
d.f. of
respectively, all under
F[O] (x) =t
o
(X .. , Xp . )
J1
for
0
0
sl =+00.
and the trivariate (if
-l-1
~j ~p,
HO in (2.1).
has a unique solution
So =_00 and
be respectively the marginal d.f.
0
~j
,l
~p
and
1
~j
~2)
p
,l
~p,
Further, we assume that
o
St' V tE(O, 1),
and
let
Let then
1
~Ot
=(1 -t)-lJ .c£O(u)du,
:tjt
"1
0
t
~t
"foo =Q, =.c£o' fa1 =Q;
<1,
~t(or= p.j(F[j] (y))dF[Ojj(X,
y), 1 <j <p,
t
(4.6)
£[0,1],
(4.7)
t
1
so that
4},o =J </>. (u)du =¢. =0
O-J
~J
~J
~
(by assumption), and let
</>'1 =0, 1
~J
~j ~p.
Also, let
~O(t)
1
=J
o
[~O(u)][£OCu)]'du +(1
~Oj(t) =[~jo(t)]'
.0
-t)"fOt"fOt '
t E[O, 1],
(4.8)
St 00
=Loo Looao(F[O](X))][~j(F[j](Y))]'dF[Oj](X, y)
+ (1 -t)fotfot '
1 ~j ~p,
t E[0, 1],
(4.9)
-9-
(4.10)
~oo(t)
~(t)
V
'"oP
~o(t)
(t)
=
~o* (t)
o
=
v
~O(t)
"'PP
~*O(t)
(t)
~t ~1,
(4.11)
~** (t)
~~O(t) = ~O(t) - ~O*(t)[~*(t)]-~*O(t),
o ~t
(4.12)
~1.
By a direct generalization of Theorem 4.1 of Sen (1979b), we have then
p* ,
under (2.1), (4.1) -(4.5) and for continuous
max{
_n
k<
where
I I~I I
Ilv"'Il, k
-~(k/n)
p
II} --
0,
as
n
(4.13)
+00,
~
stands for the maximum of the elements of
[and the
proof of (4.13) is omitted as it runs parallel to the proof of TIleorem
4.1 of Sen (1979b)].
As such, by (3.10), (4.12) and (4.13), we obtain
by some standard steps that under (2.1), (4.1) -(4.5) and for continuous
p* ,
max{
k~n
II~k&o.* -~OO(k/n) II} L
'
0,
as
n
(4.14 )
+00.
We assume that
~O(t)
is p.d., for every
(4.15)
t E(O, 1],
and denote the reciprocal matrix by
*-1
rs
~O (t) =((v* (t)))r,s=l, ... ,m'
Note that by (2.9) -(2.11), for every
{T
"'Il ,
o ~k
k' 0
~n
~k ~n}
for
n(~l),
0 <t
(4.16)
~l.
under
Pn ,
is a martingale sequance, so that by (3.7), for every
-1,
V k 1 -V
"'Il ,+
"'Il ,
k
is positive semi-definite (p.s.d.)
Also, using (3.8), (3.10) and the martingale property of
(4.17)
-10{T
"'l1 , k' 0 sk sn}, we get that
E[To
_To] [To
_To], =v(k+l) _v(k)
is p.s.d. (4.18)
"'l1,k+l "'l1,k "'l1,k+l "'l1,k
"'l1,OO·* "'l1,00·*
for every
0 sk sn -1.
continuity of
Now, by virtue of (4.4) -(4.5) and the assumed
F*, it follows that
II: 0 ss st SS +0 sl} -+ 0
-~O(s) II: 0 ss st SS +0 sl} -+
sup{ 11~(t) -~(s)
sup{ II~oo(t)
as
0 -1-0,
(4.19)
0 as 0 -1-0,
(4.20)
so that by (3.10), (4.13), (4.15), (4.19) and (4.20) -(4.21), as
max{ll~q60.*
-Yn , 00.*11:
,
lim n
-1
n~
Lo,
as n+ oo •
(4.21)
Under (2.1), (4.2), (4.4) and (4.5), for every (fixed)
Theorem 4.1.
s(21), (Os)t
Iq -k\ son}
0 +0,
l
< ... <ts(sl)
and {k, ... ,k}
satisfying
s -'
1
k. =t., 1 sj ss,
J
J
C-\T k,···,T k)
"'l1 "11 , 1
"'l1 ' s
---;n-+
V
fx\r),
'"
(4.22)
j, l =1, ... ,5.
(4.23)
N(O, I
'"
"'q v
where
"'Is
r : ] ; r· o
.
"'J -L
r"'55
Outline of the proof.
=
vet. At o ),
'"
J
-L
Note that each of the
T
is a qm(p +1)"'l1,k.
J
vector, so to prove (4.22), one may use the Cramer-Wold theorem, whereby
one needs to consider an arbitrary linear combination of these
sqm(p +1)
random variables and to prove the asymptotic normality of the same.
Once we consider such a linear combination, the case becomes very
similar to the one treated in the proof of Theorem 4.3 of Sen (1979b)
(dealing with
nicely.
q =m =1),
and the same martingale-approach works out
Therefore, the details are omitted.
Next, by (3.8), (3.10), (4.12), (4.14) and Theorem 4.1, we arrive
at the following.
-11-
Under the hypothesis of Theorem 4.1, for every (fixed)
Theorem 4.2.
s(~l),
lim n
(O~)tl <.··<ts(~l)
-1
k. =t., 1
J
J
satisfying
~j ~s,
C-~(ro
"'Il
and
"'Il , k1
, ... 'To
"'Il , k s
) ~v N(O, I Ix'r*J,
"'q '-J '"
(4.24 )
"l
where
IiI
r:*
=
"'~s
r* ] .
.
.
r*
r*
"'sl
, r~l = ~o(tj At1 ),
"'J
j, 1= 1, ... , s .
(4.25)
"'ss
By virtue of (4.15) and (4.18), we conclude that for every
E(O<E~l),
p{y(kO)O
"'Il,
*
0
Now, for every
is p.d., V k ~nd -+-1, as n
dO <E <1)
and
+00 •
(4.26)
(r, n) satisfying (4.3), we
consider a mq-variate stochastic process
W
E: "'Il , r
= {W
"'Il , r
(t), E: ~t ~l}
by letting
W
"'Il,r
(t)=[Clx'y(k)
]-~o
"'Il'-J"'Il,OOo*
Nfi,k
so that it belongs to the space
E:
fork~t<k+l tE:[E:,l],
r
qm
D
[E:, 1].
r'
Also, let
(4.27)
W ={W(t),
E:'"
be a qm-variate Gaussian function with no drift and covariance
~ t~l}
function
E{[}!(s)] [}!(t)] '}
for s, t E:[E:, 1].
Under the hypothesis of Theorem 4.1, as
Theorem 4.3.
W
-=-+
V
E:"'ll,r
i
for every
Then, we have the following.
E:
E
+00,
W in the (extended) Jl-topoZogy on Dmq[E:, 1], (4.29)
E:""
(0, 1).
Outline of the proof.
butions of
n
W
E:"'ll,r
The convergence of the finite dimensional distri-
to those of
W follows readily from Theorem 4.2,
E:'"
-12(4.14) and (4.26).
k'
=
Also, note that for
s, t £[£,1],
k = [ns]
and
[nt],
1I~,r(t) -~,r(s)11 ~11[~@~~bo.*]-\~,k-~,k,)11
+II([C @V(k)
~
~,OO'*
r~ -[C
x v(k')
~
~,OO·*
]-~)T
~n,k'
II
(4.30)
where
11·1
I
stands for the Euclidean norm.
(or
For each coordinate of
we may proceed as in the proof of Theorem 4.5
of Sen (1979b), and hence, using (4.21), (4.26) and the above, the
tightness of
{W
} follows.
£"1l,r
For intended brevity, the details
are omitted.
Let us now define
W*£
= sup{[W(t)]
~
and, in (3.13), we let
~t ~l}
'[W(t)]: £
~
nO =[£r],
where
0 <£ <1.
(4.31 )
Then, by virtue
of Theorem 4.3, under the null hypothesis (2.1),
~
max
n -<k<_r
O
n
k
~
v
and hence, in (3.13), for large
w*
a£'
where
W*, as n
(4.32)
-+00,
£
n, lea)
n
p{W*
£
~
may be replaced by
w* } =a.
(4.33)
a£
Thus, the task reduces to that of finding
w*
a£
and, toward this, some
developments will be reported in the next section.
Let us next proceed to study the non-null distribution theory.
We
confine ourselves to some local alternatives for which the asymptotic
distributions are nondegenerate.
sequence
F* .,
nl
where
1
{X\'
... ,X*
} of
~
~n
~i ~n
F*.
nl
For every
n,
we conceive of a
(p +l)-vectors, where
X*.
~l
has the d.£.
and keeping in mind (2.1) -(2.2), we consider the model
has an absolutely continuous p.d.f.
f*.
=f*(x*'
nl (x*)
~
~ ,
~d
.) '
~l
1
~i ~n,
f*.
nl
and
l
x*
=(x 0' ~')'
£EP+
~
.-
(4.34)
-B-
ff*. (x*)dX O =f . (x) =f(x), V x £EP
and as in Section 2, we have
1 $i $n.
nl
..
~
For the triangular array
nl
{d .},
~
~
we let
~1
~
D =\~
~
and
d .d .
and
'
Ll=l~l~l
assume that
n
I.
d. =0
~,
~
is p.d. for every
c.
~1
(4.35)
n(~nO)'
(4.36)
00
~k~l~k} = O.
lim{ max
n-)-OO l$k$n
Further, we define the
,
supTr(D
) <
n
~n
1=1~1
(4.37)
as in (4.1) -(4.2) and assume that
lim(I~
ld .c l . ) = .p exists.
1= ~1 "'Ill
(4.38)
n-)-OO
Let
~
and for every
(i)
(ii)
. (x*'
~
we assume that
£3,
f*(~*;~)
every
f~J
Q as an inner point
be an m-dimensional rectangle containing
3
For
lim 8
-1
j
roo
lim J...
OO
_00
[f*(~*;
I,
the limits
(0, ... ,e , 0·· '0))
j
-f*(~*;
Q)], 1 $j $m
(4.40)
exist for almost every
fOO
_00
= f...
for almost
(4.39)
8 = (8 , ... ,8m)
1
~+Q
~
~*,
ej-+O
(iii)
satisfies the following:
is absolutely continuous in
~
9J =
, .-
f*(~*;~)
and
~*,
[i*(~*; ~)][i*(~*;
_00
foo . .
[f* (~*; Q)] [f* (~*; Q)] I {f* (~*; Q)}
-1
d~*
(4.41)
_00
is p.d. and finite, where
f* =(fi, ... ,f~)'.
of alternatives in (4.34) by
{K }
n
We denote the sequence
and assume that (4.35) -(4.41)
H in (2.1) relates to 6 =0. The contiguity of
O
the sequence of probability measures under {Kn } with respect to one
hold.
Note that
under
H is insured by (4.35) -(4.41) [c.f., Hajek and Sidak (1967,
O
pp. 238-239) and Patel (1973) in this context] and will not be proved
in detail.
-14-
(4.42)
for
j =0, l, ... ,p
and
O<t~T
and let
(4.43)
for
~.
~J
° <t
(a), 1
~l,
~j ~p.
is the
pm xm
matrix with components
Then, we have the following.
Under
Theorem 4.4.
B(~)
where
in (4.34), (4.1) -(4.5) and (4.35) through
{K}
n
(4.41),
W
E-n,r
for every
-=-+0
qm
W+ E~~*, in the Jl-topoZogy on D [E, 1],
E~
~here
E >0,
Outline of the proof.
={~*(t),
u*
T
E~
E
(4.44 )
~t ~l}.
Note that the tightness of
{EWn,r}
(under
HO)'
established in Theorem 4.3 and the contiguity of the sequence of
probability measures under
of
{W
} under
E n,r
{K}
n
{K }
n
to that under
~
H insure the tightness
O
as well [c.f., Theorem 2 of Sen (1976)].
Hence, to prove (4.44), it suffices to establish the convergence of
finite dimensional distributions of
{EWn,r}
to those of
E~ +E~*'
For this, we may readily extend the proof of Theorem 5.1 of Sen (1979b)
[to the case of
q
of the
and hence, for intended brevity, the details are omitted.
W (t.),
J
~n
~l,
m ~l] by considering an arbitrary linear compound
By virtue of (3.12), (3.13), (4.33) and Theorem 4.4, we conclude
{K }
n
that under
in (4.34) -(4.41), the asymptotic power of the
proposed pes test is given by
lim p{
max I~ k l > lea) jK }
n
n
n
n-+<JO
[nE] ~k~r
= p{ [Wet)
~
+~*
~T
(t)] '[W(t) +]J* (t)] >w*
~
~T
aE
for some
t
E
[E, I]}. (4.45)
_
~
-15Also, we have for
kin -)- U; TE <u
~T,
lim P{N >klK } =p{[W(t) +l1*(t)] '[Wet) +l1*(t)] ~w* , 'rf E ~t ~UIT}, (4.46)
n-)-OO
n
~
~T
~
~T
aE
while
P{N >nTIK } =0,
noting that
o
that
by definition in (3.13) and (4.3).
n
Z~[na] -)- s~,
(s >0), V 0 ~a <1
in the s-th mean
0
0
n[sk/n -S(k_l)/n] -[f[O](sk/n)]
-1
Further,
-)- 0, V k ~r,
and
we obtain from
(4.45) and some routine steps that
lim E (ZoN IK )
n-)-OO
n
n
1
=Tf
o
1 0
p{[W(t)
frO] (sTU)
+~~(t)] '[~(t)
+~*
T
(t)]
E ~t ~u}du.
V
(4.47)
Therefore the asymptotic power as well as the expected stopping time
of the proposed pes test depends on the boundary crossing probability
of
W+ E~
11*.
E~
We shall discuss more on this in the next section.
5•
SOME GENERAL REMARKS
As has been noted after (2.2), the classical ANOeOVA model is
a special case of (2.2) where
m =1
and
q
~l.
In this case, by
virtue of (4.18), (4.28) and (4.31), we can also write
W*E =sup{t
y ={y(t), 0
where
-1
~t ~l
[Y(t)]'
[Y(t)]:
E
.~
is a vector of
standard Wiener process, so that
q-parameter Bessel process.
of
W*
E
~t ~l}
q
(5.1)
independent copies of a
{[yet)] '[yet)], 0
~t ~l}
is a
Some tabulation of the critical values
is due to t1ajumdar (1977) and a more analytical approach to
this is due to DeLong (1980).
Thus, for this special case, we have
no problem in applying the proposed pes tests.
A very similar case holds
when we have an ANOeOVA model relating to the scale parameter when we
have (2.2).
pO(y; fl(c.
-c) Ix) =po(y(l + Mc. -c)) Ix)
ro.Jl
"11
,...., "'1
'"'11
,....,
f"'o,J
~
(5.2)
-16and the
c. and 6
~1
~
satisfy the condition that
1 +6(c.
~
~1
-s).
.-
> 0,
V i.
Though this scale model has been considered for the analysis of variance
v
models [c.f., Hajek and Sidak (1967)], it has not been considered
explicitly for the ANOeOVA model, even in the case of no censoring.
m ~2
In the general case of
(e.g., joint location-scale model),
the computation of the (simulation or analytical) critical values of
W*
aE
seems to be quite involved (because it depends on the covariance
structure in (4.28)).
Simplification is of course possible when we
have for some scalar function
Yt(s),
~OO(s) =Yt(s)~OO(t)
is
!
given by (5.1) with
in
s.
q
for every
s ~t,
(5.3)
In this case, again we have the solution
being replaced by qm.
Study of the asymptotic power properties of the pes tests [viz.
(4.45) -(4.47)] demands the knowledge of the boundary crossing
probabilities for drifted Bessel processes.
The prospect for this
rests heavily on the simulation techniques (as the analytical tools
are not yet properly available).
some studies are due to
others.
~lajumdar
For the analysis of variance model,
(1977) and DeLong (1980), among
There is some pressing needs for such studies for the ANoeOVA
problem.
Further, it is usually difficult to employ the concept of Pitmanefficiency in pes tests.
Hence, some alternative, meaningful measures
of asymptotic efficiency should be explored and, if needed, numerical
studies should be made.
Finally, throughout the paper, we have considered the case of
pes
rank statistics for the ANOeOVA problem. Cox(1972,1975) has considered
some quasi-nonparametric regression models for covariate-adjustments in
survival analysis. Though Cox has not considered the case of repeated
~
-17-
significance tests arising in PCS models, his theory can be extended to
cover such repeated significance testing. In fact, for a class of simple
hypotheses relating to the Cox regression model, such repeated significance
tests are considered in Sen(198l). The theory of such tests rests on some
invariance principle similar to the ones considered in this paper. It is
intended to extend the theory to the case of composite hypotheses (which
requires a somewhat different approach as the simple unweighted random
sampling theory from a finite population will no longer be applicable for
this complicated situation) and the same will be taken up in a subsequent
communication. Unlike the Cox model, the models in the current paper do
not require the proportional hazard assumption and therefore remain valid
for a wider class of situations. If, however , the proportional hazard model
can be justified, then, one would expect that the Cox regression model based
theory would perform better.
REF ERE N C E S
1.
CHATTERJEE, S.K. and SEN, P.K. (1973).
Nonpara~etric
testing under
progressive censoring. Calcutta Statist. Assoc. Bull.
2.
COX,
D.R.
13-50.
(1972). Regression models and life tables. Jour. Royal
Statist. Soc. Sere B.
D.R.
22~
J4~
187-220.
(1975). Partial likelihoods.
Biometrika~ 62~
3.
COX,
4.
DELONG,D.H. (1980). Some asymptotic properties of a progressively
censored nonparametric test for multiple regression. Jour.
MUltivar. Anal. lQ, to appear.
5.
HAJEK, J. and SIDAK, Z.
Press, New York.
6.
~~JUMDAR,
7.
~1AJtP.1DAR,
v
~
~
(1967).
Theory of Rank Tests.
269-276.
Academic
H. (1977). Rank order tests for multiple regression for
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Assoc. Bull. ~~, 1-16.
H. and SEN, P.K. (1978). Nonparametric tests for
multiple regression under progressive censori~g. Jour. MUltivar.
Anal. ~, 73-95.
-18-
8.
PATEL, K.M. (1973). Hajek-~idak approach to the asymptotic
distribution of multivariate rank order statistics. Jour.
MUZtivar. AnaZ. ~, 57-70.
9.
PURl, M.L. and SEN, P.K. (1969a). Analysis of covariance based on
general rank scores. Ann. Math. Statist. ~Q, 610-618.
10. PURl, f1.L. and SEN, P.K. (1969b). A class of rank order tests
for a general linear hypothesis. Ann. Math. Statist. ~Q,
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11. QUADE, D. (1967). Rank analysis of covariance.
Statist. Assoc. ~~, 1187-1200.
Jour. Amer.
12. SEN, P.K. (1976). A two-dimensional functional permutational
central limit theorem for linear rank statistics. Ann.
~obabiZity !' 13-26
13. SEN, P.K. (1979a). Weak convergence of some quantile processes
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414-431.
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censoring. Sankhya~ Ser. A. 41.
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tests based on progressively censored quantile processes.
Contributions to Statistics: 60-th Birthday VoZume for
Professor C.R. Rao. (in Press).
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for some induced quantile processes and some repeated significance
tests. Ann. Statist. 9,
17. SEN, P.K. and PURl, M.L. (1970). Asymptotic theory of likelihood ratio
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Math. Statist. 4Z, 87-100.
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