A FISHERIAN DETOUR OF THE STEP-DOWN PROCEDURE
by
Pranab Kumar Sen
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1363
October 1981
A FISHERIAN DETOUR OF THE STEP-DOWN PROCEDURE*
by
PRANAB KUMAR SEN
University of North Carolina, Chapel Hill
In a Step-down procedure, apart from an hierarchy of the component
hypotheses (leading to the steps), one also needs to settle on, their
individual levels of significnace (constrained on the overall one).
The
Fisher method of combining independent tests is extended to the step-down
procedure and its Bahadur-efficiency and (asymptotic) optimality results
are considered.
1.
Introduction.
~~~
For general multivariate analysis of variance (MANOVA)
models, the exact distribution theory of the conventional likelihood ratio
~
(or other allied) test statistics, often, gets quite complicated.
This is
especially the case with incomplete multiresponse designs and hierarchical
designs [see, Roy et. al. (1971)], where the conventional methods may encounter
considerable difficulties.
Further, unlike the univariate case, in a general
multivariate model, restriction to invariant tests may not yield an unique
best test among them, leaving some room for other ad hoc procedures.
One
such procedure is the step-down procedure, proposed by J. Roy (1958), which
can be viewed as a tributary of the general union-intersection procedure of
Roy (1953), and is usually simpler to construct.
AMS Subject
Classification:
Key Words & Phrases:
According to this procedure,
62H15, 62F03
Asymptotic optimality, Bahadur slope, combination
of test statistics, MANOVA models, probability integral transformation,
step-down procedure.
'/.;
Work partially supported by the National Heart, Lung and Blood Institute,
Contract NIH-NHLBI-71-2243-L from the National Institutes of Health.
-2an overall hypothesis. viewed as a (finite) intersection of m
. teste d 1n
. steps:
ones. 1S
(~ 1)
component
T
'
f or t h
. ma d e. 1. f and
est1ng
e 'J th component 1S
only if. in all the preceding steps. the component null hypotheses were
accepted. 1
~
j
~
m, so that the overall null hypothesis is accepted only when
all the components are accepted.
Step-down testing procedures are generally
not invariant ones, and the ordering of the steps (or related component
hypotheses) may sometimes be advocated on the ground of their relative
importance (and, otherwise. in an arbitrary manner).
choose the levels of significance for the
subject to some specified
a (0 < a < 1).
Further, one needs to
m component tests (say, a •..•• a ).
1
m
Usually. the component hypotheses
test statistics (say. T ••..• T ) are quasi-independent, when the null hypothesis
1
m
holds. so that
1 - a = (l-a )"'(l-a )
(1. 1)
1
and then. the
aj
m •
may be determined suitably either according to some index
of preference for the
m components or by the equality-criterion:
a =... =a =1 - (l_a)l/m.
1
m
For combining independent tests. based on the probability integral
transformation on the test statistics. various methods are available in
the literature [see. van Zwet and Oosterhoff (1967) and the references cited
therein]; among these the omnibus test of Fisher (1932) possesses some
asymptotic optimality properties [see Littell and Folks (1971). and others].
The object of the present investigation is to focus on this probability
integral transformation on the component test statistics
T , .•• ,T , in a
1
m
step-down procedure. and to show that the theory, due to Fisher (1932),
available for independent tests works out as well in this case.
In particular,
the exact distribution theory and the Bahadur-efficiency results hold for
a step-down procedure as well.
Along with the preliminary notions, the
-3proposed procedure is formulated in Section 2.
Section 3 is devoted to
the study of the Bahadur (1960) efficiency results.
The concluding section
deals with the asymptotic optimality of the proposed procedure for regular
MANOVA models and some general comments.
For a general class of hypothesis testing
problems, Roy (1953) introduced the union-intersection (UI-) principle
where the null hypothesis
H (and the alternative
O
H) are expressed as
the intersection (and union) of some component hypotheses
i
E
J.
J
In a step-down procedure, the index set
denote its cardinality by m
(2.1)
H
o
(~
HOi (and
Hi)'
is finite [and we
1)], so that
m
= n H
i=l
and
Oi
H=
m
U
Hi
i=l
where there is an hierarchy in the formulation and testing of the component
hypotheses (HOi vs. Hi' i=l, ... ,m).
(based on the test statistic T ):
1
First,
If
terminates along with the rejection of
At the j th step, H
Oj
the second step.
on a statistic T.) only when
J
HI
HOI is tested against
HI
is accepted, then the process
H ; otherwise, one proceeds to
O
is tested against
H. (based
J
HOi' i < j, are all accepted; if
H.
J
is
accepted, the process terminates at that step along with the rejection of
H
0
and, otherwise, one proceeds to the next step, for
that
.th
J
H
is accepted only when
O
HOI' .•• ,HOm
j=l, •.• ,me
Note
are all accepted, and at the
step, one usually makes a conditional test, given the preceding steps,
for j=2, ••• ,m.
Bk
Let
be the sigma-field generated by the random elements
entering the first
k
steps, for k=l, ••• ,m
sigma-field.
Bk
is nondecreasing in
(2.2)
Then
BO
and let
k (0
:$
k
:$
be the trivial
in).
Let then
-4-
If we assume that at each step, right hand side critical regions are used,
then corresponding to some preset
a , ••. ,a ,
m
1
satisfying (1.1), we may
define
o
G.(t.)=l-a.,l::<;j::<;m
(2.3)
J
Thus, at the jth step,
every
i
pp. 41-43)], the
t?J
J
H. is accepted if
~ t ~, while
T.
J
(l : <; j : <; m).
< j
J
J
J
In regular MANOVA models [see Roy et. al. (1971,
are appropriate variance-ratio statistics, so that the
T.
J
can easily be computed by reference to standard statistical tables.
To introduce the Fisherian detour, we define
(2.4)
U. = -2 log {I - G.(T.)},
J
J
J
1::<; j : <; m ,
and let then
U; = U1 + ... + U , for 1 : <; j : <; m ;
j
(2.5)
U* = U* .
m
First, we consider the following result underlying the test procedure to
be proposed.
Theorem 1.
If the
G., 1 : <; j : <; m are all
under HO' for every reaZ
t:
0::<; t
everywhere, then,
:$ co,
p{u* ~ tlHo} = 2-m (f-n1)-1
(2.6)
aontinuol~
J
co
J
m-1
exp(-~u)du.
u
t
Proof·
of
By (2.2) and (2.4), for continuous
U., given
J
B.J- 1
does not depend on
(2.7)
G., the conditional distribution
J
(and under H ) is the simple exponential one (which
O
U., i
1.
< j), for
j = 1, ••• ,m.
As such, writing (under H )
O
E{exp(itU*)} = E[exp(itU:_1)E{exP(itUm)IB _ }]
m 1
=
(1-2it)-lE[exp(itU* 1)]'
m-
we obtain by repeated reduction that
(2.8)
E{exp(itU*)} = (1-2it)-m ,
-5-
and this implies (2.6).
u~
Let
Q.E.D.
be defined by equating the right hand side of (2.6) to
a (0 < a < 1).
Note that as in the case of combination of independent test
statistics [Fisher (1932)],
distribution with
2m
u*
is the upper 100% point of the chi square
a
degrees of freedom (DF).
Then, we propose the
following testing procedure:
At the first step, compute
with the rejection of
If
H ' while, if U~ < u~, proceed to the second step.
O
At the jth step (only when
U~
<
u~,
V
i < j), if
terminates along with the rejection of
to the next step, for j=I, ... ,m.
u*
(=
U*) <
m
u
U~ ~ u~, stop at this stage along
U~ >- u *' t h e process
a
J
H ' while for
O
Again,
H
U.* < u * , one proceeds
J
a
is accepted only when
O
*
a
Since the
U.
J
are non-negative, for
j
~
1, we obtain that
for at least one 1 ~ j ~ mlH
Type I error
= P{I~~
-J-m
o}
U~J ~ u*IH
}
a O
= P{U: ~ u~IHO} = a
•
Hence, the proposed Fisherian detour eliminates the need of choosing
[satisfying (1.1)], and uses the same critical level
case of independent test statistics.
1
~
j
~
a , •.• ,a
m
1
u * ' pertaining to the
a
Computations of the
G
j
(and hence, U ),
m, are generally simple and can be made with the aid of standard
statistical tables or existing computer programs on standard statistical
distributions.
proposed test will be studied and the same will be incorporated in the
study of the asymptotic optimality of the proposed method.
to make the dependence of the statistics
At this stage,
T , ... ,T
on the sample size (n)
m
1
j
-6-
explicit, we denote them by
T
1 ,n
, ..• ,T
,respectively.
m,n
G~n),
denote the null distribution in (2.2) by
G~n)
J
G.* (l
converge to some
(3.1)
U.
J
,n
j
~
J
~
-2 log {l_G~n) (T.
=
J ,n
J
n -+
m), as
)}
'
1
J
1
:$
j
~
Similarly, we
~ j ~ m; generally, these
Let then
00.
* =
m·, U(n)
Note that (2.6) holds for every n (for which the
m
. U.
I J=1
J ,n
are properly defined),
so that for every (such) n,
* ~ tIH } -+ 1, as t -+
-2t -1 log P{U(n)
O
(3.2)
We assume next that as
(3.3)
n
~.
where the
J
that if, for
-1
U.
J ,n
n -+
00
-1
1
-+~.,
J
j
~
~
m, with probability 1 ,
n
while for the
G~n), we have
(3.5)
_2n- 1 log {l-G~n)(nt)}-+l/J.(t), V t
T.
J ,n
-+ V., 1
J
~
j
~
m, with probability 1 ,
J
J
j
for
1
(of
t), then (3.3) holds with
~
In passing, we may remark
00
(3.4)
~
•
,
are real, nonnegative quantities.
n -+
00
m, where the
(3.6)
l/J
~.
J
E:
J
[0,(0) ,
are nonnegative and continuous functions
j
= l/J.(v.) , for j=I, ••• ,m .
J J
In MANOVA models, as we shall see in the next section, (3.3)-(3.6) holds
under fairly general conditions.
* -+
n-1 U(n)
(3.7)
*
~
=
~1
Then, by (3.1) and (3.3), as n -+
+ ••• +
~m
' with probability 1 •
By (3.2) and (3.7), we conclude that the (exact) Bahadur slope for
equal to
*
~.
00,
* is
U(n)
Finally, the Bahadur efficiency of the proposed Fisherian
detour is also equal to
*
~.
-7-
From consideration of the best average power (over suitable contours
in the parameter space) and other criteria, a likelihood ratio test for
Ln .
H vs. H may be advocated; we denote the allied test statistic by
O
Usually,
-2 log L
has asymptotically chi square distribution with an
n
-2 log L , (3.2) holds.
appropriate degrees of freedom, and hence t for
n
Further, from the results of Bahadur (1967)t the exact slope for
-2 log
Ln
can be obtained in terms of the Kulleack-Liebler information and that can
H
and
O
further be simplified if
H are close to each other.
The Bahadur efficiency of the proposed Fisherian detour with respect
to the likelihood ratio test can thus be expressed as a ratio of the two
slopes.
We intend to study this in more detail in the next section.
for MANOVA models.
We shall establish the
asymptotic optimality (in the light of the Bahadur efficiency) of the
4It
proposed Fisherian detour for regular MANOVA models.
Consider the
usual MANOVA model:
(4.1)
where
~
(of rank
is the matrix of unknown parameters,
m
~
according to
n)
p '"
~),
'"
with
is a specified matrix
£ are independently distributed
and the rows of
N (0,
~
a positive definite (but, unknown) matrix.
We intend to test for
(4.2)
where
£
(rxm)
is a specified matrix of rank
r
(~m).
(4.3)
(4.4)
(4.5)
S
e
Y'Q Y and
'"
e'"
S
H
= Y'o Y
'" ~H'"
If we define
-8-
then the likelihood ratio test for (4.2) rests on the critical region
(4.6)
where
p{w1IHO} = u, the desired level of significance, and Theorem 8.6.1 of
Anderson (1958) provides the desired chi-square approximation for
S ~-1
It is well-known that
(4.7)
n
irrespective of
-1
has the Wishart distribution (with n-m
e
DF), so that for every fixed
m, as n
~e ~
-2 log A.
~ 00,
k , with probability 1 ,
H being true or not.
O
On the other hand, if we assume
that as
(4.8)
n-
/), '"
A p d. ,
.h:'h:
- .- ~ ._,
1
.
then, i t follows by some routine steps that for any (fixed)
~,
(4.9)
with probability 1, as
n
~
Thus, the (exact) Bahadur slope for the
00.
likelihood ratio test is equal to
-1
-1-1
Trace(k ~'£'[~ £] ~)
(4.10)
.
To formulate the step-down procedure with the intent of the Fisherian
detour, we let
X= (X 1 , ••. ,Xp),
let
so that each
Xj
is an n-vector.
Similarly,
and let
(4.11)
conventionally, we let
upper
jXj
minor of
X(O) = 0,
k = «cr
follows the model
(4.12)
jj
,»,
~(O)
= O.
Further, let
for j=1, ••. ,p.
k(j)
be the
Then, conditional on
-9where
(4.13)
[For j=l. the second term on the right hand side of (4.12) drops out and
2
VI = aU·]
Thus, on letting [for each j (=I, .•. ,p)]
8.
(4.14 )
~J
we may write [see Roy et. ale (1971, page 42)]
p
(4.15)
(where
n
C. = [C, 0.]. O.
~J
~
~J
Cn.
HO ';
J
j=1
~-J
being a null matrix of order
~J
(1
and (4.12) may be written as
$
j
~
r X (j-l). 1
~
j
~
p)
p). on which the
classical ANOVA theory yields the following test statistics:
(4.16)
e
where
(4.17)
(4.18)
for
S (j)
H
j=I, •..• p.
Under
H .'
OJ
DF's (r, n-m-j+l), so that
F.
J
rF.
J
chi square distribution with
has the variance-ratio distribution with
has asymptotically (as
r DF. for j=I, •.• ,p.
n
~ 00)
the central
Q.
Note that the
J
are
all indempotent matrices, with
(4.19)
Rank of
Q.
J
Trace of
Further, conditional on !(j-l)' when
Q. = n - m - j + 1,
J
HOj
F-distribution with non-centrality parameter
(4.20)
may not hold,
1
F.
J
$
j
$
P
has a non-central
-10where
(4.21)
B .
"1lJ
Y(j-l) = ~(j-l) + ~(j-l). where
Note that
so that
Y
~(j-l)
surely, for
n
~
00,
1
for each
(Y'
Y
~(j-l)~(j-l)
~
~
j
(=I, ... ,p), under (4.8)
n
with probability 1.
~
~
is indempotent of rank (j-1), almost
'
~(j-1)
j
(4.22)
p.
)-l y
e(.J- 1) ~ N(O, "1l
I
~ L(.J- 1))'
~
As such. by some standard steps, it follows that as
-1
-2
-1-1
rF.J ~ v.J {n~C'[CA
C'] ~J
cn.}.
.:-lJ~
~
~
Since (3.5) holds for the central F-distributions (as
= t,
~.(t)
well as the central chi-square distributions) with
J
by an appeal
to (3.3), (3.6) and (3.7), we claim that for the proposed Fisherian detour
of the step-down procedure, the (exact Bahadur slope is equal to
p
- 1 Cn.}
. IV.J-2{ n~C'[cA- 1
C']
I J=
N.:-lJ
(4.23)
.:-lJ~
~
~
p
-2
-1
-1
. IV.J Trace{C'(CA C') N.:-lJ.:-l
Cn.n~}
I J=
J
~
~
-1
p
. 1Trace{C'(CA
C')
L: J=
~
~
~
-1
-2
C(v.J.:-lJ
n.n~)}
J
~
-1 C') -1 C(.
I P IV.-2 n.n~)}
Trace{C'(CA
~
~
~
~
J= J .:-lJ.:-lJ
Finally, note that by (4.13) and (4.14)
(4.24)
I~=l{lk(j_l)I/lk(j)ll(gj =
g(j-l)Yj)(gj -
rj~(j-1))
gt;-l g, ,
where the last step follows from the
Gra~Schmidt
triangular reduction
(orthogonalization) process [see Rao (1965, p. 21)].
(4.24), the Bahadur slope reduces to
Thus, from (4.23) and
-11-
(4.25)
Trace{J? 1£ 1 (~-1£ I) -1£/?L:- 1 }
Trace{L:-l(J?,£,(~-I£,)-I£J?}
•
and this agrees with (4.10).
Hence, the Bahadur-efficiency of the proposed
Fisherian detour with respect to the likelihood ratio test is equal to unity,
so that, the proposed procedure is asymptotically efficient.
For independent
test statistics this property is due to Littell and Folks (1971).
It may be remarked that for general hierarchical model or incomplete
multiresponse models [see Roy et. al. (1971)], the proposed Fisherian detour
works out well and its Bahadur slope may also be computed.
the same design matrix
(~)
However, there
does not appear in the various step-designs, and
hence, (4.23)-(4.25) may not hold.
Also, for such models, (4.10) does not
hold, and a more complicated expression arises.
I conclude this section
with the following remark on the classical step-down procedure.
has asymptotically chi square distribution with r DF, for
HOj ' rF j
1
~
j
~
Under
p.
Thus, i f
2
xm,a
be the upper 100a% point of the chi square
distribution with m DF, then (asymptotically)
(4.26)
where the
rF.
J
rF.
J
H
O
2
~ xm,a.
,for every 1 ~
is accepted
j
iff
~ p ,
J
are quasi-independent.
As such, using (4.22), it follows that
the Bahadur slope for the step-down procedure is
(4.27)
max
-2{ n.C
1 1 (CA-I C1)-1 cn.,
}
.< v.
1<
-J-P
J
NJ~
which can not be greater than (4.25).
~
~
~J
Thus, in the light of the Bahadur-
efficiency, the proposed Fisherian detour has an edge over the classical
step-down procedure.
,
-12-
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Ann. Math. Statist.
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~, 303-324.
•
Ann. Math. Statist.
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Oliver &
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~2,
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