THE USE OF A PREL1NINARY TEST FOR INTERACTIONS
IN THE ESTTh'lATION OF FACTORIAL MEANS
by
Atindraraohan Gun
Institute of Statistics
Series No. 436
July 1965
~limeograph
iii
ACKNOWLEDGEMENTS
This investigation was suggested by Dr, R. L. Anderson and was
carried out under the guidance of the chairman of my advisory commi.ttee, Dr, H. R. van der Vaart.
I owe a debt of gratitude to them
for taking a lively interest in my work and for helping me cross
many a hurdle on the way.
I am also grateful to the other members of
the committee for going through the rough draft and making some
helpful suggestions,
The late Professor S. No Roy, as a former
member of my conmittee and otherwise, had been a constant source of
inspiration.
In his death I suffered a personal loss.
My thanks are due to Mrs. Linda Kessler and Mrs. Ann Bellflower
for doing the difficult typing of this dissertation.
I am also
thankful to Mrs. Diana Smith who did the programming for most of the
numerical part of the work,
I should also put on record my
appreciation of the help I received from Miss Helen Ruffin,
Mrs, Dorothy Green and Miss Lillian Hunter on numerous occasions.
Finally, I would like to express my gratitude to the U. S.
Educational Foundation in India, the Institute of International
Education and the North Carolina State for financial assistance,
and to the Government of West Bengal, India, for granting me leave of
absence from my job in Presidency College, Calcutta.
iY
TABLE OF CONTENTS
Page
LIST OF TABLES .
vi
vii
LIST OF ILLUSTRATIONS
1.
INTRODUCTION
1
1.1. Statement of the Problem
1.2. Review of Literature
2.
3.
:1
7
THEORETICAL ASPECTS OF 2x2 CASE
10
26
FURTHER S'IUDY OF 2x2 CASE WI'IH SOME NUMERICAL RESULTS
29
3.L Computational Procedure
3.2. Percentage Pbints of x
3.3. Bias as a Proportion of 711
....
3.~ Relative Mean Square Error: MSE(T)/(oF/r)
3 . .5- Comments on MSE .
3.~ Optimal Choice of a
3.~ An Alternative Formula for MSE .
3.8. Appendix: Proof of Lemma 3.5.1
4.
10
2.1. Notation
2.2. Preliminary Test Procedure and Definition of the
Estimator T' .
2.3. Distribution of the Estimator T
2.~ Expected Value of the Estimator and Its Mean Square
Error
2.5. Some Properties of the lQ Function.
2.& General Observations
2.7. Appendix.: Monotonicity of Incomplete Beta Function
THEORETICAL ASPECTS OF pxq CASE
16
17
23
25
29
30
31
34
37
40
48
52
55
4.L Notation
4.2. Estimation of a Single Cell Mean
4.3. Simultaneous Estimation of Cell Means
4.4- Some Relations between wand w*
4. .5- Preliminary Tes t Procedure
4.6. Distribution of the Estimator of a Cell Mean
4.7. Expected Value and MSE of the Estimator of a Cell Mean, and
55
56
58
61
62
64
67
AMSE
4.8. General Observations
4.9- An Alternative Approach
10
,
.
71
73
v
TABLE OF CONTENTS (continued)
Page
5.
FURTHER STUDY OF pxq CASE WITH SOME NUMERICAL RESULTS
5·L
5·2.
5·3·
5,4.
5·5·
6.
Percentage Pbints of x
Bias of Tij_as a Proportion of Yij
Relative AMSE: JU!R,E/ ( d2 /r) .
Comments on the Values of AMSE
Optimal Choice of a
SOME DECISION-THEORETIC RESULTS
6,1,
6.2.
6.3.
6.4.
Introduction
Fixed.sample Decision Problem
Decision-theoretic Formulati.on of a TE Procedure
Bayes Rules for Simple TE Problems .
6.~ Bayes Rules for Simultaneous Estimation of Cell Means
in a Factorial Experiment
6.6. Minimax Decision Rules
7.
SUMMARY AND CONCLUSIONS
7. L Summary .
7,2. Conclusions
7.} Suggestions for Further Research
LIST OF REFERENCES
75
75
75
78
84
85
90
90
90
94
9~
100
102
104
104
106
107
111
vi
LIST OF TABLES
Page
302.1
Upper a-points of x (2x2 case)
31
303,1
Values of 7i~ E(~ll- T), r=e
32
303·2
Values of
7~~ E(~ll- T), r-5
33
3.303
33
3.4.1
Values of 7~i E(~ll- T), r=lO
2
Values of the ratio MSE(T)/(cr /r), r=e
30402
Values of the ratio MSE( T) / (i /r), r:005
36
30403
Values of the ratio MSE(T)/(l/r), r=lO .
36
30601
Levels of significance for ANOVA test corresponding to
a = .25 and .50
47
35
5,1.1 Upper a-points of x (pxq case) .
5·2.1
Values of
7il E(~ij-
5,2.2
Values of
7i3
5.301
5.3.2
Values of the ratio AMSE/(J2/r), p=q-3, r=3
2
Values of the ratio AMSE/ (cr /r), p=q=3, r-5
81
5.3.3
Values of the ratio AMSE/(i /r), p=q=5, r=3
82
5.3.4 Values of the ratio AMSE/(d2/r), p=q=5, r-5
83
Tij ), p=q=3
77
E(~ij- Tij ), p=q=5
77
~ r AMSE'!r(Tla)/cr2
5.3.5
Values of
5.5.2
Levels of significance for ANOVA test corresponding to
a-025, .50 and. 75
80
87
89
vii
LIST OF ILLUSTRATIONS
Page
3.5.1
2
Variation of rMSE/a with ~ for different significance
levels (with r=5) . • , • . • •
I,
38
1.
1.1.
INTRODUCTION
Statement of the Problem
Consider
a factorial experiment with two factors, A and B, having
,
p and q levels, respectively.
combinations has
r(~
Suppose that each of the pq treatment
2) replications.
Denoting by Yijk the response to the treatment combination (AiB j )
in the k-th experimental unit, let
Yijk
Where
= I-L
ij + €ijk
i
=
1,2, ••. • .• , p
j
=
1,2, .•• · .• , q
k
=
1,2, .•. · .. , r
,
(1.1.1 )
}
,
ij is the expected response to the combination (AiB j ) per experimental unit, and €ijk is the random component, assumed NIID(O,cr2 ).11
I-L
We shall further assume that for our experiment the 'fixed effects
model' (Eisenhart's Model I) is appropriate, so that we may write
(1.1.2)
where
I-L
= general
effect,
ai = (main) effect of Ai'
~j
= (main)
effect of B ,
j
r ij = interaction effect of Ai and Bj
,
l/NIID(O,cr2) means 'normally, independently and identically distributed
with mean
and variance cr2t .
°
2
and
q
E
P
E
a = 0
i=l 1.
j=l
t3 = 0
j
,
q
E r ij = 0
j=l
P
E r 1j = 0
i=l
for
i = 1,2, ... •.. ,p
for
j = 1,2, ...••. ,q
(1.1.3 )
,
Now suppose our objective is to estimate a cell mean, say
~ij.
Let~j
r
(i = 1,2, .. • ,p ; j = 1,2, ... ,q)
= Z yijk/r
k=l
q
Y1. = E yij/q
j=l
-
and
...
Y
..
,
(1.1.4)
P
Y. j =
,
E yij/p
i=l
q
P
= E E Yi~/pq
1=1 j=l
)'
The least squares estimators of the parameters in model (1.1.2) are
~
.
..
=Y
ai =
.
..
..
•
Y1. .. Y
..
r ij = Yij
..
t3 j
..
..
:::;: Y. j - Y
,
(1.1.5 )
..
+y
.. Y
.. Y
1.
.j
Thus the lea~t squares estimator of ~ij under model (1.1.2), which is in
consequence uqbiased ahd minimum-variance under that model, is
!/Fbr simplicity, we prefer these symbols to the more elaborate
Yij .' Y1. .' Y. j.
and
y
3
.
..
Il ij = Il + a i + t3 j + r ij
(1.1.6)
We have
E(ll" ij ) = ll
(1.1.6a)
iJ
var(ll" ij ) = (J 2 /r
and
(1.1.6b)
If interaction effects are supposed to be absent, then model (1.1.2)
reduces to
(1.1.2a)
,
,
with the same restrictions o~ a sand t3 s as in (1.1.3).
interaction model, the least squares estimator of Il
ij
Under this no-
is
...
"
"
ll iJ = Il + a i + t3j
-1.
+y-
=y
.j
-y-
which is unbiased and minimum-variance under this model.
Nqw suppose (1.1.2), and not (1.l.2a), is the appropriate model for
our experiment.
.
.
"..
Noting that Il, ai' t3
j
and r
are unbiased estimators
ij
of the corresponding parameters, that
..
2
Var(ll) = .(J /pqr
,
..
2
var(ai ) = (J (q-l)/pqr
Var(r"
ij
,
) = (J2 (p-l)(q-l)/pqr
..
Var(t3 j )
=
2
(J
(p-l)/pqr
,
(1.1.8)
,
and that these components are mutually independently distributed, we
have then
-
E(~1j)
4
= ~ij
- r ij
(1.1.9a)
(1 - (p-l)(q-l)]
pq
(1.1.9b)
and
=~
r
Suppose further that minimum mean square error (MSE),
rather than unbiasedness plus minimum variance, is our criterion of a
good estimator.
This means in effect that we have in mind a loss function
of the form
where c is a positive quantity Which may depend on the parameters, and T
is the proposed estimator of
~ij'
and our goal is to control the risk
It is seen that in this situation, even tnough (1.1.2) 1s the proper
4'
model,
~ij'
times.
For
rather than
~1j'
wil+ be the more desirable estimator at
...
4'
MSE(~ij)
= var(~ij)
= (T2/r
(1.1.10 )
and
MSE(~ij) = var(~ij)
= (T2
r
+
ri j
[1 _ (P-l)(q-l)]
pq
so that
-
4'
MSE(J.L ij ) ~ MSE(J.L ij )
+r 2
1j
,
(1.1.11)
5
as long as
2 < ~2
'Y ij - r
.
(p-l~(q-l) =
pq
..
Var( 'Y i,1)
(1.1.12)
As a consequence, if the relevant parameters woqld be known, the
following estimation proqedure would be advisable:
use
~ij
However, since these parameters are unknown, t4e natural procedure that
suggests itself is to
substit~te
the parameter inequalities.
the outcqme of a preliminary test for
This leads to the
followi~
estimation
procedure:
1)
First, test for the w:pothesie that
0+ possibly
som~
average of all pq such quantities (see
Chapter 4), is less than or equal to unity.
2)
In case significance is found in step 1), take ~ij as the
estimate of
~ij;
otherwise take
~ij·
ijence, if t(x) is the probability of rejection of the hypothesis
2
'Y ij
~
,.
var('Y ij ) (or similar hypotheses in the p x q case), the ultimate
estimator T has the form
1/
-
..
(1 - t(x)} ~~j + t(x)~iJ
1/A next
-
step, not taken in this work, might be to look into more
..
general functions of ~iJ' ~ij' and some test statistic.
6
The consequences of such an estimation procedure following a
tes~
of
significance (a TE procedure, for short) form the subject-matter of the
present investigation.
The heart of the difficulty is that the interpreta-
tion of the joint application of a test and an estimation procedure has
to be different from the interpretation that would be valid for their
individual applications.
The fact that a test and an estimation procedure
are carried out on the same data causes these two procedures to be
interdependent in the probability sense, which entails rather cumbersome
distribution problems.
We shall derive the distribution of the suggested estimator, its bias
and mean square error.
Next, we shall study how the bias and the mean
sqqare error are affected by our choice of r, the number of replications,
and 0, the level of significance of the preliminary test.
A rather detailed study will be made for the simple case Where
p = q = 2 (Chapters 2 and
3).
For the general case, ~irections in which
our findings for the 2x2 case will need to be modified will be indicated
and these modifications worked out to some extent (Chapters 4 and 5).
In Chapter 6, we shall set up the general problem of estimation
after a preliminary test as a prOblem in decision-making and shall
investigate the nature of Bayes rules and
minima~
rules.
We shall first
make some general remarks and then devote oUr attention to some special
cases, viz., the estimation of the mean of a normal population when the
variance is
~nown
as well as when it is unknown, and the estimation of
cell means in a factorial experiment.
7
1.2.
Review of Literature
We have taken as the starting point of our discussion a paper by
Anderson (1960) in which he considered the MSE's of parameters as a basis
of comparison between alternative models and alternative modes of
analysis for factorial experiments.
The problem of estimation after a preliminary test of significance
has been studied in a variety of contexts (all different from ours) by
a number of investigators.
All have examined, as we shall do, how the
biases and MSE's are affected by following a TE procedure.
Mosteller (1948), Kitagawa (1950) and Bennett (1952) dealt with a
2
TE procedure for estimating the mean of a normal population N(~l'~ ),
When samples from this and another population
variance, are given.
2
N(~2'~
1 with
the same
If we denote by n , n the two sample sizes and by
l
2
Xl' X the two sample means, the procedure consists of making a test for
2
- as the estimate in case of
the equality of ~l' ~2' and taking Xl
significance and taking the pooled value
-
nlxl + n 2x2
n + n
l
2
/
otherwise.
Bancroft (1944) considered a similar situation, Where samples of
sizes n , n drawn from normal populations N(~l,~i) and N(~2'~~) are
l
2
given and the problem is to estimate
2
~l'
The hypothesis of equality of
the two variances is first tested by means of the sample variances si
2
and s2'
2
~l;
2
In the case of significance, one takes sl as the estimate of
otherwise, the pooled value
8
is taken as the
Kit~gawa
in
e~timate.
(1950) also considered the problem or
inte~penetrating
~
x
•
Supposing
followed i$ to take
t
sampling, where actually one has a sample rrom a
bivariate normal population with means
estimate
sometimes pooling
x and
~
x and
.
y , say, and one wants to
~
yare the two sample means, the method
x or (x + Y)/2
hypothesi p of equality of
~
as the estimate, according as the
x and ~y is rejected or not in a preliminary
test of significance.
Bancroft (1944) and Kitagawa (1963) investigated the application of
TE procedures to lin~ar regression analysis.
dealt with some aspects of the same
considere~
probl~m.
Larson and Bancroft (1963)
Kitagawa (1963) also
the use of TE procedures in response surface analysis.
Asano (+960) considered the application of TE procedures to some
prob1em~
in biometrical and pharmaceutical research.
The procedures
boil down to the estimation or means after testing for variances, and
the estimation of a variance-like runction of effects in analysis of
variance after a preliminary test or these effects.
Bennett (1956) discussed TE procedures in Which ultimately One
wants interval estimators, rather than point estimators, of means and
variances.
Huntsberger (1955) generalized the TE procedure by considering a
weighted sum of simple estimators, the weights being determined by the
observed value of the test statistic.
In the case of
no~al1y
9
distributed estimators, he showed that the weighting procedure on the
whole effects a greater degree of control on the MSE than does the TE
procedure.
Kitagawa has brought together many of these findings in his recent
publication
(1963) where he has discussed some other aspects qf the TE
procedure as well.
Bancroft
A mention may also be made of the review article by
(1964).
There is one aspect in which our TE
p~ocedure
envisaged in the investigations cited above.
involved
~n
differs from those
In our case, the hypothesis
the preliminary test has the form
(parameter)2 5 A(variance of its estimator)
A being
~n
appropriate positive constant, whereas in the investigatiqns
referred to, the hypothesis is of the .usual null type:
parameter
=0
t
We are thus concerned with
ordinary significance.
t
~terial
significance
The notion of material
rather than with
~ignificance
and some
methods of testing for material significance in certain specific
problems have been discussed by Hodges and Lehmann
(1954).
10
2.
2.1.
THEORETICAL ASPECTS
Or 2x2
CASE
Notation
In the 2x2 case we may consider the problem of estimating the cell
mean ~ll'
We shall see (in Section 3.1) that the results for this mean
hold with little modification for the other cell means as well.
We shall denote by a (0 ~ a~ 1) the level of significance at which
the preliminary test is made.
T will denote the estimator of
~ll'
...
whicp. equals !l1l1 in the crse of signifieance and
For simplicity, we shall
w
z
and
r
..
=y
..
..
= YJ-l
+
(Yl .
..
..
• y
-
~ll
otherwise.
writ~
) +
- Y1. '- Y.l +Y
Cr. l
..
-..)
= ~ll
,.
,..
.. y
A
==
)'11
= ~ll
..
~ll
..
}
(2.1.1)
I"
Note that
r1l
i~
the least squares estimator of 711 in the interaction
model (1.1. 2).
~e
denot~d
sample error variance (under the interaction model) will be
by s 2 and the number of its
degree~
of freedom by v.
Thus here
" :;; 4(r-l)
and
2.2.
Preliminary Teet Procedure and Definition of the Estimator
T
From Chapter 1, p. 5, it is seen that prior to estimatiop we are to
test the hypothesis
11
or,. equivalentJ
(2.2.la)
against the
al~ernative
H:L:
2
where
~roupB
of invariance under the custOmBfY transformation
dictate that this test should be based solely on the statistic
(vide Lehmsnn
(1954)'.
re~uire
(1959)t
L~t ~be
0
1, Section 5, and
Cha~ter
= probability
~enotes t~e pa~
Note that
power
and Lehmann
the spep1f1ed level of significance, 1.~"
af
suppose
gene~l~y
un~form1y
in
HQ at the point 4rz 2/s 2 and
a corresp?nding to HO' Just
of reJeqt1~
~e pafa~~eJ;'
as 8 1 will denote the part of
t~e
Hodge~
that
where ~(4rz?/s~)
9
2
'" = 4rYll/cr
Consider~tionB
we
(~.2.lb )
",)1,
e
space
~orresponding
tes~lpg
to
81.
a hypothesis the goal is to maximize
Qver 9 1 , i.e., in our case to maximize
(2.2.4)
$ubJect to (2.2,3), and it is known that the UMP invariant test is given
l?y:
q>
o
c{1
0
if
1.f
2
4rz 2 /s > c
4rz 2 /s2 ~ C
12
Howeve~,
1n
TE
~
~~ocedure
t~e purpo~e
sq~are err~r
over 9, the mean
be to minimize, 4niformly
~hould
of the ultimate estimator (see Section
1.1)
'I'
2
2
(+ - (fI( 4rz 2/13»)W
+ cp( 4rz 2/s)(w+z)
;I:
'!hus we "ant tQ minimize (by making
e(T) = Ee(T
MSE
...
8
suitable
(2.2.6a)
choic~
of cp)
~ll) 2
• Ee(w + ~(4r~2/s~)z ... ~11]2
unito~
Note
over 8.
~het ~or
non-randomized
tests ,
f
i
I
MSEe(T) = Ee[{~ ... ~(4~z2/s2»)(w ... ~11)2 + ~(4rz2/s2)(w + z ... ~11)2] •
(2.2.6q)
We 6ho~d, therefore! look ~or an invariant test ~(4rz2/s2) which
has level a and
w~1c~
mintm1zes (2.2.6b).
test minimizing (a.2 r6b) for
,
test.
eac~ 8E 9
,
We
s~~l
call an invariant
a uniformly TE-best invariant
HOlIever, the following theorem shows thf;\t a uniformly TE·best
~nvartant
test does
Theorem I e.g.;t.
.
i
~ot ~Xi8t.
For a
>0
there is no invariant test ~(4;rz2/s2),
and such that
MSEe(T)
is a minim"4Jll for each 8e 9.
(see (2. 2.6b ) )
13
Proof:
Consider the test of level 0 (and hence of level a)
Fbr all pqints Of
e
~
for Which
=0
this is the TE-best invariant test.
if
~
=0
MSEe(T)
(implying 711
(for
defi~1tion
of
~
see (2.2.2)h
For from (2.2.6b) it follows that
= 0), then
= Ee[(W
- ~11)2 + 2 ~(4rz2/62)(w - ~ll)z + ~2(4rz2/s2)z21
= Ee(W
-
~~1)21
+
Ee(02(4r~2/s2)z2}h1
(since w, z and s are independently
distributed and E(w) ~ ~ll when 7 11
the equality holdiqg itt
tp
=0
= 0.)
a.e.
It will be enougn to show that there exists another test of level a
which fqr
~......
CD
is better than the above test.
~(4rz 2 Is 2 )
Here an6 elsewhere
certain
={
~n t~1s
non-qent~al
F
10
Put
it
if
thesis Fa stands for the upper
distribution-~see
(2.2.l3a).
~point
of a
We shall see in what
follows tnat for this test
so that it ssti&fies (2.2.7a)
8S
well
8S ~ •
0 does.
From (1.1.11) it
i)
1/since this theorem admits randomized tests, • may assume values
other th~n 0 or 1. Hence ~2 doe~ nQt redu~e to ~ here, as it did in
(2. 2.6~).
14
follows that for the estimator corresponding to the preliminary test
(2.2.8)
MSEe(T) ~ co
if 1jr
~ 00
On the other hand, Theorem 3.5.1 shows that for the estimator
corresponding to the preliminary test (2.2.9)
2
MSE (T) ~ a /r
as
e
1jr
~ 00
Consequently, it does not seem unreasonable to use the UMP invariant
test of level a as the preliminary test in our T.E procedure and to
study the properties of the resulting estimator.
Thus we shall be
using the preliminary test (2.2.5) where C is so chosen that
sup
e€ 8 0
Pr[ 4rz2 /s2 >
ci eJ
Q
III
2 2
Since the statistic 4rz /s has a distribution dependent
alone and since the power function of this test is monotonically
increasing in 1jr, the above is equivalent to the condition
Pr[ 4rz2
/s 2>1
C 1jr
2~ 2 has
Here-4rz
IJ
III
III
a
(2.2.10)
the non-central F distribution with (l,v) d.f. (or,
equivalently, the non-central t
2
distribution with v d.f.) and non-
centrality parameter
1jr =
2
2
4rh/a
.
(2.2.n)
If Fa denotes the upper a-point of the non-central F distribution with
(l,v) d.f. and 1jr
III
1, then (2.2.10) implies that
15
C
=F
(2.2.12)
a '
and then (2.2.5) coincides with (2.2.9).
= 4rz 2 /s 2, give~ e.g., in Graybill
The probability element of F
(1961), is
Hence
Fa
is given by the equation
ell
exp(-1/2)
1
.
(f/y)m-l/2
d(f/,,)
(1+f/,,),,72~+172
2m,
m. B( m+1/2,,,/2)
1:
m=o
(2.2.13a)
For notational simplicity, we shall deal with
x
=
4rz 2 / "s 2
1 +4rz 2/"s2
='~
F
1 +F"
and
(2.2.14a)
rather than witn F and F.
a
co
exp(-1/2)
where I x
1:
m;:::o
This x
a is then given by
1
I
(m+l/2, ,,/2)
m,
xa
2 m.
=1 - a ,
(r,s) is the incomplete beta function
a
1
B(r,s)
xr-l,( l-x )s-l dx
(2.2.14b)
16
2.3.
Distribution of the Estimator T
Note that the TE estimator (2.2.6a) of the cell mean
~ll'
in case
the preliminary test is defined by (2.2.9), is
if
w
T- { w+z
where F
=
if
2 2
4rz /s and F is still defined as in equation (2.2.l3a).
Ct
Hence the probability element of T is
where gl and g2 are the conditional density fUnctions of w and
w~
respectively.
Noting that w, z and s2 are mutually independently distributed (the
2
first two normally, as N(~ll- Yll,3rr2/4r) and N(y ll ,rr /4r), respectively,
and the last one as X2 rr 2 /'V, x2 being the (central) X2 statistic with 'V
'V
'V'
d.f.), we may write the expression (2.3.2) as
hl(t)
1f
h (Z) h (s2)d:?; d(s2)dt
2
3
R
= hl(t) JI'h (Z) h (s2)dZ d(s2)dt
2
3
R
+ h (t,. )dt -
4
11
R
h4"(t/Z) h (Z) h (s2)dZ d(s2)dt
2
3
17
= h4 (t,. )dt
Jf
+
(hl(t) - h:(tlz)]h2 (Z) h (S2)dZ d(S2)dt
3
R
and h are the density functi ons of w, Z and s 2 ; h4 (t, Z)
3
the joint density of t = w+z and z; h4 (t,.) the marginal density of
Here hJ., h
t
= w+z
2
and h4* the conditional
den~ity
of t given Z; and the region of
integration R is defined by
c
and R is the complement of R.
Since the marginal distribution of t
conditional distribution of t
= w+z
= w+z
given z is
is N(~11,~2/r) and the
~(~11+Z-Yll,3~2/4r),
(2.3.3) may be written as
(2n~2/r)-1/2 exp[-(t-~11)2/ 2~2Jdt
+
ff ((2n
a~2(1/2 exp(-(t-~11-+'yll)2/(2 . ,~2)}
o
R
2
2
- (2n . ~
2Q:)~1/2 exp(-(t-~11-z+r11 )2/(2 . tf)}J
h
2.4.
2
x
(Z) h (s2)dZ d(s2)dt
3
Expected Value of the Estimator and Its Mean Square Error
The result
yields from (2.3.5) the expected value of T:
18
Jt
00
Ee(T) ==
Pr(dt)
-00
= J.L
11
+
J1
((J.L
11
-r 11 ) - (J.L1l +Z-rl1 )]h2 (Z) h (S2)dZ d(S2)
3
R
(Where, a6 before, the subscript
e
serves to remind the reader that these
expected values depend on the J.L ijJ r ij and rr 2 )
==
J.Ll1 -
If
z h (Z)
2
h
3
2
(s2)dZ d(6 )
(2.4.1)
R
Again, the result
yields from (2.3.5) the MSE of T:
J
00
e(T-J.Ll1 )2
E
=
-00
(t-J.Ll1 )2 Pr( dt)
2
2
h (Z) h (S )dz d(s )
2
3
2
= rr /r + 2r
11
11
Z h 2 (Z) h (s2)dZ d(s2)
3
R
- JJ[{
R
2
2
( 2
Zh2 (Z) h (S )dz d s )
3
(2.4.2)
19
Putting
(k is a positive integer1
we thus have
(2.4.4a)
and
(2.4.4b)
Now
J
k
JJ
=
(21t
~r1/g
zk exp[-(z-r
11
)2/(2
~)]
h (s2)dZ d(s2)
3
R
= exp( -,,/2)(21t
~r1/2
JJ
zkexp[-z2/(2·
R
On making the transformation
u
we have
Hence
= 4rz 2/(j 2
,
~)) x
20
_.
J1
...,
- exp(~W/c)(2~
2
2
s F /er
-1/2J J
4r)
CD
2
0-
o
eX
exp(-u/2) ~
'0
(4
rr11
/ 2)2m+1(
0-
0( 2) I4r
du d(s2)
•
2!U
use e notatiop from Rao (1952), viz.,
G(a,p,x):;
Also
{
r~)
p
o
exp(-ax)xp-1
for
x
~
0
for
x
<
0
•
.
(2m+1)!
ffi-O
x h3 s
to
2
00
2/ m+1
uo- 4r)
21
Now
h (S2) d(s2) • G(1/2,~/2'VS2/~2) d(ys2/~2)
3
since vs2/~2 has the (central)
x2
distribution w1t~ v d.t.
Also we have
the result, given, e.g., in Raq (1952),
x
-<w
x;y ..,.
w
B{I?l~P2)JvPl-l(1;"V)P2"!1
dv
o
•
(2.4.6)
ponsequently,
and
2
J
2
• ~
exp(-W/2)
~r
CD
/
m
E ,(o/L;)
m.
~o
{Ix (m+3/2,v/2) + WIx
a
a
{m~/~,v/2)1
The change of the order of integration and summatton made here is valid
by virtue of Theorem 27B in Halmos (1950).
QJt/2, v/2, xa.)
Let us define the symbol
by the equation
(2.4.7)
Where
~
j
O.
We may then write
and
In terms of this Q function, (2.2.l4b), (2.4.1) and (2.4.2) are a8
fOllows:
(which determines the cut-off point ~a)'
(2.4.9 )
and
(12
=
r
(1 -
~/2('/2,V/2,Xa.)
4
i
.
+ ~ (2~/2('/2,V/2,Xa.) - ~/2(V/2,v/2,Xa)})
(2.4.10)
l/someti~es we
shall write simply
Q~.
23
2.5.
Some
Propertie~
of the Q ~ction
Si~ce
(a)
o -<
I
x
(m+j,v/2) < 1
-
ex
,
we also have, on multiplying each term by (~/2)m/m~ and addi~ over m
frOpl 0 to GI,
tb,e [lower
upper pounds being attained only if x
a~
a
=0
and x
a
=1
respectively.
For 0 <
X
a < 1, somewhat better bounds are obtained as ;follows:
Making the
~ransfo~tion
we have
1
Xa
m+j-l(l -x )V/2- 1 dx
[ x
= xm+j
a
J
0
> x~+j
Jz~+j-l(1_z)V/2-1
1
o
Hence
and
Again,
~king
the transformation
1 - x = (l-z )(l-x )
a
we have in a similar manner
,
dz
24
and
(b)
Other things remaining constant, I x
increases monotonically
a
inc~eases (i.e., as a decreases). Hence Qj(W/2,~/2,Xa) also
a
increases monotonically as x increases from 0 to 1 (or, equivalently,
a
as a decreases from 1 to 0), j, Wand ~ remaining fixed.
as x
(c)
As shown by Jordan (1962), p. 85, for 0 < x <:: 1
a
and
for anyj, j' wittl 0 < j < j' ~ see Section 2.7).
(d)
<0
,
because of (2.5.4), showing that other things remaining the same, Q is
a monotonically decreasing function of W, provided 0 < x < 1.
a
Sec1;ion 3.5 we shall show that actually Q ....> 0 as W...;:. CIO •
In
25
2.6.
General Observations
On the basis of the results mentioned in the preceding section,
some general comments can be made on the bias and MSE of the estimator
T (wi t,n 0 < ex < 1).
The bias of T is
(2.6.1)
and in magnitude is
Because of (2.5.1) then
(2.6.3 )
i.e., the bias of T lies between that of
(2.5.5)
~ll
and that of
-
~ll.
implies that the bias becomes smaller and smaller as
from 0 to
Further,
~
increases
co.
Again, consider the MSE of T.
We have
Hence
> 0 for 0.5 '" .5 1
This means that in the domain 0 .5
greater than that of
Also
-
~ll.
~
.5 1 the MSE Of T is necessarily
26
> 0 for 1
It follows that in the domain
1jr ~
:s 1jr
<
(2.6.6)
(Xl
1 the MSE of T is necessarily greater
lit
than that of
For
1jr =
~ll'
0
so that at this value of
value of
X
ex
(1.
e.,
W, the MSE is increased by taking a smaller
a larger value of ex).
Another result that follows from (2.5.1) is that for
1jr
< 1/2
(2.6.8)
1jr ~
so that if
2.7.
Appendix:
,.
1/2, then T is necessarily better than ~ll'
Monotonicity of Incomplete Beta FUnction
The incomplete beta function
x
I
Ix(p,~) =
yP-l(l_y )q-ldy /
o
where Q :s x
:s 1
and p, q
1
J
yP-l(l_y)q-l dy
,
0
~
0, is known to have the properties that
and
I (p,q) < I (p,q+l)
x
x
,
provided 0 < x < 1 (see, e.g., Jordan (1962), p. 85).
We shall show
that these results can be generalized, that in fact I (p,q) is, for
x
27
o < x < 1,
a monotonically decreasing (increasing) function of the real
variable p (of the real variable q).
Let p': > p > O.
Then, denoting the complete beta functJLon by
B(p,q), we have
- i'(1
1 x
yp-l(l_y)q-l . p' -l(
)q-l
z·
l-z
dy dz
x
1
J.J '
:J. J
-
yP
o
-1
(l-y)q -1 zP-1 (l-z)q-1 dy dz
0
1 x
yp-l(l_y)q... liP~-;l(i~z)q-ldY dz
x
0
Now, in the domain
{(y,z)
J0
< Y< x
,
z/y > 1
,
x < z < l}
,
we have
so that also
,
(zjy)p :p > 1
,
i. e.,
y
p-l pi -1
_-'P' -1 p-l
z
>yZ
This implies that the expression (2.7.1) is positive and hence that
if pi
> P > o.
In a similar way, one can show that
if q'
>
q
>0
•
3.
3.1.
FURTHER STUDY OF 2X2 CASE WITH SOME NUMERICAL RESULTS
Computational Procedure
Note that
1s the probability that a Poisson r.v. with parameter W/2 takes the value
I
m (m
= 0,1,2, •••.•• ).
These terms have been tabulated, e.g., in the
tables pUblished by the Defense System Department of GE (1962).
Also
values of the incomplete beta function
for various combinationsof
p and q and for x ranging from 0 to 1 have
r
been tabulated in Pearson (1934).
Using these two sets of tables and
applying, wherever necessary, the relation
we have found values of Ql/2(1/2,~/2,X) and from these, by inverse
i~terpolation,
those of the upper
x
~point,
= 1~'7~
xcf of our test statistic
,
as defined in (2.2.14).
The same procedure of combining tables of the Poisson distribution
with those of the incomplete beta function yields values of
~/2(w/2,~/2,Xa) and ~/2(w/2,v/2,Xa)' whence, using (2.4.9) and (2.4.10),
we have obtained the bias of T as a proportion of Y and rMSE(T)/~2
ll
for various combinations of
~
= 4(r-l),
a and t.
But here we had to
resort entirely to the electronic computer, since the tables are not as
30
extensive as needed for our purpose.
The computer generated series of
terms of the Poisson law and of values of the incomplete beta function
and then combined them in the desired manner.
It is to be noted that the formulas for bias and MSE are essentially
the same for each of the cell means ~ij (i
because 712
= 7 21 = -
711 and 7 22
= 711 ;
= 1,2
and j
= 1,2).
This is
while the preliminary test in
each case is based on
Hence the biases of the estimators of all four cell means have the same
magnitude but are either positive or negative, i.e., they are
while the MSE's are all equal.
All values appearing in the tables (in this chapter and in Chapter
5) are correct to the last decimal shown.
3.2.
Percentage Points of x
In order to study the variation of the bias and MaE with r, a and
2/2
V = 4rr+l
rr , we have considered three values of r, viz., 2, 5 and 10,
which correspond to ~
= 4, 16
and
36, respectively.
Fbr each of these
values of r, we have considered procedures based on tests of levels
a = .01, .05, .25 and .50.
The two trivial values, a
corresponds to making no test and taking
~ll
=0
(Which
as the estimate lnd a
=1
A
(Which corresponds to making no test and taking
~ll
as the estimate) may
also be taken into account, and for these the percentage points of x are
x
o
=1
31
and
X
1
::::
0
,
irrespective of the number of replications.
The percentage points x
a
for a
=
.01, .05, .25 and .50 are shown
in Table 3.2.1.
Table 3.2.1.
Upper
~points
of x (2x2 case)
Number of
replications
(r)
.01
Level
.05
2
.908
5
10
3.3.
oi-'t
.n..
sign.ificance ( ex)
.25
.50
.789
.492
.246
.479
.342
.160
.0670
.258
.173
.0754
.0303
Bias as a Proportion of Y
ll
We have seen in Section 2.6 that
Also O:S
~/2:S
1.
Hence Q3/2 is the proportion of Yll that constitutes
the bias of the estimator T.
The following tables show how the relative
bias
varies with
Wfor
our chosen values of r and a.
Note that Q3/2 is also
the proportion of Y that constitutes the bias of the estimator Tij of
ij
~ij for any i,j (i,j
= 1,2).
32
The values corresponding to a = 0 and a = 1 are not shown here.
these two cases, the biases are Yll and 0, irrespective of rand
the estimator T equals
Table 3·3.1.
~11
W, since
'" in the other.
in one case and ~11
Values of Y11 E(~ll-T)
(r=2)
.01
---=."....,~-';--==""=
0
0.2
0.4
0.6
1.0
1.4
2.0
3.0
4.0
6.0
10.0
For
·98.5
.983
.981
·979
·975
·971
.964
·952
·939
·912
.850
.05
.......
.25
--
--~-- ;-...~.'"""--.-:.."...,.~~.
·923
·915
·907
.899
.883
.866
.842
.800
.758
.676
.524
.,..,.,.,,..,...,..,,.,,,,,,,~-
.
·50
..._...
-,,,,~,,,.,..--~,-
...
,.,,,~
.608
.260
.586
.565
.544
·505
.468
.418
·344
.283
.189
.082
.243
.228
.213
.186
.163
.133
.095
.068
.034
.009
~-,
33
Table 3·3·2.
0
0.2
0.4
0.6
1.0
1.4
2.0
3·0
4.0
6.0
10.0
Table 3·3·3·
0
0.2
0.4
0.6
1.0
1.4
2.0
3·0
4.0
6.0
10.0
Values of y~i E(~ll-T)
(r=5 )
.01
.05
.25
.50
·987
.983
.980
.976
.967
·958
·942
·911
.874
·790
.601
·925
·912
.899
.885
.858
.830
.786
·711
.637
.498
.278
.588
.561
.534
.509
.461
.416
·357
.275
.210
.120
.037
.233
.216
.200
.186
.159
.137
.109
.074
.050
.023
.005
Values of y~i E(~ll-T)
(r:10)
.01
.05
.25
·50
.988
.984
·980
·975
.965
.954
·934
.695
.849
·744
.516
.926
·912
.897
.882
.851
.820
·770
.687
.605
.453
.229
.577
.548
.520
.494
.444
.399
·339
.256
.192
.106
.031
.228
.211
.195"
.181
.155
.132
.104
.070
.047
.021
.004
34
From the above tables it is seen that, for fixed r and a, the bias
decreases as 0/ increases from zero.
This is Just a verification of what
we proved theoretically in (2.5.3), viz., that Q3/2 is a monotoni.cally
decreasing function of 0/.
For a given a, the bias may start (at 0/ = 0) at a slightly' higher
level for a higher value of r.
But the rate at which it decreases with
increasing 0/ is higher, the higher the value of r; and, on the whole,
th.ere is a def":1ni.te gain in taking a larger number of replica t1.on.6.
As regards the effect of the level of si.gnificance a; it 1e seen
that as a increases, rand 0/ being kept fixed, the bias decreases.
This,
again, is a verification of the result in Section 2.5(b) that Q}/2 is a
monotonically decreasing function of a.
~elative
bias may be
con~'
sidered moderate (even when 0/ is small) for a value of a like .50 or even
~.
However, the main subject of our investigation is not the bias
but the MSE which will be discussed in the next section.
3.4. Relative Mean Square Er~or: MSE(T)/(rr2 /r)
Formula
(2.6.4) Shows that for fixed r,
a and
f, the MSE of the
proposed estimator T is given by
MSE(T)
rr2/r
= 1 _ Q3 / 2 +
~
i
4
(2Q..
~/2
-
Cl~
~/2
)
(3.4.1)
In this section we shall study the behaviour of this ratio for varying
r, a and 0/.
35
We first note that for a :::: 0, x
:::: 1 (the case that corresponds to
a
11 , the least squares estimator under the nointeraction model), Q3/2 : : ~/2 == 1, for all rand 1jr; hence
taking T equal to
1-l
MS~( T) I
0- /r
a=:: 0
On the other hand, for
taking T equal to I-l
model), Q3/2
:=
ll
a
==
1, x
a
== 3/4 + 1jr /4
==
0 (the case that corresponds to
, the least squares estimator under the interaction
G:>/2 :::: 0, for all rand
MSE( T)
2
0-
identically in 1jr.
/r
I
a=::1
1jr; Ilenee
:= 1
.
We have, therefore, excluded a
following tables, where we give, for r
==
:=
1 from the
2, 5 and 10 and 1jr
==
0, 0.2,
0.4, 0.6, 1.0, 1.4, 2.0, 3.0, 4.0, 6.0 and 10.0, the values of the ratio
for five levels of significance, viz., a:= 0, .01, .05, .25 and .50.
Table 3.4.1.
Values of the ratio MSE(T)/(o-2/r )
(r==2)
ex
1jr
0
0.2
0.4
0.6
1.0
1.4
2.0
3·0
4.0
6.0
10.0
0
.01
.05
.25
·50
·75
.80
.85
·90
1.00
1.10
1.25
1.50
1.75
2.25
3·25
·754
.804
.855
·905
1.005
1.105
1.252
1.495
1·731
2.183
2·993
.769
.821
.872
·922
1.020
1.115
1.252
1.463
1.652
1·966
2.135
.848
.894
.936
·976
1.047
1.109
1.185
1.273
1.324
1·352
1.272
·935
·959
.981
1.000
1.031
1.055
1.078
1.096
1.097
1.078
1.047
36
Table 3.4.2.
Values of the ratio MSE(T)/(~2/r)
(r=5 )
0:
0
1f
0
0.2
0.4
0.6
LO
1.4
2.0
3.0
4.0
6.0
10.0
Table 3.4.3.
.75
.80
.85
·90
1.00
1.10
1.25
1.50
1.75
2.25
3·25
.01
.05
.25
·50
·753
.805
.857
.908
1.011
1.113
1.264·
1.506
1.733
2.123
2.590
.769
.824
.878
·932
1.035
1.133
1.270
1.468
1.624
1.819
1.845
.853
·901
·946
.987
1.058
1.182
1.247
1.272
1.253
1.140
·942
.965
·985
1.003
1.030
1.050
1.069
1.080
1.077
1.056
1.020
1.116
Values of the ratio MSE(T)/(rr2/r)
(r=10)
0:
1f
0
0.2
0.4
0.6
1.0
1.4
2.0
3.0
4.0
6.0
10.0
0
.01
.05
.25
.50
·75
.80
.85
·90
1.00
1.10
1.25
1.50
1.75
2.25
3·25
.753
.805
.857
·909
1.013
1.117
1.269
1.511
1·734
2.098
2.436
.769
.825
.880
.934
1.039
1.138
1.275
1.468
1.615
1.177
1·727
.856
·905
.949
.989
1.059
1.115
1.178
1.237
1.256
1.229
1.118
.943
·966
.986
1.003
1.030
1.050
1.067
1.077
1.073
1.052
1.018
37
3.5.
Comments on MSE
An examination of the values tabulated in SectIon 3.4 indicates how
the MSE of T changes with r, a and
~.
First, for fixed r and a (0 < a < 1), as ~ increases from zero the
quantity rMSE(T)/rr
2
increases for a time, reaches a maximum and then
For small a, this tendency to decrease may not be
tends to decrease.
apparent from Tables 3.4.1-3.4.3, since we have not considered
sUfficiently large
~-values.
follow from Theorem 3.5.1.
However, the fact that this is so will
We first state the following lemma which
is proved in Section 3.7:
For a > 0, a a 0, lim ~aQj(~/2,q,Xa) = 0.
Lemma 3.5.1.
~~CXl
Sijnce
we immediately have from the lemna
Theorem 3.5.1.
For a >
° rMSE(Tla)/rr2 "';>1 as ~ ..,;:.
OD
Secondly, for given a and ~, the change in rMSE(T)/rr 2 is only slight
for a change in r.
Indeed, in the range from
~
=
ratio is almost the same for r = 2 as for r = 10.
that the MSE itself remains constant.
2
almost as rr /r does:
values of
~
° to
~
= 4.0, this
This is not to say
On the contrary, the MSE varies
it varies almost inversely as r.
Also, for higher
the ratio itself decreases appreciably as r increases.
Lastly; for fixed r and
in a (see Figure 3.5.1).
~,
the MSE is highly sensitive to a change
We found in Section 2.6 that for
2 • .
° < a < 1 the
2
MSE is greater than [3/4 + ~/4]rr /~ if ~ ~ 1 and greater than rr /r if
..
e
e
e
1.75
1.50
0:= .25
N
~ 1.25
LLJ
(f)
~
0:::.50
~
0::: I
7~
1.001
I
............
.75
o
2
4
3
5
6
7
8
9
10
VALUE OF Y'
"2; 1
Fi
~ gure./
..
.,,~
..
varJ.8
tolon ",.. . ...-P
::'.'
T'!''''''i:i':/r--- '"T"''+-h
............ 4..c;-(l'~P'
~1'1t
rl'10J:..I·..
·'.~v;_ <;Jf" .;0-~l!c
... .J.J..J..er':~.
;!.,,,'y\4f~J..C~,."·· ..~~"':):..
S....,•..,.
Q"l,.!...." '...
~·C:" ,01(·,;
•.·_.,
, . · .....
(\ .... -l·~h
..
,{.LV
_c=.\
r-"J;
\).J
G")
39
w > 1.
The higher the value of a, the higher generally is the difference
between MSE and [3/4 + w/4J~2/r for W~ 1 but the lower is the difference
2
between MSE and ~ /r on the larger part of the range (l,~). FUrther,
while the maximum regretl
maximum regret in
(l,~)
/ in [O,lJ increases with increasing a, the
decreases with increasing a.
The value of W
at which this second maximum occurs also draws nearer to W = 1 as a
approaches unity.
We can, however, show that this Wmust always be
greater than 5/2.
In fact, using a result from Section 2.5(d), we have
~t
(3W-3)~/2 - (2 w-5)Q3/2 - wQ7/2
> (2w-3)~/2 - (2w- 5 )Q3/2
= (5-2w)(Q3/2- ~/2)
~ 2~/2
if
W~ 5/2
+ 2~/2
,
showing that the point at which the derivative is zero (giving a maximum)
must be to the right of 5/2.
It is thus seen that, on the whole, by choosing a low level of
significance, we reduce the MSE for we(O,l], while we increase the MSE
l/ The
difference
rMSE(Tla)/~2 _ ~n rMSE(Tla)/~2
may be called our 'regret' resulting from our choice of a. It is a
function of the parameter vector ~ (in the present case of Walone).
40
for W£(l,oo).
On the other hand, a high level of significance, by and
large, leads to a slight increase in the MSE for w£[O,lJ but to a
comparatively big reduction for W£(l,oo).
'Enis would seem understandable
from the nature of the power function of the test we are using.
Note
that this power function is monotonically increasing, as a function of
either Wor a.
Because of What we said in Section 1.1, w may be called
the 'proper estimator' for W~ 1 and w + z the 'proper estimator' for
w > 1.
The probability of choosing the proper statistic then equals
l~(value
of the power function) for W~ 1 and equals the power itself.
for W > 1.
The monotonic character of the power function, therefore,
implies that the probability of choosing the proper statistic is smaller
(larger) in [0,1] and larger (smaller) in
(1,00),
the larger (smaller) the
value of a.
3.6.
Optimal Choice of a
We have seen that when the test procedure suggested in (2.2.5) is
2
used, the ratio rMSE(T)/~ depends on W, rand a.
The non-centrality
parameter, W, is generally an unknown quantity and hence is beyond the
experimenter's control.
The number of replications, r, also is in most
cases determined by the resources available to the experimenter.
However, the level of significance of the preliminary test, a, is
completely within his control, and he may choose any a that he considers
appropriate.
low level.
The aim, of course, is to keep the ratio rMSE(T)/cy
2
at a
The difficulty stems from the fact that an a that is good
41
for some 'ji-value may be bad for some other.
As always in decision
theory, the problem is how to cope with this non-uniformity.
We may
suggest four approaches which are discussed below:
(a)
Bayes Approach:
Suppose that the experimenter has some prior
knowledge of the likely magnitude of
* and that this knowledge can be
mathematically represented by a distribution function, say
s(~')
0
In
that case the value of ex (or, equivalently, of x ) that minimizes the
a
average
J'
eo
ER( 1jr) ;:
Re,+,)
o
d~ ( ~! )
where
may be said to be optimal for the purpose of the
prelimi~ry
test.
We illustrate this procedure of cllOosing ex with a gamma-type pri.or
distribution:
13 > 0, s > 0,
1
Ijr ~ 0
00
ER(V) = 1 - 4 z
m=o
r( sim+l)
mt
13
s
r(S1m+l)
m!
0
Ix (m+5/2,~/2)
ex
The derivative of ER(o/) w.r.t. x
a equals
(~~)
1
- 2
m
Q)
E
m==o
Equating this to zero and making some simplifications, we find that the
optimal x
a (hence the optimal a) is given by the equation;
x
2 Fl (\}/2-+5 /2, s+1;5 /2; 1 ~)
:::: 0
where 2Fl is a hypergeometric function.
For given
may be solved by using some method of iteration.
~
and s, this equation
Since our choice of
the prior distribution was rather arbitrary, we shall not pursue this
matter.
However, some general observations may be made without going into
the mathematics of a Bayes solution.
knowledge indicates that the probability is very high that 0/
should choose a small a.
small, then an
,
Thus, if the experimenter s prior
~
1, he
On the other hand, if this probability is
a near to unity ought to be selected.
(b)
Minimax Approach:
In the absence of any prior information
regarding the likely magnitude of ljr, one may like to guard against the
worst and choose ex in such a way that
The result (2.6.6) then shows that this
is a minimum.
a = 1.
t
t
minimax a
is
In other words, according to this approach, we should use the
estimate
~ll'
irrespective of the sample observations.
out to investigate if we could improve upon
interesting to us.
~ll'
Since we had set
this result is not very
Of course, if the nature of a concrete problem calls
for a choice of a loss function leading to (3.6.5) and for the
application of the minimax criterion, then the estimator
~ll
is the
obvious choice.
(c)
Minimizing the Ll-norm of the Regret Function:
Consider for
each ljr the regret
Min rMSE1/r(Tlex) =
a
say.
cr 2
s(ljr,a)
,
(3.6.6)
Evidently, this regret function equals
s(V,ex)
_ {rMSE( TI a)/cr
rMSE(Tlex)/cr
2
2
- (3/4 +1jr/4)
for
ljr£[O,l)
- 1
for
Ve(l,ClO) •
An approach that might seem appealing would be to choose an ex that
minimizes the total area
ClO
J
o
s(ljr,ex)dljr
44
under the regret function curve!/.
However, the following theorem
shows that this approach (of minimizing the L ~norm of the regret
l
function) again leads to the estimator ~ll' regardless of the
observations taken.
Theorem 3.6.1:
The level of significance a for which
J s(~,a)d~
ClCl
o
is a minimum is
Ct
= 1.
J s(~,a)d~
ClCl
Proof:
o
1
J
=
{rMSE/(j2
~ (3/4+~/4)} d~
o
t! {(l-~
)(1-Q3/2) +
o
~(Q3/2- ~/2)} d~
ClCl
+
tJ {(~-1)Q3/2
1
1
=
+
1
1
=
J (rMSE/(j2_1)d~
ClCl
~(Q3/2- ~/2)} d~
ClCl
t] (l-~)d~ + t]
o
+
{(~-1)~/2 + ~(Q3/2- ~/2)} d~
0
liThe difference between our approach and that of Huntsberger (1955)
in his Theorem
2 is to be noted.
Our approach would correspond to
t
'
Huntsberger s if we minimized
ClCl
J:
(~~ l)d~
(j2
= -g1
1 co [
+ 2' E
(2m+l)Ix (m+3/2,\)/2)
m=o
0;
+ 2(m+l){IX
(m+3/2, \)/2) - Ix (m+.5/2, \)/2)}]
0;
0;
>1
- '8 '
the equality holding iff
Xo;
= 0,
i.e., iff
0; ~
1.
A remark similar to
the one made at the end of Subsection (b) pertains to this result.
(d)
Minimax Regret Approach (minimizing the Chebyshev norm of the
regret function):
Another approach which might be suggested, and which
would seem quite appropriate in the absence of any prior information
regarding the variability of 'Ir, would be to choose
0;
in such a maLlner
as to minimize
An exact determination of this
t
minimax regret
d
presents some problems
because of the complicated expression for the regret function and has
not been attempted.
As for a promising suggestion for numerical work,
see the end of Section 3.7.
Anyway, it is seen from Tables 3.4.1-3.4.3 that this minimax regret
0;
lies between .25 and .50.
A value less than .25 makes the maximum in
(l,co) too large, while a value greater than .50 increases the maximum
in [0,1] to too high a level.
Apart from any theoretical criteria, it would appear from the
tables that a value of
0;
between .25 and .50 (perhaps even closer to .50)
46
also effects a satisfactory degree of control over the MSE throughout
the range of
~~values.
Moreover, for such a value of a the relative
bias of the estimator is also only of a moderate magnitude.
Note that in our discussion we have been concerned with a test for
the hypothesis
against the alternative
However, the usual tables refer to tests for
H:
o
~::;:
a
against the alternative
Noting that in both cases the critical regions have the form
4rz 2/ s 2 > C
and that the power function for such c critical region is monotonically
increasing in
~,
it is possible to convert the values of a related to
our test into levels of significance related to the usual ANaVA test.
This requires only the determination of the value of the power function
at
~
= a when
the value at
~
=1
is a.
Since the power function of our
test is
it means that we have to find the value of
when
We show these values corresponding to ex:;:;: .25 and ex:;:;: .50 in the
following table, for the set of r-values considered in the tables
for bias and MSE.
Table 3.6.1.
Level of significance for ANOVA test corresponding to
ex:;:;: .25 and .50
r
.25
·50
2
.12
·32
5
10
.10
.10
·30
·30
Thus we may conclude that for r between 2 and 10 (and likely beyond)
the preliminary test may be conducted as the usual ANOVA test, however,
at a level of significance between .10 and .30, most likely closer to
~
than to .10.
This would seem to corroborate the suggestion made
by Anderson (1960), p. 52, that the preliminary ANOVA test be made at
about the 25% level.
One thing is clear:
the customary levels of
significance, .01 and .05, are completely unsuitable for a TE procedure
in the 2x2 case, unless there is prior information that
likely to lie in [0,1].
~
is very
For such a value may reduce the MSE appreciably
in [0,1] but will make it inordinately high in (1,=).
48
3.7.
An Alternative Formula for MSE
The formula for the MSE of T in terms of
suitable for many
purpose~.
tr~e
Q function will not be
One would like to have an expression for it
in a closed form rather than in the form of an infinite series.
In this
section a general method is suggested aild the actual formula obtained
for the special case of r
= 2.
The ratio rMSE/a 2 can be written in the form:
(Xl
____1_
4J2ic
where
u
eXP(-~/2)J'exP(-u/2)COSh(J1U)J~{I
= 4rz 2 /a2 ,
Y(X,~/2)}dU,
0
= ~u/2Fa
x
and
y(x,~/2)
is the incomplete gamma
function:
x
J exp(_t)t~/2-1 dt/r(~/2)
o
and ,ff"
= J4r
yll/a .
Now putting s =
exp( -~ /2)
J
ell
exp( -u/2)
~/~/
Sinh(J~){1
o
ell
= 2r(S)![exp{o
!/~/2 = 2(r-l),
(Xl
~(JU J~ )j
=
an even integer.
f' (
-nf "2l( Ju + J~ )2}] ,exp oot)t s~·l dt
ex!1..=
SU/F~
'j"
+ J?/ )"'}
J
(2~1)~Jexp(= ~U)dU
a
00
==
rfu ][ex {= ~ ((y=1)2
+ 2Sy2/ Fa )}
p
=
exp{- ~((Y+l)2 + 2Sy2 /F'a)}]
o
r
x I
F
,
\r;c~]
a
I
by tb.e slibst;ltuti.on
to
00
rts).IexJ{~ ~
-
()'l
[(~)S-ly2S-1
if_
a
{(Y·"1)2 + 2Sy2/ F
+
'0
0
'0
0
+ (s= 1)3
J
+
(S~1)(P)S'~2 y2S~'3
ex
dy
In a si.milar way, we have
00
exp(~u/2)
eX P(-o//2U
COSh(,,(\iiU }JU{l _. I'(X
j
\l/2)}
du
o
c
X.
.
[(~.!
i ,)8=1 y28 + (S_1)(~FS')S-2 ,,'),28=2 + ."
t'
,
ex
ex
1) I Y2Jdy,
'"" + ( S-.
50
Now only even powers of yare involved i.n the polynomial, and so the
right=hand side of equation (,3. r(.4) equals
r~/~
S
J'»
....r
J
! r( y-1 )2.+ 2.-,y2/
)S-1y 28
= FrvL
,4 [(21
F
a
ex~_- 2
t
rHSE/u'- .. 1 +- ....."o •• "C""·--·f~-~(--~s·~) exp {
:?
=
~
(1 ~. x()}
j
+ (8-1
-
)(~F )S-2 y2S-2
0;
00
"'0:;
(3.7.6)
This can now be simplified by noting that
o if n 1s odd
exp( -
1
2'
2
n
'ljrt /xo;)z dz :;:
(-1-)-(n+1)/2 f(n+1)
2X
a
2
if n is even.
Thus, e.g., for tIle case r - 2, S
= 2,
formula (3,7,6) gives
51
Note that, for small v and for some analytical purposes, a formula
of this type can be more easily used than the infinite series in
e.g., i t gives a direct proof of Theorem 3 5.1.
(2.6.4);
(We have retained the
earlier proof because it hardly needs any modification in the general
context of a p x q experiment, and the generalization of the present
formula is bound to be cumbersome.)
For numerical purposes too, it
52
would be more convenient to use than formula (2.6.4), at least with the
usual number of replications, since in that case it would require the
evaluation of considerably fewer terms.
Thus it is much easier to use
in locating the point on the 1jJ'-axis for which the MSE is a maximum or
in the
detel~nation
of the minimax regret a.
An iterative procedure
for determining the minimax regret a is suggested here.
First note
that for each a the regret function has two local maXima, one for
o/£[O,lJ and the other for 0/£(1,00).
Our problem is then to determine
the a for which the larger of the two is least.
Supposing we draw a
graph for each of the local maxima as a function of a, we are to find,
proceeding from a
=1
backwards, the abscissa of the first point of
intersection of the two curves.
The idea of proceeding from a = 1
backwards is suggested by the results from Tables 3.4.1-3.4.3 which
seem to indicate that the maximum in (1,00) is a monotonically decreasing
function of a.
3.8.
Appendix:
Proof of Lemma 3.5.1
From (2. .2), for a > 0, 1. e ., for x
monotonically increasing function of q.
integer contained in
a < 1, Qj(0//2,q,x a ) is a
If qo
= [q~l],
the highest
q+l, we then have
(3.8.1)
Now
53
the change of the order of eummation and integration being admissible
by virtue of Theorem 27B in Ha;1.mos (1950).
co
I:
m=o
(1lr/2)m
ym+j -1
m~
B(m+j,qo)
=
1
00
I:
r(qo) m=o
i
_
1
- r(qo) m=o
(t/2)m
m+j-1
--- - • (m+j)(m+j+1) ...... (m+j+q -l)y
m~
0
q
(1/r/2)m [~
m~
. dY~
qo
_1
• ~
- r(qo)
dy 0
1
where
i
= 0,1, •••
Hence
where
m+j+q -lJ
Y
0
{y J +qo-1 exp(1jIy/2)}
= r(qo)
with
Also
••• q
o
•
qo
Since .E b (ljr/2)i is a polynomi.alin 1jr, tile right-hand side of
i
i=o
(3.8.2)
80
-i>
0, implying that Qj(t/2,q,xo;) also....;> 0, as ljr ......
does ljra Qj (ljr/2,q,Xo;) for a > 0.
0)
,
and
55
4.
4.1
THEORETICAL ASPECTS OF pxq CASE
Notation
As in the 2x2 case, here also we shall denote bya(O ~a ~ 1) the
level of significance at which any preliminary test is made.
T
ij will
denote the estimator of ~ij' the population mean for the cell (i,j).
This T
ij
equals
and equals
-
~ij
~ij
in the case of significance at the preliminary test
otherwise
In line with
(2.1.1), we shall write
-
w.. .y
1.J
+ cyi. - y-
-
and
-
) +cy. - Y )
.J
(4.1.1)
-
- - y . +Y
Z.. • Yij - y.
1..
.J
1.J
wij is the same as
-
~ij
~ , the least squares
and Zij is the same as 1
ij
estimator of 7 under the interaction model (1.1.2).
ij
With (1.1.2) supposed to be the appropriate model for our experiment, we shall denote by s~ the interaction mean square and by v
d.f.
The sample error variance will be denoted by s
by v.
Thus
Vl • (p-l)(q-l),
v • pq(r-l),
2
l
its
and its d.f.
(4.1.2)
and
(4.1.3)
56
4.2.
Estimation of a Single Cell Mean
When we proceed to the case of factors, A and B, at least one of
which has more than two levels, the problem of the estimation of any
particular cell mean and that of the estimation of all pq cell means
simultaneously are to be distinguished from each other.
The first
problem is discussed in the present section mainly for the sake of
completeness.
It is not to be expected that the need for it will
often arise in practice since usually one will conduct a different
type of experiment if one is interested only in one treatment combination.
As we stated in Chapter 1, we have taken low MSE as the criterion
of a good estimator, and we are making a choice between the two sets
of ~stimator~ ~ij and ~ij,on the basis of this criterion.
Looking at (1.1.12), we see that, when paq.e, the procedure for
estimating a cell mean is the same for all four cells.
In fact, in
this case, since '1, ,- + '1 , the hypotheses are the same for all four
11
lJ
cells, and since z, ,- +- z, " the corresponding tests of significance
lJ
lJ
are also the same for all four cells. For the same reasons, the
simultaneous estimation of all four means follows the same procedure.
However, the above results do not hold when p > 2 and/or q > 2,
and so the estimation of a single cell mean and that of all cell means
simultaneously will then be entirely different problems.
Suppose our problem is to estimate a single cell mean
,.."
,.
~",
lJ
Referring back to (1.1.12), we would choose either ~ij or ~ij' according
as
2
< ra
(p-l)(q-l)
pq
57
or
2
1' •.
~J
2
if 1' •. and CJ
~J
>
i
(p-l)(q-l)
pq
r
were known quantities
,
As these are unknown, we make our
0
choice on the basis of a test for
2
H :
o
pqr')' ij
<
1
2
(p-l) (q-l)CJ
against the alternati.ve
2
pqrl' ..
>1
~J
2
(p-l)(q-l)CJ
The liMP invariant test of level a for H , under appropriate transo
format ion groups (vide Lehmann (19.59), Chapter 7, Section 5), has the
rejection region
2
pqr z ..
---..:;~;u.J--""2~
> Fa (1, v)
,
(p~l)(q-l)s
where
Fa(l,v) is the upper a-point of the non-central F distribution with
(l,v) ~.!o and non-centrality parameter
pqr I'
2
<
•
_ _ _":;~;olo'J_~
= 1
(p-l) (q-l) CJ2
The problem and the expressions being similar to those in Chaptern 2
and 3, we can say off-hand that the findings of those chapters in essence
hold here also, i.f we bear in mind that here
2
pqr I'ij
replaces
(p-l)( q-l) CJ2
2
2
1jI • 4r1'11/CJ and v • pq(r-l) replaces v .4(r-l).
58
4.3. Simultaneous Estimation of Cell Means
When our problem is to estimate all pq cell means simultaneously,
we need some method of combining the pq MSE's to obtain a single criterion
of estimation.
less arbitrary.
This mode of combining the MSE's is bound to be more or
We shall consider weighted averages of MSE's.
This
means, in effect, that we have in mind loss functions of the form
p
C E
q
E
i;a1 j =1
2
w.. (T .. = 1-1 •• )
IJ 1.J
.1,J
. 1=) , w.. > 0 and
where c > 0 may depend on tb._e parameters
.
- IJ
p
E
q
E
i-l j-l
w.. • 1. The wieght w.. will then indicate the importance of
~J
~J
the cell (i,j) in the whole set.
In a real-world situation, it might
be the proportion, among all users of the factors A and B, of those who
use'the combination
(A.B.). The risk function corresponding to (4.3.1)
~
J
is the weight,ed average of MSE's (to be denoted by AMSE):
2
q
E
j-l
except for the multiplier c.
w.. E (T .. - 1-1 •• )
~J
lJ
lJ
,
Here T.. is the estimator of Il.. under a
~J
~J
given estimation procedure.
The minimization of the AMSE may then be taken as the basis of a
choice between rival estimation procedures.
For the least squares
procedure under the full model,
• t't ij
11
In the sequel c will depend on a (see Section 5.5) alone.
59
On the other hand, for the least squares procedure under the nointeraction model,
T;J'
... • Y +
•
(Y. - Y ) +
(y .-
~....J
Y)
Il ij
Now
AMSE«(!)
•
E E w. . E(~ .. - Il .. )2
•
E E w. . cr /r
i
j
~J
lJ
lJ
2
.
~
.
J
2
a/r ,
D
lJ
while
2
AMSE ('il') • E E w. .
i
j
lJ
{~
[1 -
(p-l)(q-l)
pq
J + 71.:.}
J
(p-l)( q-l}] + EEw .. 72 ·
pq
i j l J iJ
(4.3.4)
Formulae'(4.3.3) and (4.3.4) suggest the following TE procedure:
(1)
First, test the hypothesis
2
H : E E w.. 7 ..
0
i j 1 . J lJ
2
cr
< r
(p-l)(q-l)
pq
against the alternative
2
H : E E w. . 7 ..
l i j lJ lJ
(2)
r
(p-l)(q-l)
pq
In case significance is found in step (1), estimate each Il .. by
the corresponding
11
>
2
cr
lJ
~ .. j
lJ
otherwise estimate each Il . . by the corresponding
lJ
In this work we have only considered the choice between neglecting
the interactions for all cell means and including the interactions for all
cell means. A question for future research would be how to allow for cases
where interactions may be neglected for some cell means and included for
others.
60
However, a satisfactory test for H against H (for unequal weights)
o
l
Even the heuristic test, !iZ., the one based on the
is not available.
statistic
2
2
r. r. w.. z .. /s
pqr
•
1.
•
J
1.J
1.J
presents such a distribution problem, especially because the distribution
will not depend solely on
2
2
r. r. w. . I' . .fa
l.J J..J
as would make it meanIngless to be too fae LJd:itious on this poInt
In
step (1) we shall, therefore, be concerned with a test for
2
H' : r r. r. 1' •. /a
o
•
1.
•
J
2
1.J
5
(p-l)(q-l)
(4.3.5)
> (p-l)(q-l)
(4.3.6)
against
H{:
r r. r.
i
j
l./i
1.J
In this way, we shall make the preliminary test independent, as i t were,
of the choice of the weighting system.
Note that H~ and H{ correspond to the traditional ANOVA test for
interactions, except for the aspect of matert.al significance.
(In
traditional ANOVA we test the hypothesis
2
2
rr.r.w .. /a =0
i
j
lJ
against the alternative
2
2
r r. r. w. . /a
i
j
>0
. )
1.J
Although we shall apply this simpl.ification to the test procedure in
step (1), we shall not apply it to the loss function to be used in
assessing the AMSE of the estimators used in step (2); there we shall
still allow the weights attached to the different cells to be unequal.
61
A£ we shall see, the properties (the bias and AMSE) of the
estimators T will then depend on two parameters,
ij
\IF • r E E
i j
'Y~j / l-
and
2
2
'ir* .. pqr E E w.. 'Y .. / a
i
4.4. Some Relations between
j
J.J J.J
(4.3.8)
\IF and \IF*
Obviously}
2
max 'Y ••
. .
J.,J
J.J
Since
2
2
'iF > r max 'Y . . /a
-
..
].,J
J.J
we then have
(4.4.1)
In case either p or q is 2, the inequality
(4.4.1) can be improved uponj
for then there will be two pairs (i,j) for which
(4.4.2)
we shall have
From
(i)
'Y~j'S are a maximum, and
(4.4.1) it follows that:
(4.4.3)
if 'iF .. 0, then 'iF* .. OJ
and
(ii)
if
'iF*
~ 00,
then 'iF
~
(4.4.4)
(X) •
Also
where
w..
and
2'
'Y
=
~ ~ij/pq
= l/pq
]. J
2
E E 'Y ij/pq
62
We have, therefore, the following rough interpretation:
If the weights are all equal (. l/pq), then
2
larger weights go with larger 11J's , then
w* • W.
w* > W.
2 ,
If generally
On the other hand,
i f generally larger weights go with smaller liJ s, then
w* < w·
4.5. Preliminary Test Procedure
In choosing a satisfactory test for H~ against H{, our purpose
is, af course, the minimization of the AMSE (see equation (4.3.2))
uniformly over the parameter space
However, for the same reasons as
those mentioned in Section 2.2, we shall be using a test that is best from
the point of view of power rather than of AMSE.
Referring to Lehmann (1959), we see that invariance considerations
demand that a test for HI should be based on the statistic
o
2
r E E z.
i
j
'/Vl
~J
?
s-
1
.--
l
The UMP invariant test of level a has the rejection region
2
1
s2
>C
where C is so chosen that
(vide Lehmann (1959) and Hodges and Lehmann (1954)).
2
2
has the non-central F distribution with v and v ~.£.
l
and with non-centrality parameter W. Let us denote this statistic by
Now sl/s
F(vl,V;
W).
or simply Fa'
Then C must be the upper a-point of F(vl,v;v ):
l
63
As in Chapter 2, it will be convenient to deal with the statistic
1. +
say, rather than with
?2
SJ./S,
2
v1 s1.
= x,
and with
=x
a
1 + v1 F
-v ex
rather than
\tll th ji'
ex
SInce the .probabi1ity element of l!'(v ,v;1\f) is
l
v
00
exp (-1jr/2)
1
E
B(iv1+ m,iv)'
mIlO
~(vl-2)+m
(ylf)
d(~i.f),
v
i(
v +v )+m v
l
l
(l+- f)
v
(4.5.6)
(see, ,~:~'-' Graybill (1961», that of x is
exp('!r/2)
0;
L>
mllC
Hence x
a
!:~.:,
liI/2)rn
~
,v +m-1
x 1
1
m.
,v-1
- -
(l,..x)
,(4.5.7)
O<x<l.
is given by
by
Q
V1 / 2
( v 1 / 2 ' v /2; x )
a
using a notation. introduced in Section 2.4.
= 1-0
,
,
64
4.6.
Distribution of the Estimator of a Cell Mean
Considering any of the pq cell means, let US study the nature of
the distribution of its estimator, the estimation being based on the
results of the preliminary test mentioned above.
gen~lity,
is
~11
Without any loss of
we may consider the case with i-j-l, for which the cell mean
and the estimator is Tll ·
is defined by:
Note that T
ll
if
F< F
-
0:
(4.6.1)
if
~l'
where
T
ll
zll are as given by (4.1.1).
F>F
0:
,
The probabiiity element of
is then
gl(tlF <0F: ) Pr (F <0F: )dt + g2(tlF > F0: )Pr(F > F0: )dt,
where gl and g2 are the conditional density functions of w
11
(4.6.2)
and w + zll
ll
respectively.
Now w and zll are independently distributed, the former as
ll
2
N(~ll- I'll' ra
[1 _ (P-l)(q-l»))
pq
and the latter as
2
N(l'll'
~
(p-l)(q-l) ). Asmthe test statistic F • s2 / s2, we know that
pq
1
HF/i
has the (central) x
2
distribution with v d.f. and is distributed
independently of w and zll'
ll
2
Before taking up sl' let us state the
following generalization of the Cochran-Fisher theorem, first proved by
Madow (1940).
We state it in the form given by Graybill (1961):
Theorem
4.6.1: 11 Let Yl' Y2 ,··· ... , Yn be independent random variables,
Y being distributed as N(~ ,1), and let
r
r
l.'l.
=
k
E
l.'A.l.
i-l
].
Then a necessary and sufficient condition for
l.'A.l.
].
2
to be distributed as a non-central X with d.f. n i and non-centrality
parameter A. ,wbere n ,1.S the rank of A. and
1 1 1
A.s:
II'A.f.J.
,
].
I::.
),_.
and for
y'A.y
-
1-
(i~1)2,
,k) to be mutually independent is that
k
En. =
i=l ].
n .
(4.6.3)
It will be convenient to consider
2
?
"V1si/ a
to be composed of two parts:
2
pqr zll
u ==
(4.6.4)
(p-1)(q-1)l
and
u
==
(4.6.5)
Here l.' = (Y1'Y2"" ... y n ), l!:,' = (~1'~2"" . "'~n~ and
l., l!:, are' the corresponding column vectors. A. (1-1,2, ... ,k) is an rum
symmetric matrix.
].
11
66
It is known
th~t vls~/a2 has the non-central X2 distribution with vl ~.!.
and non-centrality parameter
2
2
1: 1: '1 . . /a
r
i j
l.J
•
W•
2
On the other hand, u has the non-central X distribution with 1 d.f.
and non-centrality parameter
2
pq r'1 11
(P-l)(q~l)i
=
W (say).
(4.6.6)
l
By making a suitable orthogonal transformation em the vari.ables
Yijk(i=1,2, ... ,p; j=1,2, ... ,q; k=1,2, .. "r), we can then show from
2
Theorem 4.6.1 that U has the non-central X distribution with vl-l
~.!.
and non-centrality parameter
and that U is independent of zll .
r.
Also both zll and U are independnt of wll and s
~
Hence (4.6.2) may be written as
hl(t)
+
ftf
h2(zll)h3(U)h4(s2)dZlldUd(s2)dt
SSS
h5(t,zll)h3(U)h4(s2)dZlldU d(s2)dt
C
R
•
h l (t)
JJJ
~(zll)h3(U)h4(s2)dZlldU
d( s2)dt
R
+ h (t,.)dt 5
SSS
h5(t,zll)h3(U)h4(s2)dZlldU d(s2)dt
R
• h (t,.)dt +
5
.fff
R
[hl(t) -
b;
(tlzll)]h2(zll)h3(U)h4(s2)
dZ
2
ll
dUd(s )dt
(4.6.8)
67
Here h , h , h and h 4 are the density functions of wll,zll' U and s
l
2
3
= wll + zll and zll'
and h * the conditional
5
h (t,zll) is the joint density of t
5
marginal density of t
= wl +
zll'
2
h (t,.) the
5
density of t
given zll; the region of integration R is defined by
R
={(Zll'U,
C
and R ~s t4e complement of R.
s2)lu + U ~
2
vls
2
cr
Fa}
By reductions similar to those in
Section 2.3 the above expression may be put in the form:
+
JJJ [ (23!
2
p+q-l
pq
E.... )r
t
exp
(~
2
(t-~ll + 7 11 ) /2.
p+q~l
pq
2
E....)
r
R
p+q-l
- (23! .
pq
2
E....)r
t
2
p+q-l
exp (- (t-~ll+ 711 ) /2 . pq
2
. ~ )J
x
4.7. Expected Value and MSE of the Estimator of a Cell Mean, and AMSE
From the above density function, we obtain, putting
(k is a positive integer),
and
These results afe analogous to those in Chapter 2, except for the
variate U in
(4.7.1),
wh~ch
was zero in the earlier case.
68
Now using \111' \11
2
and \II, as defined in equations (4.6.6), (4.6.7)
and (4.3.7),we make the following reductions:
x
m!
E'or k=l, terms under tbe summation sign corresponding to even values
of m vanish, and
J
80
l - exp (-\IIl/2)(2J1)-tSJS exp(-u/2)
~
(2m+l) !
R
x
h3(U)h4(S~) ~ dU d(S~)
/u
- I'n exp (-\II1/2)
SIr
R
~
(\IIl/2)m
mao
m!
Also (cf.(4.6.7»
CD
h (U) - exp (-\112/2) ~
3
nao
Hence
CD
Jl -
I'll exp( -*/2)
CD
r.r.r
~ ~
R mao n=o
If we put
U
then this reduces to
I
•
U
+ U ,
G(~,
m+
~,
U)h (U)h (s2)dU dU
4
3
d(s~
('!r/
p.,
2)P
1
v
2' P +
'2
+
~)
,
1 + 1,U ')b 4 (s 2)dU' d(s2)
G (2"' P+ '2
('!r/2)P G(l
p!
GD
:o:l
exp( -'!r/2 ) 1:
11
p=o
Y
,{;Jr/2)
pi
P
v
1
I x (p + 2 + 1,
a
in a manner similar to that in Section 2.4.
vl
1, v
1 U/)G(-2
,
~s2
(]
The validity o;f the change
of the order of integration and summation follows from Theorem 278 in
~lmos (1950).
When k.e, terms in (4.7.4) that correspond to odd values of m
vanish and so
h (U)h ( s2) du
4
3
dU d( s2)
jU
2
v (]
1
.pqr
exp( -'!r 1 / 2 )
fif
00
1:
mllQ
3
('!r1/2)m(2m+1)G(~, m + 2'
m~
u)
)
2'~·'
70
and on applying the same type of algebra as we used for J , we find that
l
this equals
2
("'/2)P
1
vl
2
p! - G( 2' P + ~ + 2 , U') h 4(S )dU'
PIIIQ
+ rl 1
00
~ -~~
J.
2)
d(~
co
exp( -\jI/2)
E
PIIIQ
Using the symbol Q, defined in
(2.4.5), we may write
4.7.7)
and
(4.7.8)
SUbstituting these in
(4.7.2) and (4.7.3), we have
71
and
~ 1)
ell
E (T 1-
2
[Vl
= ~ 1 - ---Q
2
r
pq
2
1 11
]
. + ----2 (2Q
- Q
) (4.7.10)
vi+l
a
vt +l
v~+2
r
The expectation and MSE of the estimator of any other cell mean
Generally, for any ~ .. (i=1,2, ... ,p;
lJ
j==1,2, ... ,q), whose esti.mator is T,., we have
lJ
have similar expressions.
Ee(T. ,)
l,J
and hence
p
2
q
= r.
r. w, ,E (T, . i=l j==l lJ e lJ
i-L, .)
lJ
2
== ra [1 - (p-l)(q-l) Q
pq
Vi
~
where
W*
4.8
General Observations
+
+1
L
pq
(Q
V1.
~
~
+1
)J,
Q
vJ..
~
(4.7.n)
+2
is as defined in 4.3.8).
Note that expressions (4.7.9) and (4.7 11) are similar to the
corresponding expressions in Chapter 2, viZ., (2.4.9) and 2.4.10).
The points of difference are that now we have
3/2 and
of Q
"5/2;
'Q
Vi
(p-l)( q-l) Ipq instead of 1/4, and
and Q
+1
w*
Vi
instead
+2
as well as W.
fact, the expressions in Chapter 2 may be looked upon as special forms
of (4.7.9) and (4.7.11), for in the 2x2 case
(p-1)(q-l)/pq :: 1/4 ,
In
72
and
Hence the observations made in Section 2.6 have a straightforward
generalization to the present situation.
Thus for 0 < a < 1,
(4.8.1)
and the bias tends to zero as
tends to
~
00 •
Also
_ {(l _ (p-lliq-l~
pq
;::
(p-l).(q-l)
pq
[1 _Q
=-!..-[(p-l)(q-l) pq
> 0,
1 + L*
vi + r
pq
~*] [1 -
Q
Vi
J
[2 Q
-
*
+1
pq
Q.
vi + 1 vi
~q
+
-1-* }
) +
'- lJ
+2
[Qv
- Q
]
~ + 1 vi + 2
if 0 <
~ * < (p-l)(q-l)
....,
-
(4.8.2)
and
_ 1;::
+
L*
pq
> 0, if
~* ~
This means (cf. equations
o <-
plq
[~* _ (p-l)(q-l)]Q
[Q
_ Q
v
t
+ 1
v!
+ 1
]
v +2
t
(4.8.3)
(p-l)(q-l)
(4.3.3) and 4.3.4)) that in the domain
~* < (p-l)( q-l), the set of estimators T.. is necessar,ily worse
-
lJ
73
~
than the set of estimators lJ.ij'
On the other hand, in the domain
1jr* ::: (p-l)( q-l), the set of estimators T,. is necessarily worse than
IJ
the set of estimators IJ. ..
IJ
For 1jr*
=0
,
AMSE g
2
(J
/r
=
(p-l)(q-l)
1-
(4.8.4)
Q
v~ + 1
Because of the properties of the Q function (see Section (2.5)), this
means that the ratio at \jr * :::: 0 is smaller, the larger the value of x
ex
for a particular 1jr.
However, with varying 1jr (but \jr * :::: 0), for a given
x ' the ratio increases as
increases .
\jr
a
4.9
,
An Alternative Approach
For the simultaneous estimation of all cell means, we have
considered above the minimization of the AMSE as our goal.
An
alternative approach would be to choose a procedure that keeps the
largest of the pq MSE's
Ef\ (T .. - IJ. .. )
~
IJ
2
( i :;:1 , 2, . . . , p; j =1 ,2 , . . . , q )
IJ
at a low level.
A
Now, for the estimators lJ.
A
all the MSE's are equal:
ij
MSEf\(IJ. .. )
IJ
'(1
=
2
(J
(4.9 1)
/r
for (i=l,2, ... ,q).
On the other hand, for the estimators ~ .. the ~gest MSE is
IJ
Max
i,j
[ 1 _ (p-l)(q-l) ]
pq
The expressions
(4.9.1) and (4.9.2) suggest the following TE
procedure:
(1)
First, make a test for the hypothesis
2
H '.
o
(p-l) (q-lj
pq
Ma x I 2.. < E. . l.J - r
1.,J
against the alternative
u~
2
1 ij
L"I':lX
>
lE-l)( q-l)
pq
r
i,j
(2)
2
(J
In case H is rejected in step (J. L take
o
1-1 ••
lJ
as the
estimates of ~ .. j otherwise take~.. as the estimates.
lJ
lJ
However, here too we have the problem of developing a suitable
test for H.
o
Even the heuristic test given by the rejection region
Max
i,j
2
l.J
z ..
>
C
raises too difficult a distribution problem, especially so because
the distribution of the statistic
Max
i,j
z
2
ij
is not dependent solely on the parameter
2
Max I ..
. . l.J
1,J
75
5.
FURTHER STUDY OF pxq CASE WITH
SOME NUMERICAL RESULTS
5.1
Percentage Points of x
In the pxq case, with the preliminary test given in Section 4.5,
the study of the bias for a single cell mean will go along the same
lines as for the 2x2 case.
Of greater interest is the variation of the AMSE with varying p
and <1, varying \jr and \jr* and also
w~th
varying rand 0:.
For the numerical part of the work, we have considered two pairs
of values of p and q,
vi~.,
pr;cq=3ocand p=q=5, two values of r, viz.,
r-3 and r=5, and three non-trivial levels of significance, viz., 0:=.25,
.50 and .75.
The critical values' of the test statistic x corresponding
to 0:=.25, .50 and .75, for the chosen values of p, q and r, are shown
in Table 5.1.1.
Here, again, the two trivial values of 0:, ViZ., 0:=0
and 0:-1, have not been considered, since the corresponding percentage
points are
xo
=1
(5.1.1 )
and
irrespective of p, q and r(r ~ 2).
5.2
Bias of T as a Proportion of 'Y
iJ
iJ
From (4.7.9) the estimatcilr;T of the cell mean
ij
E(TJ.' )
J
= ~ij
II
-
~'iJ Qv
...1
+1
2
(1
2' 2Y, x0: )
l-1
ij
has expectation
77
for the 2x2 case by 4 and 16, respectively, to obtain comparable
w-values for the 3x3 and 5x5 cases.
Table 5.2.1.
w
0
0.8
1.6
2.4
4.0
5.6
8.0
12.0
16.0
24.0
40.0
Values. of 1 -1
.. E( lJ. ii - T.. )
~J (p.q
Wt3) ~J
a •. 25
.661
.876
.830
.783
.734
.635
.540
.410
.243
.135
.035
·.002
Table 5.2.2.
r • 3
a • .50
·589
·522
.460
.352
.265
.168
.074
.031
.005
.000
r =5
a • .50
a • .75
·735
.626
.522
.382
.209
.105
.022
.001
.673
.596
.524
.457
.343
.251
.153
.062
.024
.003
.000
.363
.296
.241
.195
.126
.080
.040
.012
.003
.000
.000
a • .75
a •. 25
.359
.296
.243
.199
.131
.086
.044
.014
.004
.000
.000
.888
.840
.788
Values of 1i~ E(lJ. ij - Tij )
(P=<1-5)
'Ijf
0
3.2
6.4
9.6
16.0
22.4
32.0
48.0
64.0
96.0
160.0
a • .25
:r • 3
a • .50
a • .75
a •. 25
r • 5
a • .50
·987
.963
·921
.861
.697
.508
,264
.059
.009
.900
.000
.938
.865
·767
.655
.429
.247
.089
.011
.001
.000
.000
.804
.664
.522
.391
.195
.085
.020
.001
.000
.000
.000
·992
·971
·931 .
.867
.653
.360
.075
.001
.000
.000
.000
·955
.886
.787
.667
.412
.195
.036
.001
.000
.000
.000
,
a • .75
.837
.695
.544
.402
.190
.072
.011
.000
.000
.000
.000
Table 5.1.1. Upper a-points of the statistic x
Level of significance (a)
Size of
experiment
p=q=3
p-q-5
.25
·50
·75
r=3
{
.396
.290
.193
r-5
.238
.167
.107
{
r-3
.446
.386
.328
r-5
,281
.238
.198
..- -
--~--=-
is the proportion of l'iJ" that constitutes the bias of T. "
aence Q
v
t
..
~---..,." ~"""
1J
+1
We present the values of this relative bias
/' ~jl
1
(which is the same for all
E(~.. - T.. ) • Q
lJ
~j)
lJ
Vi +
1
for our chosen values of p, q, r and a
in Tables 5.2.1 and 5.2.2.
Note that for a a() , the bias of Tij is -/' ij' and for a-l, the bias
is zero, irrespective of p, q and r.
As in the
Note that a
~
2~
case, here also we have chosen eleven values of
for the 2x2 case is not comparable to the same
W.
~-value
for,
say, the 3x3 case.. . However, the values
would seem to Qe comparable (cf. formu+a for expectation of interaction
mean square; also the null-hypothesis is
~ ~
1 (see (2.2.1»
in the 2x2
case, as opposed to the null hypothesis ~ ~ (p-l)(q-l) (see (4.3.5) and
(4.3.7»
in the pxq case).
We have, therefore, multip+ied the
~-values
On examining these tables, we find that the observations we made
in connection with the bias in Section 3.3 hold here as well.
bias decreases as * increases.
only slight.
Thus the
The change in the bias for changing r is
On the other hand, a change in tpe level of significance
effects a substantial change in the bias.
The magnitude of the relative
bias seems to be dependent on the numbers of factor levels, p and q.
Thus, while for PI!(} -3 the relative bias correspondirig to a - .75 may be
said to be only moderate, for Pl!(}-D even with such a high level of
significance the relative bias remains quite high, at least towards the
beginning of the range of *-values.
It would thus seem that, if keeping
the relative bias under control is one of our objectives, then a should
be increased progressively with p and q.
5.3 Relative AMSE: AMSE/(a2 /r)
As we stated earlier, our main goal is to minimize the average mean
square error (AMSE) when we are trying to estimate all pq cell means
simultaneously.
Here we shall study the variation of the AMSE or rather
of the ratio (see equation (4.7.11»
AMSE
-r
a /r
•
1-
(p-l)(q-l) Q
(t v x~)
pq . Vi +1 2' 2' . . .
The values of this ratio corresponding to our chosen values of p,
q, r and a are shown in Tables 5.3.1 - 5.3.4.
on
Note that the AMSE depends
79
as well as on
2
2
'if- r EE7 .. /a
i j
lJ
Hence, we need, for each set of p, q, r and ex values, a two-way table,
with 'if varying between the columns, say, and 'if* varying between the
rows.
For 'if* we have taken the same eleven values as for 'if.
Not all
the 11 x 11 combinations are possible, however, as shown in Section 4.4.
The cells corresponding to the inadm.issible combinations have been left
blank in the tables.
Y
Besides a - .25, .50 and
.75, the two trivial values, a - 0 and
a - 1, may also betaken into account.
However, for a • 0,
and hence
AMSE
(J2/r .
if
+ 'if*/9
5/9
a-o:; { 9/25 + 'if*/25
,
if
irrespective of 'if and r, while for ex - 1,
Q
"'1
+1
-
Q
"'i
• 0
+1
and hence
AMSE
-r
/r
•
1,
(J
for all p, q, r, 'if and 'if* •
!/JhTables 5.3.1 - 5.3.4 the columns for 'if - 40.0 are omitted for lack
of space and because only in two columns some values are different from
1.000. These columns correspond to p-q-3, ex-.25, and even there the
largest difference is .009 for r-3 and .004 for r=5.
80
Table 5.3.1.
Values of the ratio
AMSE (02 !')
(p=q-3, r-3)
o
o
.611
.8
1.6
2.4
4.0
a-.25 5.6
8.0
12.0
16.0
24.0
40.0
o .706
.8
1.6
2.4
4.0
a-.50 5.6
8.0
12.0
16.0
24.0
40.0
o .840
.8
1.6
2~4
4.0
a-·75 5.6
8.0
12.0
16.0
24.0
40.0
4.0
.718
.785
.852
.6
.760
.818
.877
8.0 12.0 16.0 24.0
.818 .892 .940 .984
.863 .920 .956 .989
.908 ·947 ·972 .993
81
Table 5.3.2.
Values of the ratio
(p=q-3,
AMSE/(a2 /r)
r-5)
.
-~._
0
0 .605
.8
1.6
2.4
4.0
5.6
8.0
12.0
16.0
24.0
40.0
0 ·701
.8
1.6
2.4
4.0
5.6
8.0
12 .0
16.0
24.0
40.0
0 .839
.8
1.6
2.4
4.0
5.6
8.0
12.0
16.0
24.0
40.0
.6 8.0
.768 .830
.826 .873
.883 ·917
12 .0 16.0
.907 ·953
.932 .966
.956 ·979
24.0
.990
·993
.996
82
Table 5.3.3.
Values of the ratio
(p-q-5,
o
o .368
3.2
6.4
9.6
a-.25 16.0
22.4
32.0
48.0
64.0
96.0
160.0
o .400
3.2
6.4
9.6
a-.50 16.0
22.4
32.0
48.0
64.0
96 .0
160.0
o .486 .
3.2
6.4
9.6
a-.75 16.0
22.4
32.0
48.0
64.0
96.0
160.0
r-3)
AMSE/{a2 /r)
83
Table 5.3.4.
Values of the ratio
(p;:q-5,
o
o .365
3.2
6.4
9.6
22.4
32.0
48.0
o 1-"--'"-+--=:"'::'-'"
3.2
6.4
9.6
a-.50 16.0
22.4
32.0
48.0
64.0
96.0
160.0
o .464
3.2
6.4
9.6
a-.75 16.0
22.4
32.0
48.0
64.0
96.0
160.0
r=5)
16.0 22.4 52.0 48.0
·582 .770 .952 .999
.677 .827 .965 .999
.771 .884 .979 1.000
J.-...,~+--~--.-
a-.25 16.0
64.0
96.0
160.0
2
AMSE/(a /r)
64.0
1.000
1.000
1.000
96.0
1.000
1.000
1.000
84
5.4.
Comments on the Values of AMSE
Let us now examine the values tabulated in Section 5.3 to see how
the AMSE is affected by changing r, a, "1\1 and "1\1*.
Firs~
consider the variation of the AMSE with "1\1 and "1\1*,
a(o <a < 1) and r being kept fixed.
If "1\1 is kept fixed and "1\1* allowed
to vary, the AMSE increases monotonically with "1\1*.
apparent from equation (4.7.11) as well.
This, of course, is
On the other hand, if "1\1* is
kept fixed anq "1\1 allOWed to vary, then the behavior of AMSE is a bit
irregular.
It is seen that for small "1\1*, the AMSE increases monotonically
with "1\1 and tends to unity; for large "1\1*, the AMSE decreases monotonically
as "1\1 increases; while the tables seem to indicate that for some intermediate "I\I*-values, as "1\1 increases the AMSE increases for a time, reaches
a maximum and then tends to decrease.
Corresponding to Theorem 3.5.1,
here also we have the following theorem, which can be proved using
Lemma 3.5.1:
Theorem 5.4.1.
rAMSE(Tla)/i ... 1 as "1\1 or "1\1* ...
rAMSE/
i ...
Proof:
That
1 for "1\1 ...
3.5.1.
On the other hand, if "1\1*'"
00
00,
provided a > O.
00,
is obtained directly from Lemma
then "1\1 ...
00
(cf. equation
(4.4.4)) and from (4.4.1)
"I\I*aQj(f, q, xa ) ~ (pq"l\l)aQj(f, q, xa ) ... 0,
for a
> 0, so that, because of (4."1\1.11), rAMSE/a2... 1.
Secondly, as was true for the 2x2 case, the change in
2
rAMSE/a
corresponding to a change in r is rather negligible.
Lastly, for fixed r, "1\1, "1\1*, the AMSE is highly sensitive to a change
ina.
The general effect of a higher level of significance is to raise
85
the value of the AMSE toward the lower part of the range of 1jI*-values,
i'~"
for 1jI* ~ (p-l)(q-l), and to reduce the value of the AMSE toward
the other part of the range, i.e., for 1jI*
> (p-l)(q-l).
If one uses equal weights, then 1jI* .1jI, and the behavior of the
2
ratio rAMSE/a can be seen from the principal diagonals (marked for
convenience) of the above tables.
It will be found that in thi.s parti-
cular case the rati.o behaves very much like the quantity
rMSE/a2 for
a 2x2 experiment (see Section 3.5).
5.5.
Optimal Choice of a
Here, too, some suggestions may be made as to the choice of a
sUitable level of significance for the preliminary test.
One may again
consider the four approaches we talked about in Section 3.6:
(a)
Bayes Approach:
Note that since the ratio
AMSE
2
a
/r
is dependent on both 1jI and 1jI*, to use this approach we must start With a
joint prior distributiop of 1jI and 1jI*.
We do not attempt to say anything
more than the following, because the choice of the prior distribution is
bound to be artibrary:
It is apparent from Tables 5.3.1 - 5.3.4 that if
one's prior knowledge indicates that
~
is very likely to be less than
(p-l)(q-l), then a relatively small value of a should be used for the
preliminary test; while in case the prior information indicates that 1jI*
is not so likely to be small, then a high value, perhaps one near to
unity, should be selected.
(b)
Minimax Approach:
In the absence of any prior knowledge about
and 1jI*, one might like to choose a in such a way that
1jI
86
Max
r AMSE
2
\jr, \jr*
is a minimum.
significance is
choose the
The result,
~,
C1
(4.8.3) implies that this minimax level of
so that according to this approach one should
estimatom~..
l.J
, irrespective of the sample observations.
A
remark similar to the one made at the end of S.lbsection (b) of Section
3.6 pertains to this result.
te)
Minimizi~ the
!l=rJ>.).('ffi of the Regret Function:
discussed for the caGe\jr
= \jr*.
~"'e
This ap.p1"Oact ·will
The regret function analogous to (3.6.6)
1;:,
now
Min
o
r AMSE\jr(Tlo)
2
.• s(\jr,o)
C1
and it equals
- ["1 - (p-l)(q-l)
pq
+ L ] for \jr < (p-l)( q..l)
pq
-
for \jr ~ (p-l)(q-l)
- 1
As in the 2x2 case, this approach leads to the choice of 0-1 again.
(d)
Minimax Regret Approach:
weights, when \jr • \jr*.
\jr ~
We shall take the special case of equal
The result
(4.8.2) shows that for any
(p-l)( q-l)
1
whi.ch value is attained in case o.().
any \jr > (p-l)(q-l)
The result
(12- l )(q-l) + L
pq
pq
,
(4.8.3) shows that for
In£'
o
-
which value is attained in case a-l.
1
Table 5.3.5 lists the numerical
values of these expressions for p-q-3 and p-q-5 and the same el.even
w-values as were used in Tables 5.3.1
Table 5.3.5.
~
5.3.4.
2
Values of In£' r AMSE1jr(Tlo)/cr
o
1jr
p-q-3
1jr
p-q-5
0
.556
.644
0
.360
3.2
.488
6.4
.616
.744
1.000
0.8
1.6
2.4
.733
.823
4.0
1.000
5·6
8.0
1.000
9.6
16.0
22.4
1.000
32.0
1.000
12.0
1.000
48.0
1.000
16.0
1.000
64.0
1.000
24.0
1.000
96.0
1.000
40.0
1.000
160.0
1.000
1.000
Considering the differences between the entries in the principal
diagonals of Tables 5.3.1 - 5.3.4 and the corresponding entries in
Table 5.3.5, we find that the level of significance minimizing
Max s(1jr ,0)
1jr
is some value between .2 t and
.to
(closer to
.to).
On a comparison
of this result and the result in Section 3.4, it would seem that the
minimax regret 0 is at least approximately independent of p and q.
A fortune aspect, in the case of equal weights, of this ChtHCe of
o between .25 and .50 (nearer to .50) is that not only is the
ll18Ximul1t
regret minimized, but also the resulting maximum is brought to a
10\1
level, so that a sufficient degree of control over the AMSE throughout
the range of \jr-values is obtained. (The situation is similar to that
in the 2x2 case, see Section 3.6).
By looking at the tables, one can
say that such a choice would be useful even when unegual weights,
,~:~:,
:;::"e:!, provided 1a rger weights go with smaller interactions (imply ing
:J0"_ <.Ad·
But in case \jr* is much larger than \jr, such an a will mH}~e
the AMSE too high.
In a practical situation one is not likely
about the relative magnitudes of \jr and \jr*.
tC)
kno
i
,.
The proper course then
would be to control the AMSE throughout the \jr-\jr* table, by choosing a
very'
high value of O.
Evidently, in this case, the higher 6
js) t b:
hi,gher should be "the value of O.
Let us now convert the value of 0, related to our test for
against
into levels of significance to be used in corresponding ANOVA tests
for
against
'The level of significance of the ANOVA test, corresponding to an 0 used
in our test for material significance; is
These levels of significance are shown in Table 5.5.1
for the
set of valuesof p, q and r considered in this chapter.
Table 5.5.1.
Levels of significance for ANOVA test corresponding to
a •. 25, ,50 and, 75
Size of the
_____e.;..x.....e_r...;.i....m
....e....n...t_ _-+-
a
.'_2'---..4~._~ _~dQ~_
. __~=___o:.,...'7""'"2~~~~_
..
.05
.17
.ho
.04
.15
.38
.007
.04
,13
.004
,02
.10
We may conclude that for the equal-weights case our Ereli.minary
test may be conducted as an ANOVA test at about the
~
level for a
3x3 experiment and the .02 level for a 5x5 experiment.
However, when unequal weights are used, an
high a level as
~,
under proper control.
~OVA
as suggested by Anderson (lg60),may
test at
~
~
keep
as
~ ~
This situation arises presumably because the
weights are not being taken into account in the preliminary test procedure.
The development of a better test procedure is thus called for to deal with
the case of unequal weights.
6.
6.1.
SOME DEC:lSION-~RETIC RUUL'l'8
Introduction
In the course of this work, we have at various times thOught about
the question how TE procedures, in general, wpuld tit
~nto
a decision-
theoretic framework and whether some concepts of that theorY (other than
those discussed in previous chapters) might be helpful in constructing
TE procedures.
Since we feel that.our particular problem might benefit
from such an approach, we have collected some at our thougqts in this
chapter.
6.2.
Fixed-sample Decision Problem
Although the theory of statistical
de~ision~
can be treated in a
more general set-up, we shall here be concerneq with problems connected
with samples of a fixed size, so that we heve to deel only with terminal
decisions (vide Wald (1950».
We have a random variable (possibly a vector) x, teking values in
a space
~
which is the underlying sample space witn Boral field
(l"a) is defined p, a family of probal;>11i ty measures P.
a.
On
In most
situationa,
p = fP9 \ 9
where
e
is a parameter space.
eel ,
Together witq ~, we have also a space ~
of possible decisions regarding the probability measures in p.
typical member
of~
will be denoted by d.
L such that
L(e,d)
There 1s also
8
A
loss functiop
91
is the loss incurred through taking the
deci~ion
d when
e
is the true
parameter.
A decision rule (function) 6 is a measurable function from ~ to
~.
Thus 6 is a rule for making a decision on the basis of an observation
on x.
The choice between any two rival decision functions is made on
the basis of the loss function L, or rather the
J'
r( e, 6) =
L( e, 6 (x»
ri6~
function r, where
dPe(x)
I
This choice may be made from various standpoints as illustrated by
the following definitions:
Definition 6.2.1.
A decision function 6
o
is 3aid to be admissible if
there exists no other decision rule 6 such that
,
r(e,6) -< r(e,6 0 )
all
and the strict inequality holds for at
Definition 6.2.2.
is a
~-f1eld
~
Let
of subsets
e !.e ,
leas~
be a probability
of~.
one 9 t
~easure
Let the average
On
ris~
e.
(e,i), where A
~
relative to
of a
rule 6 be denoted by r*(~,8), i.e., let
r*(',6) =j'r(e,6) dl!'(A)
e
Then a decision rule 5 is said to be a BaYes rule relative to
~
~
if
for all 6.
Definition 6.2.3.
A decision function 6o is said to be minimax if
sup r(e,6 ) < sup r(e~6)
o A
e
for all 6.
Definition 6.2.4.
A decision function 6o is said to be minimax regret
(or most stringent) if
suP[r(e,6 0 )
-
e
inf r(e,a)] ~ suprr(e,6) - inf r(e,6)]
e
6
6
for all 6.
For easy reference, we now state aome theorems from Wald (1950).
Theorem 6.2.1.
w.~.t.
Suppose, for every e £
a rr-finite measure
funotion by fe(X).
~
on
(~,a)
e, Fe
1s absolutely
contin~ous
and denote the corresponding density
Assume further that:
(1) the functions L(e,6(x)) and fe(X) are measurable (C), where
is the smallest rr-field generated by the sets AxB, where A
£~,
B e Bj
(2) the loss function is non-negative: L(a,d)
If
the~e
exists a rule 6
0
~
O.
satisfying these conditions and for Which
jL(e,&o(x)) fe(X) df!'(e)
~J L(~,d)
A
~
,
fe(X) d,,(p)
~
all d
It
all d
£ ~
,
or, equivalently,
jL(fb6 0(X)) dFx(e)
®
~J' L(a,d) dP'x(~) ,
®
,
P'x being the posterior probability measure on (~,B) relative to ~, then
6o is a Bayes rule relative to
~.
~
93
Defin~~lon
TWo decision rules, 01 and
6.2.5.
x-equivalent if 6l (x)
°2,
are said to be
6 (x) a.e. [A]' where X has the same meaning
2
They are said to be risk-equivalent if
=
as in 1heorem 6.2.1.
(Note tt18t A-equivalence implies risk-equivalence. J
oF
~.
Consider, e.g., a Bayes
. rule 6r; relative to
Definition 6.2.6.
':>
is said to be unique except for x-equivalence if any other Bayes
rule relative to
1he~~~.2~g.
~
is A-equivalent to 6 .
r;
A Bayes rule relative to a prior distribution
~
is
admissible if it is unique except for A-eqUivalence.
Theorem 6.203.
Let 6 be a decision rule such t~at there exists a
0
prior distribution
~
relative to which
° is a Bayes rule,
0
r*("o
) -> sup r(e,~Vo )
..,
0
e
then
then if
,
° wLll be a minimax decision rule.
0
Coro~la~6.2.3a.
If
° be a Bayes rule relative to some prior distri~
bution ~,
, such that the risk r(e,o F ) is constant over ~,
. then
° is a
F
minimax decision rule.
Theorem 6.2.4.
Suppose that there is a sequence of prior distributions
Fk(k
~
~k'
If there exists a decision rule
1,2, .•.... ) and that Ok is a Bayes decision rule relative to
lim r*(~k'Ok)
=
k .... co
then
°o is a minimax decision rule.
° for which
0
,
603
Decision.=theor~etic
0
Formu.lation of a TE Procedure
'The px'()blem of esti1l1ati.on of a param.eter
ca:c.ce may be considered :I, n the frame'work of
l~j.
i
~
8,
... ~
6:ft.~r
d,e~:i. s1. eXi
theory
signifi'~
0
From other considera.tlor18 two r·t·val estlIr1atC)TS:i
t::' decIde whic2'! of T, (x) and ',rr,(x) 1s to be taken
®L
H8
th.e estimate of fh
c.~
..L,
values i.n
s test of
.J,.,.
a typical member of which is & dee,DE·d
where ftl(X) :is assumed to be a measurable function tsh:ing values 0 or 10
Tne fact. that we are using small VJSE as the crit;eri.on of a good
where c(e)
~
,
> 0 is independent of
T:'1e ri sk fUncti on
j
el
but may depend en other elements of
s r.9 given by
which is the MSE of & at ~j ex::ept for the mul"Ciplier
The question might very well be asked why we are
de),
""~aking
thi.s
restricted class 6 of decision rules (estimators) instead. of taking the
0
95
class of all estimatorso
The purpose is to set up a framework in
which the question raised in Section 2.2 can be discussed.
best possible function
~(x)
for
(6.3.1) (if a
~bestV
Finding the
function exists in
some sense or other) amounts to finding the optimum partition of the
sample space for a TE rpocedure.
MSE of the estimator
the
j
Since our objective is to control the
the notions of power and level of significance of
preli.minary test are clearly irrelevant.
604.
Bayes Rules for Simple TE Problems
Let x
where
~
~
(x lj x2j .o.jXn ) be a random sample of size n from
is unknown and is to be estimated.
For estimating
~
2
N(~jrr
on the basis
of this sample} we may say that we have two available estimators
and T ::
2
x.
j
T
1
Now
~
2
and
==
rr
2
/n
so that if
Il
2
~
a
2/n
one would prefer T , and if
l
~
2
2
< rr /n
one would prefer T 0 A TE procedure would be:
2
(1)
Test the hypothesis
(2)
If significance is obtained in step (1), take T (x) as the
2
estimate; otherwise take T1(x)0
),
=0
In the above notation, we now have
a(x)
= [1
t:p(~d}o + ep(x),
=
x
(6,4,1)
and
As ths prior distribution let us t,nke the conjugate prior distri=
2
bution corresponding to N(lJ.,O" ),
Some H:rgUJri€nts in support of
c:onjtl.gate prj.. ors have been given by RaLf'fa and Schlaifer (1961 L
Ct1l3,pter 30
way we can
One rnathematically quite cogent are,'UJ'JJJ,mt L" that. 1n this
11 ()rk
within a closel S\ystem (In the sense that the posterior
dist.ributlons are of the same type
Case
8S
the prior distributions),
1~
Here
® :::: (IJ. , - 00< Ii
< m}
,
dnd we need consider the prior distribution of IJ.,
The conjugate prior
distribtrti,:m of IJ. may be put in the fOJ:"xrr:
'~'00<1J.<0');
< m,
(vide RaiffB. and Schlaifer (1961), p, 54)
2
0"
> 0,
n
,
>
°
0
Tn'" post':::rior distribution of IJ. given x :1.8
,
= n9
where
n"
and
m" = (n'm'
+ n
+
11
;:)/n"
(604,4)
97
Theorem 6.2.1 implies that a Bayes solution is defined by:
J~2
m
o
5 (x)
F
if
df:)ll)
:=
x-
eo
if
]
<J (x=~)2
J\
eo
dFx(l-!)
eo
112 dF'x.(j.!) >
(X-ll)2 dP.')Il)
We have nine different si.tuations according to the values of n, n',
m', and for each of these situations 5~(X)
:=
X in those subsets of the
sample space which are listed in the last column:
(1)
n' >n
and
m' >0
o < x- < 2n'm'
,
n -n
(2)
n' >n
and
m' <0
2n'm'
<x<O
n'-n
(3 )
n' >n
and
m~
C/l ,
(4 )
n' <n
and
m' >0
x < -
2n'm'
n-n'
or
x >0
(5 )
nO <n
and
m' < 0
x< 0
or
x > =
2n'm'
n-n'
(6)
n' <n
and
m'
):
:=
:=
0
0
the empty set
-
98
(7)
n l =- n
and
m' > 0
x- > 0
(8)
nl = n
and
ml < 0
x<
(9)
nl = n
and
ml
.=
0
0
= any measurable subset of 1.
(including ¢ and L )
Same features of these results deserve special attention.
if the prior information regarding
we would choose
x aver
~
is comparatively precise (n l > n),
a small part of the real line.
On the ather hand,
i.f this is campara tive ly diffuse (n I < n), we would choose
greater part of the real line.
than that of
x
x,
When
~
x over
the
When we take any particular case in each
grcmp, the result would again seem to support common sense,
case (1) above.
Thus,
E.g"
take
has a positive mean ml and a variance smaller
a negative
x will
generally underestimate ~, while if
is much larger than ml it will be tao much of an overestimate, compared
to 0,
Nate that, for the same problem, a TE procedure, such as those
discussed in the previous chapters, would lead to the choice of 0 when
nx2 /i<
and to the choice of
x otherwise
Y?-a
(where
~
is the upper a-paint of a
non-central ~ with 1 d,f. and non-centrality parameter
n ~
Case 2:
2
/a2 = 1
. )
2
a Unknown
Here
tEl
= (~,
i I-
00
< ~ <
00,
i
O} .
99
Any con,jugate prior distribution has the following
(seE
l~iffa
and Schlaifer
(1961),
p,
normal~gamma
forIa
55):
(6,4,6)
- ro
< mv <
ro,
vt ,n9 ,v9 >0
and the corresponding posterior distri,Dutionts
2
x dll do- ,
where
,
mit
:=
(nlm l + nX)/n l !
,
v II -_ r(
" v ••
v + n • m12) +
( Vv
=2) - n II m,,2 )I'"
+ ox
V ,
and
v
Theorem
=L
1
(x.=
1
x)2/ v
6,2,1 implies that a Bayes solution is defined by:
Here F (11) is the posterior marginal distribution of
x
II
oc
)'
(v" + (ll-m,,)2 nll/vltf ( v +l /2
~
and it has density
100
Ii
Since this distribution has mean m , the above Bayes solution comes out
to be
if
5 (x)
!:'
=
if
11
2 xm
> x-2
,
(6.4.8)
and thus is the same as the Bayes solution (6.4.5) for the case of
known
(j
2
,
Note that 1n both cases the Bayes solutions are unique except for
A=equivalence (A : Lebesque measure) and hence, from Theorem 6.2,2,
are admissible.
6.5.
Bayes Rule for Simulte.neous Estimation of Cell Means in a
Factorial Experiment
Let us now take up for consideration the problem of simultaneous
estimation of the cell means of a pxq factorial experiment.
set=up and notation of Chapter 4, we have as estimators of
Using the
~!/
the two
rival sets:
A
and
We are looking for a decision rule which will, on the basis of a set of
random observations xijk(i ~ 1, .. " p j
estimate
~
according to the estimator
and according to the estimator
1/
9
r
- -Il "" \ Ill1,lllr,'
.
c:. . ,
column vector.
0' f.!l q
J
= 1, .. o,q;
j
...
k
= l, ••• ,r),
on one part of the sample space
~
on the remaining part.
~
0••
1J.2~ ,
...1....
j
)
IJ..p, q
and. IJ.
Similary' for 1::, !:, etc.
<'
i
s the corresponding
101
We are thus considering the class of decision rules 6 of which a
typical member is a vector
where
~(x)::: 0
1J..
defined by
or 1,
Suppose, as in Chapter 4, we are employing low AMSE as a criterion
of good estimators,
L(e,e) = c(~)
We then have a loss functi.on of the type:
f ~ Wijr{l - w(x)}(~1j= ~ij)2 + W(X)(~ij- ~ij)21
(6,5,2)
where c( fl) > 0 :i. s independent of the Ilij 9 s but may depend on ttle other
2
0- •
element of 8, viz"
We shall here take c(e) ::: r/0- 2 •
The risk
function is
and is the AMSE of
1J..
except for the multiplier C(fl),
The conjugate prior density of
~,
in case 0-
2
is known, is (cf,
Raiffa and Sch1aifer (1961), p. 56)
(2~0-2/r )-pQ/2
exp[- r'
~~
i
(Il
j
ij
-
m~J)2/20-2]
V
9
,r, ,,,9 >0 ,
As in Section 6.4, in either case the Bayes solution would be the
_
same.
A
It leads to the choice of Il = w or Il
-
-
-
=w
l'
+ Z ::: X ,
A
-
according as
102
l,e"
according as
This follows from Theorem 602,10
If'
equal weights are used 1"'01.' all cell means, i, e, ,
tn,en the above takes on a simpler form because
One would then choose
(,r I
~r)
L~ "t'
!±'"
r; z 2
ij
i j
'0'
~
,::H::cordi.ng
as
- 2r'
Z
L .E
1 j
0
•
l.J
m~.
> 0
IJ -...
,
/
Here again these Bayes rules are unique except for A-equivalence
(x
:=
Lebesgue measure) and hence are admissible,
6.6,
Minimax Decision Rules
Let us now study the na'ture of minimax decision rules for -the
problems discussed in Sections 6.4 and 6,5,
It seems that in general
the minimax estimators are the conventional (unbiased and minimum=
variance) estimators,
Thus for the problem of Section 6,4 the minimax estimator in the
class 6 is x,
and m = 0, i,e"
For consider a conjugate prior distribution with
case (6) of Section 6,4,
prior distribution 6(X)
r(s,x)
that
=
x is
constant (ioc"
= x is
n~
< n
As we have seen, for such a
a Bayes solution,
And the fact that
unity) implies, by virtue of Corollary 6,2,3a,
a minimax estimator in the class 6'
Being a Bayes rule unique
except for A,-equivalence, 1t 1s also admissible,
103
For the problem of simultaneous estimation of all pq cell means in
a pxq factorial experiment, di,scussed in Section 6.5, which is also the
main topic of our investigation, we can easily show that the usual
estimators
~
constitute a set of minimax
in case the weights
W
ij
are all equal.
estin~tors
in the class
For in that case we find from
t
(6,5.'7) that, for the conjugate prior distri,bution with m..
.
1'9
< rJ
.5,.(X)
:=
All,
'"
H; is a Bayes decisi,on rule.
1J
== 0,
Also, :fer this rule
r(t\j"b,) == constant (Le.) urdtY)}~lhLch imIl}i.esJ by virtu.e of Corollary
6.2.3a, that ~ is minimax,
corresponding to
,
> r,
I'
(We have not used our Bayes estimators
m~j
:=
0, since they have a risk that increases
linearly with 1jr* and hence violate the condition of Theorem 6.2.3.)
For the general case of possibly unequal weights, too, it is
conjectured that ~
:=
H; is minimax in the class 6'
From (6.5.6), it is
seen that for the sequence of prior distributions (F k ) for Which
m~j
==
° (all i,j) and r~ ~ 0,
the Bayes rules (6.5.6) also tend to
It seems plausible that the nice properties of normal distributions will
ensure that the sequence of average risks r*(~k,5k) for these Bayes
solutions will also tend to the average risk of
being the same as r(p,x), all
e
~,
viz., unity.
This
e ® , Theorem 6.2.4 would imply that x
is minimax.
liThe difference between these results and the results in Subsection
(b) of Section 3.6 and Subsection (b) of Section 5.5 is that the
functions fO in the earlier results were restricted to UMP invariant
tests, whereas in the class ~ there is no such restriction on ~ .
104
7,
7,1.
SUMMARY AND CONCLUSIONS
Summary
It .nas been our purpose in this investigation to study the
consequences (in terms of biases and mean square errors (MSE's) of
estimators) of estimating the cell means of a factorial experiment
according to either one of two different formulas, depending on the
ref.;u.lts of a prellminary test for interactions, whereby the preliminary
test and the final estirnation procedure are carried out on the same set
of data.
Such procedures combining preliminary testing and subsequent
estimation will be called test-estimation (TE) procedures; the resulting
estimators, TE estimators.
In Chapter 1 the general ideas were introduced, and it was pointed
out that we would confine our attention to two-factor experiments with
an equal number of replications per cell.
It was assumed that for our
experiments Eisenhart's Model I is appropriate and that usual normality
assumptions are valid.
Related literature was also cited in this
chapter.
The special case of factors with two levels each was considered in
Chapters 2 and 3.
In Chapter 2 the distribution of the TE estimator of
a cell mean corresponding to the UMP invariant test for interactions
was obtained, and expressions were given, in terms of a function Q, for
its bias and MSE.
In Chapter 3 numerical values of 'the relative bias'
and 1the relative MSE' were obtained.
Different approaches to the Choice
of a suitable significance level for the preliminary test were
investigated and various theorems proved, showing that two of these
105
approaches lead to a degenerate TE estimator, viz., the least squares
estimator.
An alternative formula for the MSE, whiCh, at least for
sWEller numbers of replications, will cut down substantially on tlle
numerical work, was also derived.
Chapter 4 is concerned with the general case of pxq factoria.1
Two modes (see Sections 4.3 and 4.9)
experiments (p > 2 and/or q > 2).
of combining the MSE's of the individual estimators of the pq cell meansjl
and thereby obtaining a single cri.terion for the goodness of the
estimators, were discussed.
viz"
For a special case of one of these modes,
taking the average of the MSE's (AMSE) with equal weightsjl the
distributionjl the bias and the MSE of the estimator of any particular
cell mean were obtained.
to those in Chapter 2.
The formulas were found to be closely analogous
In Chapter 5 we reported numerical results for
two special cases, viz., p
9 the
:=
q
:=
3 and p :::: q
:=
5.
The magnitude of
relative bias' was examined numerically as a function of
centrality parameter
W.
and for unequal weights.
The
A~E
D
l1on~'
was studied both for equal weights
It appeared that in the case of unequal
weights a second parameter,
W*,
had to be introduced along with
W.
The
question of what would be a suitable significance level for the
preliminary test was investigated for this more general case, and
generalizations of the theorems of Chapter 3 were presented.
Chapter 6 explored the possibility of studying TE procedures in
the framework of decision theory.
Here our findings have been tentative
but are expected to help future work on the choice of a good preliminary
test procedure.
The test procedures that we proposed in Chapters 2 and
106
4 are based on the notions of significance level and power, hence are
somewhat unsatisfactory, because these notions seem irrelevant when our
ultimate purpose is to control the MSE's or AMSE of the estimators,
7,2,
Conclusions
Now supposing the prelimina17 test suggested in Chapter 2 or
Chapter 4 is used in a TE procedure, the MSE's of the estimators can
be controlled by a proper choice of
l',
t.he nU1l!ber- of replicat:i.ons, 811d
a, the level of significance of the test,
The value of r wi.ll in
general be determined by the resources available to the experimenter, and
there is not much of a choice in this respect,
As regards a, the choice
will depend on the context of the estimation problem,
If we have some prior information regarding the likely magnitude
of the non-centrality· parameters 'If and 1jr*, we may utilize this
information to choose a suitable
(X,
very likely to be less than 'V1
(p-l)(q-l), then a small value of
==
1hus, if it is known that 1jr* is
(X
Should be chosen; while in case indications are that 'If* is not so
likely to be less than "I' then a rather high value, perhaps near unity,
ought to be selected,
In the absence of any such prior knOWledge, a minimax approacn
might be adopted,
This approach leads to the choice of the u:Y2.1
least~
.-
squares estimators,
~ijj
and so does the approach of minimizing the
L1=norm of the regret function in the equal-weights case,
However, we
set out to find alternatives to these estimators, and so these approaChes
do not seem helpful to us,
107
Another appraoch which would seem quite appropriate in the absence
of al~ prior knowledge about the variability of 0/ and 0/* would be to
choose a in such a manner as to minimize the maximum regret,
For the
2x2 case, and also for the pxq case when equal weights are used, this
approach leads to the choice of an a between 025 and .50 (nearer ,50)0
(These
a~values
refer to a preliminaF3 test for material significance;
for conversion into significance levels for customary ANOVA tests, see
:rables 3,601 and '05,10)
However" in
tttE:
pxq case, When unequal <\"elghts are used, an a of
this type does not effect a satisfactory degree of control over the
M~E
for values of 0/* that are too large in comparison to 0/ (see Tables
),3,1=),304).
It appears that in such a situation a should be
progressively increased as p and/or q increase.
Thus While an a; 075
might be suitable for the 3x3 case, an a; ,90 might be required for
the 5x5 case.
On the other hand, the use of such a high value of a
will offset any gain that might accrue from the use of a TE procedure
for values of 0/* that are small compared to %
Presumably, the
development of a better preliminary test procedure is called for to
deal with the case of unequal weightso
703.
Suggestions for Further Research
In our investigation, we have been concerned with two factors
only.
For dealing with more than two factors, there is a simple,
straightforward generalization of our procedureo
Thus (Whatever the
number of factors) we may like to use one type of estimates if the
interactions of the highest order are 'materially significant' and
108
another type of estimates otherwise.
Consider, e.g"
an experiment with
three factors A, Band C, having a, b, and c levels respectively, with
r replications for each of the abc treatment combinat,ions.
Here the
model analogous to (2.2) is
fJ.
0:.,
13"J
1
l'
k
,Co
general effect;
==
ID.ain
ef~:eects j
(C7)ik' (~Y)jk - intera(:tion effects of 1st order;
(Ol3r)ijk - interacti.on effects of 2nd order.
Let v.s write the corru3ponding sample mean as
yolJOk
'"
In
+
Do
L
+ b
j
+ c1? + (abL
L~
°
lJ
+ (ac)'k + (bC)'k + (abC)'jk
1
J
where the components are the least squares estimates of' the
parts of (7.3.1).
1
corrf,,}~londing
Taking Im1 average mean square error as tIle criterion
of a good set of estimators, we may choose as estimates of I-\jk
and
109
otherwise.
The corresponding TE procedure being similar to those
discussed in Chapters 2 and 4, our theoretical findings in those chapters
may be expected to be valid here, too.
However, in case there are more than two factors, one might want
to have a choice between more than two types of estimates.
Thus for
the case of three factors, we might like to choose among
m + a 1 + b j + ck '
m + a i +b j + ck + (ab)ij + (ac\k + (bC)jk' and
m + a i + b j + ck + (ab)ij + (ac)ik + (bc)jk + (abc\jk
'
depending on the outcome of a preliminary test concerning their AMSE's.
A TE procedure for this problem will presumably need more complicated
mathematics and, indeed, more complicated statistical thinking, for
the preliminary test now takes the form of a multiple decision problem.
The case of non-balanced factorial experiments, where the numbers
of replications for different combinations are unequal and possibly
disproportionate, has not been considered by us.
Although our findings,
e.g., regarding the optimal choice of significance level, are expected
to hold broadly in this context also, a detailed mathematical treatment
ought to be made.
Also, a more extensive decision-theoretic treatment of our problem,
than was made in Chapter 6, would seem to be in order.
Even in the more elementary treatment of Chapters 2-5, a different
family of tests, which should aim at keeping under control the MSE or
AMSE, needs to be devised.
110
In the pxq case, we should try to find a better way of handling the
case of possibly unequal weights for the AMSE.
If in the definition of
tne AMSE unequal weights are attached to the cells, a redefinition of
rr~in
effects and interactions might also be appropriate.
A generalization of Huntsberger's (1955) method of estimation to
the case of factorial experiments may be studied.
This is expected to
lead to a greater overall control of the MSE (or AMSE) here as well.
111
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