A COMPARISON OF METHODS FOR COMBINING
TESTS OF SIGNIFICANCE
BY
WILLIAM C. LOUV
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1979
Digitized by the Internet Archive
in 2009 witfn unaing from
University of Florida, George A. Smathers
Libraries
http://www.archive.org/details/comparisonofmeth00louv
ACKNOWLEDGMENTS
I am indebted to Dr. Ramon C. Littell for his guidance and
encouragement, without which this dissertation would not have been
completed. I also wish to thank Dr. John G. Saw for his careful
proofreading and many helpful suggestions. The assistance of Dr.
Dennis D. Wackerly throughout my course of graduate study is greatly
appreciated.
My special thanks go to Dr. William Mendenhall who gave me the
opportunity to come to the University of Florida and who encouraged
me to pursue the degree of Doctor of Philosophy.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS . . . . . . . .. . . . . . iii
ABSTRACT . . . . . . . .. . . . . . vi
CHAPTER
I INTRODUCTION AND LITERATURE REVIEW . . . . . . 1
1.1 Statement of the Combination Problem . . . . 1
1.2 NonParametric Combination Methods . . . . . 2
1.3 A Comparison of NonParametric Methods . . . . 5
1.4 Parametric Combination Methods . . . . . . 8
1.5 Weighted Methods of Combination . . . . .. 11
1.6 The Combination of Dependent Tests . . . . .. 12
1.7 The Combination of Tests Based on Discrete Data . 13
1.8 A Preview of Chapters II, III, and IV . . . .. 18
II BAHADUR EFFICIENCIES OF GENERAL COMBINATION METHODS . . 19
2.1 The Notion of Bahadur Efficiency . . . . .. 19
2.2 The Exact Slopes for TA) and T() . . . . . 21
2.3 Further Results on Bahadur Efficiencies . . .. 26
2.4 Optimality of T(F) in the Discrete Data Case . .. 28
III THE COMBINATION OF BINOMIAL EXPERIMENTS . . . . .. 32
3.1 Introduction . . . . . . . . . . 32
3.2 Parametric Combination Methods . . . . . .. 33
3.3 Exact Slopes of Parametric Methods . . . . .. 37
3.4 Approximate Slopes of Parametric Methods . . .. 44
3.5 Powers of Combination Methods . . . . . .. 54
3.6 A Synthesis of Comparisons . . . . . . .. 57
(F)
3.7 Approximation of the Null Distributions of T ,
(LR) (ALR)79
T T.. . . . . . . .... . 79
IV APPLICATIONS AND FUTURE RESEARCH . . . . . ... 96
4.1 Introduction . . . . . . . . . . 96
4.2 Estimation: Confidence Regions Based on
Nonparametric Combination Methods . . . . .. 96
TABLE OF CONTENTS (Continued)
CHAPTER IV (Continued) Page
4.3 The Combination of 2 x2 Tables . . . . .. 110
4.4 Testing for the Heterogeneity of Variances ..... 113
4.5 Testing for the Difference of Means with
Incomplete Data . . . . . . . ... .115
4.6 Asymptotic Efficiencies for k.. . . . ... 116
BIBLIOGRAPHY . . . . . . . . ... .... .. .118
BIOGRAPHICAL SKETCH . . . . . . . . ... . . 122
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
A COMPARISON OF METHODS FOR COMBINING
TESTS OF SIGNIFICANCE
By
William C. Louv
August 1979
Chairman: Ramon C. Littell
Major Department: Statistics
Given test statistics X( ),...,X(k) for testing the null
hypotheses H1,...,Hk, respectively, the combining problem is to select
a function of X ,...,(k) to be used as an overall test of the hypoth
esis H = H n H2 n ... n Hk. Functions based on the probability integral
transformation, that is, the significance levels attained by X(1),...,X(k)
form a class of nonparametric combining methods. These methods are com
pared in a general setting with respect to Bahadur asymptotic relative
efficiency. It is concluded that Fisher's omnibus method is at least as
efficient as all other methods whether X(),...(k) arise from contin
uous or discrete distributions.
Given a specific parametric setting, it may be possible to
improve upon the nonparametric methods. The problem of combining binom
ial experiments is studied in detail. Parametric methods analogous to
the sum of chi's procedure and the CochranMantelHaenszel procedure as
well as the likelihood ratio test and an approximate likelihood ratio
test are compared to Fisher's method. Comparisons are made with respect
to Bahadur efficiency and with respect to exact power. The power
vi
comparisons take the form of plots of contours of equal power. If
prior information concerning the nature of the unknown binomial success
probabilities is unavailable, Fisher's method is recommended. Other
methods are preferred when specific assumptions can be made concerning
the success probabilities. For instance, the CochranMantelHaenszel
procedure is optimal when the success probabilities have a common value.
Fisher's statistic has a chisquare distribution with 2k degrees
of freedom when X(1),...,X(k) are continuous. In the discrete case,
however, the exact distribution of Fisher's statistic is difficult to
obtain. Several approximate methods are compared and Lancaster's mean
chisquare approximation is recommended.
The combining problem is also approached from the standpoint
of estimation. Nonparametric methods are inverted to form kdimensional
confidence regions. Several examples for k=2 are graphically displayed.
CHAPTER I
INTRODUCTION AND LITERATURE REVIEW
1.1 Statement of the Combination Problem
The problem of combining tests of significance has been studied
by several writers over the past fifty years. The problem is: Given
test statistics X(),...,X(k) for testing null hypotheses H1,...,Hk,
respectively, to select a function of X(,...,X(k) to be used as the
combined test of the hypothesis H = H1 n H2 n ... n k. In most of the
(i)
work cited, the X are assumed to be mutually independent, and, except
where stated otherwise, that is true in this paper.
Some practical situations in which an experimenter may wish to
combine tests are:
i. The data from k separate experiments, each conducted to
(1) (k)
test the same H, yield the respective test statistics X ,...,X
It is desired to pool the information from the separate experiments to
form a combined test of H. It would be desirable to pool the informa
tion by combining the X(i) if (a) only the X(i), instead of the raw
data, are available, if (b) the information from the ith experiment is
sufficiently contained in X or if (c) a theoretically optimal test
based on all the data is intractible.
th (i)
ii. The i of k experiments yields X to test a hypothesis
H., i = l,...,k, and a researcher wishes to simultaneously test
1
the truth of l ,..., 'k. Considerations (a), (b), and (c) in the preced
ing paragraph again lead to the desirability of combining X) as a test
of H = H1 n ... n Hk.
iii. A simultaneous test of H = H1 n ... n Hk is desired, and the
data from a single experiment yield X( ) ,X) as tests of H, ...,Hk,
respectively. Combining the X(i) can provide a test of H.
In Section 1.2 several nonparametric methods of combination
are introduced. A literature review of comparisons of these procedures
is given in Section 1.3. The remainder of this chapter is primarily
a literature review of more specific aspects of the combination problem.
We make some minor extensions which are identified as such.
1.2 Nonparametric Combination Methods
(i)
Suppose that H. is rejected for large values of X Define
1
L(i) = 1 F.(X(i), where F. is the cumulative distribution function
of X under H.. If X ) is a continuous random variable, then L)
1
is uniformly distributed on (0,1) under H.. Many of the wellknown
1
methods of combination may be expressed in terms of the L Such
methods considered here are:
(1) T(F) 2QenL) (Omnibus method, Fisher [13])
(2) T(N) =_Z1(L(i)) (Normal transform, Liptak [26])
(3) (m) = min (i) (Minimum significance level, Tippett [42])
(4) T = max L (Maximum significance level, Wilkinson [44])
(5) T() = 21Ln(l L(i) (Pearson [361)
(6) T(A) EL(i) (Edgington [12]).
As the statistics are defined here, H is rejected when large values are
observed. Figure 1 (page 4) shows the rejection regions for the sta
tistics defined above when k = 2.
In the continuous case, the null distributions of these statis
tics are easily obtained. They are all based upon the fact that the
L are uniformly distributed under H. It is easily established that
this is true. The cumulative distribution function for L(i) is
P{L < } = P{1 F(x) < }
= 1 P{F(x) < 1 I }
l
= 1 P{x < F (1 )}
= 1 F{F(1 ()}
= 1 (1 9) = 9.
That T(N) has a normal distribution with mean 0 and variance k
follows trivially. The statistics T) and T are seen to be based
on the order statistics of a uniform random variable on (0,1) and
therefore distributed according to beta distributions.
(P) (F)
That T and T(F) are distributed as chisquares on 2k degrees
of freedom is established as follows. The probability density function
(i)
of L is
fL(k) = I(0,1)(P)
Let S = 2ZnL. Then
S/2 dL 1 S/2
L' dS 2
It follows that
dL 1 S/2
f (S) = f (S) = e I (2 (S).
Edgington's statistic, T(A), is a sum of uniform random variables.
As shown by Edgington, significance levels can be established for
L(2) L(2)
T(F) L) (N
L(2) L(2)
() L
L
(2)
T(m) L 1(M)
0)2)
T(D
T (P) T(A)
Figure 1. Rejection Regions in Terms of the Signific
for k=2.
ance Levels
)
values of TA on the basis of the following equation [12]:
k k k k k
(A) t k (tl) k (t2) k (t3) k (tS)
P{T >t} = ( ) k ( ) ( ) k .+ (S) (1.1i)
k! 1 k! 2 k! 3 k! S k!
where S is the smallest integer greater than t.
1.3 A Comparison of Nonparametric Methods
The general nonparametric methods of combination are rules
prescribing that H should be rejected for certain values of
(L (), L(2),..., L(k)). Several basic theoretical results for non
parametric methods of combination are due to Birnbaum [7]. Some of
these results are summarized in the following paragraphs.
(i)
Under H., L is distributed uniformly on (0,1) in the con
1
(i)
tinuous case. When H. is not true, L is distributed according to
1
a nonincreasing density function on (0,1), say gi(L(i)), if X(i)
has a distribution belonging to the exponential family. Some overall
alternative hypotheses that may be considered are:
(i)
H : One or more of the L 's have nonuniform densities
gA
gi (L(i)).
HB: All of the L(i)'s have the same nonuniform density g(L).
HC: One of the L(i) s has a nonuniform density g (L()).
H is the appropriate alternative hypothesis in most cases where prior
knowledge of the alternative densities g (L(i)) is unavailable [7].
The following condition is satisfied by all of the methods
introduced in Section 1.2.
Condition 1: If H is rejected for any given set of L 's,then
it will also be rejected for all sets of L(i)*'s such that L (i) L)
for each i [7].
It can be shown that the best test of H versus any particular
alternative in HA must satisfy Condition 1. It seems reasonable,
therefore, that any method not satisfying Condition 1 can be elim
inated from consideration [7].
In the present context, Condition 1 does little to restrict
the class of methods from which to choose. In fact, "for each non
parametric method of combination satisfying Condition 1, we can find
some alternative H represented by nonincreasing functions
gl(L(1) ,...,g k(L(k) against which that method of combination gives
a best test of H" [7].
It should be noted that direct comparison of general combining
methods with respect to power is difficult in typical contexts. The
precise distributions of the gi(L(i)) under the alternative hypothesis
are intractible except in very special cases.
(i)
When the X have distributions belonging to the oneparameter
exponential family, the overall null hypothesis can be written H:
(1) = (1),.. (k) (k)
S= 0 0 Rejection of H is based upon
0 0
(X(1),...,X(k) It is reasonable to reject the use of inadmissible
tests. A test is inadmissible if there exists another test which is
at least as powerful for all alternatives and more powerful for at
least one alternative. Birnbaum proves that a necessary condition for
the admissibility of a test is convexity of the acceptance region in
the (X(1),...,X(k)) hyperplane. For X(i) with distributions in the
exponential family, T) and T( do not have convex acceptance regions
and are therefore inadmissible [7].
Although Birnbaum does not consider Edgington's method, we see
that it is clear that T(A) must also be inadmissible. For instance,
for k=2, consider the points (O,c), (c,O), and (c/2,c/2) in the
(L(1),L(2)) plane which fall on the boundary of the acceptance region
T(A) > c. The points in the (X(1),X(2) plane corresponding to (O,c)
and (c,O) would fall on the axes at o (and "). The point correspond
ing to (c/2,c/2) certainly falls interior to the boundaries described
by the points corresponding to (c,0) and (O,c). The acceptance region
can not, therefore, be convex and hence T is inadmissible. This
argument is virtually the same as that used by Birnbaum to establish
(PM (M)
the inadmissibility of T and T
For a given inadmissible test it is not known how to find a
particular test which dominates. Birnbaum, however, argues that the
choice of which test to use should be restricted to admissible tests.
The choice of a test from the class of admissible tests is then contin
gent upon which test has more power against alternatives of interest [7].
In summary of Birnbaum's observations, since T( and T do
not in general form convex acceptance regions in the (X(1),...,X(k))
hyperplane, they are not in general admissible and can be eliminated as
viable methods. We can extend Birnbaum's reasoning to reach the same
conclusion about T By inspecting the acceptance regions formed by
the various methods, Birnbaum also observes than T) is more sensitive
(F)
to HC (departure from H by exactly one parameter) that T(F). The test
(F)
T ( however, has better overall sensitivity to HA [7].
Littell and Folks have carried out comparisons of general non
parametric methods with respect to exact Bahadur asymptotic relative
efficiency. A detailed account of the notion of Bahadur efficiencies
is deferred to Section 2.1.
In their first investigation [26], Littell and Folks compare
T(F) T(N), T and T(m). The actual values of the efficiencies are
(F)
given in Section 2.3. The authors show that T(F) is superior to the
other three procedures according to this criterion. They also observe
that the relative efficiency of T(m) is consistent with Birnbaum's
observation that T performs well versus HC.
(F)
Further, Littell and Folks show that T with some restric
tions on the parameter space, is optimal among all tests based on the
X as long as the X) are optimal. This result is extended in
(F)
a subsequent paper [28] by showing that T(F) is at least as efficient
as any other combination procedure. The only condition necessary for
this extension is equivalent to Birnbaum's Condition 1. A formal state
ment of this result is given in Section 2.3.
1.4 Parametric Combination Methods
(F)
The evidence thus far points strongly to T(F) as the choice
among general nonparametric combination procedures when prior knowledge
of the alternative space is unavailable. When the distributions of the
X(i) belong to some parametric family, or when the alternative param
(F)
eter space can be characterized, it may be possible that T(F) and the
other general nonparametric methods can be improved upon. A summary
of such investigations follows.
Oosterhoff [33j considers the combination of k normally dis
tributed random variables with known variances, and unknown means
P1l. 2,... k. The null hypothesis tested is H; P = )2 = ... = k 0
versus the alternative HA :i > 0, with strict inequality for at least
A i
one i. lie observed that many combination problems reduce to this
situation asymptotically. The difference in power between a particular
test and the optimal test for a given (p 2" ,...,i k) is called the short
coming. Oosterhoff proves that the shortcomings of T(F) and the maximum
likelihood test go to zero for all (l' .. k) as the overall signif
icance level tends to zero. The maximum shortcoming of the likelihood
(F)
ratio test is shown to be smaller than the maximum shortcoming of T
Oosterhoff derives a most stringent Bayes test with respect to
a least favorable prior. According to numerical comparisons (again
with respect to shortcomings), the most stringent test performs sim
ilarly to the likelihood ratio test. The likelihood ratio test is much
easier to implement than the most stringent test and is therefore pre
ferable. Fisher's statistic, T(F) is seen to be slightly more powerful
than the likelihood ratio test when the means are similar; the opposite
is true when the means are dissimilar. A simple summing of the normal
variates performs better than all other methods when the means are very
similar [33].
Koziol and Perlman [20] study the combination of chisquare
(i) 2
variates X ~ X (6.). The hypothesis test considered is H:
Vi 1
6 = .. = = 0 vs HA: 6.i 0 (strict inequality for at least one i)
where the 6. are noncentrality parameters. The V. correspond to the
1 1
respective degrees of freedom. An earlier Monte Carlo study by
Bhartacharya [6] also addressed this problem and compared the statistics
T(F), T(m), and EX(i) Bhattacharya concluded that X(i) and TF) were
almost equally powerful and that both of these methods clearly dominated
T(m). Koziol and Perlman endeavor to establish the power of T( and
EXi) in some absolute sense. To do this, they compare T(F) and EX(
to Bayes procedures since Bayes procedures are admissible and have
good power in an absolute sense [20].
(i)
When the v. are equal, EX is Bayes with respect to priors
giving high probability to (01,...,9k) central to the parameter space
(Type B alternatives). The test Eexp{ X ) is Bayes with respect
to priors which assign high probability to the extremes of the param
eter space (Type C alternatives). For unequal v.'s the Bayes tests
i
have slightly altered forms. The Bayes procedures are compared to
T(F) T(m) and T(N) for k=2 for various values of (1,' 2) via numerical
tabulations and via the calculation of power contours.
(i)
The statistic T is seen to have better power than the other
tests for Type C alternatives but performs rather poorly in other situa
tions. The Bayes test performs comparably to T for Type C alterna
(m)
tives and is much more sensitive to Type B alternatives than T .The
(N)
statistic, T is relatively powerful over only a small region at the
(N)
center of the parameter space. The statistic, T is seen to be dom
inated by some other procedure for each value of k investigated. The
(F) (i) (F)
statistics, T and EX are good overall procedures, with T more
sensitive to Type C alternatives and EX more sensitive to Type B
alternatives. For v E 2, T(F) is more sensitive to Type B alternatives
than EXi) is to Type C alternatives and T(F) is therefore recommended.
The opposite is true for v = 1. These observations were supported for
k>2 through Monte Carlo simulations.
Koziol and Perlman also consider the maximum shortcomings of
the tests. In the context of no prior information, they show that T(F)
minimizes the maximum shortcomings for vi. 2 while X minimizes the
maximum shortcoming for v. =1. An additional statistic can be consid
ered when vi = 1. It is T X) = Z(X(i) M the sum of chi's procedures.
For k=2, T is powerful only for a small region in the center of the
parameter space. For large k, the performance of T(X) becomes progres
sively worse. It can be said that T(X) performs similarly to T(N)
1.5 Weighted Methods of Combination
Good [14 suggests a weighted version of Fisher's statistic,
(G) (i)
T( = i .nL He showed that, if the Ai are all different, signif
icance probabilities can be found by the relationship
(C k
P{T( > x) = E A exp(x/A )
r=l
where
k1
A (r
r 1 r 2 r rl r r+l r k
Zelen [45] illustrates the use of T(G) in the analysis of
incomplete block designs. In such designs, it is often possible to
perform two independent analyses of the data. The usual analysis
(intrablock analysis) depends only on comparisons within blocks. The
second analysis (interblock analysis) makes use of the block totals
only. Zelen defines independent Fratios corresponding to the two
types of analysis. The attained significance level corresponding to
the interblock analysis is weighted according to the interblock effi
ciency which is a function of the estimated block and error variances.
A similar example is given by Pape [34 Pape extends Zelen's
method to the more general context of a multiway completely random
design.
Koziol and Perlman [20] also considered weighted methods for
the problem of combining independent chisquares. They conclude that
when prior information about the noncentrality parameters is available,
increased power can be achieved at the appropriate alternative by a
(i)
weighted version of the sum test, Eb.X if v. > 2 for all i and by
1 1
(G)
the weighted Fisher statistic, T when v i 2 for all i.
1.6 The Combination of Dependent Tests
The combinations considered up to this point have been based
on mutually independent L) arising from mutually independent statis
tics, X As previously indicated, in such cases, the functions of
the L) which comprise the general methods have nulldistributions which
are easily obtained. When the Xi (and thus the L ) are not inde
pendent the null distributions are not tractible in typical cases.
Brown [9] considers a particular example of the problem of
combining dependent statistics. The statistics to be combined are
assumed to have a joint multivariate normal distribution with known
covariance matrix f and unknown mean vector (ul ,2 2,..., k)'. The hypoth
esis test of interest is H: p. = versus H: p > p. (strict in
1 10 1 i 10
equality for at least one i). A likelihood ratio test can be derived
[31], but obtaining significance values from this approach is difficult.
Brown bases his solution on T The null distribution of T(
is not chisquare on 2k degrees of freedom in this case. The mean of
(F)
T is 2k as in the independent case. The variance has covariance
terms which Brown approximates. The approximation is expressed as
a function of the correlations between the normal variates. These
first two moments are equated to the first two moments of a gamma dis
tribution. The resultant gamma distribution is used to obtain approx
imate significance levels.
1.7 The Combination of Tests Based on Discrete Data
As noted in previous sections, the literature tends to support
(F)
T as a nonparametric combining method in the general, continuous
data framework. Those authors who have addressed the problem of com
(F)
bining discrete statistics have utilized T(F) assuming that the opti
mality properties established in the continuous case are applicable.
The problem then becomes one of determining significance
(F)
probabilities since T(F) is no longer distributed as a chisquare on
2k degrees of freedom. We describe the problem as follows. Suppose
(i)* (i)*
L derives from a discontinuous statistic, X and that a and b
are possible values of L(i)*, < a < b 1, such that a < L
is impossible. For a < E < b, P{L (i) } = P{L a} = a.
If L(i) derives from a continuous statistic, X(i), then P{L (i) =
(i)* (i)
Since a < &, L is stochastically larger than L It follows that
Fisher's statistic is stochastically smaller in the discrete case than
(F)
in the continuous case. The ultimate result is that if T(F) is com
pared to a clisquare distribution with 2k degrees of freedom when the
data are discrete, the null hypothesis will be rejected with too low
a probability.
When k, the number of statistics to be combined, is small and
(nl,n2 ...,nk), the numbers of attainable levels of the discrete sta
tistics are small, the exact distribution of T(F) can be determined.
Wallis [43] gives algorithms to generate null distributions when all
of the X(i) are discrete and when one X( is discrete. Generating
null distributions via Wallis' algorithms becomes intractible very
quickly as k and the number of attainable levels of the X) increase.
The generation of complete null distributions is even beyond the capa
bility of usual computer storage limitations in experiments of modest
size. The significance level attained by a particular value of T(F)
can be obtained for virtually any situation with a computer, however.
A transformation of T(F) which can be referred to standard tables is
indicated.
A method suggested by Pearson [37] involves the addition, by
a separate random experiment, of a continuous variable to the original
discrete variable and thus yielding a continuous variable. Suppose X(i)
can take on values 0, 1, 2,..., n. with probabilities p0 p ,..., p
ni 1 0 1 ni
Let P() = E p Note that P(i)
J x x ni ni1 1 0
If the null hypothesis is rejected for large values of X(i), then the
P j = 0,1,2,...,n are the observable significance levels for the
.th
i test; i.e., the observable values of the random variable
L(i) = FX(i) 1) under the null hypothesis. Denote by U(i)
i = 1,2,...,k, mutually independent uniform random variables on (0,1).
Pearson's statistic is defined as
L(i)(i),(i)) = L(i)(i) U) P{X)}.
P
We now establish that Lp i (X(),U i) is uniformly distributed
on (0,1) if and only if X (i) and U() are independent and Ui) is uni
formly distributed on (0,1). Omitting the superscripts for convenience,
define the random variable (X,U) by Lp = L(X) U P{X} where 0 < U < 1.
It follows that
P{X = x, U u} = P{L(x) u P(x) < Lp < L(x)}
=uP
x
= P{X = x} P{U < u}.
The statistic, 2EnL (, thus has an exact chisquare distri
p
bution with 2k degrees of freedom. Exact significance levels can be
determined for any combination problem. The concept of randomization
to obtain statistics with exact distributions has been debated by
statisticians. That a decision may depend on an extraneous source of
variation seems to violate some common sense principle. Pearson [37]
argues, however, that his randomization scheme is no more an extran
eous source of variation than is the a priori random assignment of
treatments to experimental units.
Lancaster [21] considers a pair of approximations to T(F)
Although Lancaster does not consider Pearson's method, the statistics
he introduces can be expressed in terms of the L(i) [19]:
P
i. Mean chisquare (X2):
m
E(22nL(i) = (2nL ) du
p p
= 2 {L i)(X)nL (X)
L )(X+l)nL )(X+l) }/P(X). (1.3)
'2
ii. Median chisquare (X ):
m
Median (2knL ) = 2Zn{L ( (X) + L(i)X+l)} if L (X+l) # 0
p 2
= 2 2 nL( X) if L(i)(X+) = 0.
2 2
The expectation of X is 2. The variance of X is slightly less
m m
than 4. The median chisquare is introduced because of its ease of
calculation. With the ready availability of pocket calculators with
in functions, this justification no longer seems valid. The expecta
'2 '2
tion of X is less than 2. The alternate definition of X for when
m m
(i)
L (X+1) = 0 is intended to reduce the bias (without increasing the
difficulty of calculations) [21].
David and Johnson [10] undertake a theoretical investigation of
'2
the distribution of X They prove that as n, the number of attainable
m
levels of X(i) (and hence L (i)), increases without limit, the moments of
'2
X converge to those of a chisquare distribution on 2 degrees of
m
freedom. We obtain a similar result for X by adapting David and John
m
'2
son's proof for X (superscripts are omitted for convenience). From
m
2
the definition of X given in equation (1.3), it follows that
m
2 I b1
lim E(X2)b = lim E( (2gnL )du)b
no m p
P(xi) 0
(j =1,2,...,n.)
1
Then,
2nn
lim E(X b = lim E P(x.)[ (2LnL )du]
m j=l J P
b n l b
= (2) lim E P(x.)[ n(L(x.) uP(x.))du]
j=l J 0 j J
= (2) lim EP(x.)[ {(L(x) uP(x) 1)
lim E(X2) b 1 (L(x.) uP(x.)l)2 + ...}du]b
m (2 +
1 2 1 3
(Note: kn(a) = (a1) (a1) + (a) ...)
2 3
Upon performing the integration,
lim E(X2) = (2)b lim P(x.)[(L(x.) P(x.) )
m 3J 2 j
22 1
+ ([L(x.) 1] 2 P(x.)(L(x.) 1)
1 2 b
3 J
3 [P(xj +
1b
Since all of the terms in the expansion of [ n(L(x.) uP(x))du]
0J
are multiplied by P(x.) it is evident that u(P(x.)) gives rise only to
secondorder terms in P(x.) which can be ignored in the limit. The
limit thus reduces to
n
b b
(2) lim E P(x.)[ knL(x.)du]
n 0 i=l
= (2)b lim EP(x.)[ZnL(x.)]
b b
= (2)b [nL(x)]b dL(x).
0
Letting Y = ZnL(x) yields
bY b b
(2)b(l)b e Y dY = 2bb!
0
which gives the bth moment of a chisquare distribution with 2 degrees
of freedom.
The convergence of moments does not in general imply convergence
in distribution. However, if there is at most one distribution function
F such that lim X dF = X dF, then F F in distribution [8].
n n n
Since the chisquare distribution is uniquely defined by its moments,
2 2
it follows that X X in distribution.
m 2
1.8 A Preview of Chapters II, III, and IV
In Section 2.1, the notion of Bahadur asymptotic relative
efficiency is introduced. In Section 2.2, we derive the Bahadur exact
(A) (P)
slopes for T and T The results due to Littell and Folks men
tioned in Section 1.3 are summarized in detail in Section 2.3. In
Section 2.4, we extend the optimality property for T(F) given by Littell
and Folks to the discrete data case.
Chapter III deals with a particular combination problem: the
combination of mutually independent binomial experiments. Fisher's
(F)
method, T is compared to several methods which are based directly
on the X) (rather than the L ). Comparisons are made via both
approximate and exact slopes in Sections 3.3 and 3.4. The tests are
also compared by exact power studies. These results are given in
Section 3.5. A summary of the tests' relative performances follows.
Recommendations as to the appropriate use of the methods are given.
Section 1.6 describes some proposed approximations to the null
(F)
distribution of T In Section 1.7, these methods are shown to be
less reliable than might be expected. Alternative approaches are also
o'valuated.
In Section 4.1, the combination problem is approached from the
standpoint of estimation. Confidence regions based upon T(F) and T(
are derived. The remainder of Chapter IV introduces future research
problems which are related to the general combination problem.
CHAPTER II
BAHADUR EFFICIENCIES OF GENERAL COMBINATION METHODS
2.1 The Notion of Bahadur Efficiency
Due to the intractibility of exact distribution theory, it is
often advantageous to consider an asymptotic comparison of two compet
ing test statistics. In typical cases, the significance level attained
by each test statistic will converge to zero at an exponential rate as
the sample size increases without bound. The idea of Bahadur asymp
totic relative efficiency is to compare the rates at which the attained
significance levels converge to zero when the null hypothesis is not
true. The test statistic which yields the faster rate of convergence
is deemed superior. A more detailed definition follows.
Denote by (Y1,Y2,...) an infinite sequence of independent
observations of a random variable Y, whose probability distribution P0
depends ona parameter 6 E0.
Let H be the null hypothesis H: 6 E0 and let A be the alterna
tive A: 6 E0 00. For n = 1,2,..., let X be a real valued test
0 n
statistic which depends only on the first n observations Y1,..., n'
Assume that the probability distribution of X is the same for all
6 0 0. Define the significance level attained by X by
n
Li =1F (X ) where F (x) =P IX < x}. Let {X }=X X ,....,X ,...}
n n n n 0 n n 1 2' n
denote an infinite sequence of test statistics. In typical cases, there
exists a positive valued function c(6), called the exact slope of {X },
such that for 6e 0O.
(2/n) Zn L c(6)
n
with probability one [0]. That is,
2
P{lim n L = c(O)} = 1.
n n
If {x(n)} and {X(2) have exact slopes cl(0) and c2(0), respectively
n n 1 2
then the ratio 12(0) = c (0)/c2(0) is the exact Bahadur efficiency of
{X(1)} relative to {X(2).
n n
An alternative interpretation of 12(0) is that it gives the
limiting ratio of sample sizes required by the two test statistics to
attain equally small significance levels. That is, if for E >0, N(i)()
(i)
is the smallest sample size such that L < for all sample sizes
n
n N (i), i = 1,2, then as C tends to zero [3]
(2)
lim N= ().
N (1E) 12
The following theorem due to Bahadur [3] gives a method for
calculating exact slopes. A proof is given by I. R. Savage [40].
Theorem 1. Suppose {T } is a sequence of test statistics which
n
the following two properties:
1. There exists a function b(6), O
T //nb(O) with probability one [0].
n
2. There exists a function f(t), O
continuous in some open set containing the range of
b(O) such that for each t in the open set
1
i n[lF (/n t)]f(t), where F (t) = 1P (T n t).
n n n o n
Then the exact slope of {T } is c(O) = 2f(b(9)).
n
satisfies
2.2 The Exact Slopes for T() and T(P)
In terms of the above discussion the general combination problem
can be defined as follows. There are k sequences {X (1),...,{X(k) of
n1 nk
statistics for testing H: 0 E00. For all sample sizes nl,...,n k the
statistics are independently distributed. Let Li) be the level attained
1
(ii
by X i = 1,2,...,k.
n.
n.
1
slope c.(0); that is
2 (i)
ZnL c.(O) as n. c
n. n. 1i
1i
with probability one [0]. Assume also that the sample sizes nl,...,nk
n.
satisfy n + ... + n = nk and lim = A, i = 1,...,k. Then
n +o
S+ ... = k and
1 k
2 (i)
ZnL c.( () as n m
n n. i i
1
with probability one [9]. As defined here, n can be thought of as the
average sample size of the k tests.
Two general combining methods introduced in Section 1.2 are
T (A) Edgington's additive method, and T Pearson's method. Deriva
tions of the exact slopes of T(A) and T(P) follow.
k
(A) 2 (i)
Proposition 1. Let T = n Z L The exact slope of
n /n i=l n
(A)
T is k min (c.(e)).
(A)
Proof: This proof requires the above definition of T although
n
T(A) = L(i) is a more obvious definition and is the form given in
n ni
Section 1.2 where the nonparametric methods are introduced. Nothing
is lost, however, since equivalent statistics yield identical exact
slopes. Proposition 1 is proved by using Theorem 1 (BahadurSavage).
The first step is to establish b(O) of Part 1 of Theorem 1. To accom
plish this, first suppose that
Acl(6) = min {(.c.()}.
1 i 1 1
It follows that for all > 0, there exists
n >N,
N=N(E) such that for
with probability
with probability
with probability
2 (1) 2 (i)
Qn L __ knL
n n n ni
one [6]. Then, for n >N,
L(1) (i)
n1 ni '
one [6]. It follows that
k
(1) < L(i)
n1 i=l ni n1
one [6]. Thus, for n N,
i=1,2,...,k,
i = 1,2,...,k,
for n N,
2n L(1) 2 n i) 2 n kL1)
n n n n. n nk
n n n n. n n
1 1
with probability one [0]. It follows that as n tends to infinity
2 (1) 2 (i) 2 (1)
lim n L 2 lim Zn EL lim n kL
n n n n. n n
with probability one [0]. Hence,
( >2 n
Alcl(6) > n in
n. 1 1
i
1
with probability one [0] since, as n tends to infinity,
2 (1)
lim 2 n kL(1)
n nl
= lim {2 Zn L()
n nl
 k n k} = lim nL(1)
n n nl
with probability one [0]. Thus,
T(A)
n  c (0)
r 11
with probability one [0].
The choice of A1 c(0) as the minimum was
arbitrary. Hence,
(A)
n
J^
+ min A.c.(9)
i 1
giving b(9) of Part 1 of Theorem 1. Now, as n tends to infinity,
lim  n[l F (vn t)]
n n
1 2 (i) t
= lim r n P{2 Zn L(i) /n t
n /n n.
n n. 2
1
= lim r n [exp ( k)/k!)].
n 2
The last equality if true since t is positive and therefore
nk is less than one.
exp ( ) is less than one.
lim I n[l F nn t]
n n
tk 1 tk
lim [ + 1 Yn k!] = 
2 n 2
This gives f(t) of Part 2 of Theorem 1. Thus, from Theorem 1,
C (0) = 2f(b(O)) = kmin A.c.(8).
A 1 1
Proposition 2. Let T(P = n[E n(l L )]. The exact
n F n.
n ni
slope of T is C (6)=k min A.c.(9).
n P 1i
The form of Pearson's statistic given in this proposition is
equivalent to the form given in Section 1.2. The proof of Proposition 2
entails use of the BahadurSavage Theorem (Theorem 1). The derivation
of b(0) of Part 1 of the theorem parallels the derivation of b(9) in
Proposition 1. In order to establish f(t) for part two of the theorem,
a result due to Killeen, Hettmansperger, and Sievers [18] is required.
They show that under broad conditions,
1 n f (V'n t) I n P{X Fn t} = o(l) (2.1)
n n n n
as n> where f (t) is the density function of X .
n n
Proof of Proposition 2. By a firstorder Taylor series approximation,
E n(l L (i) = E L + E I L)
n. I. n n.
1 1 1
(i)
where +0 as L 0 for all i. It follows that as n tends to infinity,
n n.
I
T
lim  = lim 2 kn[EZ n(l L(i)]
An n ni
= lim 2 kn[ L(i)]
n n.
1
= min X c (O).
i i
The last step is shown in Proposition 1. Now, to derive f(t) of
Part 2 of Theorem 1 via equation (2.1), note that the U( =1Li)
n. n.
1 i
are mutually independent uniform random variables on (0,1). Letting
V = n i it follows that
n. n.
1 1
v
f (v) = e < <
(i)
n.
1
Letting W =E V = EZn U it follows directly that
n n. n.
1 1
1f () kl w
f (w) = (k w e
n
since V is a gamma with parameters (11) Now, letting
1
f (y) = exp{ky e } (2.2)
n A
It follows that, as n tends to infinity
1 (P)
lim 1 n P{T /n t}
n n
1 2
= lim  n P{ Y > n t}
n /n n
= lim i pn P{Y > 1t
n n 2
1 ut
= lim n n fy () (2.3)
n Y 2
n
by the result given in equation (2.1). Substituting (2.2) into (2.3),
we find that
1 1 nt nt
lim n[ exp{k() exp( )}]
n 2 2
S1 knt nt
=lim [ exp( )]
n 2 2
kt 1 nt tk
= lim [k + exp( )] = t
2 n 2 2
Applying the result of Theorem 1,
C (6) = 2f(b(O)) = k min X.c.().
i
2.3 Further Results on Bahadur Efficiencies
Bahadur exact slopes are derived in Littell and Folks [26] for
other general combining methods. Their results are summarized below.
Test Exact Slope
(F)
T c F(6) = X Z.c.(6)
T(N) c, (0) E c (0)) 2
N k i i
(A) (P) [Y(i
i
(m)
T c. (0) = max ).c.(0)
m 1i 1
Thus T and T have the same exact slope as T The relationship
among these quantities is displayed in Figure 2.
X2 C2
=(i +V2)2XIOC
2C2= 2X ci
XI c = X2C2
TI'
Iy
Note: I and TI
IT and Y :
UI and _F:
CM
CM < Cm < CF
Cm
Figure 2. Relative Sizes of Exact Slopes for k=2.
The optimality property of T(F) established by Littell and
Folks [28] is mentioned in Section 1.3. The detailed result is given
here as a theorem.
Theorem 2. Let T be any function of T ,...,T(k) which is
n n1 nk
(i) ,
nondecreasing ineach of the T ', that is, t1 < t',...,tk tk
1
implies T (t ,...,t ) < T (tl ....t). Then the exact slope c(6) of
T satisfies c(O) < Z A.c.(6).
n 1 1
The nondecreasing condition is equivalent to Birnbaum's
Condition 1. (See Section 1.3.) The condition is not very restric
tive; it is satisfied by any method thus far introduced and by virtually
all other reasonable statistics.
(F)
2.4 Optimality of TF) in the Discrete Data Case
Littell and Folks established that the exact slope for T(F)
is c F() = E A c .(). This derivation is contingent on the fact that
F 1 1
(F)
T has a chisquare distribution on 2k degrees of freedom. This is
true only when the X(i) are distributed according to continuous distri
butions. A proof that the exact slope of T(F) is Z A.c.(9) when the X(i)
1 1
are discrete follows.
Suppose X(i) can take on values 0, 1, 2,...,n. with probabil
n. 1
1
nn
1
ities pO, p ...,p Let P i= p Note that
0 1 n. j x
iJ x=j
1 x=j
(i) < i) < < (i)
n. n.l 1 0
1 1
(i) (i)
for large values of X ), then the P. j = 0,1,2,...,n. are the
n. j
1
th
observable significance levels for the i test; that is, the observ
(i) (i)
able values of random variable L = 1 F(X 1) under the null
n. n.
1i
hypothesis. Assume that an exact slope exists for all tests; that is,
assume that there exist functions c l(), c2(0),...,c k() such that
2 (i)
 nL c.(0) as n 
n. n. I i
1 1
with probability one [6] for i = 1,2,...,k.
(d) (i) i
Proposition 3. Let T = (2 EZ n L )2. If
II n
1
lim  n[l F (d)n t)] exists then the exact slope of T(d) is
n n n
n o
k
cd(0) = Z c (0).
i=l
Proof: This proof utilizes the BahadurSavage Theorem (Theorem 1).
To establish the first part of Theorem 1, observe that
(1) (k)
(d) 2 (nL) 2nnL) ;
T n_ _
n n n
(A cl(a) + ... + AkCk(0))k as n + om
with probability one [0]. Consistent with the notation of Theorem 1,
denote this limiting quantity bd(0).
Now, to establish f(t) of Part 2 of Theorem 1, choose
Z.c (0,1), i = 1,2,...,k. For each i, there exists a j such that
1 [P(i) (i)
i j Pj1]
(i)
Now, since L is a discrete random variable,
n.
i
P{(i) .) = P{L (i)
n i n. j 1
1 1
Thus
P{L() e .)} P{u( ) M .}
n. i i
1
(ii
where the U are mutually independent uniform random variables on
(0,1). It follows that
P{21nL(i) >_ } < P{2n U(i) > .},
n. 1 1
1
and hence that
k k
P{ Z 2nL) >z"} P{ E 2n U(i) "}.
i=l i i=l
Thus,
1 Mi)
1 n P{[E 2nL (i)] t} n P{[ 2Zn U (i) > n t}. (2.4)
n n. n
1
The quantity Z = [Z 2Rn U (i) is distributed as the square root of
n
a chisquare with 2k degrees of freedom. It follows that the density
of Z is
n 2
f (Z 1 2k1 Z2/2
fZ (Z) k Z e
n 2 F(k)
Thus, from the result given in (2.1), the limit of the righthand side
of (2.4) can be written as
1
lim 4n f (Wn t)
n Z
Sn 2k1 1 2 n
= lim n [ (/n t)2 exp( (/n t) )]
n 21 F(k)
1 1 2
=lim  [(2k1) kn(n t)  nt]
n 2
nco
1 2
t
2
Hence, it follows from (2.4) that
1 2
f (t) > t ,
since it is assumed that the limit of the lefthand side exists.
Applying the result of Theorem 1,
Cd(e) ZiA.c.i ().
d 11
By Theorem 2,
Cd(X) 11().
Hence,
cd(9) = ic. (e).
That is, the exact slope of T(F) is EX.c.(6) regardless of whether the
11
X are continuous or discrete. The condition imposed in Proposition 3
that lim _. n[lF (d) (/n t)] exist is not very restrictive. It is
n n
n co
satisfied in most typical cases and in every example considered in this
dissertation.
CHAPTER III
THE COMBINATION OF BINOMIAL EXPERIMENTS
3.1 Introduction
Chapter III deals with the combination of binomial experiments.
That is, suppose k binomial experiments are performed. Let nl,n2,... ,nk
(1) (2) (k)
and X X ...,X he the sizes and the observed numbers of suc
n1 n2 nk
cesses, respectively, for the experiments. Denote the unknown success
probabilities as pl,p2,...,pk. Suppose one wishes to test the overall
null hypothesis H: pl = P10' p2 = P20' ". = PkO versus the alter
native hypothesis HA : p > p10' .. k PkO (with strict inequality
for at least one p.). The problem, then, is to choose the best function
of (X ... X (k) for this hypothesis test.
n I nk
1 k.
(F)
The results of Chapters I and II support T(F) as a non
parametric method with good overall power when there is no prior infor
mation concerning the unknown parameters. The method based on the
minimum significance level, T is sensitive to situations where
exactly one of the individual hypotheses is rejected. That Is, T) is
powerful versus the alternative IC of Section 1.3.
The investigations of Koziol and Perlman [20] and Oosterhoff [33]
show that the general nonparametric combining methods can be improved
32
on for certain parametric combining problems. It follows that there
may be combination methods based directly on (X ... ) that are
n nk
1 k
superior to Fisher's omnibus procedure.
Chapter III is a detailed comparison of T(F) and several
parametric combination methods.
3.2 Parametric Combination Methods
As stated in Section 1.3, no method of combination is most
powerful versus all possible alternative hypotheses. There are, however,
certain restricted alternative hypotheses against which most powerful
tests do exist.
th
Let the likelihood function for the i binomial experiment
be denoted by
n. (i) n.X(
L(p) = ( i)pX (1 p.) (3.1)
X
According to the NeymanPearson Lemma, if a most powerful test of the
null hypothesis H: pi = Pi, all i, versus the alternative hypothesis
HA: pi > io (with strict inequality for at least one i) exists, it is
to reject H if
k L(p )
iO
71 < C.
i=l L(
Upon substituting (3.1) and taking logs, an equivalent form of the test
is to reject H if
EX() M n{Pi(l Pi )/Pi (lPi)} > C. (3.2)
It follows that rejecting H when
SXi) > C (3.3)
is most powerful if p.(l p i)/p i(1p.) is constant on i.
1 i iO 1
The problem of combining 2 x2 contingency tables is closely
related to the problem being considered. The purpose of each 2x2 table
can be interpreted as testing for the equality of success probabilities
between a standard treatment and an experimental treatment. The overall
null hypothesis is that the experimental and standard success probabil
ities are equal in all experiments. The overall alternative hypothesis
is that the experimental success probability is superior in at least
one experiment.
Cochran [LO] and Mantel and Haenszel [29] suggest the statistic
E widi//Ew.p.qi (3.4)
where
S= nilni2/(nil + ni2) d = Pil Pi2
for combining 2x2 tables. Mantel and Pasternak [34] have discussed
this statistic in the context of combining binomial experiments. Each
individual binomial experiment is similar to an experiment resulting
in a 2x2 table, two cells of which are empty because the control
success probability is considered known and need not be estimated from
(CHM)
the data. The statistic defined by (3.4) will be denoted by T(
It can easily be shown that the test T(crH) > c is equivalent to the
test E X(i) > C, thus T(CMH) is the most powerful test when
p.(l p )/pi(1 p.) is constant on i.
In many practical combination problems with binomial experiments,
Pi =1/2 for all i. The null hypothesis is then H: p. =1/2, i=,2,...,k
and the general alternative hypothesis is HA: pi > 1/2 (strict inequal
ity for at least one i). This is the hypothesis testing problem under
consideration throughout the remainder of Chapter III. For pi0 =1/2,
all i, T( ) is uniformly most powerful for testing H: pi0=1/2, all i,
versus HB: pi = p2 = p3 = ... = k > 1/2.
For the hypothesis test just described, T( ) can be written
k k
(i) 1 1 ).
S(X n.)/( E n.) (3.5)
i=l 2 i 4 1
i=l j=1
This variate is asymptotically standard normal. It is of note that
this form is standardized by a pooled estimate of the standard devia
tion. An alternative statistic can be formed by standardizing each
X( yielding
(X) =1 {(x(i) 1 1 )A
T {(X n)/( n)
2 i 4 i
which also has an asymptotic standard normal distribution. The statistic,
(X)
T is analogous to the sum of chi's procedure which has been recommended
for combining 2x2 tables. The statistic T(X) is not in general equiva
lent to T(CmlI); in fact, the test T(X) > c is equivalent to the test
k (i) (x) (CMI)
E n X > c. When the n. are all equal, T and T are equivalent.
i=l
(i) (i)
Weighted sums of the X say S(g) = Z gi X form a general
class of statistics. Oosterhoff considers this class and makes the fol
lowing observations concerning their relationship with the individual
sample sizes [32]. It follows from (3.2) that if kn(p./lp.) = a gi
then the most powerful test of H versus H is E g. X( > c.
1
Let pi = + C It follows that
i 2 1
n(p./l p.) = kn(l + 2L./l 2t.)
(2e.)3 (2c.)5
= 2{2 + + ...}
1 3! 5!
= 4 + (E2)
i O+0 i
1
This implies that for alternatives close to the null hypothesis, H,
S(g) is most powerful if Ei = e gi; that is, if the deviations from the
null values of the pi are proportional to the respective g.. The sum
of chi'sprocedure, T is a special case where g. (n.) It fol
(x)
lows that the alternatives against which T) is powerful is strongly
related to the sample sizes, n,n2,....,nk.
The weighted sum, S(g), may be a viable statistic if prior
information concerning the p. is available. Under the null hypothesis,
S(g) is a linear combination of binomial random variables, each with
success probability 1/2. The null distribution of S(g) will therefore
be asymptotically normal. The proper normalization of S(g) is analogous
to that of CmllH) given in (3.5).
A wellknown generalization of the likelihood ratio test is to
reject the null hypothesis for large values of 2 kn{ sup L(O,X)/sup L(O,X)}.
c60 ~ 6
It is easily shown that for the hypothesis test being considered, the
likelihood ratio statistic is
(LR) k (i) (i) (i) X(i) X(i) 1
_i= 1 2 n. i n. n. 2
i=l1 1 1
where
1 if  1/2
.(i n
i (i)
0 if < 1/2.
i
Under broad conditions, which are satisfied in this instance, the
(LR)
statistic T(LR) has an asymptotic chisquare distribution with degrees
of freedom.
Suppose z., i = 1,2,...,k are normal random variables with
means U. and variance 1. The likelihood ratio test for H: P. = 0,
1 1
i = l,...,k versis H1: 1i i 0 (with strict inequality for at least one i)
is to reject H for large values of
k 2
E z. I{z. 0}. (3.6)
i=l
For the binomial problem, an "approximate likelihood ratio" test is then
to reject for large values of
k
(ALR) (i) 1 2 1 (i) 1
T =E (X n) / n. Ix > n.}
2 4 1 2 i
i=l
1 1 1
since (X n )/( n. ) is asymptotically a standard normal random var
iable under H. The exact null distribution of (3.6) is easily derived.
Critical values are tabled in Oosterhoff. When p = 1/2, the normal
approximation to the binomial is considered satisfactory for even
fairly small sample sizes. It follows that the exact null distribution
of (3.6) should serve as an adequate approximation to the null distribu
tion of T(R)
3.3 Exact Slopes of Parnmetric Methods
In this section, the exact slopes of r(F) T(CI) and T(LR)
are compared. We have not been successful in deriving the exact slope
for T(). A more complete comparison of methods is given in Section 3.4
with respect to approximate slopes.
Suppose X) is a binomial random variable based on n. observa
n 1
tions with unknown success probability p.. Consider testing the single
null hypothesis H: pi = 1/2 versus the single alternative hypothesis
H : p. > 1/2.
(i 1 (i) (i)
Proposition 4. Let T = X The exact slope of T is
1n. n n.
i n. i i
1
c.(9) = 2{pi n 2pi + (1 pi) n 2(1 pi)}.
Theorem 1 is used to prove Proposition 4. There are several
means by which the function f(t) of Part 2 of Theorem 1 can be obtained.
Perhaps the most straightforward way is by using Chernoff's Theorem [1].
Bahadur, in fact, suggests that the derivation of f(t) provides a good
exercise in the application of Chernoff's Theorem.
Theorem 3. (Chernoff's Theorem). Let y be a real valued random
ty
variable. Let (t) = E(e ) be the moment generating function of y.
Then, 0 < #(t) < for each t and P(0) = 1. Let P = inf{((t): t > 0}.
Let yl, Y2,..., denote a sequence of independent replicates of y and
for n = 1,2,..., let Pn = P{yi + ... + yn 01. Then
1
in P Zn P as n c .
n n
Proof of Proposition 4. For Part 1 of Theorem 1,
,(i) X(i)
T X
1 i
+ Pi
n i
1
with probability one [6] giving b(6). For the binomial problem
0 = (pl, 2 ... ,Pk)
Now, as n Lends to infinity,
lim 1
n.
lim
n.
1
= lim 
n.
1
= lim 
n.
1
= lim 
n.
1
Zn(l F (n. a))
Zn P{X()/n. > A7 a}
n. i i
(i)
n. 1
1
Zn P{X(
n.
i
 n.a > 0}.
1
The random variable
(i)
X
n.
1
(i)
X = (y
n. 1
1
 n.a can be expressed as
 a) + (y2 a) + ... (n a)
2 n
where the y. are independent replicates of a Bernoulli random variable
y with parameter 1/2. Therefore, P(t) of Chernoff's Theorem is
(t) = e (l + et).
y a 2
1
It follows that
1 at
The quantity e
T
1 at t
P = inf{ e (l + e ):
2
(1 + et) is minimized for
a
t = n 
1a
Thus,
1 a[Zn a/laj [[Zn a/1a]
S = 2e (1+ eJ /l)
1
Zn P = a Zn a/1a + Zn (1 + a/1a).
2
(3.7)
t > }.
and
Hence,
1 (i) 1
li  n P{X(i) > n.a} =a An(a/1a) n( (l +a/1a)) (3.8)
n. n. 2
1 1
giving f(a) of Part 2 of Theorem 1. Thus,
c.(0) = 2{pi kn p./lPi n (1 + p./lp.)}
1i i 2 i i
= 2{pi n 2pi + (1 p.) An 2(1 p.)}
Following the notation of Section 2.2, suppose k binomial
experiments are to be combined and the sample sizes nl,n2,..., nk
satisfy nl + n2 + ... + nk = nk and
n.
lim = X., i = l,...,k.
n 1
Then 1 + ... + k = k and
1 k
2 (i)
n L .c. (0) as n m.
n n. i i
1
According to Proposition 3, c F() = Z X c (6) in both the continuous and
F i i
discrete case if lim  n[l F (/To)] exists. The existence of this
n n
limit for a single binomial experiment is shown in (3.8) of the proof
of Proposition 4. Therefore, for the binomial combination problem, the
exact slope for Fisher's method, T(F) is
k
c ,() = X Ai{pi Zn 2pi + (1 p.i)n 2(1 pi)}.
i=l
A property of likelihood ratio test statistics is that they
achieve the maximum possible exact slope [2]. Theorem 2 states that
the exact slope for the combination problem is bounded above by
Si.c.(O). Proposition 3 shows that T(F) achieves this. If follows
11I
41
(F)
that T and the maximum likelihood test have the same exact slope;
that is,
cF(0) = CLR(6).
This relationship is true regardless of whether the data are discrete
or continuous.
(CMH) 1
Let T( C) X). This form of the CochranMantel
n n k ni
Haenszel statistic is equivalent to those previously given in (3.3)
and (3.5).
Proposition 5. The exact slope of T is
n
c (0) = 2k{p kn 2p + (1p) Zn 2(1p)} where p = E X.p..
CMH k ii
Proof. To get b(O) of Part 1 of Theorem 1,
S(CmH)
n = X i) /nk
n n
nn.
1
X(i)
1 n n
k n, n
k i i
with probability one [a]. Now, for Part 2 of Theorem 1, as n tends to
infinity,
lim 1 kn[l F(JI a)]
n
= lim r in P{ E i) ,n a}
n n k i
1 (i)
= lim k in P{E X n k a} = f(a). (3.9)
nk n.
i
(i)
Under the null hypothesis, Z X is a binomial random variable based
n.
I
on nl + ... + nk = nk trials with success probability 1/2. The quantity
(3.9) is the same as the quantity (3.7) except that n. has been replaced
1
by nk. Theorem 3 can be directly applied to line (3.9), yielding
1
lim _ n [1 F(/n a)]
n
= k{a en a n 1(1 + a/la)} = f(a)
1a 2
and therefore
cCMH(0) = 2f(b(O))
= 2k{p ln(p/lp) ln((l + plp)}
= 2k{p ln 2p + (lp) ln 2(lp)}.
A comparison of T(CME) relative to T(F) and F(LR) with respect
to exact slopes is given in the next section.
Derivation of the exact slope of the sum of chi's procedure,
(X)
T has not been accomplished. An incomplete approach to the problem
follows.
Let T(X) = Z n X ). To derive b(O) of Part 1 of Theorem 1,
T n n. n.
T(X) n.
n 1n s
,n n n
X p. as n m
i i
with probability one [8]. Now, as n tends to infinity,
1
lim Ln P{1 F(Vn a)}
1 P(n1 (1) nk (k)
= lim p{ I ( + ... ) n a).
n ( nl nk
The lefthand side of the above probability statement is a weighted sum
of independent binomial random variables based on varying sample sizes
nl,n2,...,nk each with success probability 1/2. The moment generating
function of this random variable is therefore
n n
= + e +... e+ e
2 2
(3.10)
From the form of the moment generating function given in (3.10), it is
apparent that the random variable in question can be regarded as a sum
n independent identically distributed variates each with moment generat
ing function
1 1 t)nl/n k 1 nk /n
+ Te I... + e
Then, since as n tends to infinity
n1 n(1) ^(k)
lim P1 x ( +. ... + X na
n n n nk
n n M
= lim 1{ Z X a 0},
n n. n
j=1 i
#(t) of Theorem 3 is
at 1 1 1 1 1 k
(t) = e [( + e ) ... ( + e )]
2 2 2 2
and P = inf{f(t): t a 0}
The quantity p has not been found.
3.4 Approximate Slopes of Parametric Methods
Exact slopes are defined in Section 2.1. In Section 3.3 some
comparisons among methods are made with respect to exact slopes and
corresponding efficiencies. Bahadur also defines a quantity called the
approximate slope [3]. Suppose that X has an asymptotic null distri
bution F; that is,
lim F (x) = F(x)
n
n oo
for all x. For each n, let
L = 1 F(x)
n n
be the approximate level attained. (Consistent with Bahadur's notation,
the superscript a stands for approximate.) If there exists a c(a)(
such that
2 Zn L(a) + (a)
n n
(a)
with probability one [0], then c ()() is called the approximate slope
of (X }.
n
If ca) () is the approximate slope of a sequence {x ),
1 n(
i = 1,2, then cl (a)/c2 (a ) is known as the approximate asymptotic
efficiency of {x()} relative to {x(2)
n n
A result similar to Theorem 1 is given by Bahadur [3] for the
calculation of approximate slopes. Suppose that there exists a
function b(0), 0 < b(O) < co, such that
T
n
 b(6)
with probability one [0]. Suppose that for some a, 0 < a < m, the
limiting null distribution F satisfies
1 2
Zn[l F(t)] ~ at as t c
2
(a) [b() 2
Then the approximate slope is c (() = a[b(6)]. This result is
applicable with a = 1 for statistics with asymptotic standard normal
distributions [3]. This result can be shown directly by applying the
result of Killeen et al. given in (2.1).
The approximate slope, c(a)(0), and the exact slope, c(6),
of a sequence of test statistics are guaranteed to be in agreement only
for alternative hypotheses close to the null hypothesis. Otherwise,
they may result in very different quantities. One notable exception is
the likelihood ratio statistic. When the asymptotic null distribution
is taken to be the chisquare distribution from the wellknown 2 Zn
(likelihood ratio statistic) approximation, the approximate slope of the
likelihood ratio statistic is the same as the exact slope. The approx
imate slope is based upon the asymptotic distribution of the statistic.
Equivalent test statistics may have different asymptotic null distribu
tions giving rise to different approximate slopes. This apparent short
coming does not exist with exact slopes.
In typical applied situations, the significance levels attained
by T(C0H) and T(X) will be ascertained by appealing to their asymptotic
(LR)
normal distributions. Similarly, T(R) will be compared to the appropri
ate chisquare distribution and approximate levels for T(ALR) will be
obtained from the asymptotic distribution given in Section 1.2. Approx
imate slopes based upon these asymptotic distributions would therefore
seem to afford a more appropriate comparison of the methods. In other
words, it is appealing to consider the null distribution that will be
used to obtain significance levels in practice when comparing the
statistics. The only statistic which will not usually be compared to
(CMHI)
asymptotic distributions is perhaps T .(CMH) The null distribution of
T is binomial based on nI + ... + nk trials with success probabil
ity 1/2. However, even with the availability of extensive binomial
tables, T(CMH) will often be standardized as in (3.5) and compared to
standard normal tables since the normal approximation to the binomial
when p = 1/2 is satisfactory even for fairly small sample sizes.
The asymptotic null distribution of T(F) in the discrete case
is easily shown to be chisquare with 2k degrees of freedom. This is
(F)
also the exact distribution of T(F) in the continuous case. It follows
that the approximate slope in the discrete case is the same as the
exact slope in the continuous case. In summary,
(a) (a)( k
CLR () =c =c R() =C ( =2 E X.{p kn 2p +(lp.i)n 2(lpi)}.
LR F LR F ill i i 1 1
i=l
(Cuf) (x)
In order to derive the approximate slopes for T and T
consider the linear combination Zn X of which T and T are
1 n.
(i)
special cases. The variate X has an asymptotic normal distribution
ni
1
1 1
with mean n. and variance n under the null hypothesis. It follows
2 4 i
directly that
(U) 1 ++l / 2u+l
T (Zn.Xni n2 n )/ vn
n i 2 i 2 i
is asymptotically standard normal.
Proposition 6. The approximate slope of T) is
n
c+(a) +1 2 2c+1+
c(a)() = [A (2p l)] /i
cti i
Proof. First, to get b(O),
1 C+1
Zni )
2 i
1 f2c+l
21/ Eni
1
(E()
nn
(i)
i
cE+1
n
1 i
E )
2 x
n
2"+1
2cA+l
1 i
2/ 2a
v n
X(i)
1 /( 2a+1
2 ni
Ac 1 ac+l
( (.iPi) iz
i 2 i
1 / Z2A+
2 i
with probability one [0].
(Ca)
Now, since T is asymptotically standard
n
normal,
(a)
co
= [b(O)]2
a+l 1 E+1 2 1 2a+l
i i 2 i 4 i
= [Zi (2Pi
2 2a+1
 1)] /ZA.
i
Letting a = 0 yields the approximate slope of'lT(CI)
C(a() = [EiX(2p 1)]2/k
CHH 1 i
Letting c = 2 yields the approximate slope of T ,
(a) 2
c(a() = [E(2p 1)]2/k.
X 1 i
(")
T
n
11
I aQ(i)
 (EnX M
i n.
v n 1
as n CO
By inspection of the above approximate slopes, it is apparent
(CMIH)
that T is more efficient when the p. are proportional to the A.
1 1
(X)
(the relative sample sizes) and T() is more efficient when the p. are
inversely related to the X.. The boundary of the parameter space where
(a) (a)
c (6) = c (6) is not p = p = ... = k' however. The statistic
T(CIMH) is more efficient than T(X) in more than half of the parameter
(a) (CHH) (x)
space. As a further comparison, e (T T ), the approximate
efficiency of T(C with respect to T(X), can be integrated over the
parameter space. The result is greater than one which again supports
(CMH)
use of T It should be noted, however, that when the pi are pro
portional to the A. both tests have high efficiencies relative to when
1
(X)
p. are inversely related to the A.. Therefore T() is more efficient in
a region of the parameter space where both tests have relatively low
efficiency. This is a good property for T(X)
An "approximate" likelihood ratio test is introduced in
Section 3.2. A statistic which is equivalent to the form given in
Section 3.2 is
k n
(ALR) [(i) 1 2 i (i)
T = (E [(X n) / IX n
i=l i 4 n 2
Proposition 7. The approximate slope of TALR) is
n
c (o) = eA.(2p.)1).
Proof. To find b(0) of Part 1 of Theorem 1,
(i)
,(ALRO) X
Tn 1 (i 1 2 Ix(i) 1
4 ( )n.}}
S n ni 2 2 1
{4E A.(p 1)2 as n co
i i 2
with probability one [0]. Thus,
b(O) = {4EXA(p 1)2 = {ZA.(2p l)2
I i 2 1 1
To find f(t) of Part 2 of Theorem 1, the asymptotic null distribution
of T(ALR) is required. According to Oosterhoff [33],
n
k
2 k k 2
P{z.z I{z. >0} s} = 2 Z (.)P[x2 s)
1 1 j=l J
j=1
where z. is a standard normal random variable. Since, under the null
1
hypothesis
(X 2 n.)/ / ./4 z.
n. 2 1 1 1
1
in distribution, it follows that
(ALR) (ALR) 2 2 k k 2 2
P{T (ALR) >s} = P{(T (ALR) s 2}2 k( k )P{X 2s (3.11)
n n j j
as n, oo for all s. It follows that the associated density function is
a linear combination of chisquare densities. The result of Killeen et al.
can be applied to verify that
1 2
kn[l F(t)] ~ as t *
2
(ALR)
where F is the asymptotic null distribution of T Hence,
n
c (0) = [b(0)] = Ei(2i 1)2.
ALR i 1
Before proceeding to a further comparison of approximate slopes, the
slopes are summarized in the following listing.
Approximate Slope
Fishers (T(F)
Fisher's (T )
2EA (pi Zn 2p. + (lpi) en 2(lpi)}
i 1 1 1 1
Likelihood Ratio (T(LR) 2EA {pi n 2p + (1p )n 2(lp.)}
"Approximate Likelihood Ratio"(T(ALR) Zi(2i 1)2
Sum of Chi's (T(X) [ 1 (2p 1)]2
(CM) 1 2
CochranMantelHaenzel (T ) [Ei(2p 1)]
Letting A = 5A (2i 1), it is easy to see that cALR() (
i i ALR
(ak 2 1 2 (a) (a)
c (0O) since Z A 2 [A.] It is also true that c ( c c().
X i=l1 k 1 ALR CM
i=1
Let B. = (2p. 1). It can easily be shown that
1 1
2 1 2 1 2
EX Bi [A.B] = E XX.(B. B.)2
ii k 1 1 k i ] i
i
(ALR) (CMH) (X)
given that EX. = k. Therefore T dominates both T and T
1
with respect to approximate slopes.
Approximate efficiencies of T(C) and T(ALR) with respect to
T( (and equivalently T(LR ) for A1 = 2 = 1 are given for several points
in the parameter space in Table 1. In this case of equal sample sizes,
T() is equivalent to T(C Table 2 gives efficiencies of T(CN), T(x)
and T(ALR) with respect to T (and equivalently T ( ) for A = 1/3,
A2 = 5/3. The values of A1 and A2 imply that the second test is based
on five times as many observations as the first test. When the exact
(CMH) k
null distribution of T (binomial with parameters Z n. and 1/2) is
i=1
Test
to be used to determine significance levels it is more appropriate to
employ the exact slope, Cc (0), rather than the approximate slope,
(a) (TMH) (F)
cC(a ). The exact efficiencies of T relative to T (and
equivalently T(LR)) are given in Tables 1 and 2 in parentheses.
The efficiencies listed in Tables 1 and 2 support several
previously made observations:
1. The statistic T(ALR) dominates T(CM) and T(X) with respect
to approximate slopes (efficiencies).
2. The test based on T( ) dominates the test based on T(X)
when the success parameters are proportional to the sample sizes. The
test based on T() is more efficient in the reverse case. The test
(X)
based on T is more efficient in a region of relatively low effi
ciencies for both T() and T(CH)
3. Exact and approximate slopes are not, in general, equivalent.
They are in close agreement for parameters close to the null hypothesis.
4. All of the tabled efficiencies are at most one. This is
expected from the optimality properties of T(F) and ILR) given by Theorem 2
and Proposition 3. A value of one is achieved only for the exact effi
ciency of T(C when pi = p2. This is consistent with the fact that
T(CH) is the most powerful test (and the likelihood ratio test) when
P = P2'
Table 1
Efficiencies of T(CP TX and T(L
Relative to
(LR) (F)
T( or Equivalently, to T (F) 1 = 1
.5 .6 .7 .8 .9 1.0
(0.497)
0.497
0.993
(0.489)
0.486
0.972
(0.474)
0.467
0.934
(0.447)
0.435
0.869
(0.377)
0.361
0.721
(0.497)
0.497
0.993
(1.000)
0.993
0.993
(0.892)
0.879
0.976
(0.773)
0.752
0.939
(0.674)
0.644
0.876
(0.540)
0.505
0.729
(0.489)
0.486
0.972
(0.892)
0.879
0.976
(1.000)
0.972
0.972
(0.951)
0.909
0.945
(0.856)
0.799
0.888
(0.697)
0.632
0.748
(0.474)
0.467
0.934
(0.773)
0.752
0.939
(0.951)
0.909
0.945
(1.000)
0.934
0.934
(0.964)
0.871
0.892
(0.831)
0.722
0.768
(0.447)
0.435
0.869
(0.674)
0.644
0.876
(0.856)
0.799
0.888
(0.964)
0.874
0.892
(1.000)
0.869
0.869
(0.932)
0.763
0.773
(0.377)
0.361
0.721
(0.540)
0.505
0.729
(0.697)
0.632
0.748
(0.831)
0.722
0.768
(0.932)
0.763
0.773
(1.000)
0.721
0.721
1.0
Table 2
Efficiencies of T(CMH)and (ALR) to T(LR)
Efficiencies of T T and T Relative to T
or Equivalently,
to T(F), X = 1/3, X = 5/3
toT 1 2
.9 1.0
0.166
(0.165)
0.497
0.993
0.162
(0.162)
0.486
0.972
0.156
(0.156)
0.467
0.934
0.145
(0.145)
0.435
0.869
0.120
(0.121)
0.361
0.721
0.828
(0.832)
0.497
0.993
0.993
(1.000)
0.867
0.993
0.893
(0.901)
0.981
0.984
0.737
(0.736)
0.934
0.954
0.576
(0.585)
0.830
0.896
0.420
(0.428)
0.660
0.756
0.810
(0.826)
0.486
0.972
0.935
(0.957)
0.694
0.973
0.972
(1.000)
0.848
0.972
0.932
(0.964)
0.924
0.960
0.838
(0.872)
0.921
0.924
0.673
(0.751)
0.812
0.815
0.778
(0.814)
0.467
0.934
0.867
(0.914)
0.604
0.935
0.921
(0.978)
0.925
0.937
0.934
(1.000)
0.815
0.934
0.904
(0.976)
0.861
0.916
0.805
(0.878)
0.827
0.845
0.725
(0.791)
0.435
0.869
0.790
(0.872)
0.531
0.871
0.839
(0.938)
0.623
0.874
0.867
(0.982)
0.702
0.896
0.869
(1.000)
0.759
0.869
0.823
(0.962)
0.767
0.829
0.601
(0.703)
0.361
0.721
0.646
(0.771)
0.426
0.723
0.685
(0.836)
0.490
0.727
0.714
(0.897)
0.550
0.732
0.731
(0.952)
0.601
0.736
0.723
(1.000)
0.629
0.721
1.0
3.4 Powers of Combination Methods
In the previous two sections, competing methods were compared
with respect to asymptotic efficiencies. Asymptotic efficiencies com
pare sequences of test statistics in some sense as the sample sizes tend
to infinity. Such comparisons may or may not be applicable to situa
tions when small sample sizes are encountered. Therefore, the methods
of combinations are compared in this section with respect to exact power.
As mentioned previously, exact power studies are often intractible.
For the test statistics considered here, power functions are not obtain
able in any simple form which would allow direct comparisons between
competing methods. However, through the use of the computer, it is
possible to plot contours of equal power in the parameter space. From
such plots, the relative powers of the competing methods can be surmised.
The first step in obtaining the power contours is the generation
of the null distributions for each of the five statistics: T(LR) T(F)
T(C T(X), and T(ALR). Size a = .05 acceptance regions for each of
the statistics for varying sample sizes are shown in Figures 3 8.
Acceptance regions for tests with equal sample sizes
(nl = n2 = 10, 15, 20, 30) appear in Figures 3 6. The statistics
T(LR) a (ALR)
T(LR) and T(ALR) define very similar, but not identical tests for
n1 = n2 = 10, 15, 20, 30. They define exactly the same a = .05 accep
tance regions in all four cases, and will therefore yield identical
power contours. Fisher's statistic, T(F) defines a test similar to T(LR)
and T(ALR) for n = n2 = 10, 15; in fact, T(F) defines the same a = .05
acceptance region for those sample sizes. The major difference between
(F) (LR) (ALR)
T and the two likelihood ratio statistics, T and T( is that
(F)
T (F) has many more attainable levels. For sample sizes nI =n2 =20, 30,
(F) (LR) (ALR)
T defines different a = .05 acceptance regions than T and T
The statistics T( ) and T(X) are equivalent for nI = n2.
Figures 7 8 portray acceptance regions for cases of unequal
sample sizes (nl = 10, n2 = 20 and nI = 10, n2 = 50). The difference
between T and T(C) is apparent for the case of unequal sample sizes.
The statistics T(LR) and T(ALR) define different a = .05 acceptance
(F) (LR) (ALR)
regions. In both figures, it is seen that T), T and TLR) define
similar regions.
In Section 1.3 it was stated that Birnbaum [7] has shown that
combination procedures must produce convex acceptance regions in the
(1) (2) (k)
(X X .., X ) hyperplane in order to be admissible. Each of
the acceptance regions in Figures 3 8 appear to satisfy this convex
ity condition.
The acceptance regions given in Figures 3 8 are not exact
a = .05 size regions. They are the nominal acceptance regions which are
the closest to size a = .05. In order to make a fair comparison among
the powers of the competing methods, all of the acceptance regions must
be of exactly the same size. This can be accomplished by admitting
(1) (2)
certain values of (X X ) to the acceptance region with probabil
ities between zero and one. A more precise definition of this procedure
follows. Suppose
P{T(i)
n.
1
P{T(i
n.
i
St } = .05 a,
< t } = .05 + b,
u
56
and T() does not take on any values between t and t Then all T(
S1. u n.
1 1
such that T = t are included in the acceptance region with probabil
n. L
1
(i)
ity one and T = t is included in the acceptance region with prob
n. u
1
ability a/a+b.
th(i
The power of the i test is one minus the probability that T
n.
1
fails in the acceptance region. More precisely, define the power of
th
the i test to be
Hi(p1P2) =1 [p{T) < ti(plP2)} + (a/a+b) P"T = tu (p'p)]
i 1
nI (1)
= 1 [ ( ) )p
(1) (2) (i) M ( tx
(x ,x ): T t
i
n x n2 (2)
(IP) ( )p2 (1
X(2)
(I \
(2)
n x
P2)
n (1) n x n (2) n
+ (a/a+b)E E ( 1)P (1P) (2)2 (lp
x(1) 1 (2)(2(i) x x
(x ,x ):T n.
ni U
For each test statistic, power is calculated for 2500 values of
(pp 2) in the alternative parameter space. This was accomplished with
a FORTRAN computer program. A data set consisting of these calculated
powers is then passed into SAS [4,16]. The plotting capabilities of
SAS are then exploited to portray contours of equal power in the
(p1P'2) plane.
2x
2
Figures 9 14 are .90 power contours corresponding to the
acceptance regions of Figures 3 8, respectively. The CochranMantel
Haenszel procedure, T is most powerful in the center of the
parameter space; that is, when pl and p2 are nearly equal. This is
expected since T(CM) is uniformly most powerful when pl = P2 for any
choice of sample sizes. The statistic T(C) is clearly inferior
to T(LR), TLR), and T( in the extremes of the parameter space, that
is, when pl and p2 are quite different. Further, the deficiency of
T(CMH) compared to the other methods when pl and p2 are different is
larger than the deficiency of the other methods when pi = p2. From
Figures 9 12 it can also be seen that the central wedge of the param
eter space where T( ) is more powerful shrinks as the sample sizes
(F)
increase. Fisher's statistic, T and the likelihood ratio statis
(LR) (ALR)
tics, TLR and T have similar power. Fisher's method gives
slightly more power in the central region of the parameter space while
T(LR) and T(ALR) are slightly more powerful when pl and p2 ave very
different.
(F) (LR)
For unequal sample sizes (Figures 13, 14), T and T yield
power contours too similar to be separated on the drawings. The approx
imate likelihood ratio test, T(ALR), has almost the same power as T(LR)
(F)
and T(F), having slightly more power when the experiment based on more
observations has the larger p., and slightly less power in the reverse
case. The sum of chi's procedure, T(X) is not equivalent to T(CM) when
(CMII)
n # n2. The power contours are very different with T being more
powerful when the larger experiment matches with a large p i The
statistic T() is more powerful in the opposite case.
58
Figures 15 20 are .60 power contours concomitant with the .90
power contours in Figures 9 14. The comparison of competing methods
may be more appropriately made for low powers. When all of the powers
of the tests are high it is probably unimportant which test is used.
The patterns observed in the .90 power contours are virtually the same
in the .60 power contours, however. No additional information is appar
ent except that the patterns are consistent over a wide range of powers.
T(F), T(LR) T(ALR)
T(CMH)
10
L
6  
L
2
0 I
0 2 4 6 8 10 )
Figure 3. Acceptance Regions for n = n2 = 10.
60
X(2)
S(F) T(LR) T(ALR)
14 L '
14  (CMH)
IT
12 ..
I
10 
8 
0 12 14X
6 '1
0 ,
0 2 r 4 6 8i0 12 14
Figure 4. Acceptance Regions for n1 = n2 = 15.
T(F)
T (LR) T(ALR)
......... T(CMH)
i. ...
r.51
rT.l
I
: I
''"'
""
Figure 5. Acceptance Regions for n, = n2 = 20.
X (2)
20
S8
16
14
S 2 4 6 8 10 12 14 16 18 20
0 2 4 6 8 10 12 14 16 18 20
( I)
X


62
X(2)
30 . ...... . I.. ....
.... T (F)
28 ". T (LR), T(ALR)
26 ..T(CMH)
2 6 ..., ......... T CM )
24 :
22
20  
18
14 i ....
12,
10
10 ::
8
6
6 L:
4
2
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Figure 6. Acceptance Regions for n = n2 = 30.
63
X(2)
20 .
(F) (LR)
S(F T(LR) (virtually the same)
S(ALR)
T (CMH)
3 .......... .. .... ....
6T......... .
16
.. ....... .. .
1 ........
I0
s
8
6
2 4 6 8 10
Acceptance Regions for n, = 10, n2 = 20.
Figure 7.
64
X (2)
46 
46T(F)
44 oooo T(LR)
ST(ALR)
42 ........ T(CMH)
4 2 . T(X)
. T
40
38
36
34 
324 .... i
30. o
0o
28 
26o ......
26 001 
0
24 !0
o
22 ...
: I 0
20
II
lo 
I s I
2 4 6 8 10
Acceptance Regions for nl = 10, n2 = 50.
Figure 8.
T_(F) T(LR) T(ALR)
T (CMH)
.6 .7 .8 .9 1.0 = n2 = 10.
Figure 9. .90 Power Contours for n = n2 = 10.
P
1.0
S(F), T(LR),T(ALR)
T (CMH)
.90 Power Contours for n = n2 = 15.
P
2
1.0
.9
.8
.7
.6
.5 .6 .7 .8 .9
1
1.0 I
Figure 10.
P2
2
T(F)
__ T(LR) T(ALR)
......... T(CMH)
1.0
1.0 1
.5 .6 .7 .8 .9
Figure 11. .90 Power Contours for nl = n2 = 20.
68
P
p2
1.0 T(F)
T(LR) T(ALR)
........... T(CMH)
.9
.8
.7
.6
.5 I
.5 .6 .7 .8 .9 I.0
Figure 12. .90 Power Contours for n = n= 30.
1 2
P2
1.0
.9
.8
.7
.6
N
K
.5 .6 .7 .8 .9
Figure 13. .90 Power Contours for n1 = 10, n2 = 20.
1.
1.0
T( F) T(LR)(virtually the same)
T(ALR)
......... T(CMH)
_._T(X)
69
PI
P
1.0
N
.5 .6 .7 .8 .9
Figure 14. .90 Power Contours for n1 = 10, n2 = 50.
T(F) ,T(LR)(virtually the so
__ T(ALR)
........T(CMH)
_._T(X)
70
me)
I
1.0
1.0 1
 T(F) T(LR), T(ALR)
__ T(CMH)
.60 Power Contours for nl = n2 = 10.
P
2
1.0
.9
.8
.7
.6
.5 .6 .7 .8 .9
1.0P
Figure 15.
T(F(F) ,T(LR), T(ALR)
T(CMH)
.60 Power Contours for n1 = n2 = 15.
P0
1.0
.5 .6 .7 .8 .9
1 p
1.0
Figure 16.
P
2
1.0 _F)
 T(LR) T(ALR)
......... T(CMH)
.9
.8
.7
.6
.57
.5 .6 .7 .8 .9 1.0 I
Figure 17. .60 Power Contours for n = n2 = 20.
T(F)
 T(LR) T(ALR)
......... T(CMH)
P
2
I.0
.9
.8
.7
.6
.5
. P
1.0
.5 .6 .7 .8 .9
Figure 18. .60 Power Contours for n1 = n2 = 30.
T(F) ,T(LR)(virtually the same)
__ T(ALR)
......... T(CMH)
__ WT(X)
.8
\
\
.7
.5 .6 .7 .8 .9 1.0 1
Figure 19. .60 Power Contours for n = 10, n2 = 20.
P
1.0
I.0
P2
2.0
.0O
.5 .6 .7 .8 .9
Figure 20. .60 Power Contours for n1 = 10, n2 = 50.
o P
1.0 I
 T(F) ,T(LR)(virtually the some)
_T(ALR)
......... T(CMH)
_._T(X)
\
3.6 A Synthesis of Comparisons
When detailed prior knowledge of the unknown parameters is
(F)
unavailable the class of competing methods can be restricted to T
(LR) (ALR) (x)
T T and T(. These methods are compared with respect to
various criteria in previous sections. In this section, the results
of these comparisons are synthesized to make recommendations concern
ing the optimum choice of method for various situations.
For the comparisons in the previous sections, the null hypothesis
considered is H : p = 2 = ... = = 1/2. The most general alternative
hypothesis considered is HA: p. i 1/2 (strict inequality for at least
one i). In some situations, it is reasonable to assume that the success
probability is consistent from experiment to experiment. In such cases
the alternative hypothesis of interest is HB: Pk = 2 = "' = k > 1/2.
A third alternative hypothesis of possible interest is H : p. > 1/2
Cj
(exactly one j). This alternative is appropriate if the researcher
believes that at most one p. will be greater than 1/2. The hypotheses
HA and H are probably the more frequently encountered alternatives in
practical situations.
The following recommendations are based on evidence presented
thus far in this dissertation:
(i)
1. The minimum significance level, T has good power versus
the HC alternative. It performs poorly, however, versus other alterna
tives.
2. The CochranMantelHaenszel statistic, T( forms the
uniformly most powerful test against HB. Its use is therefore indicated
Bi
whenever it can be assumed that the pi are not very different. The
(CM)
statistic T performs relatively poorly versus alternatives in the
extremes of the parameter space (Type B alternatives).
(F)
3. Fisher's combination, T(F), is not, in general, the most
powerful test versus a particular simple alternative hypothesis. Its
power, however, is never much less than that of the optimum test.
Fisher's method gives good coverage to the entire parameter space and
its use is therefore indicated whenever specification of the alternative
hypothesis cannot be made more precisely than HA.
4. There seems to be no compelling reasons to recommend the
(x)
use of the sum of chi's procedure, T unless it is known, a priori,
that the pi are inversely related to the sample sizes of the individual
binomial experiments.
(LR)
5. The likelihood ratio statistic, T and the approximate
likelihood ratio statistic, T(ALR), define tests very similar to T(
They obtain approximately the same powers throughout the parameter space.
Choosing among these three statistics then depends upon which yields
significance levels with the greatest ease and accuracy. This problem
is addressed in Section 3.7.
3.7 Approximation of the Null Distributions
(F) (JR) (ALR)
of T T T
In Section 1.6 the problem of obtaining significance levels for
(F)
Fisher's statistic, T(F), when the data are discrete is discussed.
2 '2
Lancaster's transformations, X and X are introduced. It is estab
m m
2 '2
lished in Section 1.6 that X and X both converge to chisquares with
m m
2k degrees of freedom. Although Lancaster's approach can be expected
to yield good approximate levels for large sample sizes, the degree
of accuracy has not been established for small or moderate sample sizes.
Some indication of the accuracy of significance levels obtained from
2 '2
X and X is given by observing the mean and variance of these variates.
m m
Table 3 (page 79) lists the means and variances for n = 1,2,...,20 for
2 '2
X and X when applied to one experiment. Since the altered form of
m m
T(F) will be compared to a chisquare distribution with 2k degrees of
2 '2
freedom, it is desirous that the mean and variance of X and X are as
m m
close as possible to the mean and variance of the chisquare distribution
with degrees of freedom, which are 2 and 4, respectively. For n 3,
the mean and variance of X2 are closer to 2 and 4, respectively, than
m
the mean and variance of X2. This suggests that X should, in general,
m m
'2
be a more accurate approximation than X
m
In Section 3.2, the likelihood ratio statistic, T(LR), and the
approximate likelihood ratio statistic, T(AL are introduced. The
(LR)
necessary regularity conditions can be shown to be satisfied for T(
so that the statistic can be deemed asymptotically a chisquare with
k degrees of freedom. As previously stated, the null distribution of
Table 3
'2 2
Mean and Variance of Lancaster's X and X
m m
Median chisquare (X 2)
m
Mean
1.9808
1.9531
1.9394
1.9362
1.9385
1.9430
1.9481
1.9530
1.9574
1.9612
1.9645
1.9674
1.9698
1.9719
1.9738
1.9754
1.9768
1.9781
1.9793
1.9803
Mean chisquare (X 2)
m
Mean
2.000
Variance
1.9753
2.8587
3.2223
3.3754
3.4514
3.5014
3.5428
3.5804
3.6151
3.6465
3.6748
3.6998
3.7218
3.7412
3.7582
3.7733
3.7866
3.7986
3.8092
3.8188
Variance
1.9218
2.8036
3.2419
3.4760
3.6101
3.6923
3.7462
3.7836
3.8110
3.8320
3.8486
3.8621
3.8733
3.8828
3.8909
3.8980
3.9042
3.9097
3.9145
3.9189
T(ALR) can be approximated with a distribution derived by Oosterhoff.
Significance levels are then determined by the relationship
k
(ALR) k k 2
P{T ( c} = 2 E ( ) P{X. c)
j=l J
(F) (LR) (ALR)
The null density functions of TF), T and TALR) are
plotted in Figures 21 22 for k=2, nI =n2= 6. These plots give an
indication of the difficulty of approximating the respective null
density functions. More extreme (larger) values of the statistics do
not always occur with smaller probabilities. This fact gives the jagged
appearances for the density functions. This lack of smoothness caused
difficulty in approximating a discrete density with a continuous one.
The remainder of this section contains numerical comparisons
of the abovementioned approximations. The goal is to choose the approx
imation which yields significance levels closest to the exact levels
of the respective statistic.
Tables 4 and 5 correspond to the density functions pictured
(F)
in Figures 2122. Table 4 lists the possible events as ordered by TF
2 '2
Lancaster's approximate statistics, X and X are calculated for each
m m
2 '2
event. Although it is not generally true, X and X maintain the same
m m
ordering of events as the uncorrected T(F). Significance levels obtained
2 '2
by comparing X and X to a chisquare distribution with four degrees
m m
of freedom are then compared to exact levels. The inaccuracies of these
approximations are then reflected in the columns labeled percentage error.
(LR)
Table 5 gives an evaluation of the approximations given by T and
T(ALR) These statistics define equivalent tests, but yield different
approximations to the exact densities.
The percentage errors given in Tables 4 and 5 tend to favor
(LR) (ALR)
Lancaster's approximations over the approximations to T and T
Both X and X2 yield generally conservative results in this partic
m m
2
ular case. The mean chisquare, X is somewhat more accurate than
m
'2
the median chisquare X
m
All of the approximations can be expected to improve as the
sample sizes increase. To indicate the behavior of the contending
approximations for increasing sample sizes, nominal c = .05 and a = .01
values for each statistic are given in Tables 68 for n =n2 = 3,4,5,...
The data in Tables 48 indicate that Lancaster's approxima
(LR) (ALR)
tions clearly dominate the approximations to T and T The
2 '2 (F)
optimal choice then becomes either the X or the X correction to T
m m
2 '2
Table 6 gives no clear indication as to whether X or X
m m
yields a better approximation. Both statistics give large errors for
small sample sizes. Both statistics yield errors less than 12% for
both a = .05 and a = .01 levels for n16.
The superiority of the mean chisquare, X over the median
'2
chisquare, X becomes clear for k= 3. Table 9 gives the nominal
m
2 '2
a = .05 and a = .01 values for X and X for k=3, n = n2 = n
m m 2 3
2
2,3,4...,10. The mean chisquare, X is more accurate in all cases
but two ( = .01, n and = .05, n5)
but two (0 = .01, n = 8 and 0 = .05, n =5).
I I I I I I
83
0
CO
(D
II
O:
II
0
t4'
oo
0
C!
c
40
r4
0O .,4
C:
(D 4
tO
0
I rO RJ I 0 0 O  Io o I I oJ
   o QQ oQ Q QQ QQ
(LR)
T(ALR)
(equivalent for T(LR) T(ALR)=O)
O 3 6 9 12 15
(LR) (ALR)
Figure 22. Density Functions of T and T
for n, = n2 = 6.
.40
30
.20
.0
Table 4
(F)
Lancaster's Approximations to T for k=2,
Event
6,6
6,5
6,4
6,3
5,5
6,2
6,1
6,0
n = n =6
1 2
2 '2
X X
m m
20.6355
15.8629
16.0951
13.2872
13.3847
11.7041
11.7376
11.0904
11.5546
10.8316
10.8393
10.4468
10.4477
10.3335
10.3335
8.5146
8.8442
6.9315
7.1972
6.0590
6.2988
5.9389
6.1338
5.6743
5.9072
5.5609
5.7930
4.3558
4.4867
3.4833
3.5884
3.0985
3.1968
2.9852
3.0826
Approximate
Level
.000374
.003209
.002894
.009954
.009541
.01969
.01941
.02557
.02099
.02852
.02843
.03354
.03353
.03517
.03517
.07445
.06511
.1396
.1258
.1948
.1779
.2038
.1894
.2248
.2062
.2344
.2152
.3600
.3441
.4804
.4646
.5415
.5254
.5603
.5441
Exact
Level
.0002441
.003174
.01050
.02026
.02905
.03638
.03931
.03979
.08374
.1423
.1863
.2412
.2588
.2617
.4082
.5181
.5620
.5693
4,4
5,0
Percent
Error
53.2
1.1
8.8
5.2
9.1
2.8
4.2
21.0
27.7
21.6
21.9
14.7
14.7
11.6
11.6
11.1
22.2
1.9
11.6
4.6
4.5
15.5
21.5
13.1
20.3
10.4
17.7
11.8
15.7
7.3
10.3
3.6
6.5
1.6
4.4
Table 4 (Continued)
2 '2
Event X, X Approximate Exact Percent
m m Level Level Error
2.7726 .5966 10.6
2.8397 .5850 .60 12.3
2 1.9001 .7541 7.3
1.9414 .7465 .85 8.2
,1 1.5154 .8239 5.5
1.5498 .8178 .6.2
0 1.4020 .8438 4.3
1.4356 .8380 8818 5.0
22 1.0276 .9056 3.3
1.0431 .9032 3.6
.6429 .9582 2.3
2,1 .9807
.6514 .9572 2.4
.5295 .9706 1.8
2,0 .9880
.5372 .9698 1.8
.2582 .9924 0.4
.2598 .9923 8 0.5
.1448 .9975 0.2
0 .1456 .9975 9990.2
.03142 .9999 0.0
0 .03142 .9999 1.00.0
Table 5
(LR) (ALR)
Approximations to T and T
for k=2, n1 =n2 =6
Event
(6,6)
(6,5)
(6,4)
(6,3),(6,2)
(6,1),(6,0)
(5,5)
(5,4)
(5,3), (5,2)
(5,1), (5,0)
(4,4)
(4,3),(4,2)
(4,1),(4,0)
Remainder
(LR) (ALR)
T T
16.44
12.00
11.23
8.667
8.977
6.667
8.318
6.000
5.822
5.333
3.591
3.333
2.911
2.667
1.359
1.333
.6790
.6667
0
0
Approximate
Level
.002
.000886
.0036
.004901
.0111
.01383
.0156
.0196
.0544
.02783
.1660
.08117
.2333
.1171
.5069
.2525
.7119
.3862
1.0000
1.0000
Exact
Level
.0002441
.003209
.0150
.0310
.0398
.0837
.2068
.2617
.5693
1.0000
Percent
Error
0.0
63.7
12.5
52.7
5.7
31.7
49.7
36.8
36.7
30.1
98.3
3.0
12.8
43.4
93.7
3.5
25.0
35.2
0.0
0.0
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The statistics T(F) T(LR), and T(ALR) define very similar
tests. In Section 3.6 it is concluded, therefore, that the choice
among these three statistics should depend upon which affords the best
approximation to its null distribution. The evidence of this section
2 (F)
indicates that Lancaster's mean chisquare (X ) approximation to T(F)
m
is the best choice. Even this approximation yields large errors for
small sample sizes. Tables 10 and 11 give nominal a =.05 and a = .01
levels for HLL an equivalent and more convenient form of T for
k = 2 and k=3. The exact significance levels are also given.