EXACT UNCONDITIONAL TESTS FOR
2x2 CONTINGENCY TABLES

BY

SAMY SALOMON SUISSA

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
1982

n

M^

» i

To Nicole, Daniel

and

my parents

ACKNOWLEDGEMENTS

I am deeply grateful to Professor Jonathan Shuster for
being an instrumental part of my education. His understanding
and confidence were important in my entry to this graduate
program. Throughout my studies, his patience, help and
guidance were invaluable to my work. His contribution to
this dissertation was crucial.

I am thankful to my family for their constant support:
My wife Nicole, for her patience, love and understanding,
my son Daniel, for the joys and the change of pace, and my
parents, brother and sisters who were always there with love
and encouragement.

My appreciation extends to Professors Ronald Randies,
Mark Hale, Ramon Littell and Ken Portier for their full
cooperation on such short notice.

TABLE OF CONTENTS

page

ACKNOWLEDGEMENTS iii

ABSTRACT vi

CHAPTER

1 INTRODUCTION 1

1 . 1 The Problem 1

1.2 Some Methods of Eliminating

Nuisance Parameters 3

1 . 3 Numerical Example 5

1. 4 Proposed Approach and Preview 12

2 METHODOLOGY FOR COMPUTING THE SIZE OF A TEST 14

2 . 1 Introduction 14

2 . 2 Local Bound for ir ' (p) 15

2.3 Least Upper Bound for it (p) 16

2.4 Stability of the Null Power Function 19

2.5 Choice of the Test Statistic... 20

3 THE 2x2 TABLE FOR INDEPENDENT PROPORTIONS 22

3.1 Introduction 22

3.2 Asymptotic Tests and Sample Size Formulae.. 25:

3.3 Fisher's Exact Test 2 8

3.4 An Exact Unconditional Test 31

3.5 Relation to the Chi-square
Goodness-of-fit Test 38

3.6 Power and Sample Sizes 40

4 THE 2x2 TABLE FOR CORRELATED PROPORTIONS 42

4 . 1 Introduction 42

4.2 McNemar's Test, Other Asymptotic Tests

and Sample Size Formulae.. 46

4.3 The Exact Conditional Test 48

4.4 An Exact Unconditional Test 51

4.5 Power and Sample Sizes 57

APPENDICES

A TABLES 59

B PLOTS OF THE NULL POWER FUNCTION 79

C COMPUTER PROGRAMS 91

REFERENCES 9 8

BIOGRAPHICAL SKETCH 101

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

EXACT UNCONDITIONAL TESTS FOR
2*2 CONTINGENCY TABLES

by

Samy Salomon Suissa

August 1982

Chairman: Jonathan J. Shuster
Major Department: Statistics

The so-called "exact" conditional tests are very popular
for testing hypotheses in the presence of nuisance parameters.
However, in the context of discrete distributions, they must
be supplemented with randomization to become exactly of size
a, the nominal significance level. This practice is undesir-
able since irrelevant events should not affect one's decision.
Consequently, the conditional test without randomization,
while still called "exact," becomes conservative.

As an unconditional alternative, a methodology is devel-
oped to compute the exact size of any test when the null
power function is of a given form. This approach is a way of
catering to the worst possible configuration of the nuisance
parameter by maximizing the null power function over the
domain of the nuisance parameter. As special cases, the

2x2 contingency table to compare two independent proportions
and the 2x2 contingency table to compare two correlated
proportions are considered.

For the equal sample size case, exact critical values
of the Z-test for comparing two independent proportions are
computed and tabulated for n=10 (1)150, a=.025 and oc=.05.
Sample size requirements based on the exact unconditional
one-sided Z-test, with a— .05 and 80% power, were never larger
than the corresponding sample size requirements based on the
"exact" conditional test, namely Fisher's exact test. In
fact, the proposed Z-test is uniformly more powerful than
Fisher's exact test for n=10 (1)150, a=.025 and o=.05.

For comparing two correlated proportions, exact critical
values of the Z-test, which is the appropriate square root of
McNemar's chi-square test, are computed and tabulated for
N=10(l)200, a=.025 and ot=.05, where N is the total number of
matched pairs. Here again, sample size determinations based
on the exact unconditional one-sided Z-test, with <x=.05 and
80% power, were never larger than the corresponding sample
size requirements based on the "exact" conditional test.
In fact, the proposed Z-test is uniformly more powerful than
the "exact" conditional test for the cases considered, namely
N=10 (1)200, a=.025 and a=.05.

CHAPTER 1
INTRODUCTION

1. 1 The problem

The problem of testing a simple hypothesis about a
parameter 9, the index of a known probability distribution
P_ , against a simple alternative hypothesis, for a specified
significance level a, is straightforward. Its optimal
solution is given by the fundamental lemma of Neyman and
Pearson (Lehmann, 1959:65). However, many statistical models
involve more than the parameter being tested. The distribu-
tion could be indexed by the parameters (9,v), where v is an
additional unknown parameter. If inference is required only
about 9, the parameter v is called a nuisance parameter. The
problem then becomes one of eliminating this nuisance
parameter from the model.

Over the years, many solutions to this problem have been
put forward. Basu (1977) , in reviewing two such elimination
methods, classified all solutions that have been proposed
into ten categories. In this dissertation, three of the most
common of these methods will be discussed with applications
to 2x2 contingency tables. The first method, often used in

asymptotic theory, is that of replacing the nuisance
parameter by an estimate. The second is the conditional
method, which can evolve from two different approaches. The
third method considered is the maximization, method which
essentially caters to the worst possible configuration of the
null power function of a test by maximizing this function
over the domain of the nuisance parameter. This is the
definition of the size of a test as given in Lehmann (1959)
or Ferguson (196 7) . These three methods of eliminating
nuisance parameters will be presented in section 1.2, with
an emphasis on their advantages and drawbacks. Greater
emphasis will be put on the third method, namely the
maximization method, since it is the proposed solution to the
problems considered in this dissertation. In section 1.3,
some intuitive reasons, accentuated by an example from a 2x2
contingency table, will be given as the motivating force for
considering the maximization method as a potential competitor
to the other two. In section 1.4, the proposed approach
will be discussed in the context of the two problems to which
it is applied, namely comparing two independent proportions
and comparing two correlated proportions. Furthermore, a
preview of the results will be given.

1.2 Some Methods of Eliminating Nuisance Parameters

The estimation method. A classical method of elimina-
ting the nuisance parameter v is to replace it by an estimate
v. This practice is popular in asymptotic theory where all:
the parameters, except the parameter being tested, are often
replaced by consistent estimators. This method usually leads
to simple test statistics that are referred to well -tabulated
distributions. For example, the chi-square test for com-
paring two independent proportions and McNemar's test for
comparing two correlated proportions, both have an asymptotic
chi-square null distribution with one degree of freedom.
However, because all results are asymptotic or approximate,
these methods should not be used for small samples. Further,
it is usually unclear what size constitutes a "large enough"
sample. This method should be avoided in any study where
asymptotic theory is questionable.

The conditional method. The conditional method of
eliminating nuisance parameters may evolve from two different
approaches. The first of these approaches, due to R.A.
Fisher, is described in Kendall and Stuart (1967, vol. 2) as
follows. Let (T,U) be sufficient statistics for (6,v) . If
the marginal distribution of U is independent of 9 , then U
is called an ancillary statistic. The conditionality
principle of Fisher is stated as follows: If U is an
ancillary statistic (distributed free of 8), then the
conditional distribution of Tlu is all that is needed to test

a hypothesis about 9 . This principle is clearly acceptable
when U is a sufficient statistic for v (when 0 is known)
because the likelihood will then factor into two distribu-
tions, namely that of T|U with parameter 6 and that of U with
parameter v. The problem arises, however, when U is not
sufficient for v. It is then doubtful whether this principle
leads to "optimal" tests. In fact, Welch (1939) gave an
example in which the conditional test based on T|u may be
uniformly less powerful than an alternative unconditional
test.

The conditional method can be obtained through another
approach. By restricting the test procedure to a smaller
class of tests, namely a-similar or unbiased tests, the
nuisance parameter may be eliminated. In particular, if the
distribution of (T,U) , the sufficient statistics for (8,v),
is in the exponential family, the distribution of t|u will
give UMPU (uniformly most powerful unbiased) tests for
hypotheses about' 6 (Lenmann, 1959:134). Because of the
richness of the exponential family, UMPU tests can be readily
obtained for a large number of problems through a conditional
distribution which is free of the nuisance parameter.
However, if T|U is discrete, a randomization supplementing
the experiment will be required to make the test exactly of
level a. This randomization is undesirable since irrelevant
events should not affect one's decision. Hence, the test

without randomization becomes conservative and consequently
loses power in the process.

The maximization method. The third method of elimina-
ting nuisance parameters is the maximization method. It is
based on the direct utilization of the definition of the size
of a test, as defined in Lehmann (1959:61), which requires
that the null power function of any test to be maximized over
the domain of the nuisance parameter. In other words, the
maximization method is based on the premise that the worst
possible configuration of the nuisance parameter could take
place and that the testing procedure should always be
protected against this eventuality. This method is easily
explained to laymen because they are already familiar with
the test of hypothesis for a given value of the nuisance
parameter v. Therefore, by letting v vary, they will have
an intuitive feeling for the maximization method. The major
drawback to this method is the complexity of maximizing the
null power function. With the advances in computer
technology, this method should be investigated more fully.

1.3 Numerical Example

The problem of comparing two independent proportions,
called the 2x2 comparative trial, is an example of a test of
hypothesis in the presence of a nuisance parameter. If 8,
and 8 2 are the success rates of the first and second popula-
tions respectively, then the equality of these success rates

91=62 (=e ' unspecified) involves a nuisance parameter/namely
6 , the unknown common proportion of success in the two
populations. A numerical example is given to present the
discrepancies that occur between the first two methods of
eliminating that nuisance parameter. Let the outcome of
independent random samples of size n=10 each from two binary
populations n, and II- be

Table 1.1

S F totals

nx 4 6 10
n2 8 2 10

totals 12 8 20

where S=success and F-failure. The question is: Is the
success rate of H^ less than the success rate of n2 at
significance level a=. 05 ?

In the spirit of the estimation method, 6 may be
replaced by 6 = 12/20 in the Z-test statistic with pooled
variance estimator (section 3.2) to give

/20 (8-4)

Z = — j— = 1.83

p (12x8)^

which, upon referring to the standard normal distribution,

gives an attained significance level (p-value) of .0336,
which is less than a=.0 5.

If the conditional argument is used, Fisher's exact
test (section 3.3) is the solution to both approaches. By
restricting the sample space to only those outcomes with the
same marginal totals as Table 1.1, namely

Table 1.2

S F totals

I^ 10

n2 10

totals 12 8 20

the p-value is given by the sum of the hyper geometric
probabilities of all tables with the above marginal totals
which are more extreme than the observed one (i.e. < 4
successes in 11-^) in the direction of the alternative hypoth-
esis. Accordingly, Fisher's exact test for Table 1.1 has
p-value = .0849 > .05.

The statistician is now faced with a dilemma. He would
conclude that 91<92 by the first method, but could not by the
second method at a=.05. This discrepancy raises several
questions: How accurate is the normal approximation to the
Z-test ? How conservative is Fisher's exact test without
randomization ? If the conditional method is preferred,

how does one explain to the layman that the sample space is
restricted only to the tables with 12 total successes ?

The answers to these questions lie in the null power
function, a function of the nuisance parameter 8. For each
of the two tests, the exact null power function is plotted
on the basis of the attained significance levels as the new
nominal significance levels. For the Z-test, the plot of the
exact null power function, based on the Z-value of 1.83
(nominal significance .0336), is given in Figure 1.1. It is
seen that the exact significance would be greater than the
nominal significance level of .0336 when .13<8<.87. In that
range, the Z-test is liberal in the sense that it would
reject the null hypothesis when it should not at level .0336.
However, notice that the exact null power function never
exceeds the original significance level of a=.05, the maximum
being .047. For the conditional approach, the plot of the
exact null power function based on the attained significance
level of Fisher's exact test (.0849) is given in Figure 1.2.
The conservativeness of this conditional test is obvious.
In fact, its null power function is never larger than the
original nominal significance level of a=.05, the maximum
being .045.

Further insight into the reasons for such largely
different results can be obtained by inspecting the critical
regions of each test. Once more, the Z-test will be based
on the nominal significance level .0336 and Fisher's exact

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test on the level .0849. By representing all the possible
results of the experiment in the form of points of a lattice
diagram, given in Figure 1.3, the critical regions are simply
given by marked subsets of these points.

s2 =10
9
8
7
6
5
4
3
2
1
0

® 0

10 = s.

Figure 1.3 Critical regions of Z-test and Fisher's
exact test.

In Figure 1.3, S,= number of successes from II, and S2 =
number of successes from II2. The sample points in the
lattice diagram marked by an "0" belong to the critical
region defined by the Z-test (nominal significance level
.0336) and those marked by an "x" belong to the critical
region defined by Fisher's exact test (nominal significance
level .0849). Although the nominal significance level of the
Z-test (.0336) is much smaller than that of Fisher's exact

12

test (.0849), notice that the critical region of the Z-test
contains the critical region of Fisher's exact test. This
is a flagrant example of the conservativeness of Fisher's
exact test and of the liber alness of the Z-test.

These discrepancies suggest that the maximization method
might be more appropriate, if not more exact, than the first
two methods of eliminating the nuisance parameter. Therefore,
by either adjusting the Z-test or the unconditional Fisher's
exact test, the maximization method could lead to significance
levels which are closer to the nominal levels.

1.4 Proposed Approach and Preview

In this dissertation, the maximization method will be
used in conjunction with tests derived from the estimation
method to develop an exact unconditional testing procedure'
as an alternative to the popular, but conservative, condi-
tional method. This procedure will be applied to two
problems of the 2x2 contingency table, namely that of com-
paring two independent proportions and of comparing two
correlated proportions. For this purpose, a methodology
for computing the supremum of a null power function of a
certain general form is developed in Chapter 2. The form
of this null power function includes, as particular cases,
the null power functions of the two problems considered in
this dissertation. In Chapter 3, the supremum of the null
power function of the Z-test (the size of the Z-test) for

13

comparing two independent proportions, will be computed for
the equal sample size case. Consequently, exact critical
values and minimum required sample sizes will be obtained
and tabulated for the design and analysis of such a trial.
Comparisons with the conditional method (Fisher's exact
test) will show that the exact unconditional method leads to
smaller (or equal) sample sizes for a=.05, 80% power and
common sample size n=10 (1)150.

In Chapter 4, the size of the Z-test for comparing two
correlated proportions will be computed via the methodology
developed in Chapter 2. As a result, exact critical values
and sample size determinations will be obtained and
tabulated. This exact unconditional test will produce
smaller (or equal) sample sizes than the conditional test
(the sign test) for a=.05, 80% power and the number of
paired observations N=10 (1)200.

Furthermore, a comparision of the critical regions will
show that the exact unconditional Z-tests are uniformly more
powerful than their conditional counterparts in the range
considered, namely a=.025, a=.05 and n=10(l)150 for the
independent case, and N=10 (1)200 for the correlated case.

CHAPTER 2

METHODOLOGY FOR COMPUTING
THE SIZE OF A TEST

2 . 1 Introduction

In this chapter, a methodology is developed to compute

the unconditional size of any test of hypothesis T when:

a. There exists a nuisance parameter p such that 0<p<l
and

b. The null power function of T is a linear combination
of a finite number of binomial terms in p.

Thus, we assume that the null power function of T can
be written as

tt(p) = S a. pbi (l-p)ci , (2.1.1)

ieC

where a^, b. and c. are >0, i is an indexing subscript over
the whole sample space S defined by the sampling scheme, C is
the set of subscripts for which the related sample points
belong to the critical region defined by the testing procedure
T and p is a nuisance parameter on the unit interval.

The size of the test T is given by sup ir(p) and, because
only discrete null distributions will be considered, is
restricted to a finite collection of possible values, namely,

14

15
the natural levels of test T as referred to as in Randies and
Wolfe (1979) . The remainder of this chapter deals with the
technique of computing sup it (p) . The method used is based on

XT

the mean value theorem of differential calculus applied to
successive subintervals of the unit interval.

The use of this methodology will be illustrated in two
important cases. The case of comparing two independent
proportions is given in Chapter 3 and that of comparing two
correlated proportions in Chapter 4.

2.2 Local Bound for tt ' (p)

The applicability of the mean value theorem of differ-
ential calculus, as stated in Courant and John (1965),
requires primarily a bound on the derivative of it (p) for
each subinterval. The task of finding such a bound is
facilitated by the form of tt (p) , namely that it is a linear
combination of binomial terms. In this section, a method of
computing this bound for any subinterval is given. First,
note that the derivative of ir(p) ,

tt'(p) = z a.{ b. PDi "-(l-pr1 - c. p^u-pr1"-1- } ,

ieC x x 1

(2.2.1)

is also a linear combination of binomial terms of the form

h(p) = pr (l-p)S_r

16
so that for any given subinterval I=(a,b) with 0<a<b<l,

sup h(p) = h(b) if p > b
pel

= h(a) if p < a

= h(p) if p e I (2.2.2)

and

inf h(p) = min ( h(a), h(b) ) (2.2.3)

pel

where p = r/s.

An upper bound for it' (p) can be obtained on (a,b) by
substituting the right hand side of (2.2.2) for each
positive term of (2.2.1) and the right hand side of (2.2.3)
for each negative term of (2.2.1). Similarly, a lower bound
for tt'(p) can be obtained on (a,b) by reversing these substi-
tutions. Finally, a bound M for |ir'(p)| on (a,b) is taken as
the larger of the two bounds, in absolute value.

2.3 A Least Upper Bound for tt (p)

Since local bounds for |ir*(p)| can now be computed for
any subinterval (a,b) , the mean value theorem can be applied
to the successive subintervals I,=(0,.01), I_=( .01, .02) , . . . ,
I, _ =(.99,1) of the unit interval to obtain an upper bound
for ir(p) in each I.. An upper bound for ir(p), 0<p<l, is
simply the maximum of these local upper bounds.

17

First, for each I., j=l (1)100, it is now possible to

find a M. such that
3

1 7T » (9 .) | < M. , for all 6 . e I. ,

where the M.'s are obtained by the method of section 2.2.
By the mean value theorem of differential calculus, it can
be concluded that

ttOj) e ( 7r(p )-.005M.. , tt (p . ) + . 005M. ), (2.3.1)

for all 6.e I., where p .=( j-.5)/100, the midpoint of I.
For each I., j=l (1)100, the local maximum is given by

ir+(p.) = ir(p.) + .005M.
3 3 3

It can then be deduced that, for p on the unit interval (0,1) ,

ir(p) < max { Tr+(p.); j=l (1)100 }. (2.3.2)

J J

Therefore, the right hand side of (2.3.2) is greater than
sup it (p) , the size of T.

Because Mj can be quite large for some values of j ,
it is possible that the bound on the right hand side of
(2.3.2) will be a conservative upper bound for the function
Tr(p). This bound can be improved upon to produce a least
upper bound of precision 6 in the following manner. Any

18

interval I. for which

ir (p.) > max { ir(p.); i=l (1)100 } + 6
-1 i 1

m.

must be iteratively subdivided into 2 ^ subintervals

m.

{ I., ; k.=l,2,..., 2 -1 , m.£l }, where m. is the smallest
JKj J D 3

integer such that

m.
wtP-iv ) + .005 M.. /2 : < max{max{.ir (p. ) } , max{ir (p . . ) }} + 5

3 3 33 i X Uj :uj .

(2.3.3)

for all k . , and where

ra.-l
P^v = (j-D/100 + (2k.-l)/(100x2 D )
3 3 3

is the midpoint of I., and M., is the local bound for

Dkj 33
• ir' (p) in I.. . The inequality (2.3.3) is clearly

attainable since, from (2.2.1),

M. <; Z a. max(b. ,c.) . (2.3.4)

DKj ieC x x 1

Moreover, using the right hand side of (2.3.4) rather than

M., is inefficient and would make computing costs

Dkj
prohibitively large.

By the methodology developed in this section, it is

now possible to compute the size (with precision 6) of any

test for which the null power function is of the form (2.1.1)

19

These computations will then shed a light on the extent
of conservativeness or liberalness of the tests that are
used in the elimination of a nuisance parameter in the
present context.

2.4 Stability of the Null Power Function

In the ideal case of the absence of a nuisance
parameter, the null power function is constant over the
null parameter space. However, when a nuisance parameter
is present, the magnitude of its effect on the null power
function ir (p) can be of interest. A relevant feature is
the stability or flatness of ir(p). An indication of this
stability can be created using

inf { ir(pj; pe(p ,p,)} (2.4.1)

P

where p and p, are chosen appropriately for each problem,
a jd

The difference between sup ir (p) and (2.4.1) is then an
indicator of this stability. The whole unit interval is
not used in (2.4.1) because it (0)=tt (1)=0. A lower bound
for the set in (2.4.1) can be computed from (2.3.1) by

min { Tt(p.) - .005M. ; J=Jad)Jb >

where j . = int(100p ) + 1,

a. 3.

jb = int(100pb) + 1
and int(.) is the integer function, p- being as in (2.3.1)

unconditional sample
an acceptance region
optimal procedures

20

2.5 Choice of the Test Statistic

The choice of the testing procedure is quite arbitrary
since the goal here is to compare unconditional to
conditional tests. The test statistic that is derived from
a testing procedure will simply be a means of dividing the

space S into a critical region C and
S-C. The choice should then be based on
that produce test statistics that are
powerful, simple to compute, and intuitively appealing.
Three such procedures are the likelihood ratio test
criterion, the chi- square goodness-of-fit test and a Z-test
based on the asymptotically standardized maximum likelihood
estimator, often a function of the sufficient statistic, of
the parameter being tested.

The likelihood ratio criterion is given by

R =

where L(6) is the likelihood function of the sample. If
large sample theory applies, the more convenient equivalent
statistic

X2 = -2 log(R) (2.5.1)

can be used because of its limit, chi-squared null distribution.
The chi-square goodness-of-fit test, based on the statistic

sup

Ho

L(8)

sup

L(8)

21

2 (O-E^2
* = E X (2.5.2)

i E.

i

is especially apropos because of its applicability to
multinomial data which lead to null power functions of the,
form (2.1.1). The asymptotic null distribution of (2.5.2)
is also chi-square. The third test is based on the
asymptotic normality of 9 , the maximum likelihood estimator,
if one exists, of the parameter, call it 6, being tested.
The statistic

9 - e

Z= ". , (2.5.3)

s(6 )
n

2 ~ ~ ■

where s (9 ) is the asymptotic variance of 9 , or a

consistent estimator thereof, has a standard normal asymptotic
null distribution under the regularity conditions of likeli-
hood theory. For the two problems considered in Chapters- 3
and 4, the maximum likelihood estimator is a function of the
sufficient statistic. This statistic and the chi-square
goodness-of-fit statistic are the most appealing since (2.5.1)
could be computationally laborious.

A further advantage of using the test statistics (2.5.1),
(2.5.2) and (2.5.3) is their well-known and well-tabulated
asymptotic distributions. These tables can be used to find
reasonable starting points for the critical values. Moreover,
these values can be compared to the percentage points of the
asymptotic distributions and thus provide a study of the
accuracy of the large sample approximations.

CHAPTER 3

THE 2x2 TABLE FOR
INDEPENDENT PROPORTIONS

3.1 Introduction

A classical problem is the one of comparing two
proportions from independent samples . This seemingly simple
problem that involves only four numbers, has generated a
large amount of .literature and has been the subject of much
controversy about the use of conditional tests. Since
Fisher (1935) proposed the "exact" test, Barnard (1947) and
Pearson (1947) started a conflict that has not yet been
resolved, as can be seen in the recent articles by Berkson
(1978), Barnard (1979), Basu (1979), Corsten and de Kroon
(1979) and Kempthorne (1979). Because of the complexity of
the power function, only partial attempts have been made in
order to resolve the argument. The statement of the problem
follows.

Let X and Y be independent binomial random variables
with parameters (n,?^ and (n,p2) respectively. An experi-
ment that compares p, and p_ is called a 2x2 comparative
trial by Barnard (1947) , the outcome of which is represented
in the form of Table 3.1,

22

23

Table 3.1

S

F

totals

Tl

X

n-x

n

T2

y

n-y

n

where p^ = P'(S|T^) = l-PfFl^), i=l,2. The labels S and F
represent the binary outcomes ( S=success, F=failure) and . T,
and T2 represent the two populations being compared.

The problem is to test, at level a, the null hypothesis
H0:]?l=]?2 against the alternative hypothesis H :p.<p_. Because
the other one-sided alternative H :p_>p„ and the two-sided
alternative Ha:P^P2 are treated in a similar manner, only
this one-sided case will be considered here. Furthermore,
only the case of equal sample sizes will be considered because
of its optimality under equal sampling costs (Lehmann,
1959:146). The probability of observing the outcome in
Table 3.1 is

P(X=x,Y=y) = (x} px (1-Pl)n-X {P p* (l-p2)n-y

and is, under the null hypothesis H0:p.=p2 (=p say),

P(X=x,Y=y) = {x)(y} px+* (l-p)2""*^ ,

a function of the nuisance parameter p, the unspecified
common value of p. and p_ under H_.

24

Because of this dependence on a nuisance parameter,
either approximate tests based on asymptotic results or
exact conditional tests are used. Few attempts, however,
have been made to compute the exact unconditional size of
any of these tests. Barnard (1947) proposed an uncondi-
tional test based on sup tt (p) , the size. The criterion
that he suggested was intricate and no methodology for
computing the size was given. McDonald, Davis and Milliken
(1977) tabulated critical regions based on the unconditional
size of Fisher's exact test for n<;15 and a=.01 and .05.
Again, no formal methodology for computing the size was
given. Furthermore, no sample size tables or power
calculations based on an exact unconditional test exist.

In this chapter, the most common tests are presented.
For the asymptotic case, two normal tests and some sample
size formulae are given. For the general case, and in
particular for small sample sizes, two derivations that
both lead to Fisher's exact test are presented. As an
alternative to these tests, the results of chapter 2 are
used to compute and tabulate the exact unconditional size
of two simple statistics as well as the required sample
sizes for a significance level of a=.05 and a power of
1-8=. 80. It is also shown that these tests are uniformly
more powerful than Fisher's exact test in the range
considered, namely a=.025, a=.05 and n=10 (1)150.

25
3.2 Asymptotic Tests and Sample Size Formulae

A way of circumventing the effect of the nuisance
parameter is through the use of asymptotic tests. These
approximate tests are appealing because they are usually
based on simple test statistics for which the limiting
distributions are well tabulated. They are, however,
approximations and should not be used when the sample sizes
ar.ev small.

When n is relatively large, the most widely used tests
for the hypothesis of interest are the normal tests. The
first one, based on the inversion of the asymptotic
confidence interval for P2-Pi/ is tne Z-test with an
unpooled estimator of the variance and is given by

= ^ (P2 - Pl} (3.2.2)

(p2q2 + Pi<3i>

where p1 = x/n = l-q1, p2 = y/n = l-q2, with x, y and n as
in Table 3.1. The second one, based on the asymptotic null
distribution of P2~P1f is the Z-test with a pooled variance
estimator and is given by

/n (P, - Pi)

Z = - — (3 2--21

P ( 2 p q >% (3*2'2)

where p1 and p2 are as in (3.1.1) and p = (x+y)/2n - 1-q.

The limiting distribution of both Z and Z is the standard .

u p

normal distribution and an approximate test of size a is
based on the percentage points of $, the standard normal

26

distribution function. The test statistic Z is most

P

frequently used through an equivalent test statistic, the
chi-square goodness of fit statistic given by

Xp=Zp (3.2.3)

2
which has a X.. limiting distribution. Because this chi-
square test deals with two-sided alternatives, the statistic
Z is preferred in the present context of a one-sided test.

The accuracy of the approximation was studied by
various authors. The nominal significance level a was
compared to the actual significance level by computing
(3.4.3) for some values of p. Between the papers by
Pearson(1947) and Berkson (1978), numerous studies have
shown that for Z_, the actual level could be larger than
the nominal level for some values of p, making this test a
liberal one.

To determine the sample size required in each group,
two formulae are often used. The first one, based on Z and
derived in Fleiss (1980) , determines the sample size in each
group by

[ z Up^)* - z, (p ,q +p q )h ]2
n = - ^ -^± — £_J (3.2.4)

(P2-pl)2

where z is the upper 100y percentile of the standard normal
distribution, a and 8 are the type I and type II error

27
probabilities, p. and p2 are the desired alternatives, arid
P = (Pi+Pt)/^ = 1~P- The second formula, based on the
variance stabilizing property of the arcsine transformation
on proportions, is given in Cochran and Cox (1957) by

n = (za + z3) (3.2.5)

aS -1 -1 2
2 (sin /p, - sin /p_)

Other formulae have been derived and are mostly corrected
versions of (3.2.4). Kramer and Greenhouse (1959), arguing
that the test based on Z was too liberal, adjusted (3.2.4)
and found

{ 1+[1+8(P2-Pl)/Vi5 }2 , (3.2.6)

nc np 2

4(p. - p, )

where n is the sample size found from (3.2.4). More recent-

XT

ly, namely since sample size tables based on the exact
conditional test were computed, a further adjustment to
(3.2.4) was suggested by Casagrande, Pike and Smith (1978a)
in order to arrive closer to results based on the exact
conditional test. They proposed the formula

{ l+tl+Mp^/np]* }2 \ {32.7)

nr np 2

4(P2 - Px)

28

the derivation of which was based on a slight deviation
from the derivation of (3.2.6).

3.3 Fisher's Exact Test

The "exact" method of eliminating the nuisance parameter
is based on a conditional argument and can be obtained via
two different approaches. The first approach, put forward by
Fisher (1935), is that of a permutation test. The permu-
tation test argument is a conditional one in that the
critical region is1 constructed on a space conditional on some
information from the data. Fisher argues that, because the
marginal totals of Table 3.1 alone do not supply any
information about the equality of p, and p2, it is reasonable
to test conditionally. Thus, given x+y and under HQ:p,=p2,
the probability of Table 3.1 is given by the hypergeometric
distribution, namely

(n) (n)
x y

P(x,y|x+y) = • (3.3.1)

(2n )
x+y'

This is Fisher's exact test, the size of which is based on
the tail areas of (3.3.1).

The second approach is based on the Neyman-Pearson lemma
for. testing hypotheses, a thorough treatment of which is
given in Lehmann (1959:134) in the case for which a nuisance
parameter is present. In the current case, the probability
of Table 3 . 1 is given by

29

p(x,y) = 0 Px d-Pl)n-x (y} 4 d-p2)n-y

(x)(y} (1-Pl)n (l-p,)n

x exp{x log[p1/(l-p1)]+y log[p2/ (l-p2) ] }

= (x)(y} (1-Pl)n (l-p2)n

x exp{x logL /(1_p )j +(x+y)log[p2/(l-p2)]}.

By Lemma 2 of Lehmann (1959:139), the uniformly most powerful
unbiased (UMPU) level a test for comparing p, and p2 is based
on the conditional distribution of X(=x) given T=X+Y(=t)
and has the form

<j>(x,t) =1 when x < C(t)
= y (t) when x = C(t)
= 0 when x > C(t)

where C and y are determined by

E„ [ (j»(X,T) |T=t ] = a
H0

for all t, that is

30

a = P [X<C(t)|T=t] + y(t) P„ [X=C(t)|T=t]

no "o

The conditional distribution is

P(X=x|T=t)

(x} (t-x> px

t ,n» , n v

2, U t-U p
U=0

Pl/(1-Pl)

where p = is the odds ratio,

P2/(l-p2)

and. under Hn, the distribution is given by

(n) ( n )
PH (X=x|T=t) = , x=0,l,...,t,

the same hypergeometric distribution found by Fisher's
permutation method. Here, C(t) is taken to be the largest
value such that

(n) ( n V
C(t)-1 V ^t-x;

Z £ a.

x=0 .2n.

In practice, the nonrandomized version of <{> is used,
that is $ without the random element y(t) . Therefore, the

3i

conditional test always has size ^ a and, unlike <J> , is not
UMPU of level a.

3.4 An Exact Unconditional Test

In this section, the methodology of Chapter 2 is used
to compute the size of Z , the normal test statistic with
unpooled variance estimator. This statistic was chosen on
the basis of its computational simplicity and its intuitively
appealing form. It is given by

•n (p2 - px)
>2q2 + Pxqr

Z„ = = i — 5- (3.4.1)

where p, = x/n = 1-q-, / P2 = Y/n = I'^o ' with x, y and n as
in Table 3.1. The asymptotic null distribution (the
standard normal) of Z is frequently used in this problem
to approximate its actual size. The results of this chapter
can thus be used to verify the accuracy of this approximation.

Since x and y are outcomes of independent binomial
random variables with parameters (n,p,) and (n,p2) respective-
ly •■/. the power function of any test is given by

n(p,,P2) = E E (x} px (l-p,)n"X Vp* (l-p-)n"y
1 Z (x,y)eC L L 2 2

(3.4.2)
and under HQ:p,=p~ (=p say), the null power function, also

32
denoted by tt, is given by

ir(p) = E E (x)(y} pX+y (l-p)2""^ (3.4.3)

(x,y)eC

where p is the nuisance parameter and C is the critical
region defined by the test statistic. For the one-sided test
of interest, the critical region defined by Z is given by

C = {(x,y): Zu > zu; x,y=0(l)nf z^O }. (3.4.4)

For an a level test, the critical value of Z , namely z ,

u' J u

satisfies the equation

*

zu = inf {zu: sup tt (p) £ a }. (3.4.5)

P

Since (3.4.3) has the form of (2.1.1), the methodology of

*
Chapter 2 can be utilized to find a , a value at most 6 above

sup tt (p) in (3.4.5) .

First, to simplify the computations, (3.4.3) can be

reduced to a single summation by solving the inequality

zu > zux namely

(-y(n-y) + x(n-x)^
y - x > z„ . (3,4.6)

After squaring both sides of (3.4.6), the larger root for
y in

33

n (y-x) = z2 [y(n-y) + x(n-x)]

is found to be

y = h(x) = 5 t (b2-^

2a

2
where a = 1 + z /n ,
u

b = 2x + z2 ,

' ■ - 2 2

and c = ax - xz

Hence (3.4.4) reduces to?

C = {(x,y): y > h(x) ; x,y=0(l)n }. .. (3.4.7)

Next, (3.4.3) can be written as

w(P) -I E (x"y' px+y (l-p)2"-^
x=0 y>h(x)

= Z (x} px (l-p)n-x Z (y> Py (l-p)^
x=0 y>h(x)

v

Z f(x) [ 1 - F(h(x)) ] (3.4.8)

x=0

2 2
where v = int[ n /(n+z ) ], int[.] is the integer function,

f (.) is the binomial probability mass function with parameters

(n»p) and F(.) is its cummulative distribution function.

34

The derivative of ir (p) , which can also be reduced
significantly to simplify the computations, is given by

ir'(p) = E #<?> (x+y) p^"1 (l-P)2n-x-y

,nx ,n
C

,nx ,n

-()()/ . \ x+y , , x 2n-x-y-l

- E x y (n-x+n-y) p 2 (1-p)

C

where E denotes the double summation E E

C (x,y)eC

It can be rewritten as

./ x «. (n~h (n) x+y-1 ,, ,2n-x-y
ir'(p) = E n vx-l' vy' p ■* (1-p) x ■

C

C

(n_1) (n) x+y ,, , 2n-x-y-l

- E n x y p 2 (1-p) -1

C

( ) ( ) x+y ,, x 2n-x-y-l /r) . ft«

- E n x' y p 2 (1-p) J , (3.4.9)

C

(n-l} (n-l}
where v -V = v n ' = 0

Consider the boundary of the critical region C defined by

W .'= {(x,y) : (x,y)eC and (x+l,y)£C }.

The sum of the first and third terms Of (3.4.9) becomes,
after cancellation of opposing signed identical contributions,

35

Tr-(p) = - I n (n;1)(y} px+y (l-p)211"^'1 . (3.4.10)
1 W

The other boundary of C is defined by

V = { (x,y) : (x,y)eC and (x,y-l)jzfc }.

Then the sum of the second and fourth terms of (3.4.9)
becomes

ir'(p) = En (x)(?:i} p***"1 (l-p)2n-X^ . (3.4.11)
V

Upon combining (3.4.10) and (3.4.11), the derivative of "rr (p)
is given by

it- (p) = ir£(p) + ir^(p) , .

and can be further reduced by noticing, from (3.4.4) and
(3.4.6) , that

(x0,y0)eC iff (n-y0,n-x0)eC

so that

(x,,y,)eV iff (n-y^n-x^ eC and (n-y1+l,n-x1) £C
iff (n-y, ,n-x, ) eW.

36
The derivative of it (p) can finally be written as

ir'(p) = En [<5)(y"l) p^'1 (l-p)211"^
V

(n-l} ( n ) 2n-x-y ,, ,x+y-l ,
n-y' n-x p J (1-p) ]

v (n) (n"h r x+y-1,, , 2n-x-y
E n vx y-1 [ p J (1-p)

V

2n-x-y (l-pjW-1 ] f

- P

a summation over the set of sample points that form a
boundary of the critical region C and which are directly
obtained from the reduction (3.4.7) .

The methodology developed in Chapter 2 can now be

*
utilized to find a , the size (of precision 6=.001) of Z

for any value z . For a test of significance of level a,

*
the critical value z can then be obtained by equation

(3.4.5). This is done by using the 100a percentile point of

the standard normal distribution as a starting of z . This

*
value is then incremented or decremented until a < a and

*

z is taken as the smallest value which satisfies this

inequality. This procedure was implemented in a FORTRAN

computer program, listed in Appendix C.l.

*
For n=10 (1)150 and a=.05 and .025, z , the exact

*
critical values and a , the size (of precision <5=.001) of Z ,

were computed and are given in Table A.l. Furthermore, Table

A.l also contains a,, a lower bound for {ir(p); .05<p<.95}

37

and a., a lower bound for {ir (p) ; .10<p<.90}, as indicators
of the stability of the null power function. The critical
values of Z , the Z statistic with pooled variance estimator

discussed in the next section, are also given in Table A. 1

*
and are denoted by z
P

The null power function, tt (p) , of each test procedure
was plotted for some values of n and a nominal significance
level of a=.05. For n=10, Figure B.l contains the plot of
ir (p) based on the normal approximation of Z (z =1.645) and
Figure B.2 is based on Z (z =1.645). Although both graphs

IT ST

exceed the ideal value of a=.05 for some values of p, showing
that the normal approximation produces a liberal test in this
case, it is clear that Z gives a better approximation than
Z when referred to a standard normal distribution. This
point is discussed in length in the next section. Figure B. 3
is based on the unconditional critical region defined by
Fisher's exact test, the criterion used by McDonald, Davis
and Milliken (1977) in their unconditional approach. The
conservativeness of Fisher's exact test is evident; its

actual size in this case being approximately .02. In Figure

* *

B.4, the value z =1.96 (or equivalently z =1.80) of Table A.l

is used to plot tr (p) . This plot is seen to perform the best

at approaching the nominal level a=.05 without exceeding it.

For larger values of n, the null power function of the

exact unconditional test Z (or equivalently Z ) is seen to

behave better. For n=20, Figure B.5 is the plot of ir(p)

38

* *

based on z = 1.85 (or z = 1.78) and Figure B.6 is based on

n=30 and z* =1.77 (or z* = 1.73).
u P

3.5 Relation to the Chi-square Goodness-of-fit Test

The other test statistic given in (3.2.2), namely Z ,
the square of which is the chi-square goodness-of-fit test
statistic, has a functional relationship with Z in the
present equal sample size case. This relationship can be
derived by first noticing that

/n (y - x)

Z - . (3.5.1)

[y(n-y) ,+ x(n-x)]*

and

i/n (y - x)

Z = , . (3.5.2)

P [%(x+y) (2n-x-y)]^

The square of the denominator of Z

yn — y+xn-x (3.5.3)

can be compared to the square of the denominator of Z ,

yn + xn - %(x+y)2 ■'. (3.5.4)

By rewriting (3.5.3) as

39

yn + xn - ^(x+y)2 -h (y.-x) 2 ,

it is clear that Z and Z satisfy the relation
p u

■1- = _i_ - -i_ (3.5.5)

Z^ Zz 2n
u p

which can be rewritten -as

0 2n Z2
Z2

p 2n + Z2

u

*
so that the critical value z of Z„ given in Table A.l are

P P

obtained by

r 2n Z2 -jJs
Z„ = sgn(y-x) ^-\ , (3.5.6)

p hn + Zz^

where

sgn(u) = 1 if u > 0
=0 if u = 0
= -1 if u < 0 .

Robbins (1977) has noted that |Zu| > |z | for the
equal sample size case and has posed the question as to
which of Z or Z is more powerful. This question was

40

investigated by Eberhardt and Fligner (1977) . They noticed,
via a computational argument, that the increase in the
significance level for Z is compensated fairly well by an
increase in power. Moreover, they suggested that Z should
not be used for small samples because Z is closer to a
standard normal random variable . In view of the relation
(3.5.5), Z and Z are monotonic increasing functions of
each other and are therefore equivalent in the sense that
Z , with some nominal significance level a, is equivalent
to Z with some lower level a. Thus, for the same nominal
level a, Z will reject H- more often than Z will.

3.6 Power and Sample Sizes.

Given the critical values of Table A.l, it is now
possible to compute the exact power by (3.4.2) for a=.025
and a=. 05 and various values of p. and p2« The minimum
sample size required per group to attain a power of 1-g
and significance level of a can thus be computed by solving
the equation

n = min {n: II (p,,p2) >l-g}

where the critical region that defines K (p. ,p2) is based on

*
z , a function of n.
u

This equation was solved for a=.05 and l-&=.80 and the

results are given in Table A. 2 for various combinations of

*
p, and p-. Table A. 2 also contains the critical values ?; ,

41

*
the size a (of precision 6=.001) and the attained power

*
1-6 . This table is thus sufficient for both the design

and analysis of the 2x2 comparative trial.

Table A. 3 compares the results of Table A. 2 to the

exact conditional test sample sizes [n ] found in Gail and

Gart (1973) , Haseman (1978) and Casagrande, Pike, and Smith

(1978b). Furthermore , the approximate formulae given in

*
section 3.2 are also computed and compared to n and n

in Table A. 3. For the configurations considered, it is

*
seen that n tend to be smaller than n , the sample sizes

determined by Fisher's exact test. Furthermore, the sample

sizes based on the arcsine formula [n 1 and those based on

as

Z , the pooled Z-test [n ] , tend to co-agree quite well and

*
to be, in general, slightly smaller than n . The other

formulae discussed in section 3.2, namely n„ and n_ are

c r

*

seen to exceed n and n^.

e

A direct comparision of the critical regions defined by
Fisher's exact test and the exact Z-tests was performed numer-
ically. It showed that, for all the cases considered, the
critical region defined by Fisher's exact test is contained in
the critical region defined by the exact Z-tests. Therefore,
the exact Z-tests are uniformly more powerful than Fisher's
exact test for the cases n=10 (1)150 and a=.05 and .025.

CHAPTER 4

THE 2*2 TABLE FOR
CORRELATED PROPORTIONS

4 . 1 Introduction

When the dichotomous responses for each of two regimens
are sampled in pairs, either by measuring the same experimen-
tal unit under each regimen or by pairing experimental units
with respect to some common characteristic, the problem of
comparing the success rate of these two regimens involves
two correlated proportions. Prior to 1947, this type of data
was incorrectly analyzed as if they were independent binomial
samples. McNemar (1947) derived the variance of the differ-
ence between two correlated binary random variables under the
null hypothesis of equal success rate and consequently, using
an asymptotic approach, derived the well-known "McNemar' s
test". This problem, like the independent binomial case,
falls into the realm of testing a hypothesis in the presence
of a nuisance parameter. Analogous to the independent
binomial case (Chapter 3) , the most common methods of tack-
ling this problem are based on asymptotic approximations or
on the conditional approach. The problem is formulated as
follows.

42

43

Let (R,S) represent a pair of binary random variables
with joint distribution

P(R=i,S=j) = pi. , i,j=0,l, ZZ p±. = 1.

ij J

The outcome of a random sample of N such matched pairs is
usually displayed in the form of a 2x2 contingency table
such as Table 4.1,

Table 4

1

S

0

l

totals

u

X

u+x

y

V

y+v

totals u+y x+v N

where {u,x,y,v} are the frequencies.

The problem is to test, at level a, the null hypothesis

HQ:P(R=1) = P(S=1) against one of the alternative hypotheses

H :P(R=1) < P(S=1), H :P(R=1) > P(S=1) or H :P(R=1) ^ P (S=l) .
a a a

For the sake of illustration, only the alternative hypothesis
H :P(R=1) < P(S=1) will be considered. Note that

a.

P(R=1) = P(R=1,S=0) + P(R=1,S=1)
and P(S=1) = P(R=0,S=1) + P(R=1,S=1)

44

so that the problem becomes that of testing Hn:pnl=p1Q
against H :p 1>p.fl. The likelihood of the sample is given by

( N )
P(u,x,y,v) = {u x y v' pjjg p^ p^0 pi;L

the quadrinomial distribution with probabilities {p.. ;
i,j=0,l}. Under the null hypothesis ho:Pqi=Pio (=P saY) i
the likelihood of the sample becomes

r> / \ ( ) u x+y , , o \ v
PH (u,x,y,v) = 'u x y v' PQ0 P J (l-p0Q-2p) ,

a function of the unspecified common proportion p and an
unknown probability P00^

The problem was first tackled by McNemar (1947) who used
the asymptotic approach of the standardized sufficient
statistic. Cochran (1950) , by an intuitive argument, reduced
the problem to a sign test, which is the exact conditional
test obtained by the Neyman-Pearson approach to the elimina-
tion of nuisance parameters. Bennett (1967) has computed the
chi-square goodness-of-fit test statistic and observed that
it coincides with McNemar 's test. A point to note about
these asymptotic tests is that they are also conditional in
the sense that they only involve x and y, and not N. It
turns out that they simply evolve from the asymptotic null
distribution of the exact conditional test. No attempts have
been made to compute the size of any of these tests, although
Bennett and Underwood (1970) , in assessing the adequacy of

45

McNemar's test against its continuity-corrected form, have
computed their null power functions for three values of the
nuisance parameter p, namely p=.10, .50 and .90. Beyond this
investigation, researchers have completely relied upon these
asymptotic approximations and the conditional test. It is
surprising that conditional tests were not contested in this
problem, in light of the fact that they are solely based on
the number of discordant pairs x and y, and not at all on the
number of concordant pairs u and v. That these tests do not
involve N could be disturbing. Lehmann (1959:147), discuss-
ing in the context of the sign test with ties, has hinted
that N enters the picture through the parameter p0Q when the
unconditional power is computed.

Approximate power calculations and derivations of sample
size formulae were made by Miettinen (1968). Bennett and
Underwood (1970) compared the exact and approximate powers of
McNemar's test and its continuity-corrected form for alterna-
tives close to the null state. Schork and Williams (1980)
tabulated the required sample sizes based on the exact power
function of the conditional test.

In this chapter, McNemar's test and other asymptotic-
type tests will be presented. The approximate sample size
formulae will also be given. The exact conditional test will
be derived via the Neyman-Pearson approach. The results of
Chapter 2 will then be used in section 4.4 to compute and
tabulate the size of McNemar's test for the one-sided case.

46
In section 4.5, the exact unconditional critical values
obtained in 4.4 will be used to tabulate the required sample
sizes for a significance level of a=.05 and a power of
1-3=- 80. It is also shown that this exact unconditional test
is uniformly more powerful than the exact conditional sign
test for the cases considered, namely a=.05, a=.025 and
N=10 (1)200.

4.2. McNemar's Test, other Asymptotic Tests and Sample
Size Formulae \

McNemar (1947) derived the mean and variance of S-R

(as defined in Table 4.1) under the null hypothesis and thus

proposed the asymptotic test statistic

2 (x - y)2

* = (4.2.1)

x +; y

for the two-sided alternative. This statistic has an asymp-

2
totic X^ null distribution. Cochran (1950) reduced the

problem to a sign test, using the statistic

j (x - hn) 2 (y - %n) 2

XZ = +

%n ^n

(x - y)2

x + y

where n=x+y, the total number of discordant pairs. Bennett
(1967) used the chi-square goodness-of-f it test, applied to

47

the quadrinomial frequencies of Table 4.1, to find

(u- Np0Q)2 (x - NpQ1)2 (y - Np1Q)2

x = + +

NPnn Npnl Np

(v - NPll)

00 "^01 "F10

2

+

Np

11

(x - y)2
x + y

where the P-;^,s are the maximum likelihood estimators of
the Pj^'s under HQ. The three methods lead to the same test
statistic, namely McNemar' s, and therefore have the same
asymptotic null distribution.

To determine the required sample size, Miettinen (1968)

derived two formulas. The first one, based on an approxima-

2
tion to the asymptotic unconditional power function of X

(McNemar's test statistic) gives, for a one-sided test of

significance level a and power 1-&, the required sample size

as

{ zj + z (ij;2 - A2)*5 }2

Na = 2 § (4.2.2)

1 2

where ^=Poi+Pio' A=p10~p01 and zy ^s the uPPer 10°Y percen-
tile of the standard normal distribution. The second formula

48

is based on a more precise approximation to the asymptotic

2
unconditional power function of X and is given by

{ z Jf + zR [^2 - %A2(3+i|;)]Js }2

N = 2 5 . (4.2.3)

2 2

For the purpose of comparision with exact conditional
and exact unconditional results, these formulas were computed
and are given in Table A. 6. These comparisions are discussed
in section 4.5.

4 . 3 The Exact Conditional Test

The exact conditional test is obtained by the Neyman-
Pearson approach described in Lehmann (1959) . The
probability of the sample is given by

( N ) x

P(u,x,y,v) = u x y v; pj}0.P01 P^0 P^

and can be written in the exponential family form as

( N )
P(u,x,y,v) = vu x y v1 exp{ u log(p0Q) + x log(pQ1)

+ (N-u-x-v) log(p1Q) + v log(pi:L)}

It can be reparametrized as

49

( N )
P(u,x,y,v) = vu x y v' exp{ u log(p00/p1())

+ x log(p01/p10) + v log(Pll/p10)

+ N log(p10) }.

The new parameters are, in the notation of Lehmann (1959) ,

e "■- iog(p01/p10)

v = ( log(p00/plp) , log(pi;L/p10) ),
and the hypothesis to be tested becomes Hn:6=0 against

0

Ha:6>0. The sufficient statistics" are

X = Z (l-R.)S.
i-1 X X

T= (U,V) = ( Z (1-R. ) (1-S.) , Z R. S. ).

i=l x X i=l X X

Therefore the UMPU test is given by

<j>(x,t) = 1 when x > C(t)
= y ( t) when x = C(t)
= 0 when x < C(t)

where C and y are such that

50

E { <|>(X,U,V) | U=u, V=v } = a , all u,v.

"o

To find this conditional expectation, first notice that the
distribution of (U,V) is

P(u.v) = <u v N-u-v> p", pjx d-P00-Pll)N-U-V

so that the distribution of X given U=u and V=v is

P(x|u,v) = <NxT} P*x P?0 / (P01+P10)N-U"V

-'^ (poi/(Poi+Pio))X (Pio/(Poi+Pio))Y

= (x> px (l-p)n-x

where n=x+y and p = pQ1/ (P01+P1()) ' Therefore, the null
hypothesis HQ:p01=p10 reduces to HQ:p=%, the usual sign test
problem based only on n, the total number of discordant
pairs. Because this conditional distribution of X is
discrete, the test <{> needs the randomization element y to
become UMPU of level a. However, since the practice of
using y is rare, the test without randomization will be a
conservative one and not UMPU of level a.

51
4.4 An Exact Unconditional Test

As in the case of two independent proportions, the
choice of the test statistic is based on the standardization
of the sufficient statistic for the parameter being tested,
namely Pqi'Pio' Tnis statistic is the square root of
McNemar's test statistic and is given by

x - y

Zc = r (4.4.1)

. (x+yp

where x and y are as in Table 4.1. This statistic is often
written in terms of n (=x+y) , the total number of discordant
pairs, as

x - %n
Z„ = , (4.4.2)

c

Jj/ti

the approximation to the sign test, referred to the standard
normal distribution. In this section, the methodology of
Chapter 2 is used to compute the size of Z and the exact
critical values based on Z . These values will then provide
a means of assessing the accuracy of the normal approximation.
The power function of Z is given by

^Poi^io1 = (xyeC l (u x y v> Poo Poi Pio pIi '

a function only of pQ, and p,Q since it is based on the
marginal distribution of (X,Y) which is obtained as

52

p<x'y) = "27 <u *Ny v> p»o poi pjo <>i~ry-u

- <x y N-x-y) p*x py, (1-P01-P10)N-^

The power function of Z can then be written in terms of
n (=x+y) as

f N )

n(poi'pio) = , l I x n"x N"n Poi p?oX

(x,y)eC

x (l-p01-p10)N"n , (4.4.3)

where C is the critical region defined by Z . Under the null

hypothesis Ho:Poi=pio *=e say* ' the nul1 power function of Z
is given by

l N .

A(6) = Z Z {x n-x N-n; 9n (l-29)N"n

(x,n)eC

or equivalent ly, if p = 26, by

, N .
ir(p) = S S lx n-x N-n; hn pn (l-p)N~n , (4.4.4)
(x,n)eC

a function of the nuisance parameter p, the probability of a
discordant pair. For the one-sided test of interest, namely
for the alternative hypothesis H :P01>P10j the critical

53

region C defined by Z is given by

C = {(x,n): Zc > zc; x=0(l)n, n=0(l)N, z >0}. (4.4.5)

*
For an a level test, the critical value of Z , namely z y

satisfies the equation

z = inf {z : sup it (p) < a} . (4.4.6)

Note that ir (p) , as defined in (4.4.4), is a function of p as
well as of z through (4.4.5). Since (4.4.4) has the form of

(2.1.1), the methodology developed in Chapter 2 can be

*
utilized to solve (4.4.6) and thus find a , a value at most

6 above sup tt (p) .

First to simplify the computation of (4.4.4), notice

that the inequality Z > z is in fact

x > Jjz /n + %n

so that the critical region C reduces to

C = {(x,n): x > h(n); x=0(l)n, n=0(l)N} ,

where h(n) = %{z /n + n}. The null power function (4.4.4)
becomes

54

ir(p) = -E S V'x' %n pn (l-p)N"n
n=0 x>h(n)

I n p (1-p) E x %
n=0 x>h(n)

E (n} pn (l-p)N"n [1 - F (i )■]

n=k

n n

where k = int[zc + 1], in= int[h(n)], int[.] is the integer
function and F_(.) is the binomial cumulative distribution
with parameters (n,%) . Notice that, since i > Jjn, it is
more efficient to compute

n ,n,

E (x> hn

x=i +1
n

instead of 1-Fn(in). Then, by the symmetry of the binomial
distribution with (n,^), the null power function of Z can
be rewritten as

N ,N.
ir(p) = E V pn (l-p)N"n F (n-i -1), (4.4.7)

n=k n n

in the form of (2.1.1). The derivative of the null power
function is

55

tt'(p) = I F (n-in-l) V [ n pn_i (l-p)N~n
n=k n n

(N-n) p (1-p) ]

so that the methodology developed in Chapter 2 can now be

*
utilized to find a , the size (of precision S=.001) of Z

for any value z .

In this problem, the size was taken as sup{iT (p) :0<p<.99}

because of the behaviour of tt (p) when p approaches 1. The

null power function is dominated by the last term of the

N
summation, namely p FN(N-iN~l) , when p tends to 1. From

the practical point of view, the fact that p tends to 1

implies that almost no concordant pairs will be observed, so

that the problem virtually reduces to a problem with no

nuisance parameter, namely the sign test with no ties. It

then seems reasonable to compute the size on the interval

P e(0,.99).

For a test of significance of level a, the critical

*
value z can then be obtained by equation (4.4.6). Because

Z is asymptotically normal, the 100a upper percentile point

of the standard normal distribution can be used as a starting

*
value of z . This value is then incremented until a <a and
c

* *

z„ xs taken as the smallest value of z which satisfies a <a.

c c - .

The FORTRAN computer program used to implement this procedure
is given in Appendix C.2.

56
For N=10(I)200 and a=.05 and .025, z , the exact
critical values and a , the size (of precision 6=.001) of Z ,
were computed and are given in Table A. 4. Furthermore, Table
A. 4 also contains a,, a lower bound for {ir(p) : p > 5/N} and
a2, a lower bound for {ir(p): p > 10/N} as indicators of the
stability of the null power function. These lower bounds on
p are obtained for expected number of discordant pairs of at
least 5 and 10 respectively.

The null power function, tt (p) , of the exact conditional
test, as well as of the exact and approximate unconditional
tests, was plotted for some values of N and a nominal level
of significance of a=.05. For N=10, Figure B. 7 contains the
plot of it (p) based on the normal approximation of Z
(critical value z =1.645). It is apparent that using the
normal approximation in this case induces a liberal test
for that range of the nuisance parameter p where the null
power function exceeds the nominal level a=.05, namely
• 30<p<.47 and p>.74. Figure B.8 is the plot based on the
unconditional critical region defined by the exact
conditional test, namely the sign test. Here, the test is

very conservative, its actual size being approximately .013.

*
In Figure B.9, the exact unconditional critical value z =1.90

of Table A. 4 is used to plot tt (p) . From these plots, the

exact Z-test (Figure B.9) is seen to perform best at

approaching the nominal significance level without exceeding

it, although, because of the sparsity of its natural levels

its size is only .0265.

57
The null power function of the exact unconditional test
Z is seen to behave better for larger values of N. For

N=30, Figure B.10 is the plot of tt (p) based on z =1.74

*
and Figure B.ll is the plot of tt (p) based on z =1.68 and N=40.

4.5 Power and Sample Sizes

Now that the exact critical values of Z have been

c

computed (Table A. 4) , the exact power in (4.4.3) can be
readily obtained for a=.025, a=.05, N=10 (1)200 and various
values of pQ1 and P10- Consequently, the minimum sample
size required to achieve a power of 1-3 and a significance
level of a for a combination of (Pqi/Pi0) can be computed by
solving the equation

N* = min { N: n(p01,p1{)) > 1-3} , (4.5.1)

where the critical region that defines n(p01,p,0) ^s based

*

on z , a function of N.
c

Because all other sample size results are given in terms
of the parameters ijj=pQ1+p10 and A=p,Q-p0, ' equation (4.5.1)
was solved in terms of these parameters for the purpose of
comparability. For a=.05 and 1-B=. 80, and various combina-
tions of ty and A, the minimum sample sizes from (4.5.1) are

given in Table A. 5. This table also contains the critical

*
values z , the size (of precision 6=.001) of Z and the

*
attained power 1-3 . Therefore, Table A. 5 is sufficient for

58

both the design and the analysis of the 2x2 table for

comparing two correlated proportions.

*
In Table A.6f the exact unconditional sample sizes [N ]

of Table A. 5 are compared to the exact conditional sample

sizes [N ] found in Schork and Williams (1980) . Furthermore,

the approximate formulae derived by Miettinen (1968) , namely

N and N of section 4.2, are computed and also compared

12
to N and N in Table A. 6. The exact unconditional sample

*
sizes N are seen to be smaller than N , the sample sizes

based on the exact conditional test, for all except some

combinations of ty and A. This seems to happen for larger :

values of \|> and A. The approximate sample sizes N and N

al a2
are almost equal to each other, much smaller than N and

*
slightly smaller than N .

Because these results suggest that the exact uncondi-
tional test might be more powerful than the exact conditional
test, the critical regions of each test were compared
numerically. This comparision showed that, for all the
cases considered, the critical region defined by the exact
conditional test (sign test) is contained in the critical
region defined by the exact Z-test. Therefore, the exact
Z-test is uniformely more powerful than the exact conditional
test for the cases considered, namely N=10(l)200, a=.025 and
ot=.05.

APPENDIX A
TABLES

These tables contain critical values and sample size
determinations for the problems of comparing two independent
proportions and of comparing two correlated proportions.
For one-sided tests, the tables of critical values are
produced for significance levels ot=.05 and .025, and the
sample size tables for a level of a=.05 and 80% power. The
legend for these tables is given below.

Legend for Tables A.l, A. 2 and A. 3; two independent
proportions

n = sample size in each group

a = nominal significance level

a, = lower bound for {ir(p): .05<p<.95}

a2 = lower bound for {ir(p): .10<p<-90}

ir(p)= null power function

*
z = exact one-sided critical values of Z , the Z-test

with unpooled variance estimator

z = exact one-sided critical values of Z , the Z-test
P P

with pooled variance estimator
p. = probability of success in group i, i=l,2.

59

60

*

n = sample size determined by exact Z-tests

* * *
a = size of Z-tests for n and z or z

up

* * * *

1-e = attained power for n , z or z ,• p,. and p2

The following are the sample sizes of Table A. 2 as
determined by:

ne = Fisher's exact test, Casagrande et al. (1978b)

2
nc = the corrected X approximation (3.2.6)

2
nr = the recorrected X approximation (3.2.7)

2
n = the uncorrected X approximation (3.2.4)

n = the arcsine formula (3.2.5)

*

n = the exact Z-tests.

Legend for Tables A.4, A. 5 and A. 6; two correlated
proportions

N = sample size (number of matched pairs)

a = nominal significance level

a, = lower bound for {iT(p) : p > 5/N}

a2 = lower bound for {it (p) : p > 10/N}

Tr(p)= null power function

*

z„ = exact one-sided critical values of Z , the Z-test
c c

A = P1(TP01
* = P10+P01
p01 = p(R=0'S=1) <see section 4.1)

p10 = p(R==1'S=°) (see section 4.1)

*
N = sample size determined by the exact Z-test

* *
a = size of Z-test for N and z

c

* * *

1-B = attained power for N , z , A and ^

61

The following are the sample sizes of Table A. 5 as
determined by:

N3 = McNemar's test, first approximation (4.2.2)

al
N = McNemar's test, second approximation (4.2.3)

2
Nq = the exact conditional test (sign test), Schork

and Williams (1980)

*
N = the exact Z-test.

62

Table A.l Critical Values and Sizes of Z-tests for
Comparing Two Independent Proportions.

a = .05

a = .025

*

*

*

*

*

*

n

al

a2

a

z
u

z

P

al

a2

a

z

u

z

P

10

.0068

.0251

.0476

1.96

1.80

.0005

.0044

.0212

2.17

1.96

11

.0086

.0292

.0504

1.84

1.72

.0007

.0058

.0208

2.40

2.14

12

.0105

.0329

.0471

1.86

1.74

.0010

.0073

.0225

2.26

2.06

13

.0125

.0363

.0484

1.81

1.71

.0014

.0089

.0200

2.26

2.07

14

.0146

.0394

.0495

1.77

1.68

.0018

.0106

.0209

2 .19

2.03

15

.0168

.0422

.0505

1.74

1.66

.0024

.0122

.0218

2.14

2.00

16

.0188

.0289

.0421

1.92

1.82

.0030

.0137

.0252

2.10

1.97

17

.0209

.0308

.0426

1.90

1.81

.0036

.0152

.0233

2.21

2.07

18

.0230

.0335

.0430

1.88

1.80

.0042

.0166

.0241

2.14

2.02

19

.0251

.0368

.0435

1.86

1.78

.0049

.0174

.0246

2.14

2.03

20

.0271

.0339

.0438

1.85

1.78

.0057

.0192

.0252

2.10

2.00

21

.0290

.0351

.0445

1.84

1.77

.0064

.0183

.0251

2.17

2.06

22

.0308

.0360

.0484

1.83

1.77

.0071

.0175

.0245

2.17

2.07

23

.0325

.0367

.0447

1.84

1.78

.0078

.0179

.0239

2.17

2.07

24

.0341

.0373

.0458

1.81

1.76

.0086

.0178

.0222

2.15

2.06

25

.0342

.0342

.0448

1.80

1.75

.0094

.0184

.0233

2.13

2.04

26

.0361

.0361

.0449

1.79

1.74

.0102

.0183

.0217

2.12

2.04

27

.0369

.0369

.0450

1.79

1.74

.0109

.0191

.0225

2.11

2.03

28

.0386

.0386

.0458

1.78

1.74

.0117

.0194

.0233

2.10

2.03

29

.0393

.0393

.0458

1.78

1.74

.0124

.0195

.0242

2.09

2.02

30

.0395

.0395

.0467

1.77

1.73

.0034

.0128

.0219

2.15

2.08

31

.0398

.0398

.0494

1.77

1.73

.0138

.0198

.0243

2.11

2.04

32

.0387

.0387

.0462

1.80

1.76

.0145

.0187

.0236

2.09

2.03

33

.0395

.0395

.0476

1.77

1.73

.0151

.0193

.0245

2.07

2.01

34

.0404

.0405

.0493

1.75

1.72

.0157

.0193

.0238

2.06

2.00

35

.0376

.0376

.0476

1.75

1.72

.0163

.0204

.0243

2.06

2.00

36

.0380

.0380

.0467

1.75

1.72

.0168

.0206

.0245

2.06

2.00

37

.0398

.0398

.0467

1.74

1.71

.0173

.0208

.0249

2.05

2.00

38

.0408

.0414

.0476

1.74

1.71

.0063

.0141

.0225

2.13

2.07

39

.0409

.0410

.0469

1.74

1.71

.0169

.0169

.0233^

2.10

2.05

40

.0410

.0419

.0470

1.73

1.70

.0186

.0203

.0242

2.08

2.03

41

.0411

.0422

.0491

1.73

1.70

.0186

.0186

.0237

2.07

2.02

42

.0406

.0406

.0481

1.78

1.75

.0193

.0193

.0244

2.05

2.00

43

.0411

.0413

.0487

1.76

1.73

.0196

.0197

.0241

2.05

2.00

44

.0412

.0420

.0498

1.74

1.71

.0199

.0208

.0243

2.04

2.00

45

.0396

.0396

.0492

1.73

1.71

.0202

.0217

.0249

2.03

1.99

46

.0383

.0383

.0482

1.73

1.71

.0204

.0215

.0247

2.03

1.99

47

.0395

.0395

.0492

1.72

1.70

.0176

.0177

.0251

2.06

2.02

48

.0404

.0404

.0478

1.72

1.70

.0097

.0174

.0233

2.09

2.05

49

.0417

.0417

.0486

1.72

1.70

.0176

.0180

.0241

2.07

2.03

Table A. 1 — continued

63

a = .05

a = .025

*

*

*

*

*

*

n

aT

a_

a

z

z

a.

<*-,

a

'iz

z

1

2

u

P

1

2

u

P

50

.0419

.0429

.0494

1.71

1.69

.0177

.0181

.0248

2.05

2.01

51

.0420

.0426

.0487

1.72

1.70

.0177

.0182

.0245

2.04

2.00

52

.0435

.0435

.0489

1.71

1.69

.0178

.0184

.0242

2^04

2.00

53

.0440

.0444

.0499

1.71

1.69

.0178

.0185

.0247

2.04

2.01

54

.0394

.0394

.0481

1.76

1.74

.0178

.0187

.0236

2.04

2.01

55

.0396

.0396

.0482

1.75

1.73

.0178

.0189

.0249

2.03

2.00

56

.0413

.0413

.0498

1.73

1.71

.0178

.0191

.0246

2.03

2.00

57

.0398

.0398

.0495

1.72

1.70

.0178

.0192

.0248

2.03

2.00

58

;0389

.0389

.0494

1.72

1.70

.0126

.0180

.0222

2.09

2.06

59

.0398

.0398

.0499

1.71

1.69

.0178

.0196

.0236

2.07

2.04

60

.0405

.0405

.0500

1.71

1.69

.0178

.0207

.0250

2.05

2.02

61

.0416

.0416

.0501

1.70

1.68

.0178

.0198

.0249

2 .04

2.01

62

.0409

.0409

.0502

1.72

1.70

.0178

.0194

.0247

2.03

2.00

63

.0428

.0428

.0503

1.70

1.68

.0178

.0197

.0246

2.03

2.00

64

.0406

.0406

.0485

1.72

1.70

.0178

.0204

.0249

2.02

1.99

65

.0406

.0406

.0486

1.72

1.70

.0178

.0208

.0249

2.03

2.00

66

.0407

.0407

.0491

1.72

1.70

.0178

.0207

.0239

2.04

2.01

67

.0327

.0378

.0473

1.75

1.73

.0179

.0211

.0240

2 . 03

2.00

68

.0406

.0406

.0483

1.74

1.72

.0179

.0212

.0247

2.03

2.00

69

.0383

.0383

.0483

1.73

1.72

.0147

.0175

.0231

2.08

2.05

70

.0387

.0387

.0487

1.72

1.71

.0152

.0195

.0243

2.06

2.03

71

.0392

.0392

.0489

1.71

1.70

.0179

.0196

.0242

2.05

2.02

72

.0393

.0393

.0486

1.71

1.70

.0197

.0199

.0247

2.04

2.01

73

.0400

.0400

.0485

1.71

1.70

.0193

.0193

.0248

2.03

2.01

74

.0409

.0409

.0486

1.71

1.70

.0194

.0194

.0249

2.03

2.01

75

.0412

.0412

.0486

1.71

1.70

.0198

.0200

.0251

2.02

2.00

76

.0412

.0417

.0487

1.70

1.69

.0198

.0204

.0250

2.02

2.00

77

.0413

.0418

.0488

1.70

1.69

.0198

.0210

.0251

2.02

2.00

78

.0414

.0419

.0489

1.70

1.69

.0198

.0211

.0252

2.02

2.00

79

.0415

.0420

.0489

1.70

1.69

.0198

.0210

.0238

2.03

2.01

80

.0415

.0437

.0490

1.70

1.69

.0198

.0213

.0246

2.03

2.01

81

.0430

.0430

.0496

1.70

1.69

.0161

.0186

.0243

2.07

2.05

82

.0327

.0378

.0480

1.74

1.73

.0162

.0187

.0240

2.06

2.04

83

.0327

.0378

.0480

1.73

1.72

.0160

.0176

.0240

2.05

2.03

84

.0396

.0396

.0491

1.72

1.71

.0194

.0194

.0244

2.04

2.02

85

.0398

.0398

.0491

1.71

1.70

.0196

.0196

.0249

2". 03

2.01

86

.0393

.0393

.0491

1.71

1.70

.0191

.0191

.0242

2.03

2.01

87

.0399

.0399

.0496

1.70

1.69

.0195

.0195

.0246

2.03

2.01

88

.0403

.0403

.0492

1.70

1.69

.0200

.0200

.0243

2.03

2.01

89

.0410

.0410

.0494

1.69

1.68

.0200

.0205

.0248

2.02

2.00

Table A.l — continued

64

a = .05

u

90 .0418 .0418 .0504 1.69 1.68

91 .0423 .0423 .0496 1.69 1.68

92 .0425 .0425 .0494 1.69 1.68

93 .0431 .0431 .0495 1.69 1.68

94 .0436 .0436 .0500 1.68 1.67

95 .0442 .0442 .0503 1.68 1.67

96 .0440 .0440 .0499 1.68 1.67

97 .0445 .0445 .0501 1.68 1.67

98 .0450 .0450 .0504 1.68 1.67

99 .0329 .0389 .0486 1.72 1.71

a = .025

al

a2 a*

*
z
u

*

ZP

,0200 .0207 .0249 2.02 2.00

,0201 .0211 .0250 2.01 1.99

,0201 .0208 .0242 2.02 2.00

,0202 .0209 .0245 2.02 2.00

,0170 .0184 .0228 2.07 2.05

,0171 .0185 .0232 2.06 2.04

,0171 .0186 .0236 2.05 2.03

,0153 .0185 .0244 2.04 2.02

0191 .0191 .0249 2.03 2.01

0153 .0186 .0244 2.03 2.01

100 .0382 .0391 .0504 1.71 1.70

101 .0382 .0392 .0491 1.71 1.70

102 .0382 .0393 .0503 1.70 1.69

103 .0382 .0393 .0496 1.70 1.69

104 .0382 .0394 .0493 1.70 1.69

105 .0407 .0407 .0495 1.69 1.68

106 .0413 .0413 .0495 1.69 1.68

107 .0419 .0419 .0495 1.69 1.68

108 .0421 .0421 .0495 1.69 1.68

109 .0422 .0422 .0495 1.69 1.68

,0191 .0191 .0247 2.02 2.00
,0192 .0192 .0246 2.02 2.00
.0192 .0192 .0244 2.02 2.00

0192 .0192 .0247 2.02 2.00

0193 .0193 .0250 2.01 1.99
0193 .0193 .0249 2.01 1.99

0211 .0211 .0250 2.01 2.00

0212 .0212 .0245 2.01 2.00
0176 .0176 .0225 2.07 2.05
0183 .0183 .0232 2.06 2.04

110 .0427 .0427 .0495 1.69 1.68

111 .0426 .0426 .0496 1.69 1.68

112 .0437 .0437 .0497 1.68 1.67

113 .0435 .0435 .0495 1.68 1.67

114 .0439 .0439 .0496 1.68 1.67

115 .0443 .0443 .0499 1.68 1.67

116 .0391 .0400 .0484 1.72 1.71

117 .0380 .0380 .0484 1.72 1.71

118 .0392 .0398 .0486 1.71 1.70

119 .0402 .0402 .0500 1.70 1.69

120 .0396 .0396 .0494 1.70 1.69

121 .0405 .0405 .0520 1.69 1.68

122 .0405 .0405 .0502 1.69 1.68

123 .0407 .0407 .0497 1.69 1.68

124 .0411 .0411 .0497 1.69 1.68

125 .0418 .0418 .0502 1.68 1.67

126 .0419 .0419 .0496 1.69 1.68

127 .0426 .0426 .0500 1.68 1.6.7

128 .0427 .0428 .0497 1.68 1.67

129 .0424 .0424 .0497 1.69 1.68

.0184 .0184 .0236 2.05 2.03

,0185 .0192 .0243 2.04 2.03

,0182 .0193 .0247 2.03 2.02

,0182 .0189 .0243 2.03 2.02

,0182 .0194 .0249 2.02 2.01

,0182 .0196 .0248 2.02 2.01

,0199 .0199 .0251 2.01 2.00

,0203 .0203 .0250 2.01 2.00

,0205 .0205 .0248 2.01 2.00

,0206 .0206 .0245 2.01 2.00

,0209 .0209 .0247 2.01 2.00

,0209 .0209 .0247 2.01 2.00

,0212 .0212 .0247 2.01 2.00

,0215 .0215 .0250 2.00 1.99

,0213 .0213 .0248 2.01 2.00

0181 .0181 .0232 2.05 2.04

,0187 .0187 .0239 2.04 2.03

0185 .0193 .0243 2.03 2.02

0186 .0200 .0250 2.02 2.01
0186 .0193 .0249 2.02 2.01

Table A.l — continued

65

a = .05

a = .025

*

*

*

*

*

*

n

a.

a_

a

z

z

a.

ot_

a

z

z

■

1

2

u

P

1

2

u

P

130

.0427

.0432

.0497

1.68

1.68

.0187

.0193

.0243

2.02

2.01

131

.0427

.0431

.0497

1.68

1.68

.0187

.0197

.0251

2.01

2.00

132

.0427

.0435

.0498

1.68

1.68

.0187

.0198

.0242

2.02

2.01

133

.0427

.0438

.0499

1.68

1.68

.0188

.0199

.0241

2.02

2.01

134

.0428

.0436

.0500

1.68

1.68

.0188

.0204

.0246

2.01

2.00

135

.0383

.0383

.0486

1.72

1.71

.0189

.0205

.0244

2.01

2.00

136

.0402

.0402

.0486

1.71

1.70

.0189

.0208

.0246

2.01

2.00

137

.0388

.0388

.0485

1.71

1.70

.0189

.0208

.0244

2.01

2.00

138

.0403

.0403

.0499

1.70

1.70

.0190

.0207

.0245

2.01

2.00

139

.0394

.0394

.0500

1.70

1.70

.0190

.0210

.0244

2.01

2.00

140 •

.0395

.0395

.0499

1.70

1.70

.0191

.0208

.0245

2.01

2.00

141

.0402

.0402

.0500

1.69

1.69

.0191

.0210

.0249

2.01

2.00

142

.0406

.0406

.0500

1.69

1.69

.0185

.0185

.0236

2.04

2.03

143

.0409

.0409

.0501

1.69

1.69

.0191

.0191

.0243

2.03

2.02

144

.0413

.0413

.0500

1.69

1.69

.0192

.0203

.0251

2.02

2.01

145

.0415

.0415

.0500

1.69

1.69

.0191

.0191

.0246

2.02

2.01

146

.0422

.0422

.0501

1.68

1.68

.0192

.0192

.0247

2.02

2.01

147

.0423

.0423

.0498

1.68

1.68

.0194

.0194

.0242

2.02

2.01

148

.0427

.0427

.0500

1.68

1.68

.0194

.0197

.0246

2.01

2.00

149

.0427

.0427

.0502

1.68

1.68

.0195

.0199

.0246

2.01

2.00

150

.0427

.0427

.0501

1.68

1.68

.0195

.0203

.0252

2.00

1.99

66

Table A. 2 Minimum Sample Sizes to Achieve 80% Power
and a<.05 for One-sided Z-tests for
Comparing Two Independent Proportions.

*

*

*

*

*

Pi

P2

n

V

z

P

a

1-.&

.05

.15

107

1.69

1.68

.0495

.8009

.20

56

1.73

1.71

.0498

.8016

.25

38

1.74

1.71

.0476

.8098

.30

28

1.78

1.74

.0458

.8095

.35

22

1.83

1.77

.0484

.8095

.40

18

1.88

1.80

.0430

.8190

.45

13

1.81

1.71

.0484

.8142

.10

.25

79

1.70

1.69

.0489

.8026

.30

49

1.72

1.70

.0486

.8071

.35

35

1.75

1.72

.0476

.8063

.40

26

1.79

1.74

.0449

.8088

.45

21

1.84

1.77

.0445

.8057

.50

17

1.90

1.81

.0426

. 8213

.55

13

1.81

1.71

.0484

.8016

.60

10

1.96

1.80

.0476

.8016

.15

.30

95

1.68

1.67

.0503

.8023

.35

59

1.71

1.69

.0499

.8115

.40

40

1.73

1.70

.0470

.8077

.45

29

1.78

1.74

.0458

.8001

.50

23

1.84

1.78

.0447

.8134

.55

18

1.88

1.80

.0430

.8033

.60

14

1.77

1.68

.0495

.8016

.65

13

1.81

1.71

.0484

.8366

.20

.35

111

1.69

1.68

.0496

.8005

.40

68

1.74

1.72

.0483

.8038

.45

44

1.74

1.71

.0498

.8017

.50

32

1.80

1.76

.0462

.8060

.55

26

1.79

1.74

.0449

.8298

.60

20

1.85

1.78

.0438

.8153

.65

15

1.74

1.66

.0505

.8273

.70

13

1.81

1.71

.0484

.8239

.25

.40

123

1.69

1.68

.0497

.8017

.45

71

1.71

1.70

.0489

.8017

.50

48

1.72

1.70

.0478

.8061

.55

33

1.77

1.73

.0476

.8051

.60

26

1.79

1.74

.0449

.8006

.65

20

1.85

1.78

.0438

.8091

.70

15

1.74

1.66

.0505

.8232

.75

13

1.81

1.71

.0484

.8205

67

Table A. 2 — continued

*

*

*

*

*

pl

P2

n

z

u

z

P

a

1-8

.30

.45

132

1.68

1.68

.0498

.8015

.50

77

1.70

1.69

.0488

.8021

.55

50

1.71

1.69

.0494

.8029

.60

37

1.74

1.71

.0467

.8145

.65

27

1.79

1.74

.0450

.8064

.70

20

1.85

1.78

.0438

.8075

.35

.50

136

1.71

1.70

.0486

.8016

.55

79

1.70

1.69

.0489

.8070

.60

51

1.72

1.70

.0487

.8085

.65

37

1.74

1.71

.0467

. 8111

.40

.55

144

1.69

1.69

.0500

.8021

.60

79

1.70

1.69

.0489

.8057

68

Table A. 3 Comparision of Sample Sizes to Achieve 80%
Power and a<.05 for One-sided Tests for
Comparing Two Independent Proportions.

pl

P2

n
e

n
c

n

r

np

nas

n

.05

.15

126

148

130

111

105

107

.20

67

84

72

59

55

56

.25

45

57

48

39

35

38

.30

34

42

36

28

25

28

.35

25

33

28

21

19

22

.40

20

27

22

17

15

18

, .45

17

23

19

14

12

13

.10

.25

89

104

92

79

76

79

.30

56

67

58

49

47

49

.35

39

49

42

34

32

35

.40

30

37

31

25

24

26

.45

24

30

25

20

19

21

.50

19

25

20

16

15

17

.55

16

21

17

13

12

13

.60

13

18

14

11

10

10

.15

.30

106

120

108

95

94

95

.35

65

76

67

57

56

59

.40

46

54

46

39

38

40

.45

34

40

35

28

28

29

.50

26

32

27

22

21

23

.55

22

26

22

17

17

18

.60

17

22

18

14

13

14

.65

15

18

15

11

11

13.

.20

.35

121

134

122

109

108

Ill

.40

73

.. 83

74

64

64

68

.45

49

58

50

43

42

44

.50

36

43

37

31

30

32

.55

27

34

28

23

23

26

.60

23

27

23

18

18

20

.65

17

22

18

14

14

15

.70

15

19

15

12

12

13

.25

.40

132

145

133

120

119

123

.45

78

89

79

70

69

71

.50

54

61

53

46

46

48

.55

37

45

39

32

32

33

.60

30

35

30

24

24

26

.65

23

28

23

19

19

20

.70

18

23

19

15

15

15

.75

15

19

15

12

12

13

69
Table A. 3 — continued

Pi ,

?2

n
e

nc

nr

n
P

nas

*

n

.30

.45

142

154

141

128

128

132

.50

84

92

83

74

73

77

.55

55

63

55

48

48

50

.60

41

46

40

33

33

37

.65

31

35

30

25

25

27

.70

23

28

23

19

19

20

.35

.50

143

159

147

134

134

136

.55

85

95

86

76

76

79

.60

56

64

56

49

49

51

.65

41

46

40

34

34

37

.40

.55

144

162

149

136

136

144

.60

85

96

86

77

77

79

Table A. 4 Critical Values and Sizes of Z-test for
Comparing Two Correlated Proportions.

70

a = .05

a = .025

N

al

a2

*

a

*
z
c

10

.0114

.0265

1.90

11

.0322

.0345

.0395

1.74

12

.0205

.0205

.0373

1.74

13

.0316

.0328

.0430

1.74

14

.0303

.0303

.0371

1.74

15

.0187

.0187

.0370

1.81

16

.0312

.0349

.0454

1.74

17

.0261

.0261

.0413

1.74

18

.0313

.0353

.0446

1.74

19

.0307

.0333

.0399

1.74

20

.0220

.0220

.0395

1.79

21

.0310

.0356

.0459

1.74

22

.0278

.0278

.0425

1.74

23

.0309

.0357

.0427

1.74

24

.0307

,0332

.0412

1.74

25

.0305

.0357

.0494

1.74

26

.0301

.0357

.0434

1.74

27

.0275

.0275

.0413

1.74

28

.0307

.0349

.0407

1.74

29

.0302

.0315

.0407

1.74

30

.0303

.0357

.0450

1.74

31

.0303

.0357

.0407

1.74

32

.0262

.0262

.0407

1.77

33

.0301

.0357

.0458

1.74

34

.0299

.0299

.0435

1.74

35

.0298

.0356

.0429

1.74

36

.0304

.0327

.0425

1.74

37

.0302

.0356

.0449

1.74

38

.0299

.0355

.0419

1.74

39

.0400

.0411

.0501

1.68

40

.0399

.0400

.0500

1.68

41

.0305

.0305

.0500

1.72

42

.0303

.0354

.0433

1.74

43

.0298

.0323

.0428

1.74

44

.0375

.0375

.0500

1.68

45

.0352

.0352

.0501

1.68

46

.0395

.0395

.0501

1.68

47

.0382

.0382

.0501

1.68

48

.0290

.0300

.0427

1.74

49

.0396

.0408

.0501

1.70

,0114 .0208

,0062 .0062 .0197

,0133 .0204 .0233

,0120 .0120 .0213

.0069 .0069 .0126

,0127 .0186 .0211

,0114 .0114 .0209

,0125 .0175 .0225

,0128 .0163 .0208

,0103 .0103 .0208

,0126 .0187 .0246
,0142 .0142 .0251
0200 .0200 .0249
0182 .0182 .0247
,0121 .0121 .0213
,0124 .0166 .0211
,0154 .0154 .0246
,0204 .0204 .0246

0185 .0185 .0246
,0120 .0128 .0207

,0126 .0166 .0207

0155 .0155 .0246

0196 .0196 .0246

0179 .0179 .0246

0125 .0128 .0207

0208 .0208 .0249

,0123 .0151 .0226

0127 .0182 .0226

0119 .0168 .0226
0123 .0192 .0242

0192 .0192 .0250

0120 .0143 .0226
0208 .0208 .0250
0163 .0163 .0247

0186 .0186 .0247
,0175 .0175 .0248
,0198 .0198 .0248

0202 .0202 .0248
0115 .0150 .0229
0118 .0167 .0228

2.01
2.12
2.01
2.01
2.14
2.01
2.01
2.01
2.01
2.07

2.01
1..97
1.97
1.97
2.05
2.05
1.97
1.97
1.97
2.05

2 .05
1.98
1.98
1.98
2.06
1.98
2.01
2.0.1
2.01
2.01

1.98
2.04
1.98
1.99
1.99
1.99
1.99
1.99
2.03
2.03

Table A. 4 — continued

71

a = .05

a = .025

*

*

*

*

N

al

a2

a

z

c

al
.0167

a2
.0167

a
.0248

z
c

50

.0325

.0325

.0500

1.70

1.99

51

.0352

.0352

.0500

1.70

.0183

.0183

.0247

1.99

52

.0333

.0333

.0500

1.70

.0174

.0174

.0247

1.99

53

.0365

.0365

.0500

1.70

.0194

.0194

.0246

1.99

54

.0359

.0359

.0501

1.70

.0200

.0200

.0246

1.99

55

.0387

.0387

.0501

1.70

.0120

.0151

.0230

2.03

56

.0299

.0360

.0453

1.74

.0123

.0163

.0227

2.03

57

.0309

.0309

.0501

1.73

.0187

.0187

.0248

1.99

58

.0390

.0407

.0501

1.68

.0177

.0177

.0248

1.99

59

.0311

.0311

.0501

1.73

.0168

.0168

.0247

1.99

60

.0347

.0347

.0501

1.70

.0186

.0186

.0247

1.99

61

.0329

.0329

.0501

1.70

.0190

.0190

.0246

1.99

62

.0294

.0359

.0445

1.74

.0197

.0197

.0250

1 . 99

63

.0298

.0359

.0427

1.74

.0200

.0200

.0246

1.99

64

.0298

.0358

.0470

1.74

.0125

.0185

.0234

2.01

65

.0294

.0358

.0453

1.74

.0116

.0189

.0231

2.01

66

.0297

.0356

.0442

1.74

.0119

.0186

.0230

2.01

67

.0382

.0395

.0498

1.70

.0201

.0201

.0248

1.99

68

.0378

.0378

.0501

1.70

.0200

.0200

.0247

1.99

69

.0334

.0334

.0499

1.70

.0200

.0200

.0247

1.99

70

.0381

.0398

.0499

1.68

.0199

.0200

.0247

1.99

71

.0346

.0346

.0495

1.70

.0119

.0180

.0234

2.02

72

.0378

.0393

.0501

1.68

.0199

.0199

.0250

1.99

73

.0377

.0401

.0500

1.68

.0194

.0194

.0247

1.99

74

.0374

.0374

.0501

1.68

.0198

.0199

.0247

1.99

75

.0376

.0384

.0500

1.68

.0191

.0191

.0247

1.99

76

.0375

.0391

.0501

1.68

.0197

.0198

.0247

1.99

77

.0374

.0395

.0501

1.67

.0181

.0181

.0247

1.99

78

.0373

.0396

.0501

1.67

.0186

.0186

.0247

1.99

79

.0371

.0393

.0500

1.69

.0196

.0197

.0251

1.98

80

.0370

.0397

.0499

1.69

.0112

.0185

.0234

2.02

81

.0369

.0388

.0501

1.67

.0195

.0195

.0250

1.99

82

.0369

.0388

.0500

1.67

.0194

.0195

.0247

1.99

83

.0368

.0389

.0501

1.67

.0194

.0195

.0247

1.99

84

.0367

.0390

.0500

1.67

.0192

.0192

.0247

1.99

85

.0365

.0390

.0501

1.67

.0193

.0194

.0247

1.99

86

.0364

.0391

.0500

1.67

.0192

.0193

.0247

1.99

87

.0362

.0380

.0500

1.67

.0192

.0192

.0247

1.99

88

.0361

.0381

.0500

1.67

.0191

.0192

.0251

1.98

89

.0360

.0381

.0500

1.67

.0190

.0192

.0251

1.98

Table A. 4 — continued

72

a =

05

a = .025

*

*

*

*

N

al

a2

a

z

c

al
.0190

a2
.0191

a
.0248

V

90

.0359

.0382

.0500

1.69

1.99

91

.0358

.0382

.0499

1.69

.0189

.0191

.0247

2.00

92

.0357

.0383

.0500

1.67

.0189

.0190

.0247

2.00

93

.0356

.0383

.0500

1.67

.0189

.0189

.0247

2.00

94

.0354

.0384

.0499

1.67

.0188

.0190

.0246

2.00

95

.0353

.0385

.0499

1.67

.0188

.0189

.0247

2.00

96

.0351

.0371

.0499

1.67

.0188

.0188

.0251

1.99

97

.0350

.0372

.0500

1.67

.0187

.0187

.0250

1.98

98

.0348

.0372

.0499

1.67

.0187

.0187

.0249

1.98

99

.0347

.0372

.0500

1.67

.0187

.0187

.0248

1.98

100

.0346

.0373

.0500

1.67

.0117

.0194

.0236

2.01

101

.0344

.0371

.0500

1.70

.0118

.0190

.0236

2.01

102

.0343

.0371

.0500

1.70

.0185

.0185

.0248

1.99

103

.0342

.0410

.0499

1.68

.0199

.0199

.0248

1.99

104

.0341

.0396

.0500

1.68

.0192

.0192

.0247

1.99

105

.0341

.0383

.0500

1.68

.0198

.0198

.0247

1.99

106

.0340

.0394

.0500

1.68

.0199

.0199

.0247

1.99

107

.0339

.0396

.0500

1.67

.0199

.0199

.0247

1.99

108

.0338

.0409

.0499

1.67

.0198

.0198

.0246

1.99

109

.0338

.0410

.0499

1.67

.0198

.0198

.0250

1.99

110

.0337

.0411

.0499

1.67

.0198

.0198

.0249

1.99

111

.0336

.0412

.0499

1.67

.0203

.0203

.0246

2.00

112

.0395

.0413

.0499

1.67

.0190

.0190

.0246

2.00

113

.0395

.0413

.0499

1.67

.0203

.0203

.0251

1.99

114

.0395

.0399

.0499

1.69

.0202

.0202

.0248

1.99

115

.0382

.0382

.0499

1.69

.0192

.0192

.0248

1.99

116

.0394

.0404

.0499

1.68

.0197

.0197

.0251

1.99

117

.0394

.0409

.0499

1.68

.0203

.0203

.0247

1.99

118

.0380

.0380

.0499

1.68

.0205

.0205

.0247

1.99

119

.0388

.0388

.0499

1.68

.0205

.0205

.0251

1.98

120

.0390

.0390

.0499

1.68

.0205

.0205

.0251

1.98

121

.0392

.0403

.0499

1.68

.0205

.0205

.0251

1.99

122

.0391

.0406

.0499

1.68

.0204

.0205

.0245

2.00

123

.0391

.0406

.0500

1.67

.0204

.0204

.0245

2.00

124

.0391

.0407

.0499

1.67

.0182

.0182

.0245

2.00

125

.0390

.0407

.0499

1.67

.0204

.0204

.0250

1.98

126

.0390

.0408

.0499

1.67

.0204

.0204

.0250

1.98

127

.0390

.0409

.0498

1.69

.0198

.0198

.0251

1.98

128

.0389

.0409

.0499

1.69

.0199

.0199

.0251

1.98

129

.0389

.0410

.0498

1.68

.0203

.0203

.0251

1.98

Table A. 4 — continued

73

a = .05

a = .025

*

*

*

*

N

aT

a->

a

z

Oti

a-i

a

Z

1

2

c

1
.0203

2
.0203

.0251

C '

130

.0388

.0403

.0499

1.68

1.98

131

.0388

.0403

.0498

1.68

.0203

.0203

.0251

1.98

132

.0387

.0404

.0498

1.68

.0202

.0203

.0251

1.98

133

.0387

.0389

.0499

1.68

.0202

.0203

.0245

2.00

134

.0386

.0397

.0499

1.68

.0202

.0202

.0245

2.00

135

.0386

.0397

.0500

1.67

.0202

.0202

.0245

2.00

136

.0386

.0398

.0499

1.67

.0202

.0202

.0250

1.98

137

.0385

.0398

.0499

1.67

.0202

.0202

.0250

1.98

138

.0385

.0399

.0499

1.67

.0201

.0202

.0250

1.98

139

.0385

.0399

.0499

1.67

.0201

.0201

.0250

1.98

140

.0384

.0400

.0499

1.67

.0199

.0199

.0250

1.98

141

.0383

.0400

.0498

1.67

.0201

.0201

.0250

1.98

142

.0383

.0401

.0497

1.68

.0200

.0201

.0251

1.98

143

.0382

.0401

.0497

1.68

.0200

.0201

.0251

1.98

144

.0382

.0402

.0499

1.67

.0094

.0197

.0241

2.01

145

.0381

.0402

.0499

1.67

.0095

.0197

.0237

2.01

146

.0381

.0403

.0499

1.67

.0200

.0200

.0249

1.99

147

.0380

.0403

.0498

1.67

.0199

.0200

.0246

1.99

148

.0380

.0404

.0498

1.67

.0199

.0200

.0246

1.99

149

.0379

.0395

.0498

1.67

.0199

.0199

.0246

1.99

150

.0379

.0395

.0498

1.67

.0199

.0199

.0246

1.99

151

.0379

.0395

.0498

1.67

.0198

.0199

.0250

1.98

152

.0379

.0396

.0499

1.67

.0198

.0199

.0250

1.98

153

.0378

.0396

.0499

1.67

.0198

.0199

.0250

1.98

154

.0377

.0387

.0498

1.67

.0198

.0198

.0250

1.98

155

.0376

.0387

.0498-

1.67

.0198

.0198

.0250

1.98

156

.0376

.0388

.0499

1.67

.0197

.0197

.0250

1.98

157

.0375

.0388

.0497

1.68

.0197

.0197

.0246

2.00

158

.0374

.0388

.0496

1.68

.0197

.0197

.0246

2.00

159

.0374

.0389

.0500

1.67

.0197

.0197

.0246

2 . 00

160

.0373

.0389

.0498

1.67

.0196

.0197

.0249

1.99

161

.0373

.0389

.0498

1.67

.0196

.0197

.0247

1.99

162

.0372

.0390

.0498

1.67

.0196

.0197

.0246

1.99

163

.0372

.0390

.0498

1.67

.0196

.0196

.0246

1.99

164

.0372

.0390

.0498

1.67

.0195

.0196

.0246

1.99

165

.0371

.0390

.0498

1.67

.0195

.0196

.0246

1.99

166

.0371

.0391

.0498

1.67

.0195

.0196

.0246

1.99

167

.0370

.0391

.0498

1.67

.0195

.0196

.0246

1.99

168

.0370

.0391

.0498

1.67

.0194

.0195

.0246

1.99

169

.0369

.0392

.0498

1.67

.0194

.0195

.0247

1.99

Table A. 4 — continued

74

a = .05

a = .025

*

*

*

*

N

al

a2

a

zc

al

a2

a

zc

170

.0368

.0392

.0498

1.67

.0194

.0195

.0251

1.98

171

.0367

.0393

.0498

1.67

.0194

.0195

.0245

2.00

172

.0367

.0393

.0498

1.68

.0193

.0195

.0245

2.00

173

.0366

.0393

.0498

1.68

.0193

.0195

.0251

1.99

174

.0365

.0382

.0498

1.67

.0193

.0194

.0249

1.99

175

.0365

.0382

.0498

1.67

.0193

.0193

.0247

1.99

176

.0364

.0383

.0498

1.67

.0192

.0193

.0246

1.9.9

177

.0363

.0383

.0498

1.67

.0192

.0193

.0246

1.99

178

.0363

.0383

.0498

1.67

.0192

.0193

.0246

1.99

179

.0362

.0383

.0498

1.67

.0191

.0193

.0246

1.99

180

.0362

.0383

.0498

1.67

.0191

.0193

.0251

1.98

181

.0361

.0384

.0498

1.67

.0191

.0192

.0251

1.98

182

.0361

.0372

.0498

1.67

.0191

.0191

.0247

1.99

183

.0360

.0372

.0498

1.67

.0191

.0191

.0250

1.99

184

.0360

.0372

.0498

1.67

.0190

.0190

.0245

2.00

185

.0360

.0372

.0498

1.67

.0190

.0190

.0250

1.99

186

.0359

.0373

.0498

1.67

.0190

.0190

.0251

1.99

187

.0359

.0373

.0499

1.67

.0190

.0190

.0249

1.99

188

.0359

.0373

.0501

1.67

.0190

.0190

.0247

1.99

189

.0358

.0373

.0497

1.68

.0189

.0189

.0246

1.99

190

.0358

.0373

.0498

1.68

.0189

.0189

.0246

1.99

191

.0358

.0373

.0498

1.68

.0189

.0189

.0246

1.99

192

.0357

.0374

.0500

1.67

.0189

.0189

.0251

1.98

193

.0357

.0374

.0498

1.67

.0189

.0189

.0250

1.98

194

.0357

.0374

.0498

1.67

.0189

.0189

.0250

1.98

195

.0356

.0374

.0498

1.67

.0188

.0188

.0250

1.98

196

.0356

.0374

.0498

1.67

.0188

.0188

.0251

1.98

197

.0355

.0374

.0498

1.67

.0188

.0188

.0250

1.98

198

.0355

.0374

.0498

1.67

.0188

.0188

.0251

1.98

199

.0354

.0374

.0498

1.67

.0188

.0188

.0247

1.99

200

.0354

.0375

.0498

1.67

.0187

.0187

.0247

1.99

Table A. 5 Minimum Sample Sizes to Achieve 80% Power
and a <.05 ; for One-sided Z-test for
Comparing Two Correlated Proportions.

75

1-e

.10

.30

185

1.67

.0498

.8006

.22

134

1.68

.0499

.8014

.14

80

1.69

.0499

.8038

.20

.98

153

1.67

.0499

.8003

.94

146

1.67

.0499

.8014

.90

139

1.67

.0499

.8014

.86

135

1.67

.0500

.8066

.82

129

1.68

.0498

.8018

.78

122

1.68

.0499

^8018

.74

116

1.68

.0499

.8042

.70

108

1.67

.0499

.8016

.66

103

1.68

.0499

.8026

.62

96

1.67

.0499

.8027

.58

89

1.67

.0500

.8013

.54

82

1.67

.0500

.8025

.50

77

1.67

.0501

.8070

.46

70

1.68

.0499

.8005

.42

67

1.70

.0498

.8219

.38

58

1.68

.0501

.8193

.34

51

1.70

.0500

.8058

.30

44

1.68

.0500

.8186

.26

38

1.74

.0419

.8081

.22

30

1.74

.0450

.8124

.30

.88

63

1.74

.0427

.8029

.84

58

1.68

.0501

.8081

.80

57

1.73

.0501

.8088

.76

52

1.70

.0500

.8033

.72

49

1.70

.0501

.8035

.68

46

1.68

.0501

.8029

.64

44

1.68

.0500

.8117

.60

39

1.68

.0501

.8020

.56

39

1.68

.0501

.8324

.52

35

1.74

.0429

.8003

.48

33

1.74

.0458

.8120

.44

30

1.74

.0450

.8123

.40

26

1.74

.0434

.8075

.36

22

1.74

.0425

.8076

.32

19

1.74

.0399

.8195

.40

.98

39

1.68

.0501

.8209

.94

38

1.74

.0419

.8089

.90

35

1.74

.0429

.8037

.86

34

1.74

.0435

.8046

Table A. 5 — continued

76

1-6

.40

.82

33

1.74

.0458

.8090

.78

31

1.74

.0407

.8114

.74

29

1.74

.0407

.8117

.70

27

1.74

.0413

.8108

.66

25

1.74

.0494

.8067

.62

23

1.74

.0427

.8003

.58

22

1.74

.0425

.8120

.54

20

1.79

.0395

.8134

.50

17

1.74

.0413

.8002

.46

15

1.81

.0370

.8030

.42

14

1.74

.0371

.8365

.50

.88

21

1.74

.0459

.8029

.84

21

1.74

.0459

.8201

.80

19

1.74

.0399

.8062

.76

18

1.74

.0446

.8049

.72

17

1.74

.0413

.8031

.68

16

1.74

.0454

.8119

.64

14

1.74

.0371

.8020

.60

13

1.74

.0430

.8166

.56

12

1.74

.0373

.8325

.52

11

1.74

.0395

.8517

.60

.98

18

1.74

.0446

.8522

.94

16

1.74

.0454

.8414

.90

16

1.74

.0454

.8525

.86

15

1.81

.0370

.8196

.82

13

1.74

.0430

.8023

.78

12

1.74

.0373

.8084

.74

11

1.74

.0395

.8107

.70

11

1.74

.0395

.8504

.66

11

1.74

.0395

.8935

77

Table A. 6 Comparision of Sample Sizes to Achieve 80!
Power and a<.05 for One-sided Tests for
Comparing Two Correlated Proportions .

A.

*

N

N

N

*

N

al

a2

e

.10

.30

179

180

199

185

.22

127

128

146

134

.14

70

74

94

80

.20

.98

150

150

159

153

.94

143

143

152

146

.90

137

137

147

139

.86

131

131

141

135

.82

125

125

135

129

.78

118

118

129

122

.74

112

112

122

116

.70

106

106

116

108

.66

99

99

110

103

.62

93

93

103

96

.58

86

87

96

89

.54

80

80

90

82

.50

73

74

83

77

.46

67

67

76

70

.42

60

61

70

67

.38

53

54

63

58

.34

46

48

56

51

.30

39

41

50

44

.26

31

33

44

38

.22

22

25

36

30

.30

.88

58

59

65

63

.84

56

56

61

58

.80

53

53

59

57

.76

50

50

56

52

.72

47

47

53

49

.68

44

44

50

46

.64

41

41

47

44

.60

38

38

44

39

.56

35

35

41

39

.52

32

32

38

35

.48

29

29

35

33

.44

25

26

32

30

.40

22

23

30

26

.36

18

20

27

22

.32

14

16

23

19

.40

.98

36

36

42

39

.94

34

35

38

38

.90

33

33

37

35

.86

31

31

35

34

78

Table A. 6 — continued

A

*

N

N

N

N

al

a2

e

.40

.82

29

30

34

33

.78

28

28

33

31

.74

26

26

31

29

.70

24

25

29

27

.66

23

23

27

25

.62

21

21

25

23

.58

19

19

24

22

.54

17

18

22

20

.50

15

16

21

17

.46

13

14

19

15

.42

10

11

17

14

.50

.88

20

20

23

21

.84

19

19

22

21

.80

17

18

21

19

.76

16

16

20

18

.72

15

15

19

17

.68

14

14

18

16

.64

13

13

17

14

.60

11

12

16

13

.56

10

10

14

12

.52

8

9

13

11

.60

.98

15

15

18

18

.94

14

14

17

16

.90

13

13

16

16

.86

13

13

15

15

.82

12

12

15

13

.78

11

11

14

12

.74

10

10

13

11

.70

9

9

12

11

.66

8

8

11

11

APPENDIX B
PLOTS OF THE NULL POWER FUNCTION

In this appendix, plots of it (p) , the null power
function, are given for the two problems considered here.
The dotted line represents the nominal significance level
on which tt(p) is based. These plots are referred to in
section 3.4 for the two independent proportions case, and
in section 4.4 for the two correlated proportions case.

79

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APPENDIX C
COMPUTER PROGRAMS

The listing of the FORTRAN computer programs used
to compute the size of the Z-test for each of the two
problems consdered is given in this Appendix. For the
case of two independent proportions, Appendix C.l gives
the exact p-value for any n=10 (1)150 and any value of the
Z-test statistic with unpooled variance estimator. In
Appendix C.2, the case of two correlated proportions, the
exact p-value for N=10 (1)200 and any value of the Z-test
statistic can be obtained.

91

92

APPENDIX C.l

HEAL A,B,C.TBY(151)

INTEGER X,U,Y,UU,B0UNDYn51) , PHI (151)

DOUBLE PRECISION LFAC (15 1) ,P, Q, LP, LQ, PX (151 ) ,FX (151) ,

♦PVALDE.DEHBND^ESU-DE^L^COMlPFJAT^pftATi. ' ' l ' '

*PaAXl,|MAX2,PHINl,Pi3IN2,PL,P&,PLl.PL2,PIJ^PU2,SGP,INF,
*C1,C2fT.T3,T4,T5,T6,D1.D2,D3,5u,D$,D6'LPX'LFx'

*MAXP,MAXIM,MI&IM5,MINl1o
DOUBLE PRECISION DLOG, DLOG10, DEXP, DMAX1 ,DMIN1 /DABS
WRITE (6 . 40)
40 FORMAT (« »,///,

* ' N = SAMPLE SIZE IN EACH GROUP,'/

* ' Z = NORMAL STATISTIC WITH UNPOOLED VARIANCE.'/

* 'ENTER N, Z, IN FREE FORMAT'/)

C ."-.'.

99 READ(9,*)N,Z

MAXP=0.0

N1=N+1

LFAC(1)=0.D0

DO 1 J=2.N1
1 LFAC(J)=LFAC(J-1)+DLOG(DBLE(FLOAT(J-1)))

C

C SIMPLIFY THE COMPUTATIONS AND FORM THE

C BOUNDARY OF THE CRITICAL REGION

C

C

X=0
A=1 ._

A=1+ (Z**2)/N

C=(X**2)*A-(Z**2) *X

TBY (X+1) = (B+SQRT < (B**2) - (4*A*C) ) ) / (2*A)

IF (TBY(X+1)-N) 3,3,4 I ) ) / \± *>

3 X=X+1

GO TO 2

4 U=X-1
UU=U+1

DO 5 X=1,UG

BOUNDY(X) =INT(TBY(X) +1)

PHI(X)=INT(TBY(X))

L=X- 1 '

5 CONTINUE
MAXIM=0.D0
MINIM5=1.D0
MINI10=1.D0
19=0

15=0
P=-.005D0

6 IFfP.GT.0.49D0) GO TO 76

72 TFfL.BQ. 1) P=P+.01D0

IFJL.EQ.1) PP=P+.005D0

Q=1-P

LP=DLOG(P]

LQ=DLOG(Q)
DO 7 J=1,N1
X=J-1

T=LFAC (N-H)-LFAC (J) -LFAC (N+1-J+1 ) +X*LP+ (N-X) *LQ
IF (T..LT.-180.0) PX(J)=0.D0 l ' W

IF (T.GE. -180.0) PX(J)=DEXP(T)

93

IF (X 8,9.8

9 FX(J)=PX(JJ
GO TO 10

8 FX(J)=FX(J-1)+PX(J)

10 CONTINUE
7 CONTINUE

PVALUE=0

DEEU=0

DERL=0

DO 30 K=1,UU

X=K— 1

IF (PX(K) .LE.0.D0) GO TO 11

LPX=DLOG10(PX(K))

IF {(1~FX(PHIJK) +

LFX=DLOG10(1-FXJ[P

IF (7LPX+LFX) .LT.-7g.b) GO TO 11

PVALOE=PVALOE+PX{K)* (1-FX(PHI{K) +1))

11 Y=BOUNDY(K)
LCOM=LFAC "

(N+1)-LFAC(K)-LFAC(N + 1-K + 1)

+ LFAC(N+1)-LFAC(Y)-LFAC(N+1-Y-1 + 1)

C

C

C LOCAL BOUND FOR THE DERIVATIVE OF

C THE NULL POWER FUNCTION

C

C

C1=FLOAT(X+Y-1)

C2=FLOAT(2*N-X-Y)

PHAT1 = C1/{C1+C2)

PHAT2=1-PHAT1

PL=P- (. 005D0/ (2** (L- 1) ) )

PU=P+L005D0/J2**]L-1)))

IF (PHAT1.GT.PD) GO TO 12

IF ((PHAT1.LE.PU) .AND. (PHAT1 . GE. PL) ) GO TO 13

IF (PHAT1.LT. PL) GO TO 14

12 PHAX1=PU
PMIN1=PL
GO TO 15

13 PMAX1=PHAT1

IF (PL.NE.O.DO) GO TO 21

PMIN1=PL

GO TO 15

21 PL1=C1*DL0G(PL)+C2*DL0G(1-PL)
PU1=C1*DLOGJPU)+C2*DLOG(1-P0)
IF (PL1.LT. PU1J — '"- -*
IF JPL1.GE.PU1
GO TO 15

14 PMAX1=PL
PHIN1=PU

15 CONTINUE

IF (PHaT2.GT.PU) GO TO 16

IF (jfPHAT2.LE.PU) .AND. (PHAT2. GE. PL) ) GO TO 17

IF JPHAT2.LT. PL) GO TO 1 8 "

16 PMAX2=PU
PHIH2=PL
GO TO 19

17 PMAX2=PHAT2

IF (PL.NE.O.DO) GO TO 22

PMIN2=PL

GO TO 19

22 PL2=C2*DLOG (PL) +C1 *DLOG ( 1-PL)
PU2=C2*DLOGJPU5+C1*DLOG(1-PU)
IF (PL2.LT. PU2) PKIN2=PL

IF JPL2.GT.PU2) PHIN2=PU
GO TO 19

18 PMAX2=PL
PHIN2=PU

19 CONTINUE
T3=LCOi3+Cl*DLOG{paAXl)+C2*DLOG(1-P8AX1)
IF (T3.LT. -180.0) D3=0.D0

IF (T3.GE.-180.0[ D3=DEXP(T3)
IF (PHIN2.EQ.0.D0) GO TO 23

J)+C2*DLOG(
I)+C2*DLOG{'
I) PMIN1=PL
I) PMIN1=PU

94

T4=LC0M+C2*DL0GfPMIN2) +C1*DL0G (1-PMIN2)
IF (T4.LT. -180.0) D4=0.D0
IF JT4.GE. -180.0) D4=DEXP(T4)
GO TO 24
23 D4=0.D0

24 IF (PHIN1.EQ.0.D0) GO TO 25
T5=LC0M+C1*DL0G(PMIN1)+C

_. _. *--— .,+C2*DLOG(1-PMIN1)

IF (T5.LT.-180.d) D5=0.D0

IF JT5.GE. -180.0) D5=DEXP(T5)

GO TO 26

25 D5=0.D0

26 CONTINUE
T6=LC0M+C2*DL0G{PHAX2)+C1*DL0G(1-PMAX2)
IF (T6.LT. -180.0) D6=0.D0

IP (T6.GE. -180.0) D6=DEXP(T6)
DERU=DERU+D3-D4
DERL=DERL+D5-D6
30 CONTINUE

DERBND=DBAX1 (DABS (DERU) , DABS (DERL) )

C

C COMPUTE THE SIZE

C

C

SUP=PVALUE+ (.005D0/(2**(L-1)) ) *DERBND

INF=PVALUE-(.005D0/(2**(L-1)))*DERBND

HAXP=DMAX1 (MAXP,PVALUE)

IF (P.GE.0.1D0) 19=1

IF (P.GE.0.05D0) 15=1
C
C CHECK FOB PRECISION OF SIZE, HERE DELTA=.001

C

33

IF (I
L=L+1

SOP-MAXP) .LE. 0.001) GO TO 74

P=P-(.005D0/(2**(L-1)))
GO TO 72

74 MAXIM=DMAX1 (MAXIM, SUP)

IF (I5.EQ.1) MINIK5=DMIN1 (MINIM5,INF)

IF (I9.EQ.1) MINI10=DMIN1 (HINI10,INF)

IF (L.EQ.1) GO TO 6

IF { (PP-PUJ.LT.0.00001D0} GO TO 75

P=P+].005D0/(2**{L-1)))*2

GO TO 72

75 P=PP-.005D0
GO TO 6

76 CONTINUE
HRITEJ6.41)

41 FORMAT]1 ' 'PVALUE FOR COMPARING TWO INDEPENDENT1,
* 'PROPORTIONS' /'FOR: N«,
" 4X, »Z',4X,' I5:',2X,»MIN5« . 6X. ' MIN10 ' ,5X, ' P7ALUE'/)

WRITE (6.33) N,Z,HINIM5, MINI 10, MAXIM
FORMAT (' ' ,I4,F8.3,3F11.5)
GO TO 99
END

95
APPENDIX C.2

INTEGER T.T1.T2

DOUBLE PRECISION LFAC {20 1

■*wHipH|#.PL1FPD1fPL2#P02,DLOG#6EXP;DABSjDHAXlJpHINl"
40 FORMAT(* ',///,

* 'THIS PROGRAM COMPOTES THE P-VALOE OF THE Z-TEST'/

* 'FOR COMPARING TWO CORRELATED PROPORTIONS FOR'/

* • H = TOTAL NUMBER OF MATCHED PAIRS,'/

* ' Z = NORBAL STATISTIC.'/

* 'ENTER N, Z, IN FREE FORMAT'/)

LP5=DLOG(0.5D0)
99 READ {9,*)N,Z
N1=N*1

LFAC(1)=0.D0
DO 1 J=2,N1

1 LFAC (J)=LFAC(J-1)+DLOG(DBLE(FLOAT(J-1)) )

C

C SIMPLIFY THE COMPUTATIONS

C

C

LN=INT((Z-. 0000001) **2)+1

DO 2 M=LN,N

M1=M+1

T=INTJ (Z*SQRT (FLOAT(M) ) +H) /2)

T1=T+l

T2=T+2

PM=0.D0

AM=0.DO

DO 3 J=T1.M

C=LFAC(MM)-LFAC{J+1)-LFAC(H-J + 1)+M*LP5
IF (C.LT.-180.) PM=0.D0
IF (C.GE.-180.) PM=DEXP(C)
3 AH=AH+PH

2 LAHJM)=DLOG(AM)
MAXP=0.D0
MAXIM=0.D0
MINIM5=1.D0
MINIH9=1.D0
19=0

15=0

P=-.005DO
6 IF (P.GT.0.98) GO TO 76

L=1
72 IF (L.EQ.1) P=P+.01D0

IF (L.EQ.1) PP=P+.005D0

Q=1-P

LP=DL0G (P)

LQ=DLOG(Q)

DERU=0.D0

DERL=0.D0

PVALUE=0.D0
C
C

C LOCAL BOUND FOR THE DERIVATIVE
C OF THE NOLL POSER FUNCTION

96

C
C

PL=P-(.005D0/(2**(L-1) ))

PU=P+(.005D0/(2**(L-1)))

DO 4 R=LH..N

PHAT1=FL0AT(M-1) /FLOAT (N-1)

PHAT2=FLOATJM) /FLOAT (N-1)

IF (PHAT1.GT.PU) GO TO 12

IF {7PHAT1.IE.PU) .AND. (PHAT1.GE. PL)) GO TO 13

IF JPHAT1.LT. PL) GO TO 14

12 PMAX1=PU
PMIN1=PL
GO TO 15

13 PMAX1=PHAT1

IF (PL.NE.O.DO) GO TO 21

PMIN1=PL

GO TO 15

21 PL1=(M-1)*DLOG(PL) + (N-M) *DLOG(1-PL)
PU1=JM-1) *DLOG (PU) + (N-M) *DLOG (1-PU)
IF (PL1.LT.PU1) PMIN1=PL

IF JPL1.GE.PU1) PMIN1=PU
GO TO 15

14 PMAX1=PL
PHIN1=PU

15 CONTINUE

IF (PHAT2.GT.PU) GO TO 16

IF ((PHAT2.LE.PU) .AND. (PHAT2.GE. PL)) GO TO 17

IF (PHAT2.LT. PL) GO TO 1 8

16 PHAX2=PU
PMIN2=PL
GO TO 19

17 PMAX2=PHAT2

IF (PL.NE.O.DO) GO TO 22

PHIN2=PL

GO TO 19

22 PL2=H*DLOG (PL) + (N-M- 1) *DLOG (1-PL)
PU2=H*DLOG (PUJ + (N-M-1)*DLOG{1-PUJ
IF (PL2.LT.PU2) PHIN2=PL

IF JPL2.GT.PU2) PMIN2=PU
GO TO 19

18 PHAX2=PL
PHIN2=PU

19 CONTINUE

LCOEF=LAH(H) +LFAC (N+ 1 V-LFAC (13+1) -LFAC(N-H+1)
FU1=LCOEF+(H-1)*DLOG f PMAX1) + (N-H) *DLOG( 1-PHAX1)

IF (FU1.LT. -180.0) GU1=0.D0

IF (FU1.GE. -180.0) GU1=M*DEXP(FU1)

FL2=LCOEF+a*DLOGJPMAX2)+ (N-M-1) *DLOG (1-PHAX2)

IF (FL2.LT. -180.0) GL2=0.D0

IF (FL2.GE. -180.0) GL2=(N-!1) *D3XP (FL2)

IF (PHIN2.EQ.0.D0) GO TO 23

FU2=LCOEF+Jl*DLOG (PHIN2) + (N-M-1) *DLOG (1-PHIN2)

IF (FU2.LT. -180.0) GU2=0.D0 .

IF JFU2.GE. -180.0) G02= (N-H) *DEXP (FU2)

GO TO 24 '

23 GU2=0.D0

24 IF (PMIN1.EQ.0.D0) GO TO 25

FL1=LCOEF+ (H-1)*DLOG (PHIN1 ) + (N-M) *DLOG ( 1-PIUN1)

IF (FL1.LT. -180.0) GL1=0.D0

IF JFL1.GE. -180.0) GL1=M*DEXP (FL 1)

GO TO 26

25 GL1 = 0.D0

26 CONTINUE
DERU=DERU+GU1-GU2
DERL=DERL+GL1-GL2
LPVAL=LCOEF+M*LP+ (N-M) *LQ

IF (LPVAL.LT.-180.) P7AL=0.D0
IF (LPVAL.GE.-180.) PVAL=DEXP (LPVAL)
PVALUE=PVALUS+P7AL
4 CONTINUE

DERBND=DHAX1 (DABS (DERU) , DABS (DERL) )

97

C

C COMPUTE THE SIZE

C

C

SOP=PVALOE+(.005D0/(2**(L-1) ) ) *DER3ND
INF=PVALUE-(.005D0/J2**{L-1))) *DERBND

maxp=dmaxi(Haxp,pvalue)

IF " '

IF
C
C CHECK FOE PRECISION OF SIZE, HERE DELTA=.001

C

P=DMAX1 (HAXP.PVALUE)
(P.GE. (10./FLOAT{N)J) 19=1
(P.GE. (5.AlOAT(N))) 15=1

IF ((SUP-MAXP) .LE. 0.001) GO TO 74
L=L+1

P=P- (. 005D0/ (2** (L-1 ) ) )
GO TO 72
74 MAXIM=DMAX1 (MAXIM, SUP)

IF (I5.EQ.1) MINIM5=DMIN1 (MIMIM5,INF)
IF (I9.EQ.1) MINIM9=DMIN1 (MIHIM9,INF)
IF (L.EQ.1) GO TO 6
IF ((PP-PUJ.LT.0.00001D0) GO TO 75

IF (L.EQ.1) GO TO 6
IF ((PP-PUJ.LT.0.00001D0)
P=P+].005DO/(2**(L-1)) )*2
GO TO 72

75 P=PP-.005D0
GO TO 6

76 CONTINUE

SRITEjf6.41)

41 FORMAT (* ','PVALUE FOR COMPARING TWO CORRELATED •
* 'PROPORTIONS',/* FOR :N'f

*o^*lZ,!'?i?f,IS:,*2x^!1TN5,f6X,'MINl0«,4X,«PVALaEV)
WRITE (6.33) N,Z,MtNlfi5.MINIM9, MAXIM
FORMATP »,I4,F8.3,3Fll.5)

GO TO 99
END

REFERENCES

Barnard, G.A. (1947). Significance Tests for 2*2 Tables.
Biometrika 34 , 123.

Barnard, G.A. (1979). In Contradiction to J. Berkson's
Dispraise: Conditional Tests can be more Efficient.
Journal of Statistical Planning and Inference 3, 181.

Basu, D. (1977) . On the Elimination of Nuisance Parameters.
Journal of the American Statistical Association 72, 355.

Basu, D. (1979) . Discussion of Joseph Berkson's Paper "in
Dispraise of the Exact Test" . Journal of Statistical
Planning and Inference 3, 189.

Bennett, B.M. (1967). Tests of Hypotheses Concerning Matched
Samples. Journal of the Royal Statistical Society B 29,
468. —

Bennett, B.M., and Underwood, R.E. (1970). On McNemar's Test
for the 2x2 Table and its Power Function. Biometrics
26, 339. -■•■

Berkson, J. (1978) . In Dispraise of the Exact Test. Journal
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Casagrande, J.T., Pike,.M.C, and Smith, P.G. (1978a). An
Improved Approximate Formula for Calculating Sample
Sizes for Comparing Two Binomial Distributions.
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Casagrande, J.T., Pike, M.C., and Smith, P.G. (1978b). The
Power Function of the "Exact" Test for Comparing Two
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Cochran, W.G. (1950) . The Comparision of Percentages in
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Cochran, W.G., and Cox, G.M. (1957). Experimental Designs,
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98

99

Corsten, L.C.A., and de Kroon, J. P.M. (1979). Comment on
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Courant, R., and John, F. (1965). Introduction to Calculus
and Analysis, Volume 1. Interscience, New York:
John Wiley and Sons.

Eberhardt, K.R., and Fligner, M.A. (1977). A Comparision of
Two Tests for Equality of Two Proportions.
The American Statistician 31, 151.

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Fisher, R.A. (1935) . The Logic of Inductive Inference.
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Proportions , 2nd ed. New York: John Wiley and Sons.

Gail, M. and Gart, J.J. (1973). The Determination of Sample
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Haseman, J.K. (1978). Exact Sample Sizes for Use with the
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Kempthorne, 0. (1979). In Dispraise of the Exact Test:
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Sample Size and Selection of Cases, Psychopharmacology :
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100

McNemar, Q. (1947). Note on the Sampling Error of the
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Illustrated on the Interpretation of Data Classed in a
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Randies, R.H., and Wolfe, D.A. (1979). Introduction to the
Theory of Nonparametric Statistics. New York: John
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Robbins, H. (1977). A Fundamental Question of Practical
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Schork, M. A., and Williams, G.W. (1980). Number of

Observations Required for the Coraparision of Two
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Annals of Mathematical Statistics 10, 58.

BIOGRAPHICAL SKETCH

Samy Suissa was born on July 9, 1954, in Berrechid, a
suburb of Casablanca, Morocco. In 1964, Samy's family moved
to Montreal, Canada, where they still reside. There, he
attended Baron Byng High School, famous for its role in "The
Apprenticeship of Duddy Kravitz" by Mordechai Richler, until
1968 and Outremont High School until 1970. He then entered
McGill University where he received a Bachelor of Science
degree in mathematics in June 1976, and a Master of Science
degree in mathematical statistics in November 1977.

Samy came to the University of Florida in the fall of
1977 to pursue a doctoral degree in statistics. While
studying at the University of Florida, Samy has worked as a
graduate assistant performing statistical consulting duties
in the Department of Orthopaedics among other places.

On July 1, 19 82, Samy Suissa will join the "Service
d' Epidemiologic Clinique" of the Montreal General Hospital's
Research Institute in the capacity of biostatistician. He
will also have an appointment as assistant professor in the
Department of Epidemiology and Health at McGill University.
He has recently been awarded a research grant from "Les Fonds
de Recherche en Sante du Quebec" for 19 82-19 83.

He is married to the former Nicole Bonenfant of Montreal,
Canada, and has a son, Daniel Moshe.

101

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of* Philosophy .

<W ^LsW'

Jonathan J. Shuster, chairman
Professor of Statistics

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.

GLJJ, & fcuJLft

<L»L_

Ronald H. Randies
Professor of Statistics

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.

YlfaV P ipLcU

Mark P. Hale

Associate Professor of Mathematics

This dissertation was submitted to the Graduate Faculty of
the Department of Statistics in the College of Liberal Arts
and Sciences and to the Graduate Council, and was accepted
as partial fulfillment of the requirements for the degree of
Doctor of Philosophy.

August, 1982

Dean for Graduate Studies
and Research

fillip