sup h(p) = h(b) if > b
peI
= h(a) if < < a
= h(t) if E I (2.2.2)
and
inf h(p) = min ( h(a), h(b) ) (2.2.3)
pCI
where 0 = r/s.
An upper bound for i' (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 T'(p) can be obtained on (a,b) by reversing these substi
tutions. Finally, a bound M for it''(p) on (a,b) is taken as
the larger of the two bounds, in absolute value.
2.3 A Least Upper Bound for r(p)
Since local bounds for JI'(p) can now be computed for
any subinterval (a,b), the mean value theorem can be applied
to the successive subintervals Il=(0,.01), I2=(.01,.02),...,
I100=(.99,1) of the unit interval to obtain an upper bound
for ir(p) in each I.. An upper bound for r(p), 0
simply the maximum of these local upper bounds.
First, for each I., j=l(1)100, it is now possible to
find a M. such that
Il'( .)I < M. for all 89 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
T (e ) e ( ir(p ).005Mj r(pj)+.005M. ), (2.3.1)
for all 8.e I., where p.=(j.5)/100, the midpoint of I..
For each I., j=1(1)100, the local maximum is given by
n+(pj) = n(pj) + .005M .
It can then be deduced that, for p on the unit interval (0,1),
r(p) < max { n (pj); j=l(l)100 }. (2.3.2)
J )
Therefore, the right hand side of (2.3.2) is greater than
sup i(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
S(p). This bound can be improved upon to produce a least
upper bound of precision 6 in the following manner. Any
interval I. for which
+
+ (pj) > max { r(pi); i=l(1)100 } + 6
1
m.
must be iteratively subdivided into 2 subintervals
m.
SIjk ; k.=l,2,..., 2 J, mj.l }, where m. is the smallest
integer such that
m.
n(Pjk )+.005 Mjk /2 < max{max{i(p.)}, max{I (pj )}}+ 6
j j i Uj j
(2.3.3)
for all k., and where
m.1
Pjk. = (jl)/100 + (2kjl)/(100x2 )
is the midpoint of Ijk. and Mjkj is the local bound for
7''(p) in Ijk,. The inequality (2.3.3) is clearly
attainable since, from (2.2.1),
Mjk. iC ai max(bi,ci). (2.3.4)
j iEC 1
Moreover, using the right hand side of (2.3.4) rather than
Mjkj is inefficient and would make computing costs
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).
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 i(p) can be of interest. A relevant feature is
the stability or flatness of i(p). An indication of this
stability can be created using
inf { i(p); PE(paPb)} (2.4.1)
p
where pa and pb are chosen appropriately for each problem.
The difference between sup n(p) and (2.4.1) is then an
indicator of this stability. The whole unit interval is
not used in (2.4.1) because i(0)=I(1)=0. A lower bound
for the set in (2.4.1) can be computed from (2.3.1) by
min { r(pj) .005M.; j=ja()jb
where ja = int(100pa) + 1,
jb = int(100pb) + 1
and int(.) is the integer function, pj being as in (2.3.1).
'.a
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
unconditional sample space S into a critical region C and
an acceptance region SC. The choice should then be based on
optimal procedures that produce test statistics that are
powerful, simple to compute, and intuitively appealing.
Three such procedures are the likelihood ratio test
criterion, the chisquare goodnessoffit test and a Ztest
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
sup L(8)
H0
R =
sup L(6)
where L(e) 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 chisquared null distribution.
The chisquare goodnessoffit test, based on the statistic
2 (0iEi)2
X = EO (2.5.2)
i E.
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 chisquare. The third test is based on the
asymptotic normality of 6n, the maximum likelihood estimator,
if one exists, of the parameter, call it 6, being tested.
The statistic
Z n
2 (2.5.3)
s(n,)
where s2(6 ) is the asymptotic variance of n 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 chisquare
goodnessoffit 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 wellknown and welltabulated
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,pl) and (n,p2) respectively. An experi
ment that compares p, and p2 is called a 2x2 comparative
trial by Barnard (1947), the outcome of which is represented
in the form of Table 3.1,
_I~
Table 3.1
S F totals
T1 x nx n
T2 y ny n
where pi = P(SITi) = 1P(FITi), i=1,2. The labels S and F
represent the binary outcomes ( S=success, F=failure) and T1
and T2 represent the two populations being compared.
The problem is to test, at level a, the null hypothesis
H0:Pl=P2 against the alternative hypothesis Ha:P1
the other onesided alternative Ha:p1>p2 and the twosided
alternative Ha:p#3p2 are treated in a similar manner, only
this onesided 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 (1 n ( ((p y
and is, under the null hypothesis H0:pl=P2 (=p say),
P(X=x,Y=y) = x)( px+y (1p)2nxy
a function of the nuisance parameter p, the unspecified
common value of pl and p2 under H0.
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 r(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 Fishek'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
1B=.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.
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
are. 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 p2Pl, is the Ztest with an
unpooled estimator of the variance and is given by
/Z n (P2 1 (3.2.2)
u 1^ 1 1
(P2 2 + Plql)
where Pl = x/n = 1ql, P2 = y/n = 1q2, with x, y and n as
in Table 3.1. The second one, based on the asymptotic null
distribution of p2P1, is the Ztest with a pooled variance
estimator and is given by
/n (p2 1
Z = (3.2.2)
P (2 p q )
where P1 and p2 are as in (3.1.1) and p = (x+y)/2n = lq.
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 4, the standard normal
distribution function. The test statistic Z is most
p
frequently used through an equivalent test statistic, the
chisquare goodness of fit statistic given by
X = Z (3.2.3)
P P
which has a X1 limiting distribution. Because this chi
square test deals with twosided alternatives, the statistic
Z is preferred in the present context of a onesided 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
[ za (2pq) zl (plq1+p2q2) ]2
n = 2 (3.2.4)
(p2) 2
where z is the upper 100y percentile of the standard normal
distribution, a and B are the type I and type II error
probabilities, p, and p2 are the desired alternatives, and
P = (pl1+2)/2 = 1p. The second formula, based on the
variance stabilizing property of the arcsine transformation
on proportions, is given in Cochran and Cox (1957) by
(za + z ) (3.2.5)
n
as 2(sin/ l sin 1/2)2
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
n { 1+[1+8(p2P1)/np }2 (3.2.6)
n =n
c p 4(22 12
4 (P2 Pl)2
where n is the sample size found from (3.2.4). More recent
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
n =n { 1 4( /n 2 (3.2.7)
r p (2 2
4 (P2 Pl)
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 is constructed on. 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 HO:Pl=P2,
the probability of Table 3.1 is given by the hypergeometric
distribution, namely
n n
P(x,ylx+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 NeymanPearson 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
P(x,y) = px (1plnx (n) p (lp2)nY
= (n) )(lpl)n (l2)n
x exp{x log[pl/(lpl)]+y log[p2/(1p2)]}
= x) (lpl) (lP2)n
fPl/(Pl 1
x exp{x log 2/(1p +(x+y)log[p2/(lp)]}.
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
4(x,t) = 1
= y(t)
= 0
when x < C(t)
when x = C(t)
when x > C(t)
where C and y are determined by
EHo [ ((X,T)IT=t ] = a
"0
for all t, that is
a = P [X
The conditional distribution is
n n
P(X=xlT=t) =
t n n
S(u) tu Pu
u=0
Pl/(1pl)
where p= is the odds ratio,
P2/(1P2)
and.under H0, the distribution is given by
n n
x tx
PH (X=xlT=t) = x=0,...t,
O 2n
(t )
t
the same hypergeometric distribution found by Fisher's
permutation method. Here, C(t) is taken to be the largest
value such that
C(t)l x tx
x=0 2n
t
In practice, the nonrandomized version of 4 is used,
that is 4 without the random element y(t). Therefore, the
conditional test always has size s a and, unlike (, 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 Zu, 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 p1)
Z = ( (3.4.1)
(P2 2 + 11)
where pl = x/n = 1ql = y/n = 1q2 with x, y and n as
in Table 3.1. The asymptotic null distribution ( the
standard normal) of Zu 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,pl) and (n,p2) respective
ly ,. the power function of any test is given by
(p ) = E E x nx ( p (lP2)n
1(P1'P2) = S S x p1 (lp1)
(x,y)eC
(3.4.2)
and under H0:Pl=p2 (=p say), the null power function, also
denoted by i, is given by
7(p) = E x) (y pX+y (lp)2nx (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 onesided test
of interest, the critical region defined by Zu is given by
C = {(x,y): Zu > zu; x,y=0(l)n, zu O }. (3.4.4)
For an a level test, the critical value of Zu, namely zu,
satisfies the equation
zu = inf {zu: sup r(p) < a }. (3.4.5)
P
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 r(p) in (3.4.5).
First, to simplifythe computations, (3.4.3) can be
reduced to a single summation by solving the inequality
Zu > z namely
y x> Z(y(ny) + x(nx) (3.4.6)
After squaring both sides of (3.4.6), the larger root for
After squaring both sides of (3.4.6), the larger root for
y in
I
2 2
n (yx) = z [y(ny) + x(nx)]
u
is found to be
y = h(x) = b + (b4ac
2a
where a = 1 + z/n,
2
b = 2x + z
2 2
and c = ax xz2
Hence (3.4.4) reduces to,
C = {(x,y): y > h(x) ; x,y=0(1)n }.
Next, (3.4.3) can be written as
v
T (p) = E E
x=O y>h(x)
v
x=0
v
=
x=0
(n) n) y p)2nxy
x y p (lp)
(n) p (lp)nx
x p (lp) E
y>h(x)
f(x) [ 1 F(h(x)) ]
2 2
where v = int[ n /(n+z ) ], int[.] is the integer function,
u
f(.) is the binomial probability mass function with parameters
(n,p) and F(.) is its cumulative distribution function.
(3.4.7)
(3.4.8)
() pY (lp)ny
The derivative of (p), which can also be reduced
significantly to simplify the computations,is given by
i'r(p) = E y (x+y) px+y (1p)2nxY
C
E ) (nx+ny) pX+y (1p)2nxyl
C
where Z denotes the double summation Z Z
C (x,y)eC
It can be rewritten as
p S (nl n) x+y1 2nxy
(p) = Z n x1 y p (1p)xy
C
+ n ( yl px+1 (lp)2nxy
C
n (nxl) n) x+y (lp)2nxy
C
s n ( nl ) p+y (lp)2n1 (3.4.9)
C
nl nl
where 1 = n = .
Consider the boundary of the critical region C defined by
W = {(x,y) : (x,y)eC and (x+l,y)gC }.
.The sum of the first and third terms of (3.4.9) becomes,
after cancellation of opposing signed identical contributions,
35
wlj(p) = E n x ) p+y (lp)2nxy1 (3.4.10)
W
The other boundary of C is defined by
V = {(x,y) : (x,y)eC and (x,y1)yC }.
Then the sum of the second and fourth terms of (3.4.9)
becomes
n ni
2(p) = E n yl p+1 (lp)2nxy (3.4.11)
V
Upon combining (3.4.10) and (3.4.11), the derivative of N(p)
is given by
i '(p) = i (p) + IT(p)
and can be further reduced by noticing, from (3.4.4) and
(3.4.6), that
(x0,Y0)EC iff (ny0,nx0)EC
so that
(xYl1)eV iff (nyl,nx1)EC and (nYl+l,nx1) C
iff (ny1,nx1) W.
I
The derivative of n(p) can finally be written as
n ni
(p) = E n [ y p (lp)2nxy
V
(ni (nx) 2nxy 1p)+y
ny nx p (lp)X+Y1 ]
(= n )( yi px+yl p)2nxy
= n (x y [ p (lp)
V
Sp2nxy (lp)+y1 ]
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 zu. For a test of significance of level a,
the critical value zu 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 S a and
zu is taken as the smallest value which satisfies this
inequality. This procedure was implemented in a FORTRAN
computer program, listed in Appendix C.I.
For n=10(1)150 and a=.05 and .025, zu the exact
critical values and a the size (of precision 6=.001) of Zu,
were computed and are given in Table A.1. Furthermore, Table
A.1 also contains al, a lower bound for {r(p); .05
and a2, a lower bound for {' (p); .10
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
The null power function, i(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.1 contains the plot of
S(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
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.1
is used to plot n(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 Zu (or equivalently Z ) is seen to
behave better. For n=20, Figure B.5 is the plot of w(p)
based on zu = 1.85 (or z = 1.78) and Figure B.6 is based on
n=30 and zu = 1.77 (or z = 1.73).
3.5 Relation to the Chisquare Goodnessoffit Test
The other test statistic given in (3.2.2), namely Z ,
the square of which is the chisquare goodnessoffit test
statistic, has a functional relationship with Zu in the
present equal sample size case. This relationship can be
derived by first noticing that
/n (y x)
Z = (3.5.1)
S [y(ny) + x(nx)
and
/n (y x)
Z (3.5.2)
Zp = [(x+y)(2nxy)] (3.5.2)
The square of the denominator of Z
u
yn y2 + xn x2 (3.5.3)
can be compared to the square of the denominator of Z ,
yn + xn (x+y2 (3.5.4)
yn + xn (x+y) (3.5.4)
By rewriting (3.5.3) as
2 2
yn + xn (x+y) (yx)
it is clear that Z and Z satisfy the relation
p u
1 1 1
222
Z Z2 2n
u p
which can be rewrittenas
2n Z2
2 u 2
P 2n + 2
u
so that the
obtained by
critical value z of Z given in Table A.1 are
P P
2n Z2 2
Z = sgn(yx) n + Z
P2n + Z
u
where
sgn(u) = 1
= 0
= 1
(3.5.6)
u > 0
u = 0
u < 0
Robbins (1977) has noted that IZuI 2 IZ  for the
equal sample size case and has posed the question as to
which of Z or Z is more powerful. This question was
u p
(3.5.5)
investigated by Eberhardt and Fligner (1977). They noticed,
via a computational argument, that the increase in the
significance level for Zu is compensated fairly well by an
increase in power. Moreover, they suggested that Zu 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
u p
each other and are therefore equivalent in the sense that
Zu, 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 H0 more often than Z will.
3.6 Power and Sample Sizes.
Given the critical values of Table A.1, 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 18
and significance level of a can:thus be computed by solving
the equation
n = min {n: I(pl'P2)1B}
where the critical region that defines 13(p1,P2) is based on
Zu, a function of n.
This equation was solved for a=.05 and 1B=.80 and the
results are given in Table A.2 for various combinations of
pl and p2. Table A.2 also contains the critical values ,
*
the size a (of precision 6=.001) and the attained power
18 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 ne
in Table A.3. For the configurations considered, it is
*
seen that n tend to be smaller than ne, the sample sizes
determined by Fisher's exact test. Furthermore, the sample
sizes based on the arcsine formula [nas] and those based on
Z the pooled Ztest [n ], tend to coagree quite well and
to be, in general, slightly smaller than n The other
formulae discussed in section 3.2, namely nc and nr are
*
seen to exceed n and ne.
A direct comparison of the critical regions defined by
Fisher's exact test and the exact Ztests 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 Ztests. Therefore,
the exact Ztests are uniformly more powerful than Fisher's
exact test for the cases n=10(1)150 and a=.05 and .025.
CHAPTER 4
THE 2x2 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 wellknown "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.
Let (R,S) represent a pair of binary random variables
with joint distribution
P(R=i,S=j) = pij i,j=0,l, ZE pij = 1.
ij
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 1 totals
0 u x u+x
R
1 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
H0:P(R=1) = P(S=1) against one of the alternative hypotheses
Ha:P(R=1) < P(S=1), Ha:P(R=1) > P(S=1) or Ha:P(R=1) $ P(S=1).
For the sake of illustration, only the alternative hypothesis
H :P(R=1) < P(S=1) will be considered. Note that
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)
so that the problem becomes that of testing H0:P01=P10
against Ha:P01>P10 The likelihood of the sample is given by
P(u,x,y,v) = xu X y v) p0u X p v
P00 P01 P10 Pll
the quadrinomial distribution with probabilities { pij ;
i,j=0,1}. Under the null hypothesis H0:p01=p10 (=p say),
the likelihood of the sample becomes
N
N u ) u x+y v
P (u,x,y,v) = (ux y v p00 p (1p002p)v,
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 NeymanPearson approach to the elimina
tion of nuisance parameters. Bennett (1967) has computed the
chisquare goodnessoffit 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
McNemar's test against its continuitycorrected 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 p00 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 continuitycorrected 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 NeymanPearson 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 onesided case.
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
1B=.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 SR
(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 twosided alternative. This statistic has an asymp
totic Xnull distribution. Cochran (1950) reduced the
problem to a sign test, using the statistic
X 2 +
X2 = (x n)2 + (y n)
n n
(x y)2
x + y
x+y
where n=x+y, the total number of discordant pairs. Bennett
(1967) used the chisquare goodnessoffit test, applied to
the quadrinomial frequencies of Table 4.1, to find
2 (u Noo2 (x N01) 2 (y Np)2
X + +
NOO0 NPO1 NP10
(v Np11
+
Np11
(x y)2
x+y
x + y
where the P..'s are the maximum likelihood estimators of
the Pij's under H0. 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
tion to the asymptotic unconditional power function of X2
(McNemar's test statistic) gives, for a onesided test of
significance level a and power 1B, the required sample size
as
{ z a + z (i2 2) }2
N = (4.2.2)
1 A2
where i=P01+Pl0, =p 10p01 and z is the upper 100y percen
tile of the standard normal distribution. The second formula
is based on a more precise approximation to the asymptotic
unconditional power function of X2 and is given by
{ z % + z8 [(2 A 2(3+fl)] }2
N = (4.2.3)
a2 2
For the purpose of comparison with exact conditional
and exact unconditional results, these formulas were computed
and are given in Table A.6. These comparisons 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
P(u,x,y,v) = x y v)00 p01 p py
and can be written in the exponential family form as
N
P(u,x,y,v) = (u x y V exp{ u log(p00) + x log(p0l)
+ (Nuxv) log(pl0) + v log(pll)}.
It can be reparametrized as
N
P(u,x,y,v) = u x y v exp{ u log(p00/P10)
+ x log(p01/10) + v log(p1l/pl0)
+ N log(pl0)
The new parameters are, in the notation of Lehmann (1959),
I = log(p01/p10)
v = ( log(p00/p10) log(pll/p10) )
and the hypothesis to be tested becomes H0:8=0 against
H a:>0. The sufficient statistics are
N
X = E (1R.)Si
i=l
N N
T = (U,V) = ( Z (1R ) (1S.) Z R. S. ).
i=l i=l 1
Therefore the UMPU test is given by
((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
EH { X(X,U,V) I U=u, V=v } = a all u,v.
To find this conditional expectation, first notice that the
distribution of (U,V) is
P(u,v) = (u v Nuv p (l00pll)Nuv
so that the distribution of X given U=u and V=v is
P(xlu,v) = (xy p1 p / (Po0 o) Nuv
= (pO/(P01+PO)) (10/(P01+Pi1))
= x( px (1p)nx
where n=x+y and p = pO1/(P01+P10). Therefore, the null
hypothesis H0:P01=P10 reduces to H0: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 0 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.
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 p01P10. This statistic is the square root of
McNemar's test statistic and is given by
x y
Z = (x+y) (4.4.1)
(x+y)
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 i n
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 Zc. These values will then provide
a means of assessing the accuracy of the normal approximation.
The power function of Zc is given by
N
(P1P10) = Z s s (u x y v) p00 px1 0
(x,y)cC u
a function only of p01 and p10 since it is based on the
marginal distribution of (X,Y) which is obtained as
Nxy N
P(x,y) = u x y ) p0 p p Nxyu
u=0 P00
N
= (x y Nxy) p p0 (1P01Pl0Nxy
=xyN P01 lo(olpo
The power function
n (=x+y) as
H(P01o'p0) =
of z can then be written in terms of
c
E C
(x,y)eC
N
x nx Nn Pl0 nPx
x (p ) Nn
(1P lPlo)
where C is the critical region defined by Zc. Under the null
hypothesis H0:P01=p10 (=e say), the null power function of Zc
is given by
A(6) = E
(x,n)EC
N
x nx Nn 8n (12e)Nn
or equivalently, if p = 28, by
1 (p) = z E
(x,n)EC
N
x nx Nn) n pn (l_p)Nn
a function of the nuisance parameter p, the probability of a
discordant pair. For the onesided test of interest, namely
for the alternative hypothesis Ha:P01>Pi0, the critical
(4.4.3)
S (4.4.4)
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 Zc, namely z ,
satisfies the equation
c = inf {zc : sup (p) < al. (4.4.6)
p
Note that r(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 7(p).
First to simplify the computation of (4.4.4), notice
that the inequality Z > z is in fact
x > z /n + n
so that the critical region C reduces to
C = {(x,n): x > h(n); x=0(l)n, n=0(1)N} ,
where h(n) = {z c/n + n}. The null power function (4.4.4)
becomes
N
ir(p) = 0 x
n=O x>h(n)
N
n=0
N
= S
n=k
() ( n pn (p)Nn
n x p (l)
() pn (1p)Nn
n p (l)
N p (
n pn (1p)Nn
x>h(n)
[1 Fn (in)]
where k = int[z2 + 1], i = int[h(n)], int[.] is the integer
function and F (.) is the binomial cumulative distribution
with parameters (n,). Notice that, since in n, it is
more efficient to compute
n
S
x=i +1
n
n
( ,) n
x3
instead of lFn(in). Then, by the symmetry of the binomial
distribution with (n,), the null power function of Z can
c
be rewritten as
N N
r(p) = p (lp) F (ni ),
n=k n n
(4.4.7)
in the form of (2.1.1). The derivative of the null power
function is
() n
x
N N
N N pnl Nn
T'(p) = S F (nin1) n [ n p (lp)Nn
n=k
(Nn) pn (1p)Nnl
so that the methodology developed in Chapter 2 can now be
utilized to find a the size (of precision 6=.001) of Z
c
for any value z .
In this problem, the size was taken as sup{w(p):0
because of the behaviour of w(p) when p approaches 1. The
null power function is dominated by the last term of the.
summation, namely pN FN(NiN1) 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
Zc 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
* *
c is taken as the smallest value of zc which satisfies a sa.
The FORTRAN computer program used to implement this procedure
is given in Appendix C.2.
For N=10(1)200 and a=.05 and .025, zc 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 al, a lower bound for {1(p): p > 5/N} and
a2, a lower bound for {r(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, i(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 i(p) based on the normal approximation of Z
(critical value Zc=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
.30.74. 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 '(p). From these plots, the
exact Ztest (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.
The null power function of the exact unconditional test
Zc is seen to behave better for larger values of N. For
N=30, Figure B.10 is the plot of r(p) based on z =1.74
c
and Figure B.11 is the plot of n(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
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 p01 and pl0. Consequently, the minimum sample
size required to achieve a power of 18 and a significance
level of a for a combination of (p01,p10) can be computed by
solving the equation
N = min { N: H(p01,p10) > 18 } (4.5.1)
where the critical region that defines H(po01,p0) is based
*
on z a function of N.
Because all other sample size results are given in terms
of the parameters i=P01+P10 and A=pl0P01 equation (4.5.1)
was solved in terms of these parameters for the purpose of
comparability. For a=.05 and 18=.80, and various combina
tions of p and A, the minimum sample sizes from (4.5.1) are
given in Table A.5. This table also contains the critical
values zc the size (of precision 6=.001) of Zc and the
attained power 18 Therefore, Table A.5 is sufficient for
both the design and the analysis of the 2x2 table for
comparing two correlated proportions.
In Table A.6, 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
1 2
to N and N in Table A.6. The exact unconditional sample
sizes N are seen to be smallerthan N the sample sizes
based on the exact conditional test, for all except some
combinations of 4 and A. This seems to happen for larger
values of i and A. The approximate sample sizes N and N.
a1 a2
are almost equal to each other, much smaller than N and
e
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 comparison 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 Ztest. Therefore, the exact
Ztest is uniformely more powerful than the exact conditional
test for the cases considered, namely N=10(1)200, a=.025 and
a=.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 onesided tests, the tables of critical values are
produced for significance levels a=.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.1, A.2 and A.3: two independent
proportions
n = sample size in each group
a = nominal significance level
al = lower bound for {i(p): .05
a2 = lower bound for {i(p): .10
n(p)= null power function
zu = exact onesided critical values of Z the Ztest
with unpooled variance estimator
p = exact onesided critical values of Zp, the Ztest
with pooled variance estimator
pi = probability of success in group i, i=1,2.
n = sample size determined by exact Ztests
*
a = size of Ztests for n and zu or z
*
18 = attained power for n zu or z Pl and P2
The following are the sample sizes of Table A.2 as
determined by:
ne = Fisher's exact test, Casagrande et al. (19
nc = the corrected X2 approximation (3.2.6)
nr = the recorrected X2 approximation (3.2.7)
78b)
n
n =
as
n =
Legend for
proportions
N
a =
a1 =
a2 =
7(p)=
z =
C
A
1 =
P01 =
Pl0 =
N
*
a =
18 =
2
the uncorrected X approximation (3.2.4)
the arcsine formula (3.2.5)
the exact Ztests.
Tables A.4, A.5 and A.6: two correlated
sample size (number of matched pairs)
nominal significance level
lower bound for {r(p): p > 5/N}
lower bound for {r(p): p > 10/N}
null power function
exact onesided critical values of Z the
P10P01
P10+P01
P(R=0,S=1) (see section 4.1)
P(R=1,S=0) (see section 4.1)
sample size determined by the exact Ztest
size of Ztest for N and z
c
attained power for N z A and q
Ztest
The following are the sample sizes of Table A.5 as
determined by:
N = McNemar's test, first approximation (4.2.2)
a1
Na = McNemar's test, second approximation (4.2.3)
Ne = the exact conditional test (sign test), Schork
and Williams (1980)
N = the exact Ztest.
Table A.1 Critical Values and Sizes of Ztests for
Comparing Two Independent Proportions.
a = .05
a1 a2 a zu z
.0068
.0086
.0105
.0125
.0146
.0168
.0188
.0209
.0230
.0251
.0271
.0290
.0308
.0325
.0341
.0342
.0361
.0369
.0386
.0393
.0395
.0398
.0387
.0395
.0404
.0376
.0380
.0398
.0408
.0409
.0410
.0411
.0406
.0411
.0412
.0396
.0383
.0395
.0404
.0417
.0251
.0292
.0329
.0363
.0394
.0422
.0289
.0308
.0335
.0368
.0339
.0351
.0360
.0367
.0373
.0342
.0361
.0369
.0386
.0393
.0395
.0398
.0387
.0395
.0405
.0376
.0380
.0398
.0414
.0410
.0419
.0422
.0406
.0413
.0420
.0396
.0383
.0395
.0404
.0417
.0476
.0504
.0471
.0484
.0495
.0505
.0421
.0426
.0430
.0435
.0438
.0445
.0484
.0447
.0458
.0448
.0449
.0450
.0458
.0458
.0467
.0494
.0462
.0476
.0493
.0476
.0467
.0467
.0476
.0469
.0470
.0491
.0481
.0487
.0498
.0492
.0482
.0492
.0478
.0486
1.96
1.84
1.86
1.81
1.77
1.74
1.92
1.90
1.88
1.86
1.85
1.84
1.83
1.84
1.81
1.80
1.79
1.79
1.78
1.78
1.77
1.77
1.80
1.77
1.75
1.75
1.75
1.74
1.74
1.74
1.73
1.73
1.78
1.76
1.74
1.73
1.73
1.72
1.72
1.72
1.80
1.72
1.74
1.71
1.68
1.66
1.82
1.81
1.80
1.78
1.78
1.77
1.77
1.78
1.76
1.75
1.74
1.74
1.74
1.74
1.73
1.73
1.76
1.73
1.72
1.72
1.72
1.71
1.71
1.71
1.70
1.70
1.75
1.73
1.71
1.71
1.71
1.70
1.70
1.70
a= .025
a1 a2 a u z
.0005
.0007
.0010
.0014
.0018
.0024
.0030
.0036
.0042
.0049
.0057
.0064
.0071
.0078
.0086
.0094
.0102
.0109
.0117
.0124
.0034
.0138
.0145
.0151
.0157
.0163
.0168
.0173
.0063
.0169
.0186
.0186
.0193
.0196
.0199
.0202
.0204
.0176
.0097
.0176
.0044
.0058
.0073
.0089
.0106
.0122
.0137
.0152
.0166
.0174
.0192
.0183
.0175
.0179
.0178
.0184
.0183
.0191
.0194
.0195
.0128
.0198
.0187
.0193
.0193
.0204
.0206
.0208
.0141
.0169
.0203
.0186
.0193
.0197
.0208
.0217
.0215
.0177
.0174
.0180
.0212
.0208
.0225
.0200
.0209
.0218
.0252
.0233
.0241
.0246
.0252
.0251
.0245
.0239
.0222
.0233
.0217
.0225
.0233
.0242
.0219
.0243
.0236
.0245
.0238
.0243
.0245
.0249
.0225
.0233"
.0242
.0237
.0244
.0241
.0243
.0249
.0247
.0251
.0233
.0241
2.17
2.40
2.26
2.26
2.19
2.14
2.10
2.21
2.14
2.14
2.10
2.17
2.17
2.17
2.15
2.13
2.12
2.11
2.10
2.09
2.15
2.11
2.09
2.07
2.06
2.06
2.06
2.05
2.13
2.10
2.08
2.07
2.05
2.05
2.04
2.03
2.03
2.06
2.09
2.07
1.96
2.14
2.06
2.07
2.03
2.00
1.97
2.07
2.02
2.03
2.00
2.06
2.07
2.07
2.06
2.04
2.04
2.03
2.03
2.02
2.08
2.04
2.03
2.01
2.00
2.00
2.00
2.00
2.07
2.05
2.03
2.02
2.00
2.00
2.00
1.99
1.99
2.02
2.05
2.03
Table A.1  continued
a = .05
O1 a2 a z zp
.0419
.0420
.0435
.0440
.0394
.0396
.0413
.0398
.0389
.0398
.0405
.0416
.0409
.0428
.0406
.0406
.0407
.0327
.0406
.0383
.0387
.0392
.0393
.0400
.0409
.0412
.0412
.0413
.0414
.0415
.0415
.0430
.0327
.0327
.0396
.0398
.0393
.0399
.0403
.0410
.0429
.0426
.0435
.0444
.0394
.0396
.0413
.0398
.0389
.0398
.0405
.0416
.0409
.0428
.0406
.0406
.0407
.0378
.0406
.0383
.0387
.0392
.0393
.0400
.0409
.0412
.0417
.0418
.0419
.0420
.0437
.0430
.0378
.0378
.0396
.0398
.0393
.0399
.0403
.0410
.0494
.0487
.0489
.0499
.0481
.0482
.0498
.0495
.0494
.0499
.0500
.0501
.0502
.0503
.0485
.0486
.0491
.0473
.0483
.0483
.0487
.0489
.0486
.0485
.0486
.0486
.0487
.0488
.0489
.0489
.0490
.0496
.0480
.0480
.0491
.0491
.0491
.0496
.0492
.0494
1.71
1.72
1.71
1.71
1.76
1.75
1.73
1.72
1.72
1.71
1.71
1.70
1.72
1.70
1.72
1.72
1.72
1.75
1.74
1.73
1.72
1.71
1.71
1.71
1.71
1.71
1.70
1.70
1.70
1.70
1.70
1.70
1.74
1.73
1.72
1.71
1.71
1.70
1.70
1.69
1.69
1.70
1.69
1.69
1.74
1.73
1.71
1.70
1.70
1.69
1.69
1.68
1.70
1.68
1.70
1.70
1.70
1.73
1.72
1.72
1.71
1.70
1.70
1.70
1.70
1.70
1.69
1.69
1.69
1.69
1.69
1.69
1.73
1.72
1.71
1.70
1.70
1.69
1.69
1.68
a = .025
a1 a2 a zu z
.0177
.0177
.0178
.0178
.0178
.0178
.0178
.0178
.0126
.0178
.0178
.0178
.0178
.0178
.0178
.0178
.0178
.0179
.0179
.0147
.0152
.0179
.0197
.0193
.0194
.0198
.0198
.0198
.0198
.0198
.0198
.0161
.0162
.0160
.0194
.0196
.0191
.0195
.0200
.0200
.0181
.0182
.0184
.0185
.0187
.0189
.0191
.0192
.0180
.0196
.0207
.0198
.0194
.0197
.0204
.0208
.0207
.0211
.0212
.0175
.0195
.0196
.0199
.0193
.0194
.0200
.0204
.0210
.0211
.0210
.0213
.0186
.0187
.0176
.0194
.0196
.0191
.0195
.0200
.0205
.0248
.0245
.0242
.0247
.0236
.0249
.0246
.0248
.0222
.0236
.0250
.0249
.0247
.0246
.0249
.0249
.0239
.0240
.0247
.0231
.0243
.0242
.0247
.0248
.0249
.0251
.0250
.0251
.0252
.0238
.0246
.0243
.0240
.0240
.0244
.0249
.0242
.0246
.0243
.0248
2.05
2.04
2.04
2.04
2.04
2.03
2.03
2.03
2.09
2.07
2.05
2.04
2.03
2.03
2.02
2.03
2.04
2.03
2.03
2.08
2.06
2.05
2.04
2.03
2.03
2.02
2.02
2.02
2.02
2.03
2.03
2.07
2.06
2.05
2.04
2.03
2.03
2.03
2.03
2.02
2.01
2.00
2.00
2.01
2.01
2.00
2.00
2.00
2.06
2.04
2.02
2.01
2.00
2.00
1.99
2.00
2.01
2.00
2.00
2.05
2.03
2.02
2.01
2.01
2.01
2.00
2.00
2.00
2.00
2.01
2.01
2.05
2.04
2.03
2.02
2.01
2.01
2.01
2.01
2.00
Table A.1  continued
a = .05
aO a2 a zu
n
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
.0418
.0423
.0425
.0431
.0436
.0442
.0440
.0445
.0450
.0329
.0382
.0382
.0382
.0382
.0382
.0407
.0413
.0419
.0421
.0422
.0427
.0426
.0437
.0435
.0439
.0443
.0391
.0380
.0392
.0402
.0396
.0405
.0405
.0407
.0411
.0418
.0419
.0426
.0427
.0424
.0418
.0423
.0425
.0431
.0436
.0442
.0440
.0445
.0450
.0389
.0391
.0392
.0393
.0393
.0394
.0407
.0413
.0419
.0421
.0422
.0427
.0426
.0437
.0435
.0439
.0443
.0400
.0380
.0398
.0402
.0396
.0405
.0405
.0407
.0411
.0418
.0419
.0426
.0428
.0424
.0504
.0496
.0494
.0495
.0500
.0503
.0499
.0501
.0504
.0486
.0504
.0491
.0503
.0496
.0493
.0495
.0495
.0495
.0495
.0495
.0495
.0496
.0497
.0495
.0496
.0499
.0484
.0484
.0486
.0500
.0494
.0520
.0502
.0497
.0497
.0502
.0496
.0500
.0497
.0497
1.69
1.69
1.69
1.69
1.68
1.68
1.68
1.68
1.68
1.72
1.71
1.71
1.70
1.70
1.70
1.69
1.69
1.69
1.69
1.69
1.69
1.69
1.68
1.68
1.68
1.68
1.72
1.72
1.71
1.70
1.70
1.69
1.69
1.69
1.69
1.68
1.69
1.68
1.68
1.69
a = .025
a1 a2 a zu p
1.68
1.68
1.68
1.68
1.67
1.67
1.67
1.67
1.67
1.71
1.70
1.70
1.69
1.69
1.69
1.68
1.68
1.68
1.68
1.68
1.68
1.68
1.67
1.67
1.67
1.67
1.71
1.71
1.70
1.69
1.69
1.68
1.68
1.68
1.68
1.67
1.68
1.67
1.67
1.68
.0200
.0201
.0201
.0202
.0170
.0171
.0171
.0153
.0191
.0153
.0191
.0192
.0192
.0192
.0193
.0193
.0211
.0212
.0176
.0183
.0184
.0185
.0182
.0182
.0182
.0182
.0199
.0203
.0205
.0206
.0209
.0209
.0212
.0215
.0213
.0181
.0187
.0185
.0186
.0186
.0207
.0211
.0208
.0209
.0184
.0185
.0186
.0185
.0191
.0186
.0191
.0192
.0192
.0192
.0193
.0193
.0211
.0212
.0176
.0183
.0184
.0192
.0193
.0189
.0194
.0196
.0199
.0203
.0205
.0206
.0209
.0209
.0212
.0215
.0213
.0181
.0187
.0193
.0200
.0193
.0249
.0250
.0242
.0245
.0228
.0232
.0236
.0244
.0249
.0244
.0247
.0246
.0244
.0247
.0250
.0249
.0250
.0245
.0225
.0232
.0236
.0243
.0247
.0243
.0249
.0248
.0251
.0250
.0248
.0245
.0247
.0247
.0247
.0250
.0248
.0232
.0239
.0243
.0250
.0249
2.02
2.01
2.02
2.02
2.07
2.06
2.05
2.04
2.03
2.03
2.02
2.02
2.02
2.02
2.01
2.01
2.01
2.01
2.07
2.06
2.05
2.04
2.03
2.03
2.02
2.02
2.01
2.01
2.01
2.01
2.01
2.01
2.01
2.00
2.01
2.05
2.04
2.03
2.02
2.02
2.00
1.99
2.00
2.00
2.05
2.04
2.03
2.02
2.01
2.01
2.00
2.00
2.00
2.00
1.99
1.99
2.00
2.00
2.05
2.04
2.03
2.03
2.02
2.02
2.01
2.01
2.00
2.00
2.00
2.00
2.00
2.00
2.00
1.99
2.00
2.04
2.03
2.02
2.01
2.01
Table A.1  continued
a = .05
a1 I 2 C zu z
n
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
.0432
.0431
.0435
.0438
.0436
.0383
.0402
.0388
.0403
.0394
.0395
.0402
.0406
.0409
.0413
.0415
.0422
.0423
.0427
.0427
.0427
.0497
.0497
.0498
.0499
.0500
.0486
.0486
.0485
.0499
.0500
.0499
.0500
.0500
.0501
.0500
.0500
.0501
.0498
.0500
.0502
.0501
1.68
1.68
1.68
1.68
1.68
1.72
1.71
1.71
1.70
1.70
1.70
1.69
1.69
1.69
1.69
1.69
1.68
1.68
1.68
1.68
1.68
1.68
1.68
1.68
1.68
1.68
1.71
1.70
1.70
1.70
1.70
1.70
1.69
1.69
1.69
1.69
1.69
1.68
1.68
1.68
1.68
1.68
S= .025
a a2 a zu z
.0187
.0187
.0187
.0188
.0188
.0189
.0189
.0189
.0190
.0190
.0191
.0191
.0185
.0191
.0192
.0191
.0192
.0194
.0194
.0195
.0195
.0193
.0197
.0198
.0199
.0204
.0205
.0208
.0208
.0207
.0210
.0208
.0210
.0185
.0191
.0203
.0191
.0192
.0194
.0197
.0199
.0203
.0243
.0251
.0242
.0241
.0246
.0244
.0246
.0244
.0245
.0244
.0245
.0249
.0236
.0243
.0251
.0246
.0247
.0242
.0246
.0246
.0252
2.02
2.01
2.02
2.02
2.01
2.01
2.01
2.01
2.01
2.01
2.01
2.01
2.04
2.03
2.02
2.02
2.02
2.02
2.01
2.01
2.00
2.01
2.00
2.01
2.01
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.03
2.02
2.01
2.01
2.01
2.01
2.00
2.00
1.99
.0427
.0427
.0427
.0427
.0428
.0383
.0402
.0388
.0403
.0394
.0395
.0402
.0406
.0409
.0413
.0415
.0422
.0423
.0427
.0427
.0427
Table A.2 Minimum Sample Sizes to Achieve 80% Power
and ns.05 for Onesided Ztests for
Comparing Two Independent Proportions.
l P2 n zu z p 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
Table A.2  continued
*
P P2 n Zu Z
1.68
1.70
1.71
1.74
1.79
1.85
.35 .50
.55
.60
.65
136 1.71
79 1.70
51 1.72
37 1.74
.40 .55 144 1.69 1.69
.60 79 1.70 1.69
.0500 .8021
.0489 .8057
18
1.68
1.69
1.69
1.71
1.74
1.78
1.70
1.69
1.70
1.71
.0498
.0488
.0494
.0467
.0450
.0438
.0486
.0489
.0487
.0467
.8015
.8021
.8029
.8145
.8064
.8075
.8016
.8070
.8085
.8111
Table A.3 Comparision of Sample Sizes to Achieve 80%
Power and ca.05 for Onesided Tests for
Comparing Two Independent Proportions.
S P2 n nc nr n 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 .35 121 134 122 109 108 111
.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
Table A.3  continued
Pl P2 ne nc nr np nas n
132
77
50
37
27
20
136
79
51
37
.30 .45
.50
.55
.60
.65
.70
.35 .50
.55
.60
.65
.40 .55
.60
144 162 149 136 136
85 96 86 77 77
Table A.4 Critical Values and Sizes of Ztest.for
Comparing Two Correlated Proportions.
a = .05
a* *c
a1 a2 a z
.0114 .0265
.0322 .0345 .0395
.0205 .0205 .0373
.0316 .0328 .0430
.0303 .0303 .0371
.0187 .0187 .0370
.0312 .0349 .0454
.0261 .0261 .0413
.0313 .0353 .0446
.0307 .0333 .0399
.0220 .0220 .0395
.0310 .0356 .0459
.0278 .0278 .0425
.0309 .0357 .0427
.0307 .0332 .0412
.0305 .0357 .0494
.0301 .0357 .0434
.0275 .0275 .0413
.0307 .0349 .0407
.0302 .0315 .0407
.0303 .0357 .0450
.0303 .0357 .0407
.0262 .0262 .0407
.0301 .0357 .0458
.0299 .0299 .0435
.0298 .0356 .0429
.0304 .0327 .0425
.0302 .0356 .0449
.0299 .0355 .0419
.0400 .0411 .0501
.0399 .0400 .0500
.0305 .0305 .0500
.0303 .0354 .0433
.0298 .0323 .0428
.0375 .0375 .0500
.0352 .0352 .0501
.0395 .0395 .0501
.0382 .0382 .0501
.0290 .0300 .0427
.0396 .0408 .0501
1.90
1.74
1.74
1.74
1.74
1.81
1.74
1.74
1.74
1.74
1.79
1.74
1.74
1.74
1.74
1.74
1.74
1.74
1.74
1.74
1.74
1.74
1.77
1.74
1.74
1.74
1.74
1.74
1.74
1.68
1.68
1.72
1.74
1.74
1.68
1.68
1.68
1.68
1.74
1.70
a = .025
"1 "2 a Zc
.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
2.01
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
a= .05
N 2 *
N 1 a2 a Zc
.0325 .0325 .0500
.0352 .0352 .0500
.0333 .0333 .0500
.0365 .0365 .0500
.0359 .0359 .0501
.0387 .0387 .0501
.0299 .0360 .0453
.0309 .0309 .0501
.0390 .0407 .0501
.0311 .0311 .0501
.0347 .0347 .0501
.0329 .0329 .0501
.0294 .0359 .0445
.0298 .0359 .0427
.0298 .0358 .0470
.0294 .0358 .0453
.0297 .0356 .0442
.0382 .0395 .0498
.0378 .0378 .0501
.0334 .0334 .0499
.0381 .0398 .0499
.0346 .0346 .0495
.0378 .0393 .0501
.0377 .0401 .0500
.0374 .0374 .0501
.0376 .0384 .0500
.0375 .0391 .0501
.0374 .0395 .0501
.0373 .0396 .0501
.0371 .0393 .0500
.0370 .0397 .0499
.0369 .0388 .0501
.0369 .0388 .0500
.0368 .0389 .0501
.0367 .0390 .0500
.0365 .0390 .0501
.0364 .0391 .0500
.0362 .0380 .0500
.0361 .0381 .0500
.0360 .0381 .0500
1.70
1.70
1.70
1.70
1.70
1.70
1.74
1.73
1.68
1.73
1.70
1.70
1.74
1.74
1.74
1.74
1.74
1.70
1.70
1.70
1.68
1.70
1.68
1.68
1.68
1.68
1.68
1.67
1.67
1.69
1.69
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
a = .025
a *
ai a2 a Zc
.0167 .0167 .0248
.0183 .0183 .0247
.0174 .0174 .0247
.0194 .0194 .0246
.0200 .0200 .0246
.0120 .0151 .0230
.0123 .0163 .0227
.0187 .0187 .0248
.0177 .0177 .0248
.0168 .0168 .0247
.0186 .0186 .0247
.0190 .0190 .0246
.0197 .0197 .0250
.0200 .0200 .0246
.0125 .0185 .0234
.0116 .0189 .0231
.0119 .0186 .0230
.0201 .0201 .0248
.0200 .0200 .0247
.0200 .0200 .0247
.0199 .0200 .0247
.0119 .0180 .0234
.0199 .0199 .0250
.0194 .0194 .0247
.0198 .0199 .0247
.0191 .0191 .0247
.0197 .0198 .0247
.0181 .0181 .0247
.0186 .0186 .0247
.0196 .0197 .0251
.0112 .0185 .0234
.0195 .0195 .0250
.0194 .0195 .0247
.0194 .0195 .0247
.0192 .0192 .0247
.0193 .0194 .0247
.0192 .0193 .0247
.0192 .0192 .0247
.0191 .0192 .0251
.0190 .0192 .0251
1.99
1.99
1.99
1.99
1.99
2.03
2.03
1.99
1.99
1.99
1.99
1.99
1.99
1.99
2.01
2.01
2.01
1.99
1.99
1.99
1.99
2.02
1.99
1.99
1.99
1.99
1.99
1.99
1.99
1.98
2.02
1.99
1.99
1.99
1.99
1.99
1.99
1.99
1.98
1.98
Table A.4  continued
a= .05
a* *
a1 a2 a z
.0359 .0382
.0358 .0382
.0357 .0383
.0356 .0383
.0354 .0384
.0353 .0385
.0351 .0371
.0350 .0372
.0348 .0372
.0347 .0372
.0346 .0373
.0344 .0371
.0343 .0371
.0342 .0410
.0341 .0396
.0341 .0383
.0340 .0394
.0339 .0396
.0338 .0409
.0338 .0410
.0337 .0411
.0336 .0412
.0395 .0413
.0395 .0413
.0395 .0399
.0382 .0382
.0394 .0404
.0394 .0409
.0380 .0380
.0388 .0388
.0390 .0390
.0392 .0403
.0391 .0406
.0391 .0406
.0391 .0407
.0390 .0407
.0390 .0408
.0390 .0409
.0389 .0409
.0389 .0410
.0500
.0499
.0500
.0500
.0499
.0499
.0499
.0500
.0499
.0500
.0500
.0500
.0500
.0499
.0500
.0500
.0500
.0500
.0499
.0499
.0499
.0499
.0499
.0499
.0499
.0499.
.0499
.0499
.0499
.0499
.0499
.0499
.0499
.0500
.0499
.0499
.0499
.0498
.0499
.0498
1.69
1.69
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.70
1.70
1.68
1.68
1.68
1.68
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.69
1.69
1.68
1.68
1.68
1.68
1.68
1.68
1.68
1.67
1.67
1.67
1.67
1.69
1.69
1.68
a = .025
S* *
a1 a.2 a Z
.0190 .0191 .0248
.0189 .0191 .0247
.0189 .0190 .0247
.0189 .0189 .0247
.0188 .0190 .0246
.0188 .0189 .0247
.0188 .0188 .0251
.0187 .0187 .0250
.0187 .0187 .0249
.0187 .0187 .0248
.0117 .0194 .0236
.0118 .0190 .0236
.0185 .0185 .0248
.0199 .0199 .0248
.0192 .0192 .0247
.0198 .0198 .0247
.0199 .0199 .0247
.0199 .0199 .0247
.0198 .0198 .0246
.0198 .0198 .0250
.0198 .0198 .0249
.0203 .0203 .0246
.0190 .0190 .0246
.0203 .0203 .0251
.0202 .0202 .0248
.0192 .0192 .0248
.0197 .0197 .0251
.0203 .0203 .0247
.0205 .0205 .0247
.0205 .0205 .0251
.0205 .0205 .0251
.0205 .0205 .0251
.0204 .0205 .0245
.0204 .0204 .0245
.0182 .0182 .0245
.0204 .0204 .0250
.0204 .0204 .0250
.0198 .0198 .0251
.0199 .0199 .0251
.0203 .0203 .0251
1.99
2.00
2.00
2.00
2.00
2.00
1.99
1.98
1.98
1.98
2.01
2.01
1.99
1.99
1.99
1.99
1.99
1.99
1.99
1.99
1.99
2.00
2.00
1.99
1.99
1.99
1.99
1.99
1.99
1.98
1.98
1.99
2.00
2.00
2.00
1.98
1.98
1.98
1.98
1.98
Table A.4  continued
a= .05
au a2 a Zc
.0388
.0388
.0387
.0387
.0386
.0386
.0386
.0385
.0385
.0385
.0384
.0383
.0383
.0382
.0382
.0381
.0381
.0380
.0380
.0379
.0379
.0379
.0379
.0378
.0377
.0376
.0376
.0375
.0374
.0374
.0373
.0373
.0372
.0372
.0372
.0371
.0371
.0370
.0370
.0369
.0403
.0403
.0404
.0389
.0397
.0397
.0398
.0398
.0399
.0399
.0400
.0400
.0401
.0401
.0402
.0402
.0403
.0403
.0404
.0395
.0395
.0395
.0396
.0396
.0387
.0387
.0388
.0388
.0388
.0389
.0389
.0389
.0390
.0390
.0390
.0390
.0391
.0391
.0391
.0392
.0499
.0498
.0498
.0499
.0499
.0500
.0499
.0499
.0499
.0499
.0499
.0498
.0497
.0497
.0499
.0499
.0499
.0498
.0498
.0498
.0498
.0498
.0499
.0499
.0498
.0498.
.0499
.0497
.0496
.0500
.0498
.0498
.0498
.0498
.0498
.0498
.0498
.0498
.0498
.0498
1.68
1.68
1.68
1.68
1.68
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.68
1.68
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.68
1.68
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
a= .025
2
al a2 a ac
.0203
.0203
.0202
.0202
.0202
.0202
.0202
.0202
.0201
.0201
.0199
.0201
.0200
.0200
.0094
.0095
.0200
.0199
.0199
.0199
.0199
.0198
.0198
.0198
.0198
.0198
.0197
.0197
.0197
.0197
.0196
.0196
.0196
.0196
.0195
.0195
.0195
.0195
.0194
.0194
.0203
.0203
.0203
.0203
.0202
.0202
.0202
.0202
.0202
.0201
.0199
.0201
.0201
.0201
.0197
.0197
.0200
.0200
.0200
.0199
.0199
.0199
.0199
.0199
.0198
.0198
.0197
.0197
.0197
.0197
.0197
.0197
.0197
.0196
.0196
.0196
.0196
.0196
.0195
.0195
.0251
.0251
.0251
.0245
.0245
.0245
.0250
.0250
.0250
.0250
.0250
.0250
.0251
.0251
.0241
.0237
.0249
.0246
.0246
.0246
.0246
.0250
.0250
.0250
.0250
.0250
.0250
.0246
.0246
.0246
.0249
.0247
.0246
.0246
.0246
.0246
.0246
.0246
.0246
.0247
1.98
1.98
1.98
2.00
2.00
2.00
1.98
1.98
1.98
1.98
1.98
1.98
1.98
1.98
2.01
2.01
1.99
1.99
1.99
1.99
1.99
1.98
1.98
1.98
1.98
1.98
1.98
2.00
2.00
2.00
1.99
1.99
1.99
1.99
1.99
1.99
1.99
1.99
1.99
1.99
Table A.4  continued
a= .05
al 2 a Zc
.0368
.0367
.0367
.0366
.0365
.0365
.0364
.0363
.0363
.0362
.0362
.0361
.0361
.0360
.0360
.0360
.0359
.0359
.0359
.0358
.0358
.0358
.0357
.0357
.0357
.0356
.0356
.0355
.0355
.0354
.0354
.0392
.0393
.0393
.0393
.0382
.0382
.0383
.0383
.0383
.0383
.0383
.0384
.0372
.0372
.0372
.0372
.0373
.0373
.0373
.0373
.0373
.0373
.0374
.0374
.0374
.0374
.0374
.0374
.0374
.0374
.0375
.0498
.0498
.0498
.0498
.0498
.0498
.0498
.0498
.0498
.0498
.0498
.0498
.0498
.0498
.0498
.0498
.0498
.0499
.0501
.0497
.0498
.0498
.0500
.0498
.0498
.0498
.0498
.0498
.0498
.0498
.0498
1.67
1.67
1.68
1.68
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.68
1.68
1.68
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
a= .025
a1 "2 a z
.0194
.0194
.0193
.0193
.0193
.0193
.0192
.0192
.0192
.0191
.0191
.0191
.0191
.0191
.0190
.0190
.0190
.0190
.0190
.0189
.0189
.0189
.0189
.0189
.0189
.0188
.0188
.0188
.0188
.0188
.0187
.0195
.0195
.0195
.0195
.0194
.0193
.0193
.0193
.0193
.0193
.0193
.0192
.0191
.0191
.0190
.0190
.0190
.0190
.0190
.0189
.0189
.0189
.0189
.0189
.0189
.0188
.0188
.0188
.0188
.0188
.0187
.0251
.0245
.0245
.0251
.0249
.0247
.0246
.0246
.0246
.0246
.0251
.0251
.0247
.0250
.0245
.0250
.0251
.0249
.0247
.0246
.0246
.0246
.0251
.0250
.0250
.0250
.0251
.0250
.0251
.0247
.0247
1.98
2.00
2.00
1.99
1.i99
1.99
1.99
1.99
1.99
1.99
1.98
1.98
1.99
1.99
2.00
1.99
1.99
1.99
1.99
1.99
1.99
1.99
1.98
1.98
1.98
1.98
1.98
1.98
1.98
1.99
1.99
Table A.5 Minimum Sample Sizes to Achieve 80% Power
and a 5.05: for Onesided Ztest for
Comparing Two Correlated Proportions.
AN z a 1a
.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
A N zc 1a
.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
Table A.6 Comparision of Sample Sizes to Achieve 80%
Power and a.05 for Onesided 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 a N N N N
a1 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 7(p), the null power
function, are given for the two problems considered here.
The dotted line represents the nominal significance level
on which ir(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.
80
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APPENDIX C
COMPUTER PROGRAMS
The listing of the FORTRAN computer programs used
to compute the size of the Ztest for each of the two
problems considered is given in this Appendix. For the
case of two independent proportions, Appendix C.1 gives
the exact pvalue for any n=10(1)150 and any value of the
Ztest statistic with unpooled variance estimator. In
Appendix C.2, the case of two correlated proportions, the
exact pvalue for N=10(1)200 and any value of the Ztest
statistic can be obtained.
APPENDIX C.1
C
C
C
C
C
C
REAL AB C TBY(151)
INTEGER ~ Y UU,BOUNDY(151 ,PHI(151)
DOUBLE PRSCISON LFAC(151)1,P,Q LPLQ PX(151),FX(151),
VALUEE DERBND,DERU,DERL,LCOM P.AT1 PHAT2
*PMAX, PMAX2,PMIN1 PBIN2,PL, PPL1 .L2,PUi,PU2,SUP,INF,
*C1,C2,T T3, T4T5T6 D1 D2,D3,D4,D5,D6,LPX,LFX,
*MAXP MAXIMlMINIIM5, MNI0O
DOUBLE PRE VISION DLOG,DLOG10,DEXP,DMAX1,DMINI,DABS
WRITE(6,40)
40 FORMAT( ///,
C
C
'THIS PROGRAM COMPUTES THE PVALUE OF THE ZTEST'/
'WITH UNPOOLED VARIANCE ESTIMATOR, TO COMPARE'/
'TWO INDEPENDENT PROPORTIONS FOR'/
*' N = SAMPLE SIZE IN EACH GROUP,'/
Z = NORMAL STATISTIC WITH UNP60LED VARIANCE.'/
'ENTER N, Z, IN FREE FORMAT'/)
C
C
99 READ(9,*)N,Z
MAXP=0.0
N1=N+1
LFAC(1)=0.DO
DO 1 J=2 N1
1 LFAC(J)=LFAC(J1)+DLOG(DBLE(FLOAT(J1)))
C
C
C SIMPLIFY THE COMPUTATIONS AND FORM THE
C BOUNDARY OF THE CRITICAL REGION
C
C
X=0
A=1+ Z**2)/N
2 B=2*X+2**2
C=(X**2 *A(Z**2) *
TBY(X+1)=(B+SQRT (B**2)(4*A*C)))/(2*A)
IF (TBY(X+1)N) 3,4 i
3 X=X+1
GO TO 2
4 U=X1
UU=U+1
DO 5 X=1 ,U
BOUNDY (X =INT TBY(X) + 1)
PHI(X)=INT(TBY(X))
L=X1
5 CONTINUE
MAXIM=O.DO
MINIM5=1.DO
MINI10=1.DO
19=0
15=0
P=.005DO
6 IFPP.GT.0.49DO) GO TO 76
72 IF L.RQ.1) P=P+.01DO
IF L.EQ.1) PP=P+.005DO
Q= P
LP=DLOG(P)
LQ=DLOG(Q)
DO 7 J= ,N1
X=J1
T=LFAC(N+1)LFAC(J)LACC(N+1J+1)+X*LPt(NX)*LQ
IF (T.LT.180.0) PX(J)=0.DO
I? (T.GE.180.0) PX(J =DEXP(T)
IF (X) 8,9 8
9 FX(J=PX( J
GO TO 10
8 FX(J)=FX(J1)+PX(J)
10 CONTINUE
7 CONTINUE
PVALUE=0
DERU=0
DERL=0
DO 30 K=1,UU
X=K1
IF (?X(K).LE.0.DO) GO TO 11
LPX=DLOG10(PXK)
IF ((1FX(PHI K +1) .LE.O.DO) GO TO 11
LFX=DLOG10(.1 (PHK)+1))
IF ((LPX+LFX).LT.7o.0) GO TO 11
PVALUE=PVALUE+PX(K)* (1FI(PHI(K)+1))
11 Y=BOUNDY(K)
LCOM=LFAC(N+1)LFAC(K)LFAC(N+1K+1)
+LFAC(N+l)LAC(Y)LFAC(N+1Y1+1)
C
C
C LOCAL BOUND FOR THE DERIVATIVE OF
C THE NULL POWER FUNCTION
C
C
C1=FLOAT X+Y1)
C2=FLOAT 2*NXY)
PHAT1=C1 /C1+C2)
PHAT2=1 PAT1
PL=P(.005D0/ 2** L1 I
PU=P+ .005DO C2**( L1))
IF (P AT1.GT.PU GO TO 12
IF ((PHATI.LE.PU).AND.(PHAT1.GE.PL)) GO TO 13
IF (HAT.LT.PL) GO TO 14
12 PMAX1=PU
PHIN1=PL
GO TO 15
13 PMAX1=PHATI
IF (PL.NE.0.DO) GO TO 21
PHIN1=PL
GO TO 15
21 PL1=C1*DLOG PL)+C2*DLOG 1PL)
PU1=C1*DLOG PU +C2*DLOG 1PU)
IF PL1.LT. PUl PINI=PL
IF (PL1.GE.PU1) PBIN=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 CPHAT2.LT.PL) GO TO 18
16 PMAX2=PU
PMIN2=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 1PL)
PU2=C2*DLOG(PU +C1*DLOG 1PU)
IF (PL2.LT.PU2 PMIN2=PL
IF PL2.GT.PU2) PMIN2=PU
GO TO 19
18 PMAX2=PL
PHIN2=PU
19 CONTINUE
T3=LCOM+C1*DLOG(PIAX1 +C2*DLOG(1PMAX1)
IF (T3.LT.180.d0 D3=0.D0
IF (T3.GE.180.0[ D3=DEXP(T3)
IF (?IN2.EQ.O.DO) GO TO 23