for i, j

= 0, 1.
Consider a random sample of size n matched-pairs data from this
distribution. This type of data is frequently displayed in a 2 x 2 table as follows:

Total where

for i, j = 0,1 is the ijth observed cell count. The distribution of the vector of

Li,j Pij

random cell counts (Xoo , X OI , X lO , Xu) is multinomial with probability mass function will be denoted by

The multinomial

Li,j Xij
Consider the problem of testing

H o : PI

2: P2 versus HI: PI < P2

where PI

= P(YI = 0) = POO + POI

H o : POI

and P2

= P(Y2 = 0) = POO + PlO,

or equivalently

2: PlO versus HI:

< PlO'
A challenge in the construction of tests for this hypothesis is the presence of two nuisance parameters under H o, that is, the discordant cell probabilities POI and PlO' The discordant pair parameter space is II = {(POI,PlO): 0 ~ POI ~ 1, 0 ~ PlO ~ 1, and POI
Exact tests for (1) will be considered. By showing that certain probabilities are maximized on the boundary of H o and HI, we will propose exact unconditional tests using two different concepts of a p-value, the standard and the confidence interval p-values. 2 ,1 The p-value, size, and power computations use the exact multinomial distributions of the data. Because of the discrete nature of the data, the exact tests do not have sizes exactly equal to the specified a.
Rather they are level-a tests with exact sizes less than or equal to a. In this problem exact size a tests must be randomized tests, but randomized tests are seldom used in practice. We consider only nonrandomized tests. Hsueh, Liu, and Chen 6 considered exact unconditional tests for (1). Their RMLE test corresponds to the test we define in (8). But, they provided no specific information for the b = 0 case (their notation) which corresponds to (1). They did not provide size and power comparisons for the tests we will discuss. The data in these types of problems are usually summarized as (XOl , X lO , n-XOI -XlO ). That is, the cell counts X oo and Xu are summed. The heuristic idea is that the individual values X oo and Xu do not give information about the relative sizes of POI and PlO. Tests could be defined that depend on the individual values X oo and Xu, but we do not know of any such tests that have been proposed. So we too will summarize the data in this way and simply denote the data as (XOl ' X lO ), because the third count is a function of the first two. The trinomial pmf of (XOl , X lO ) will be denoted by

( mXOl,XlO,n,pOl,plO ) -

XlO n ~

),POlPlO

-POI-PlO

)n-X01-xlO

2: 0 and
n. The parameter space is given in (2).

Example

Here is a typical biomedical example of matched-pairs data for which testing of (1) might be of interest. A regulatory agency sometimes checks the analyses of a medical laboratory. The laboratory knows when it is being checked. An experimenter thinks that the laboratory might be more careful when it knows its results are being scrutinized. To confirm this the experimenter sends two samples from the same person to the laboratory for an antibody analysis. In one case the laboratory is told the sample is part of a check (OPEN case); in the other case the
laboratory is not told the sample is part of a check (CLOSED case). The experimenter also has a "gold standard" for each sample and thus knows if the laboratory analysis is correct or incorrect. n such pairs of samples are sent, resulting in matched-pairs data of this form: OPEN CLOSED correct incorrect Total correct

incorrect

In this situation the experimenter might want to test the one-sided hypothesis (1), because

HI : POI

says that the probability that the laboratory analysis is correct is smaller in
the CLOSED situation than in the OPEN situation. Other biomedical examples of matched-pairs data often involve comparison of standard and innovative procedures or drugs. Hsueh, Liu, and Chen 6 give an example of this type comparing diagnostic procedures for liver lesions.
Monotonicity of a Joint Distribution
To maximize certain probabilities in the calculation of p-values, we will use a monotonicity property described in this section.
Definition 1 In two dimensions, a set R is a Barnard convex set if (x, y) E R, x' ::; x, and
y' ~ y imply (x', y') E R.
A Barnard convex set R contains all those points that lie above and to the left of a point
(x, y) if (x, y) E R. Note that a Barnard convex set is not necessarily convex in the sense of

the usual mathematical definition of a convex set. The word "convex" is adopted because the shape property of the set is vaguely related to a convex set as described by Barnard. 8
Sidik and Berger 7 proved the following theorem about the monotonicity of the joint distribution of random variables X and Y over a Barnard convex set.

Theorem 1 Let

Pfh,(h (x,
y) be a joint probability model for random variables X and Y

indexed by parameters on

Suppose that the marginal distribution of X depends only
and the marginal distribution of Y depends only on
If for each y the family of
conditional distributions of X given Y
is stochastically increasing in

and for each x

the family of conditional distributions of Y given X = x is stochastically increasing in then, for any Barnard convex set R, when

01 and

This theorem presents sufficient conditions for achieving the distributional monotonicity (4) over a Barnard convex set. The monotonicity may also be seen as a type of multivariate stochastic order in parameters over a two-dimensional set. For discussion of multivariate stochastic orders, see Shaked and Shanthikumar. 9
Four Exact Unconditional Tests
In this section we define four exact unconditional tests of (1). The tests are defined by their p-values. The first two tests use the standard definition of a p-value, namely,
p(x) = sup Po(T(X) ~ T(x)),
where T(x) is the observed value of the test statistic. By Theorem 8.3.27 in Casella and Bergerl l , this defines a valid p-value in that the test that rejects H o if and only if p( x) ~ a is a level-a test of H o. The two tests we define use two different statistics, McNemar's Z statistic and the likelihood ratio (LR) statistic. The second two tests are defined by confidence interval p-values, namely,

= supPo(T(X)

T(x)) + 13,
where C = C(x) is a 100(1 - /3)% confidence set for () under Ho. By the Lemma in Berger and Boos 1 , this also defines a valid p-value. Again, the two tests use the Z and LR statistics. The suprema in these definitions are over two dimensional sets for our matched pairs problem. But, we show in each case that the calculation can be reduced to a one dimensional maximization, thereby greatly simplifying the numerical burden. The tests are exact unconditional tests because the exact trinomial distribution (3) of (X01 , X lO ) is used in the calculation of the p-values, and, hence, the size of the tests is guaranteed to be less than or equal to the nominal value a. Berger and Boos 1 noted that the standard p-value (5) may be very conservative if the supremum occurs at a point far from the true parameter value. To address this potential problem for matched pairs data, Hsueh, Liu, and Chen6 mentioned the possibility of using the method proposed by Storer and Kim 12 and Kang and Chen 13, namely, replace the supremum in (5) by the single probability calculation at the maximum likelihood estimate of the parameter under H o. Unfortunately, as these authors noted, this method does not necessarily produce a valid p-value. The confidence interval p-value in (6) addresses the conservativeness of the standard p-value by considering only parameter values supported by the data, the values in the confidence set C(x). But, it does this in such a way as to yield a valid p-value.

Standard p-value tests

The p-value using McNemar's test statistic. Consider the signed square root of McNemar's test statistic 3

Z(XOl' XlO)

= viXOI + XlO.
Because large values of Z give evidence against H o, the p-value for testing (1) using Z(XOl' XlO) by the standard definition of p-value is

pz (XOl , XlO)

{(POI ,PlO): POI :::::PlO}

PpOl,PIO(Z(X01 , X lO )

2': Z(XOl, XlO))

{(POl,PlO): POl~PlO}

(u,v)ERz(XOl,XlO)

m(u,V;n,pOI,plO)

where RZ(XOI, XlO) = {(u, v) : Z(u, v) ~ Z(XOl, XlO)}.
The supremum in this p-value is
typically calculated numerically, and this may be difficult due to the maximization over a two-dimensional nuisance parameter space. By finding the partial derivatives of Z(XOI, XlO) with respect to

and XlO, Sidik lO

showed that RZ(XOI, XlO) is a Barnard convex set. (The partial derivative with respect to is positive and the partial derivative with respect to
is negative.) In a 2 x 2 matched-pairs is binomial, i.e.,
design, the conditional distribution of X OI given X lO

' b( XOI n ,

POI XlO
(n - XlO)! (POI) XOI XOl!((n - XlO) - XOl)! 1 - PlO

POI) (n-XlO)-XOl

Similarly, the distribution of X lO given X OI =
is binomial, b(XlO; n - XOI,PlO/(l - POI))'
Using the result of Casella and Berger (Exercises 8.28 and 8.29)1l, it can be concluded that the family of conditional distributions of X OI given X lO =
is stochastically increasing
in POI for any fixed PlO, and the family of conditional distributions of X lO given X OI = is stochastically increasing in
for any fixed POI' Therefore, the joint distribution of X OI
and X lO satisfies (4) of Theorem 1. For any P such that POI ~ P ~

and (p,p) E II (e.g.,

P = PIO)'

and, hence, sup

PpOl,PlO(Z(XOI , X lO ) ~ Z(XOl, XlO)) = sup Pp,p(Z(XOI ' X lO ) ~ Z(XOI, XlO)).

O::;p::;~

Thus, the standard p-value for testing (1) using McNemar's test statistic (given in (7)) can be simplified to sup Pp,p(Z(XOl ' X lO ) ~ Z(XOl, XlO)) sup

o::;p::;~

o::;p::;~ (u,v)ERz(XOl,XlO)

m(u,v;n,p,p).

The p-value

PZ(XOl' XIO)

can be computed as the maximum probability over one nuisance
parameter rather than two under H o. The p-value in (8) indicates that the supremum over the null parameter space occurs on the boundary of H o and HI, that is

= PIO = p.

The p-value using the likelihood ratio test (LRT) statistic. In a 2 x 2 matched-pairs sample, the multinomial log likelihood function is

where ~i,j Xij =

Following Robertson, Wright, and Dykstra l4 , the order restricted max-
imum likelihood estimators (MLEs) of the multinomial parameters under the constraint of

H o are IO

= 0, 1,

+XlO.'

if XOI ~ , l, f XOI < 1
Similarly, the MLEs under the constraint of HI are lO MLEs

= {~01

~O ~O~~XlO;

i = 0, 1, if XOI ~

n ' 2, J

- 0, 1,

if XOI <
Consider the following form of the LRT statistic:

'( ) /\ XOO,XOl,XIO,Xll

= sUPH! m (POO,POl,PI0,Pll;XOO,XOl,XIO,Xll ).
sUPHo m(POO,POl,PIO,Pll;XOO,XOI,XIO,Xll)
The log of the LRT statistic can be expressed as IO

_ { L( XOl, XIO ) -

log (Xo~~~lO) x

(XO~~~IO)

if XOI if XOI
+XlO) +XIO 01 log (XQI2XOl + 10 log (XQI2XlO )

< XI0.

Note, although we started with the full data likelihood, this LRT statistic depends only on X 01 and XIOo This gives some justification to the summarization of the data that is usually made. To define an exact unconditional test using L(XOl' XIO), because small values of L(XOl' XIO) support HI, consider the set

RdXOl,XIO)

= {(u,v): L(u,v)::; L(XOl,XIO)}.
The set contains all the data points whose test statistic is at most as large as the observed test statistic. By finding the partial derivatives with respect to

and XlO, it can be shown

that RdxOl, XlO) is a Barnard convex set. lO Therefore, arguing as we did for the Z statistic, an exact unconditional p-value for testing (1) using the LRT statistic is

PdXOl' XlO)

o::;p::;t
sup Pp,p(L(XOl , X lO ) ~ L(XOl' XlO))

(u,v)ERL(xOI,XlO)

m(u, v; n,p,p).
Confidence interval p-value tests
In this section, we define two more exact unconditional tests, now using confidence interval
p-values as defined in (6) and again using the statistics Z and L.
The confidence interval p-value using McNemar's test statistic. Suppose C,B(XOl' XlO) is
a 100(1- (3)% confidence set for the parameters (POl,PlO) calculated from the observed data under H o. Then, the confidence set p-value using McNemar's test statistic is

supremum in (13) is taking over a one-dimensional interval 1/3(XOl, XlO) rather than over a two-dimensional set C/3 (XOl , XlO).
The confidence interval p-value using log LRT statistic.
Similarly, we can derive a
confidence interval p-value using L(XOI' XIO) that requires maximization only on the boundary of H o and HI' An exact unconditional confidence interval p-value for testing (1) using the log LRT statistic is sup

pEI{3(XOl,XlO)

Pp,p(L(X01 ' X lO )

L(XOl, XlO)) + f3

SUP PEI{3(XOl,XlO) (u,V)ERL(XOl,XlO)
where 1/3(XOI, XlO) is the Clopper and Pearson interval calculated from (10) and RdxOI, XlO) is the same as in PdXOI, XlO). The confidence interval p-values
depend on the error probability f3. For
testing (1), Sidik lO tried several values of f3 with different values of a and concluded that
= 0.0005 yielded good level-a = 0.05 tests. This is the value of

f3 used in the remainder

of this paper. All four ofthe exact unconditional p-values defined in this section are valid p-values. The tests that reject H o if and only if the respective p-values are less than or equal to a specified a are level-a tests for (1). The exact unconditional p-values must be calculated numerically. But, all four p-values have been expressed in terms of a one-dimensional maximization of a polynomial in p which is not difficult.
Exact Size and Power Comparison
In this section, we compare the exact sizes and powers of six tests of (1). We consider the
four exact unconditional tests defined in the previous section. We denote these tests by Z,
Ze, L, and L e corresponding to the p-values Pz, PZc ' PL, and PLc' respectively. We also
consider two more common tests, McNemar's asymptotic test, which we denote by M, and the exact conditional binomial test which is defined by conditioning on the total number of discordant cell counts and which we denote by CB. For testing (1) the p-value of CB is
+ X lO = t used to calculate PCB conditional distribution assuming POI = PlO. The p-value of M is
The distribution of X OI given X 01

is binomial(t,1/2), the

where Z* has a standard normal distribution. CB is both conditionally and unconditionally a level-a test of (1), but M is only approximately a level-a test. Exact sizes and powers of these six tests are computed using the trinomial distribution (3).
Size and power computations
Consider first the exact sizes of the tests Z, L, Ze, and L e. For a given value of a, the level-a rejection region of Z is

{(u,v): Pz(u,v)

where pz is defined in (8). Define Z' = min{ Z(XOl' XlO) : PZ(XOl' XlO) ~ a}. Then R

{(u, v) : Z (u, v)

Z'}. By definition, the exact size of Z is

size(R~) =

{(POl ,PlO): POl2:PlO}

PpOl,PlO((X01 ,

X lO ) E R~).
By the same argument as in Section 4.1, R is a Barnard convex set. Therefore, by Theorem 1 the exact size of Z is

size(R~)

Pp,p((X01,X lO ) E R~)

Ospst (u,v)ER

Similarly, the size of L is calculated by replacing R in (14) with the level-a rejection region
R L = {(u, v) : PL(U, v) ~ a}.
For the confidence interval tests Ze and L e , it is not obvious that the level-a rejection regions of these tests are Barnard convex sets. For a given n and a we can examine the rejection regions of Ze and L e. If they are Barnard convex sets, then the tests' sizes can be calculated by following (14). For everyone of the sample sizes in our comparisons and
a =.05, the rejection regions of Ze and L e are Barnard convex sets and the exact sizes were
computed as in (14). Consider computing the exact sizes of GB and M. For GB, (X~l' x~o) E ReB if X~l ~ and x~o ~

XIO XOl

(XOI,XlO)

E ReB' where ReB

= {(u, v)

: P(U ~ ulU

+ v = u + v)

This is because

Hence, the level-a rejection region of CB is a Barnard convex set. The rejection region

R z = {(u, v) : Z(u, v)

zo,} of the asymptotic test M is also Barnard convex set by the
same argument as in Section 4.1. (za is the 100(1 - a)% percentile from a standard normal distribution.) Therefore, the exact sizes of both GB and Z are computed similarly to (14). Finally, the exact powers of these tests are calculated based on the trinomial distribution of the data. For example, the power of Z for (POl,PlO) E HI is

power(pOI,PlO;R~)

(U,V)ER

m(u,v;n,POl,PlO).

Size and power comparisons
The sizes and powers of the six tests, Z, L, Ze, L e , GB, and M, were computed as described in Section 5.1. For Ze and L e , f3 interval.
= 0.0005 was used as the error probability for the confidence In this study all comparisons were carried out using a = 0.05. The first five tests
are level-a tests. M is asymptotically level-a. In Table 1 we list the exact sizes of the tests for 15 sample sizes, n = 10(5)40(10)100(50)200. First, consider the four exact unconditional tests, Z, L, Ze, and L e. For n

= 10, the tests

are identical and the sizes are equal. For n = 15, 20, and 25, the size of L is closest to a =.05; in some cases the sizes of Ze and L e equal the size of L. In all but one case, for all n
30 the sizes of all four tests are between.0484 and.05. So, all four exact unconditional
tests do a good job of attaining a size close to but no more than the nominal level of a =.05. Sidik lO examined the sizes of the tests for 39 sample sizes and found that in cases when the sizes of Z and Ze differed greatly, Ze had the larger size, closer to a. The same was true when comparing Land L e ; L e had the size closer to a. On the other hand, in Table 1 the sizes of the asymptotic test M are larger than a = 0.05 for all the sample sizes. Clearly, M is liberal for testing (1). The sizes of G B are small, rising above.045 in only three cases in Table 1. The size of
G B is smaller than all the other tests for all sample sizes except n = 30, for which its size is
slightly larger than the sizes of the four exact unconditional tests. As expected, GB is very conservative because of the conditional nature of the test. To better understand the sizes of the tests, we plotted the size functions of the six tests for n
= 50 and n = 100 in Figure 2.
The size function is the function that is maximized in
computing the exact size of a test, for example, the size function of Z is

f(p;R~) =

(u,v)ER'Z

m(u,v;n,p,p)

for 0 < P < -. - 2
The size function of M exceeds the a = 0.05 line over some regions of p for both sample 14

sizes. The curve for CB is always much lower than the line a

0.05 over the complete

region of p. On the other hand, the size function curves of Z, L, Zc, and L c are very close to and below the line a
= 0.05 over most of the region of p.
In particular, the curves of Z,
Zc, and L c for n = 100 are close to a =.05. Note, the tests Z and L are identical, as are Zc and L c , for n = 50 and a =.05 in Figure 2.
To compare the powers of the tests with a = 0.05, we considered the nine sample sizes, n
= 10,25,35,50,60,80,100,150,200.
The exact powers were calculated for the grid

of 100 pairs of and PlO

and PlO under

which are determined by

= 0.025(0.05)0.475

= [POI + 0.05](0.05)[1 -

POI]. The average powers are given in Table 2, and these
relationships can be noted. For all nine sample sizes, the average power for M is the highest. But, of course, this is because M is a liberal test, and its size exceeds a

Among the

five level-a tests, Zc always has the highest average power or is tied for the highest. For all cases except one (n

= 25 compared to

L) CB has the lowest average power, confirming
the conservativeness of the conditional test. In most cases in Table 2, the average powers of the four exact unconditional tests, Z, Zc, L, and L c , are very close, but Zc has a slight advantage. Another summary of the pairwise comparisons for the same nine sample sizes is presented in Table 3. Each block of nine entries represents a comparison of the row test and the column test, and the nine positions in each block correspond to the nine sample sizes in this pattern, 35 80
200. The symbol "=" indicates the power function of the two tests are exactly equal because the rejection regions of the two tests are identical. Notation "<" means the column test is uniformly more powerful than the row test because the rejection region of the row test is a proper subset of the rejection region of the column test. Symbol ">" indicates the row test is uniformly more powerful than the column test because the rejection region of the column 15
test is a proper subset of the rejection region of the row test. In cases where none of the uniform comparisons apply, the powers were computed for all the 100 paired points (POI,PlO)' The proportion of the points at which the column test's power exceeds the row test's power is listed as a percent. These comparisons show that M is uniformly more powerful than all the other tests because of the incorrect, liberal size of the test. The four exact unconditional tests are uniformly more powerful than CB in all cases except the comparison with L when

n = 25. Zc is identical to or uniformly more powerful than Z and L for seven of the nine
sample sizes. L c is identical to or uniformly more powerful than L for eight of the nine sample sizes. In addition, L c is identical to or uniformly more powerful than Z for six of the nine sample sizes, and its power is higher than Z more frequently for the other three sample sizes. As far as the comparisons of Zc and L c are concerned, Zc is uniformly more powerful than L c for five of the nine and identical to L c for another three of the nine sample sizes. In all five cases when Zc is not the same as or uniformly more powerful than another level-a test, the power of Zc exceeds the power of the other test over more than 50% of the alternative points. Thus, for the cases considered in Tables 2 and 3, Zc appears to be the level-a test with the best power properties.

Conclusions

In this paper, we introduced four exact unconditional tests for the problem of testing the
one-sided hypothesis about two paired proportions. By considering the monotonicity of the joint distribution, these tests can be defined by considering one nuisance parameter on the boundary of H o and HI' This simplifies the computation of p-values for these tests. The size and power of the four exact unconditional tests, an asymptotic test, and a conditional test were compared. We found that the exact unconditional tests, Z, L, Zc, and
L c , have accurate size properties; their exact sizes are less than and very close to the level of
the tests. For sample sizes like n
= 100, the size function curves suggested that

Z, Zc, and

L c are approximately unbiased for testing (1) with the curves being very close to a

= 0.05

over almost the whole region of p on the boundary between H o and HI. Also, we found that the exact size of the asymptotic test M is always larger than the nominal level of the test. Therefore, it is not appropriate to use M for testing (1). In addition, it has been shown that the exact conditional binomial test C B is conservative, and its size is usually much smaller than the level of the test. Furthermore, the results of the power comparisons indicate that the confidence interval tests Zc and L c are generally more powerful than the non-interval tests Z and L. Among the four tests, Zc, L c , and Z generally have better power than L. The exact unconditional tests are almost always uniformly more powerful than the exact conditional binomial test C B. The asymptotic test M is uniformly more powerful than all the other five tests because of its incorrect, liberal size. Overall, in this comparison, Zc appears to be the level-a test with the best power properties.

Acknowledgement

We thank Dennis D. Boos, David A. Dickey, and William H. Swallow for their helpful comments.

References

1 Berger RL, Boos DD. P values maximized over a confidence set for the nuisance parameter. Journal of the American Statistical Association 1994; 89: 1012-1016. 2 Bickel PJ, Doksum KA. Mathematical Statistics: Basic Ideas and Selected Topics. San Francisco: Holden-Day, 1977. 3 McNemar Q. Note on the sampling error of the difference between correlated proportions or percentages. Psychometrika 1947; 12: 153-157. 4 Cochran WG. The comparisons of percentages in matched samples. Biometrika 1950;

37: 256-266.

5 Suissa S, Shuster JJ. The 2 x 2 matched-pairs trials: exact unconditional design and analysis. Biometrics 1991; 47: 361-372. 6 Hsueh HM, Liu JP, Chen JJ. Unconditional exact tests for equivalence or noninferiority for paired binary endpoints. Biometrics 2001; 57: 478-483.
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sided comparisons of two parameters in discrete data. Technical Report. Raleigh, NC: North Carolina State University, Statistics Department; 1997.
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2002. 18

12 Storer BE, Kim C. Exact properties of some exact test statistics for comparing two
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Table 1 Exact Sizes of the Tests with a = 0.05
(/3 = 0.0005 in Zc and Lc)
0.0208 0.0304 0.0339 0.0342 0.0498 0.0448 0.0404 0.0373 0.0462 0.0397 0.0465 0.0401 0.0443 0.0439 0.0441

0.0652 0.0592 0.0577 0.0539 0.0558 0.0528 0.0527 0.0595 0.0524 0.0598 0.0523 0.0567 0.0522 0.0521 0.0521
0.0265 0.0369 0.0393 0.0382 0.0494 0.0499 0.0499 0.0495 0.0493 0.0492 0.0492 0.0491 0.0495 0.0489 0.0488
0.0265 0.0369 0.0393 0.0478 0.0494 0.0486 0.0484 0.0495 0.0493 0.0492 0.0492 0.0491 0.0491 0.0493 0.0495
0.0265 0.0498 0.0485 0.0478 0.0450 0.0499 0.0497 0.0495 0.0493 0.0492 0.0492 0.0491 0.0491 0.0489 0.0489
0.0265 0.0304 0.0485 0.0478 0.0494 0.0485 0.0484 0.0495 0.0493 0.0492 0.0492 0.0491 0.0491 0.0493 0.0495
Table 2 Average (over 100 points) Power with a

= 0.0005 in Zc

M 0.439 0.618 0.676 0.733 0.757 0.794 0.820 0.860 0.884

and L c )

0.287 0.555 0.635 0.700 0.733 0.775 0.804 0.849 0.876
0.316 0.577 0.662 0.723 0.751 0.789 0.817 0.858 0.880
0.316 0.588 0.663 0.725 0.753 0.791 0.817 0.858 0.882
0.316 0.554 0.662 0.723 0.751 0.787 0.813 0.854 0.879
0.316 0.581 0.659 0.725 0.752 0.790 0.816 0.858 0.882
Table 3 Pairwise Power Comparison of the Tests for a = 0.05

= 0.0005 in Zc and L c )

< < < < < < < < < < < < < < <

< < <

< <

> > >

> >

< < < <

> 32 > > > >
sample size in each block

50 100

Note: = means row and column tests are the same. < means column test is uniformly more powerful than row test.
> means row test is uniformly more powerful than column test.
Numeric value is percentage of HI on which column test's power exceeds row test's power.
U()(Ol,)(lO) ------------------~------------~

l()(Ol, )(10)

Figure 1 The confidence interval 113 and confidence set under H o
The size function for n = 50

Q) C')

---'-"-'-"----- --CB M

Z Zc L

The size function for n = 100

.' (V.

- - _--.----uu.--.--h.mh. uum

_?:.?2l-.

---------~--_/
--~----------------------

~-, \)

"

CB M Z Zc L

Figure 2 The size functions of the tests with a
= 0.05 ((3 = 0.0005 in Zc