Power function comparisons in inequality hypothesis testing

161

Economics Letters 9 (1982) 161-167 North-Holland Publishing Company

POWER FUNCTION COMPARISONS HYPOTHESIS TESTING *

IN INEQUALITY

Thomas

A. YANCEY,

Robert

and George

Unicersr!,~

o/ Illimis,

IL

Received

10 December

Urbunu

BOHRER

61801,

G. JUDGE

USA

1981

This paper evaluates the power of the inequality likelihood power with that of other tests normally used in practice.

ratio

tests and compares

its

1. Introduction In a recent issue of this journal, we developed for the orthonormal linear statistical model a framework for hypothesis testing with multiple inequalities or with mixtures of equalities and inequalities. In particular, a likelihood ratio test was given, rejection regions were illustrated and an abridged table of critical values for an F test at 5 percent significance level was presented. Building on those results, in this paper we evaluate the power of these tests using the bivariate t distribution and analyze how its power compares with that of (i) a sequence of individual tests on parameters concerning the same inequality hypothesis, (ii) a test involving a mixture of inequality-equality hypotheses, and (iii) the traditional likelihood ratio test involving equality hypotheses.

2. Joint inequality hypotheses Consider the K mean or orthonormal linear statistical model y = Z8 + e, where y and e are (TX 1)normal random vectors with means Z6 and 0, respectively, and covariance a21,, 0 is a K dimensional vector of unknown parameters, Z is a (TX K) known matrix and Z’Z = I,.

* Research

for this paper

was supported

0165-1765/82/0000-0000/$02.75

in part by a National

Science Foundation

0 1982 North-Holland

grant.

T.A. Yancey et al. / Power comparisons

162

in inequahty hypothesis testing

Suppose, as is often the case in economics, we wish to test inequality hypotheses about two or more elements of the unknown K dimensional vector 8 where the maximum likelihood estimator 6 = Z’y - N( 0, u ‘I,), with independent (6, - 0,)/dt(.-,, and 82 = ( y - Zd)‘( y - Zb)/(r -K) with (T-K) c?~/u’-x:~_~). As an illustration consider the null hypothesis ZZ,,:8, 2 0, B,a 0 with alternative hypothesis that at least one 6, < 0, i = 1, 2. Two possible rejection regions can be imagined. First, one could run a single inequality (single tail) I test of a giyen size for each 19,a 0 against 8, < 0, ZZ,, is then rejected if either of the 0,/(i = t, G co, we call this a combined individual inequality test (CII), u,. Each of the t, has a t distribution with (T- 2) degrees of freedom (df), and c0 is the critical value of the test. The boundary of the rejection region is shown in fig. 1 as a solid line, and the acceptance region is the L shaped convex set in the sample outcome space. A second test possibility would be to use the likelihood ratio inequality test (LIR) with the test statistic U, = Zc_-a,O)(~,)tf + Zc_,,0)(t2)ti. where t, and t, are as defined above and have a bivariate distribution. Zc ,( t,) is an indicator function which is one if t, falls in the interval ( .) and zero otherwise. In the LRI test, ZZ, would be rejected if u2 > c:. With LRI and CII tests of the same size and df, c, > c,,. For example, for (Yof 0.05 and 10 df, c0 = 2.212 and c, = 2.326. The boundary of the rejection region for the LRI test is shown by the dotted line in fig. 1. Both tests are unbiased. We are interested in the power of each test

I+2 I I

Accept &>O

I

a2 to

I

-2.326

; I I \

-2.212

0 \ ‘\

Reject 8,>0

e2t0 Fig. I. Acceptance

‘I

-2.212 ~~____-__-2.326

I and rejection

regions for the LRI and CII tests

T.A. Yancey et al. / Power comparrsons in inequality hypothesrs testing

and the amount by which the power differs between hypothesis tests at various points in the 8 space.

3. Power function

163

these and equality

evaluations

To compare the power for the LRI test, the CII test, the conventional joint likelihood ratio equality test (LRE) and the combined individual equality test (CIE), we use the two parameter, two hypothesis space. In making the power comparisons for tests of the same size and df, we (i) determine the critical values, c0 and c,, for the tests, (ii) choose values for the non-centrality parameter X = (0: + r32)/2a2, and (iii) compute the power along the vector 0, = 68,. To obtain values of the power function for different points in the 6 space, use is made of a numerical integration program developed by Milton (1972) and extended by Bohrer and Schewish ( 1% 1). Let us first consider the power function on the vector in the 8 space where 6, = 8,. Thus the probability of rejecting the hypothesis 0, 2 0, 8, > 0 is obtained along the vector 8, = 6, = * fi. The graphs of the power functions for the LRI, CII, LRE and CIE tests are shown in fig. 2.

I -----

I

1

I

I

Inequality, LR, U1 Inequality, Combined Individual Test,& Equality, LR, U3 Equality, CombinedIndividual Test,&,

0.2 -

-3

0

-2

1

%k Fig. 2. Power functions IO degrees of freedom.

along 8, =0, ~0 for the LRI, CII, LRE, CIE tests for size 0.05 and

164

T.A.

YO~ICC~et al. / Power comparisons m inequa1it.v hypothesis testing

The power functions for each test when 8, < 0 and 8, = 0, = - fi, are tabled in the appendix. As is clear from fig. 2 when 8, = 19, and a = 0.05 and 10 df, the LRI test dominates the CII test. Fig. 2 also shows that with a = 0.05 and 10 df, the LRE test ( u3) dominates the CIE test ( u4) along the line 8, = 6,. These same patterns for power functions on the vector 0, = 8z < 0 hold for other levels of a and other degrees of freedom. In fact. as reported in the appendix, along the line 8, = 0, < 0, the LRI test dominates the mixed equality-inequality likelihood ratio test which dominates the LRE test. For a wide range of 8 values in the parameter space when X G 1 and (8,
4. Numerical

integration’ procedures

The significance levels for these tests were calculated using binary research programs with IMSL routines for t and F distributions. The transformation of critical values from the bivariate tv, to the 4;2,V, distribution for the same size and degrees of freedom are c, and cf/2, respectively. Powers were calculated using an adaptation to r-probabilities of Milton’s (1972) algorithm for the multivariate normal probabili-

T.A. Yancey et al. / Power compurisons

in rnequulity hypothesis testtng

165

ties. That algorithm is subject to the troubles diagnosed and solved by Bohrer and Schewish (1981) in a related problem, and one of us proceeds to make modifications to reduce the problems in future use of these algorithms for higher dimensional hypotheses.

5. Conclusions

and extensions

Each of the tests considered guards against, to a different degree, failing to discover that parameter values are in a particular region of the parameter space. The inequality likelihood ratio test is best when one places the highest priority on discovering that both claims in the null hypothesis are incorrect. The equality likelihood ratio test is best when testing 8, = 0, 8, = 0 and the priority is on discovering that 0, a 0 and 0, < 0. Since in econometrics we may have strong a priori reasons for thinking that 8, < 0 and 13~< 0 is true if the hypothesis, 8, 2 0 and 8,a 0, is not true, the multiple inequality hypothesis test seems useful. Given these results a useful extension would be a procedure that selects among the hypotheses (i) 8, 3 0, 8, > 0, (ii) 6, b 0, 8, < 0, (iii) 8, < 0, 8, > 0, and (iv) 8, < 0, e2 < 0 and accepts the one with the highest average probability of being correct. One of the authors has made some progress on this multiple comparison problem. The computer programs used for the orthogonal case can also be used for the non-orthogonal design matrix. Bohrer and Chow (1978) have an algorithm which computes the weights for F distribution probabilities for the non-orthogonal design matrix case. Bartholomew et al. (1972) have considered inequality tests for .0, = 0, 0, = 0 for the linear model. Given the availability of computer programs we found it just as easy, in determining the c, value for a given size test, to work with the multivariate t distribution as it is to work with a weighted average of F distributions as we did in the hypothesis testing paper [see Yancey et al. (1981)]. In either case no tables are feasible for the linear model with a general design matrix and numerical integration programs must be used.

References Bartholomew, D.J., R.E. Barlow, J.M. Bremner and H.D. Brunk, 1972. Statistical inference under order restrictions (Wiley, New York). Bohrer, Robert and Winston Chow, 1978, Weights for one-sided multivariate inferences, Applied Statistics 27, Algorithms AS 122, lOO- 104.

166

T.A.Yanceyetal.

/

Powercomparisons

in inequality h,rpothesrs testrng

Bohrer, Robert and Mark. J. Schewish, 1981. An error-bounded algorithm for normal probabilities of rectangular regions. Technometrics 23. 297-300. Milton, Roy C., 1972. Computer evaluation of the multivariate normal integral. Technometrics 14, 881-889. Yancey. T.A.. G.G. Judge and M.E. Bock. 1981. Testing multiple equality and inequality hypotheses in economics, Economics Letters 7. 2499255.

Appendix Values of the power functions measured along specified lines in the parameter space for Likelihood Ratio (LR) and Combined Individual (CI) tests at the 5 percent level with 10 degrees of freedom for mixtures of equality and inequality hypotheses about two parameters. Non-centrality parameter value x

0.0

0.0312 0.0625 0.1250 0.2304 0.5063 1.0404 1.5 2.0 4.0 6.0 8.0

Probability

of rejecting

On the line 8, = b’,

H,: e,=o,e,=o H,: H, not true

H,,: e,=O.e>aO H,: H, not true

H,:e,a0.e2>0 H,: H, not true

LR c-=2.8636

CI c=2.6088

LR c=2.613

CI c =2.4472

LR c-=2.326

Cl cx2.212

0.050 0.054. 0.057 0.064 0.077

0.050 0.053 0.057 0.064 0.076 0.107 0.166 0.226 0.285 0.506 0.679 0.801

0.050

0.050 0.060 0.065 0.076 0.092 0.132 0.207 0.270 0.337 0.569 0.737 0.847

0.050 0.074 0.086 0.105 0.134 0.199 0.313 0.402 0.490 0.750 0.888 0.953

0.050 0.07 I 0.082 0.099 0.123 0.178 0.272 0.347 0.422 0.662 0.814 0.903

0.111 0.178 0.248 0.319 0.576 0.760 0.873

0.062 0.069 0.08 1 0.100 0.147 0.238 0.314 0.394 0.659 0.826 0.916

T.A.

Ho 6’,20,

Yancey et al. / Power comparisons

B, 20 H,: H,

in inequahty

hyporhesls testmg

167

is not true

On theline 0,=-J%;

8, =2e,

LR c=

2.326

0.050

0.072 0.084 0.103 0.131 0.195 0.308 0.397 0.486 0.798 0.887 0.953

8,=-o,

e,=o

CI

CI c=2.212

LR c=2.326

CI c=2.212

LR c =2.326

cx2.212

0.050 0.070 0.081 0.097 0.122 0.178 0.277 0.356 0.437 0.690 0.842 0.924

0.050 0.067 0.077 0.092 0.115 0.172 0.277 0.363 0.451 0.721 0.872 0.945

0.050 0.066 0.075 0.090 0.113 0.169 0.276 0.364 0.455 0.731 0.881 0.951

0.050 0.052 0.055 0.060 0.068 0.092 0.138 0.180 0.226 0.408 0.565 0.692

0.050 0.053 0.056 0.062 0.073 0.100 0.155 0.202 0.253 0.445 0.605 0.727