COMMUN. STATIST.-THEORY METH., 26(1), 129-138 (1997)

NONPARAMETRIC TESTS BASED ON CONTRASTS FOR D-STAR, BIPARTITE AND BROOM TREE ALTERNATIVES

NAOTO HOSHINO and YOICHI SEKI

Department of Computer Science, Gunma University, 1-5-1 Tenjin-cho, Kiryu, Gunma, 376, Japan

Key Words and Phrases: isotonic regression; maximin efficient rank test; Pit- man efficacy.

ABSTRACT

We consider a rank test statistic based on contrasts for testing against

d-star, bipartite and broom tree alternatives. Contrast coefficients are deter-

mined so that the minimum asymptotic power can be maximized. Finally, the

power of the proposed test is compared with that of Chacko's x:,2,k test.

1. INTRODUCTION

Let X,j ( i = 1. . . . , k ; j = 1, . . . , n,) be k independent random samples

having continuous distribution function F ( x - B ; ) . This paper considers prob-

lems of testing the null hypothesis Ho : O1 = . . . = Bk against the foliowing

alternatives:

130 HOSHINO AND SEKI

D-star : [el , . . . , 8,-11 5 6, <_ [Om+1,. . . ,8k] (1 < m 5 k ) , (1.1)

Bipartite : [e l , . . . ,@,I _< . . ,Ok] ( I 5 m < k), (1.2)

Broomtree: B 1 < _ ~ ~ ~ _ < O m < [ 6 m + 1 , . . . , B k ] ( l _ < m _ < k ) , (1.3)

where nl is known and at least one inequality is strict. In an earlier

paper (Hoshino et al.. 1995), we have developed a normal theory of testing

against these alternatives which are useful in many practical situations.

In section 2, we give contrast statistics for testing against the above alter-

natives, and discuss asymptotic distributions of the statistics. In section 3, we

determine the contrast coefficients for each alternative so that the minimum

asymptotic power can be maximized. Such tests are called maximin efficient

linear rank tests by Shi (1988). He has proposed the test for an umbrella al-

ternative: B1 5 . . . 5 Oh 2 . . . 2 4 . In section 4, the powers of the proposed

tests are compared with that of the z:;2,k test, which is a nonparametric ver-

sion of the likelihood ratio test x2 in the normal case and has been developed

for a simple order alternative: Q1 5 . . . 5 by Chacko (1963) first. Robertson

et al. (1988) have shown that for large samples the x:,~ test statistic has

the same null distribution of the z2. For the d-star, bipartite and broom tree,

these null distributions are given by Hoshino et al. (1995) in the case of equal

sample sizes: n2 = . . . = n~ .

2. TEST ST.4TISTICS BASED ON CONTRASTS

Let Rtj be the rank of Xi, when all k samples are pooled and let Rj =

CY:, Ri,/n,. Let N = zfz1 n, and n i / N -+ A, E ( 0 , l ) as iV -t co. For

simplicity throughout this paper, we assume that A, = n, /N for i = 1 , . . . , k.

For testing Ho against the d-star, bipartite and broom tree alternatives,

we adopt a test statistic Trank = ~ f = , XiaiRi, where a = (a l , . . . , a,k)' E A =

{ r E RkI ~ f = ~ Aiz, = 0).

From the result in Hettmansperger and Norton (1987), we have the fol-

lowing theore~n which givcs the asymptotic null distribution of Trmk.

NONPARAMETRIC TESTS 131

Theorem 2.1. If Ha is true, T L k = 4 1 2 / ( N q ~ ~ , ~ / J m has

the standard normal distribution as iV -+ cc. This theorem shows that for large ili we can obtain a critical value t , of

Tfank at a significant level cr from the expression P r ( T L k > t , ) = 1-@(t,) =

a , where @ is the standard normal distribution function.

Let C denote a convex cone in R~ determined by the d-star, bipartite

or broom tree ordering. For a vector c E C, we define a hypothesis Hc by

Hc : 0, = O0 + @ c , / f l for p > 0 and r = 1,. . . , k . Suppose that F has a

density f with J f2(x)dx < m. Then by the result in Hettmansperger and

Norton (1987), we have the following.

Theorem 2.2. If Hc is true, T:,k has a normal distribution with mean

pe and variance 1 as iV -+ co, where e is called the Pitman efficacy of the test

based on TLk and given by

It is easily seen from Theorem 2.2 that for large N if the unknown pa-

rameter 0, equals to Bo + B G / f l for some c E C. the test based on T&,k has

asymptotic power 1 - @(t , - Be). Hence the test has asymptotically higher

power as r ( a , c - 3) increases for a given c.

3. COMPUTATION OF CONTRAST COEFFICIENTS

In this section, we determine a contrast coefficient vector a for each of

the three altmiatives so as to rnaxirnize the minimum value of r . The contrast

tests with such coefficients are called maxirrii~ efficient linear rank tests, and

the drfinitio~i is as follows.

132 HOSHINO AND SEKI

Definition. A test Lased on Trmk = ~ f ; = ~ X,a;R, is said to be maximin

effic~ent linear rank test if

mill r(aO, c - S) = max min r (a , c - CEC ~ E A CEC

Let B be a convex cone determined by a partial ordering which has p

i~iequalities. Any vector in B satisfies each of the inequalities as "<" or as "=".

A (0,l)-vector in B is called a corner if it provides the maximum number (less

than p) of inequalities that are "=". A corner is simple if only one inequality

is ^<", while all the others are "=". Thus each simple corner corresponds to

one inequality. A corner is complex if more than one inequality is .'<".

Ahelson and Tukey (1963) have shown that the maximin a0 is a vector

in B representable as a nonnegative linear combination of all corners cj as

a' = x?j(~J - ' J ) , bJ > O1 (3.2) j

where r(aO, c, - E J ) takes the same value, r', for each corner c, appearing

witah Pj > 0 and r(ao, c, - C j ) > ,re for each corner cj appearing with Pj = 0.

Let z be any vector in B. If all corners are simple, z is representable

uniquely as a linear combination of all the corners as follows:

where each corner cj corresponding to inequalities that are satisfied by z as

"<" appears with /3, > 0 and each corner cj corresponding to inecjualities that

are satisfied by z as "=" appears with Pj = 0. Suppose that a E A is a vector

which makes r (a , c - 5) equal for all corners. The vector a is the solution of

the equations

NONPARAMETRIC TESTS 133

for the corners and the contrast-ensuring equation ~ f = , X a , , - - 0. Then we

can show as follows that the maximin a0 is the isotonic regression a* of a

with weight vector X = (A1,. . . , X k ) ' and the given partial ordering, that is

Since the isotonic regression a' E B is representable as the linear combination

(3.3), it is sufficient to show that r(a*, cj - c,) = r' for each corner c, corre-

sponding to inequalities that are satisfied by a* as "<" and r(a*, cj - C j ) 2 T*

for each corner c, corresponding to inequalities that are satisfied by a* as "=".

From Theorem 1.3.2 of Robertson et al. (1988,p.17), for all c E B,

which is an equality for the corners corresponding to inequalities which are

strictly satisfied by a". Hence, since a* is representable as the linear combi-

nation (3.2), we have that a0 = a*.

On the other hand, if B has complex corners, we might have to repeat

tedious work some times such that we choose a subset of corners and check

whether the vector a obtained from (3.4) for the subset satisfies (3.1).

3.1. D-star alternative. Let 1 be an all 1's vector, and let ej be an unit

vector with the j t h component equal to 1. The convex cone C determined by

the d-star ordering ha5 k - 1 simple corners cj = 1 - ej, j = 1 , . . . , m - 1, and

cj = e,+l, j = in,. . . , k - 1. Solving the equations (3.4) for a11 the corners,

we have that

For reasons mentioned above, a0 is the isotonic regression of a given by (3.5).

134 HOSHINO AND SEKI

3.2. Broom tree alternative. The convex cone C determined by the broom

tree ordering has k - 1 simple corners cj = 1 - xi=, el, j = 1 , . . . , m - 1, and

C j = e,+l, j = m , . . . , k - 1. Solving the equations (3.4),

where A. = 0 and Ai = XI + . . . + X i for i = 1,. . . , m - I. Hence a" is the

isotonic regression of a given by (3.6).

3.3. Bipartite alternative. The convex cone C determined by the bipar-

tite ordering has k complex corners, so in general we must try some subsets

of corners until we get aO. Here, we suppose that nl = -. . = n, = n and

nm+l = . . . = nk = n' (n' may equal n). Then A, = X = n/{mn + ( k - m)nl)

for i = 1 , . . . , m and X i = A' = nr/{mn + ( k - mjn') for i = m + 1 , . . . , k. In

this case, we can easily find a".

Let us denote the k corners by cj = 1 - ej for j = I , . . . , m and C, = e,

for j = m + 1, . . . , k. If we solve the equations (3.4) for the corners cl, . . . , ern,

we have that ai = -,/=/A for i = 1, . . . , m and ai = m \ i m /

(k-m)Xr) for i = m+l , . . . , k. Since this vector a E C is alinear combination

of these corners with positive weights, it satisfies (3.1) when c:=, X,a,c, 2

d m ' for ern+, , . . . , ek or equivalently

On the other hand, if we use the corners em+,, . . . , ck, we can see that

a, = -(k - m ) j m / ( m X ) for i = 1 , . . , rn and a, = 4-/X for

i = m + 1,. . . , k satisfy (3.1) when the inequality (3.7) does not hold.

NONPARAMETRIC TESTS 135

4. THE POWER OF THE TEST BASED ON T L k

We compare the power of the test based on TLk with that of the &k

test. The j&k statistic is given by

where w; = &ni and is the isotonic regression of R; with weight vector

w and a partial order determining the alternative hypothesis.

We performed a Monte Carlo power study for a normal case, F;(x - 1 0.) = exp (-& (r - B ~ ) ~ ] , and for an exponential case, Fi(x - 0;) =

pexp (-pfx - 0;)). In this study, we computed power estimates of TLk and

x:ank for the d-star, bipartite and broom tree alternatives of k = 5 and 6,

with equal sample size cases: n1 = ... = n k n and various alternative

configurations corresponding to values of B;, i = 1,. . . , k. In each setting,

we generated normal and exponential random numbers by the Box-Muller

method and the inverse transform method, respectively, using the uniform

random number generator of Park and Miller (1988). We used lo5 replications

to obtain one estimate for the powers of these tests.

TABLES I, I1 and 111 give the results for the d-star, bipartite and broom

tree alternatives of k = 5 and in = 3, respectively. We assumed that a2 = 1

and p = 1, and simulated the powers for the sample size n = 2,5,10 and the

significant level a = 0.05. Critical values at a = 0.05 for the T L k statistic

were obtained from its asynlptotic null distribution, which was shown to be the

standard normal distribution in Theorem 2.1, and critical values at a = 0.05

for j&nk were obtained from its asymptotic null distribution given by Hoshino

et al. (1995). So, when the sample size n is small, the actual significant level is

not necessarily equal to 0.05. We also give estimates for the actual significant

HOSHINO AND SEKI

TABLE I

The powers of &k and Trmk for the d-star alternative:

[01,02] 5 O3 < [84r85], when nl = . . . = nk E n and a = 0.05

xiank T,',k

81 O2 83 04 85 n normal exponential normal exponential 0.5 1.0 1.5 2.0 2.5 2 0.362 0.513 0.539 0.660

Actual significant level 2 0.024 0.024 0.044 0.045 5 0.044 0.045 0.049 0.050

10 0.048 0.047 0.051 0.051

TABLE I1

The powers of x : , ~ and T L k for the bipartite alternative:

[01,02,03] < [84,85], when nl = . . . = nk - n and a = 0.05

Xrank 'rank

8, O2 O3 84 O5 n normal exponential normal exponential 0.5 1.0 1.5 2.0 2.5 2 0.251 0.396 0.512 0.646

Actual significant level 2 0.0'20 0.019 0.057 0.057 5 0.044 0.043 0.051 0.052

10 0.048 0.046 0.050 0.049

NONPARAMETRIC TESTS

TABLE I11

T h e powers of ?(:ank and T L k for the broom tree alternative:

& n k Tr*ank 81 02 03 04 05 n normal exponential normal exponential 0.5 1.0 1.5 2.0 2.5 2 0.414 0.564 0.537 0.665

5 0.902 0.958 0.914 0.971 10 0.997 0.999 0.997 0.999

Actual significant level 2 0.026 0.026 0.049 0.049 5 0.046 0.047 0.050 0.051

10 0.048 0.047 0.051 0.050

level in these tables. Thus ,& is more conservative than TLk , in the case

of small sample size. T h e tables shows tha t the power of T;mk is superior to

that of ,v:ank in many cases. Especially when the sample size n is small, since

Y:,~ is conservative, the power is somewhat lost. We obtained similar results

for k = 6.

From these results, we can recognize for the d-star, bipartite and broom

tree alternatives that the power of Treank is satisfactory in many situations.

Furthermore, since the asymptotic null distribution of Tr*a,k is standard nor-

mal. the test based on T;a,k can be easily performed. 011 the other hand, the

asymptotic null distribution of ;Y:,,,~ has obtained onlv in the case of equal

sample sizes. Hence we recommend the trst Tr*mk, especially in the case of

uneql~al sample sizes.

HOSHINO AND SEW

The authors are grateful to the referee for his useful comments and sug-

gestions which led to several improvements.

BIBLIOGRAPHY

Abelson, R. P. and Tukey, J . W. (1963). "Efficient utilization of non-numerical information in quantitative analysis: general theory and the case of simple order," Ann. Math. Statist., 34, 1347-1369.

Chacko, V. J . (1963). "Testing homogeneity against ordered alternatives," Ann. Math. Statist., 34, 945-956.

Hettmansperger, T. P. and Norton, R. M. (1987). "Tests for patterned alternatives in k-sample problems," JASA, 82, 292-299.

Hoshino, N., Miyazaki, H. and Seki, Y. (1995). "On the level probabilities for useful partially ordered alternatives in the analysis of variance," Comrn. Statist.-Theor. Meth., 24, 2059-2071.

Park, S. K. and Miller, K. W. (1988). "Random number generators: good ones are hard to find," Communications of ACM, 31, 1192-1201.

Robertson, T., Wright, F. T . and Dykstra, R. L. (1988). Order Restricted Statistical Inference, Wiley, New York.

Shi, N.-Z. (1988). "Rank test statistics for umbrella alternatives," Comm. Statist.-Theor. Meth., 17, 2059-2073.

Received A p r i l , 1995; Revised June, 1996.