on power functions of the likelihood ratio tests for the simple loop order in normal means: unequal...

Statistics & Probability Letters 14 (1992) 253-267

North-Holland 17 July 1992

On power functions of the likelihood ratio tests for the simple loop order in normal means: Unequal sample sizes

Bahadur Singh Department of Statistics, UniL~ersity of Iowa, Iowa City, LA, USA

Michael J. Schell * Department of Biostatistics, St. Jude Children’s Research Hospital, Memphis, TN, USA

Received February 1990 Revised August 1991

Abstract: Likelihood ratio tests are considered for two testing situations; testing for the homogeneity of normal means against the

alternative restricted by a simple loop ordering trend and testing the null hypothesis that the means satisfy the trend against all alternatives. The exact expressions are given for the power functions for the simple loop order, k = 4 and unequal sample sizes, both for the case of known and unknown variances.

Keywords: Simple loop order, likelihood ratio tests, power functions.

1. Introduction

We consider experimental situations where one wishes to compare several treatment means with a control or standard, assuming that all of the treatment means are at least as large as the control mean and that one of the treatment means is known to be as large as the remaining treatment means. In a clinical study the effect of several chemotherapeutic regimens may be compared to an untreated control, where additionally one regimen is known to be at least as effective as all others, e.g., cancer patients may be given one of four regimens: a standard chemotherapeutic regimen (control), the standard regimen plus either drug A or B, or the standard regimen plus both drugs A and B.

Assuming that the observations are normally distributed with a common variance u’, let H, denote the hypothesis that P, G [pu2, pS,. . . , ,uk_ll G pk where pL1 denotes the control mean and pi, i = 2, 3,. * . ) k, denote the treatment means. The hypothesis H, imposes a partial order on the parameters pr, pcLz,. . . , pu,, which is referred to as the simple loop ordering in Robertson, Wright and Dykstra (1988, p. 84). Much of the information concerning the simple loop in the above’text can be found in Barlow et al. (1972) as well. For k = 3, it is a simple ordering and for k = 4, it corresponds to a 2 X 2 matrix

Correspondence to: Dr. M.J. Schell, Department of Hematology/Oncology, 375-B Med Surge II, University of California, Irvine, CA 92717, USA. * This work was supported by the American Lebanese Syrian Associated Charities (ALSAC).

0167.7152/92/$05.00 0 1992 - Elsevier Science Publishers B.V. All rights reserved 253

Volume 14, Number 4 STATISTICS & PROBABILITY LETTERS 17 July 1992

ordering. The likelihood ratio test (LRT) of homogeneity of means, i.e., H,,: p, = p-L2 = . . . = pu, versus H, - H,, is developed in Bartholomew (1959) and the LRT for testing H, as a null hypothesis against all other alternatives is developed in Robertson and Wegman (1978).

Robertson et al. (1988, pp. 86-98) demonstrate that the gain in power obtained by using the LRT for order restricted alternatives rather than the omnibus LRT can be quite substantial for means which satisfy, or nearly satisfy, the specified order restrictions. Robertson et al. (pp. 176-188) also compare the restricted LRTs with contrast tests for testing homogeneity with an order restricted alternative. They conclude that unless additional information concerning the spacings among the means is available, the LRTs are preferred.

For the case of known u and equal sample sizes, Bartholomew (1961) studied the power function of the restricted LRT. He gave exact formulae for k = 3 and 4, conjectured which mean spacings in H,, a fixed distance A (A is defined in equation (2.6) below) from H,,, would maximize and minimize the power functions. (We shown in Section 4 that the mean spacing conjectured to yield the minimum power is incorrect.) Knowing the mean spacing that gives the minimum power is quite useful since then one can obtain conservative estimates of the common sample size needed to obtain a desired power at a specified point in H, - H,,. Although in practice u is usually unknown, often the degrees of freedom associated with its estimator are sufficiently large than the results for known (T* provide reasonable approximations.

In some applications, researchers (see Chase, 1974) wish to place a larger sample size on the control than on the treatment populations. Unequal sample sizes also arise when some data are missing. Power functions have already been derived under unequal sample sizes for the related simple order and simple tree restrictions (Singh and Wright, 1989; and Singh, Schell and Wright, 1989). In this article we obtain the power functions fo the LRTs for the simple loop alternative in the case of unequal sample sizes.

If the order restriction H , is in question, one might wish to test H , versus H 2 - H ,, where H 2 places no restrictions on the population means. For this hypothesis, Eeden (1958) considered several ad hoc tests and Robertson and Wegman (1978) studied the order restricted LRTs.

In Section 2, we study the power functions of the LRTs of H,, versus H, - H,, versus H, - Hi for the case of known (T and possibly unequal sample size. Exact expressions are given for k = 4. The analogous results for unknown u are presented in Section 3. In Section 4 a search to determine the mean spacings that provide the minimum and maximum power is described. Powers are calculated for various sample size allocation plans for the simple loop order in Section 5 and the powers obtained are compared to powers of LRT tests under other order restriction assumptions in Section 6.

2. The case of known variance a2

Let Xi, X2,..., x, be the means of independent random samples from normal populations with a

common variance, v2; that is, for i = 1, 2,. . . , k, the x, are independent and x, - N(pu,, a*/n,). Let

w = (w,, w*, . . .) w,> with wi=ni/u2 and let jZ=(jZi, jZ2,..., jZLk) denote the maximum likelihood

estimator (MLE) of p = (p,, p2,. . . , pk) subject to the restriction p E H,. The restricted MLE, jY, can be readily obtained by the algorithm described in three steps as follows:

(i) First apply the simple tree algorithm as described in Robertson, Wright, and Dykstra (1988, p. 19) with the first sample mean as the control mean and the next (k - 2) sample means as the treatment means. Let i denote the number of (k - 2) treatment means in the middle that are pooled with x,. These i treatment means are omitted in the next step.

(ii) Second, apply the above simple tree algorithm identifying minus one times the k th sample mean as the control mean and minus one times of the remaining (k - 2 - i) sample means in the middle as the treatment means. The resulting estimates are then multiplied by minus one.

(iii) Finally, if the first (control) population estimate is less than or equal to the kth treatment (last) population estimate, then one has the desired MLE of I_L subject to /.L E H,. Otherwise, ii, = b,

254


i=l,2 , . . . , k, where b is the weighted average of the k sample means defined after equation (2.1) below.

If a* is known, then the LRT of H,, versus H, - H,, rejects H,, for large values of

(2.1)

where fi = C~,,w,x,/C~, ,w,, and the LRT of H, versus H, - H, rejects H, for large values of

(2.2)

Of course, the LRT of H,, versus H, - H,, is the usual chi-square test statistic and rejects H,, for large values of xi2 = Cf= ,w,<X, - fi)*. Using fundamental properties of the projections p (cf. Theorem 1.3.2 of Robertson et al., 19881, it follows that J& = X,“, + Ft.

Robertson and Wegman (1978) show that H,, is least favorable within H, for ,Yf, and hence, critical

values for Xi, and XF2 may be determined from their distributions under H,,. Assuming p E H,,, let

P,(I, k; w) be the probability that there are I distinct values among p,, p2,. . . ,jTk. (Note that the subscript L denotes the simple loop order.) It has been shown that under H,, with C, > 0 and C, > 0,

p[~;, ac,] = &,(L k; W’[x:-I Cl (2.3) I=2

where x,’ denotes a chi-square variable with v degrees of freedom and

k-l

P[j3,z=.c,] = c f’,(l, k; 44x;-&2]

I= 1 (2.4)

(see Bartholomew, 1961; and Robertson and Wegman, 1978). The level probabilities P,(f, k; w> for the simple loop ordering can obtained by numerically integrating (2.4.14) in Robertson et al. (1988) and using their recursive relation, (2.4.4), but this approach is quite complicated for even moderate values of k. For k = 3 and 4, tables of critical values for Xi, and Xf2 are given in Robertson et al. (1988). The distributions determined by (2.3) and (2.4) are referred to as X2 distributions because they are mixtures of chi-squares distributions.

Applying the algorithm mentioned above to (x, + a, x2 + a , . . . , Xk + a) gives (F, + a, jZ2 + a,. . . , jlik + a). Thus, the power functions of Xi, and Xf, are invariant under shifts by a constant vector, i.e., their powers are the same at w and p + ae where e is a k-dimensional vector of ones and a is some constant. Hence, without loss of generality we may restrict attention to

B= i /_elRk:

the orthogonal complement of H, under the inner product (p., CL’), = CT= ,wjpjpJ. The distance from p to H, is determined by

1

l/2 k k > with fi = c wjpj/ c wj. j=l j=l

(2.5)

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2.1. Expression for the power function when k = 4

The expression for the power functions with unequal weights and k = 4 populations are derived by using the techniques of Bartholomew (1961) for the case of known variance. With C, > 0 and C, > 0, let ~T&L) = P&x,‘, 2 C,) and T&L) = P,<& 2 C,), where P,(A) denotes the probability that the event A occurs given population mean vector p. While the formulae for the power functions presented below are quite complex, they provide useful insights into the power of these tests. Moreover, researchers commonly allocate larger sample sizes to the first population (i.e., control) compared to the treatment populations, giving rise to unequal weights. For k = 4, H, n B is three-dimensional cone with edges e, = d,(-w,, -w,, -w4, w1 + wz + ws), e2 = d,(--(w, + w,>, -(w, + w,), wl + w2, w1 + w,), e3 = d,(-(w, + w,), WI + w3, -(w, + w,), W, + w,) and e4 = d4(-(~2 + ~3 + w,), wl, w,, w,) with d,, d,, d,, d, > 0.

The following notation is used in the power functions of the simple loop order for k = 4:

~(x c) = (x + vqb(x - dq + (1 +x2)@(x - Jq 7

44x1 >

A,= w3 l/2

(w,+w2)(w,+w2+W3) 1 [twl + w2)p3 - wlpl - w21-L21)

A, = w4

(wl+w2+w3)w 1 l/2

[(WI + w2 + W3)P4 - w11-L1- W2P2 - W3F3] 9

w2w3

[ 1 10

A,= ~ w2 + w3

(P3 -cL2),

Wl l/2

A, = (w2+w3)(wl+w2+w3) 1 [w2/?+2 + W3P3 - (w2 + W3hl~

A, = Wl

(w2+w3+w4)w 1 l/2

[ W2P2 + W3P3 + W4P4 -(W2+W3+W4h],

w3w4

[ 1 l/2

A,= ~ w3 + w4

(cL4-k3)5

1 l/2

A, = w2

(w3+w4)(w2+w3+w4) Iw3p3 + w4p4 - cw3 + w4)l-L2] 9

1 l/2

A, = w4

(w2 + w3)(w2 + w3 + w4) Kw2 + W3)P4 - W2P2 - W3P31)

(2.6)

AlO = cwl + w2)(w3p3 + w4p4) - cw3 + w4)(wlh + w2p2)

[cwl + w2)(w3 + w4)w]1’2

9

256


A = (A; +A;+A;)“*, A' = (A;2 + A;2 + A;)“*, A =A,cos8+A,sinOsinq +h,sinOcosq, A’=A’,cos8+A~sin6sin~+A3sinf3cosr],

A,=A,cosO+A,sin8, A’, =A;cosO+A,sinO,

A,=AlcosO+A,,sinO, A~=A~cos13+A’,,sinO,

A,=A,cosO+A,sin8, A;=A’,cosO+h’,sinO,

A4 = A, cos 0 + A, sin 19, A5 = A, cos 0 + A, sin 8,

b = [ w2wj/{w1( w, + wz + w,)}] “*, b’ = [ w2w3/{w,( w2 + ~3 + Wq)}] I’*,

y = [W*(W, + W2 + W,)/(WlW3)l’/‘~ Y’ = [W3(Wl + W2 + W,)/(WlW,)l 1’2,

6 = [ wq( w, + w*)/( w3w)p2, 6’= [W&l +w3)/(w2w)]“*,

7= [wl(W3+w~)/(w2w)]l~2, 7’ = [ Wl( W* + w,)/( wy)] 1’2,

,, _ 7 -

[

w,zw3w+ WA Wl + W*)(W1+ W3)

2 l/2

W,Wz(W1+ W2 + W3)W 1 ’

al=SiIl-1[Wl(Wi+W2+W3)/{(Wl+W2)(Wl+W3)}]1’2,

a~=sin-1[w,(w2+w,+w4)/{(w3+w4)(w2+w4)}]1’2,

a2 = sin- ‘k(w1 +w,)/I( W3 + Wq)( Wl + W2 + W3)jl l’*,

w2w3 ff

wl(wl+w2+w3)

A,=ctn-‘[-7sin(q +q)], A,=ctn-1[-7ncos(~ +(Y~)],

and A',, A;, A;, A;, are obtained from A,, A,, A,, A,, by interchanging w2 with w3, and p2 with p3, respectively. The proof of the next result is found in the Appendix.

Theorem 1. For k = 4, a positive weight vector w and I_L E lR4,

e-Nw-3v2

+@(-A*) ZT / =” tan-‘6

+(A,, C,) de

e-(“;Z+A:)/2

+ @(-A;) 2= jT” $(A’,, C,) de tan-‘8’

e -(A: + A:“)/2

+@(-A,) ZT / =‘2 tan-’ 7

(cl( AZ, C,) de

+ @(-A’,) e-‘*‘;;“/2 /r-;;] ,$( A;, C,) de T

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Volume 14. Number 4 STATISTICS & PROBABILITY LETTERS 17 July 1992

+ @‘( -A,)@( -W(A,, - g) + @( -A;>@( -W’(& - ;cl)

+@(A3-G)/” jby4(x-A,)+(y-A~)dxdy --m --m

and

+@(A6+$?,)/” jbry4(x-AMY-A,)dxdy -rn --m

e-(A;z+Aq)/2

+ @(A’,“) ZT / 3T’2~( A;, C,) de ?r

e-(A:-A:V

+@(A,) 2r / r+ol,

+(4, C,) do ?r

e-(“:+Am

+@(M ** / .TT+(Y;

+(A,> C,> de Tr

+ @(-A’, - G)l;+@[ A;O - T’( x + A;)]+(x) dx.

(2.7)

(2.8)

3. The case of unknown variance u2

Let X1, X2,..., %,, H,, H,, H,, w, ii, and h be defined as in Section 2, and let S2 be an estimator of u2 which is independent of x = (x1, x2,. . . , x,> where Q = uS2/a2 N ,yz where v is a positive integer.

The LRT of H, versus H, - H, rejects H, for large values of

so, = ~C~=,ni(Fi-fi)2 VX,‘,

Ez-

C;=l"i(xj-F;)2+vS2 Z+Q (3.1)

258


and the LRT of HI versus H 2 - H, rejects H , for large values of

(3.2)

(cf. Robertson et al., 1988, pp. 63-65). These tests are equivalent to the ones obtained by Bartholomew (1961) and Robertson and Wegman (1978), but have the advantage that S,,, + Xi, and S,, + Xf, as v + 00. (Note that the convergence referred to here is the convergence in distribution.)

Robertson and Wegman (1978) showed that H, is least favorable within H, for S,,, and that under Ha, conditional on I distinct values in p, Xi, - xf_,, & -xi_,, and j$, and Xf2 are independent. Hence, under H,, with C, > 0, C, > 0,

k-1+vC P[S,,>C,]= &(LkW)P F,--l,k-l+Y~ l_1 [ __-A

/=2 v 1

and

k-l

p[s,,>c,] = c p,(l, k; W)p I=1

(3.4)

where F, ,, is an F variable with a (b) degrees of freedom in the numerator (denominator). Since the P,(l, k; i> are the same level probabilities as in Section 2, the formulas, tables and the approximations mentioned there are useful in this case also. Sasabuchi and Kulatunga (1985) describe a two-moment beta distribution approximation for (3.3) when the weights are equal. Singh and Wright (1988) extend their work to (3.4) and possibly unequal weights.

3.1. Expressions for the power functions when k = 4

In addition to the notation, symbols and definitions described above, let h(u) be the density function of u = Q/v. Then, we have the following theorem concerning the power functions n-l,(p) and r;*(p) of S,,

and S,,, respectively. The proof of this theorem is given in the Appendix.

Theorem 2. For k = 4, positive weight vector w and p E R4,

259


and

(3.6)

where the probability P,[Xf2 2 UC,] is obtained from T,~(P) by replacing c2 by UC, in (2.8).

4. Minimal and maximal powers

Robertson et al. (1988, p. 97) notes that, for a fixed value of A and under equal weights, the maximum power for the simple loop is conjectured to occur at p such that p2 = pcL3 and p2 - p1 = p4 - p2. The minimum power is conjectured to occur for p such that p, < ,FL~ = ICL~ = I_L~ or p, = p2 = ~~ < pq. We examined these conjectures by using a searching algorithm as described in Singh and Wright (1989). Without loss of generality one can search for mean spacings such that H, f’ B and characterize the spacing in terms of A and two angles. For the equal weights case with A = 1, 2, 3, 4 and a significance level of 0.05, a one-degree search was conducted over H, to find the extreme powers of X,“,, i.e. powers were computed for every possible combination of angle measurements in one-degree increments. No spacing was identified that yielded a power larger than that of the spacing conjectured above to have maximum power. However, the spacings pFLI = p2 < p3 = p4 and p, = p., < p2 = p4, which are two edges of the cone, had the minimum power identified by the searching algorithm. The power for both of these spacings is 0.1873 for A = 1, while the spacings that had been conjectured to have minimum power have a power of 0.1983.

5. Simple size allocation

Powers have been calculated under selected weightings and are presented in Tables 1 and 2. The weightings used in Tables 1 and 2 were chosen for the reasons which follow. The weighting 49:1:1:49 is nearly an equally weighted comparison of the two extreme treatment groups, the 33:33:1:33 weighting is nearly an equally weighted comparison on two extreme and one intermediate treatment groups and the 1:l:l:l weighting gives equal weights to all four treatments. The 2:l:l:l weighting was observed to

260


Table 1

Power of the likelihood ratio test of X0, * for the simple loop with k = 4, p, = 0 and X4= ,w, = 1

Y4 P2 P3 Relative weights - wt:w2:w3:w4

1:l:l:l 4:1:1:4 49:1:1:49 2:l:l:l 33:33:1:33

5.00 0.00 0.00 0.5788 0.7305 0.7955 0.5287 0.6913

0.00 2.50 0.5642 0.7168 0.7948 0.5510 0.6925

0.00 5.00 0.6832 0.7424 0.7955 0.6805 0.6957

2.50 2.50 0.4875 0.6961 0.7942 0.5355 0.6154

2.50 5.00 0.5642 0.7168 0.7948 0.6362 0.6165

5.00 5.00 0.5788 0.7305 0.7955 0.6928 0.6913

Average a 0.5591 0.7162 0.7948 0.5904 0.6544

% of max power b 70.3 90.1 100 74.3 82.3

7.00 0.00 0.00 0.845 1 0.9403 0.9653 0.7951 0.9209

0.00 3.50 0.8278 0.9313 0.9650 0.8120 0.9216

0.00 7.00 0.9243 0.9468 0.9654 0.9203 0.9238

3.50 3.50 0.7364 0.9166 0.9647 0.7877 0.8601

3.50 7.00 0.8278 0.9313 0.9650 0.8850 0.8613

7.00 7.00 0.845 1 0.9403 0.9653 0.8336 0.9209

Average a 0.8161 0.9304 0.9650 0.8308 0.8913

% of max power b 84.6 96.4 100 86.1 92.4

a Average power over the alternative space was computed by the method described in Section 5.

b Ratio of the average power to the maximum power (in %).

Table 2

Power of the likelihood ratio test of Sat for the simple loop with k = 4, CL, = 0 and X:= tn, = n = 20

\r nCL4 \r nF2 hF3

5.00 0.00 0.00

0.00 2.50

0.00 5.00

2.50 2.50

2.50 5.00

5.00 5.00

Average a % of max power b

7.00 0.00 0.00

0.00 3.50

0.00 7.00

3.50 3.50

3.50 7.00 7.00 7.00

Sample size vector

(5, 5,5,5) (6,4,4, 6) (7, 3, 3,7) (8, 2,2, 81

0.5249 0.5866 0.6395 0.6859

0.5167 0.5731 0.6262 0.6768

0.6204 0.6430 0.6680 0.6967

0.4549 0.5278 0.5971 0.6619

0.5167 0.5731 0.6262 0.6768

0.5249 0.5866 0.6395 0.6859

0.5130 0.5709 0.6252 0.6765

64.5 71.8 78.7 85.1

0.7907 0.8455 0.8843 0.9127

0.7771 0.8290 0.8709 0.9049

0.8812 0.8934 0.9063 0.9203

0.6926 0.7752 0.8410 0.8917

0.7771 0.8290 0.8709 0.9049

0.7907 0.8455 0.8843 0.9127

Average a 0.7677 0.8241 0.8688 % of max power b 79.6 85.4 90.0

a Average power over the alternative space was computed by the method described in Section 5. b Ratio of average power relative to the maximum power (in percentages) for cr* known case in Table 1.

0.9043

93.7

provide good power over a range of spacings in testing for the presence of a simple tree ordering (Singh, Schell and Wright, 19891, while the weighting 4:1:1:4 weights the two intermediate treatments and the two extreme treatments equally, but giving greater weight to the extreme treatments. Among these

261


A G 0 1

l-42

Fig. 1. Distribution of selected points in the alternative space for the simple loop order restriction where I*, = 0 and kq = 1.

weight&., 49:1:1:49 has the highest power for all spacings considered followed by the weighting 4:1:1:4. The weighting 33:33:1:33 has uniformly higher power than the 1:l:l:l weighting, while the relative power of the weighting 2:l:l:l and the previous two weightings depends upon the spacing considered. These results suggest that a simple loop design should be abandoned in favor of a test of pcL1 vs. pd. While this is preferable from a power standpoint, it does not allow for estimation of all four treatment groups. An alternative rationale for treatment allocation is to allow for estimation of mean levels of the treatment groups with equal error for each of the means under H,. Such an allocation would require that w, = w, and w2 = wa, due to symmetry. Although the exact details are under investigation, it seems certain that this is achieved for some vector w, where w, > w2.

We can estimate the average power over the entire alternative space where p1 and pq differ by a fixed distance by taking weighted averages of six powers obtained using the mean spacing vectors chosen in Tables 1 and 2. We now describe our method for weighting these powers. Figure 1 shows a distribution of points in the alternative space of the simple loop where pcL1 = 0 and pq = 1. The spacings given in Tables 1 and 2 represent vector multiples of these labelled points. The upper triangle ACF has been divided into the square BDEC and two triangular regions ABD and DEF. Due to the symmetries of the simple loop order restriction, the lower triangle AFG has the same average power if either w, = w, or w2 = w3. All weight vectors in Tables 1 and 2 satisfy at least one of the equalities above. Estimating the power in these three regions by averaging the vertices and weighting the square BDEC twice that of each triangle (ADB and DEF) yields the formula: average power = (2A + 5B + 3C + 7D + 5E + 2F)/24. Comparison of the average powers demonstrates that the differences mentioned above can be quite substantial. For example, when u2 is known and p.4 = 5, the 4:1:1:4 weighting has an average power that is 28% greater than the equal weights power (Table 1). Under the same conditions, we can see the reduction in power that results from adding additional treatments to an equal weights design. The average power of the equal weight design on three treatment is 82% of the power for a comparison of two treatments and the equal weights design for four treatment groups has 70% of the power of the two group comparison. Comparing the first columns in Tables 1 and 2 we note that for the equal weights case with a sample of size 5 per group and a2 unknown, the power is roughly 7% lower and when a2 is known.

262


6. Comparison with other order restrictions

The use of order restricted analyses depends on the partial order assumptions made. In the discussion below let H; = {pcLi: p, G pL2 G ,LL~ G p4}, that is, Hi is the set of means that satisfies the simple order restriction. Note that Hi c H, and that H, - Hi = (II,: p, < p.3 G pz < p4). It is already well established (Robertson et al., 1988, pp. 86-98) that, if the partial order assumption is true, restricted tests are more powerful than unrestricted chi-square tests. There is a hierarchy of order restrictions, with the simple loop order being in between the two most widely studied order restrictions: the simple tree and the simple order. Thus it is not surprising that the power for a given spacing in Hi and weight vector will exceed the less restrictive simple tree order (which assumes that p, < pj, i = 2,. . . , k) and be less than that of the simple order. We will use the average power obtained from Table 1 as a rough way to compare the power of the three order restrictions over Hi. Table 3 presents these average powers for the three order restrictions and the chi-square test. When pq = 5, the average power of the simple tree design exceeds that of the chi-square test by lo-12% for various weight vectors. The simple loop design outperforms the simple tree by 7-14%, but is within 2% of the average power of the simple order design. The differences in power are somewhat lower when pq = 7. For k = 4 treatments, the simple loop has 5

restrictions (i.e. p1 G p2, P, G pl, P, G pcLq, p2 G pu,, pL3 G ~~1, compared to 6 for the simple order and 3 for the simple tree. Hence, it is not surprising that the power of the simple loop design is closer to the simple order power than to the simple tree power.

Table 3

Comparison of the average power a of Xi, for three different order restrictions ’

cp=

and different weight vectors with k = 4, CL, = 0, ,w, = 1

IL4 Order restriction Relative weight vector

1:l:l:l 4:1:1:4 49:1:1:49 2:l:l:l 33:33:1:33

5.00 Chi-square 0.3791 0.4839 0.5319 0.3966 0.4270

Simple tree 0.4829 0.6085 0.6560 0.5184 0.5426

Simple loop 0.5591 0.7162 0.7948 0.5904 0.6544

Simple order 0.5824 0.7255 0.7958 0.6100 0.6587

7.00 Chi-square 0.6630 0.7973 0.8443 0.6870 0.7302

Simple tree 0.7561 0.8773 0.9124 0.7876 0.8222

Simple loop 0.8161 0.9304 0.9650 0.8308 0.8913

Simple order 0.8313 0.9345 0.9653 0.8538 0.8935

a Average power over the alternative space was computed by the method described in Section 5

b The powers for the individual mean spacings are partly given in Singh, Schell and Wright (1989) for the simple tree. The results for the simple order are unpublished.

Appendix

The proofs of Theorem 1 and 2 are like those given for the case of simple order or simple tree for k = 4 and unequal weights (Singh and Wright, 1989; Singh, Schell and Wright, 1989). Since they are somewhat involved, we provide sketches of the proofs here.

Proof of Theorem 1. Assume that the random variable L denotes the number of distinct values in ii. The power function, V&L), is obtained by computing C;‘=,P,(L = 1, j?,‘, a C,) and ~&.~u) is obtained in an

263


analogous manner. Let

5 = [ (w~~~2)]“2(x”‘);

v, = w4 l/2

(wl+w2+w3)w 1 [ (w1+w2+w3)x4-wlxl-w2x2-w3x3 ) 1

(A.11

(A-2)

(A.3)

and for i = 4, 5,. . . , 10, let V be defined as Ai (see (2.6)) with pj replaced by zj for j = 1, 2, 3, 4. It should be noted that V, m N(A,, 1) for i = 1, 2,. . . , 10; VI, V, and V, are independent; V,, V, and V, are independent; VI, V, ‘and V,0 are independent; and Vt + V; + V,’ = ,y&.

Note that the case [L = 41 is the union of two disjojnt subcases: (i) [_?, - x1 > 0, x3 - x2 > 0, ~,-~,~0],and(ii)[~,-~,>0,~,-~,>0,~,-~,>01.1ncase~i~,~f2=0and~~,=V~+V~+ Vz. The contribution to X,‘, is given by P<V, > 0, V2 - VI/y > 0, V3 - SV2 > 0, Vt + V: + V: > Cl).

Now, we make a transformation to spherical coordinates:

V, =Rcose, I/,=Rsinesinq, I/,=Rsintlcosq, O<R<m, O<t?<nandO<q<2T.

(A.4)

Then, ,& = R2, J = R2 sin 8, and [V,>O, V2-VJy>O, V,-SV,>O, V,2+V~+V~~C,l=[cos~~ 0, tan e > l/(y sin 71, tan 77 < l/6, R2 > C,]. The remaining steps in the proof are like those in Bartholomew (1961). The limits of integration are: (i) 0 < 77 < tan-‘(l/6), and (ii) tan-‘(l/(r sin 7)) < 0 < $r; and the required probability is the first term in (2.7). The contribution to the power function of

Xi1 from the second subcase is obtained by interchanging (w2, p2> and (wg, ,u3) in the first subcase, and the required probability is the second term in (2.7).

The case [L = 31 is the union of four disjoint cases, namely, (i> jZ1 = jJiZ < j& < jX4, (ii) j& = FL3 < E2 < c4,_(iii) F1 <CL2 <_jZ3 = jZ4 and (iv) FcLI < ,ELg < jZZ = jZTi4. In case (i), [X2 -Xi < 0, (W, + W7,)Z3 - WIXl - w,X,>O, X4-X,>O]. ,&= Vz+ V32 and consequently ,Fr2- 1. 2 - V2 The contribution to the power function of X,‘, in this case is P[ V, G O]P[V, > 0, V, - SV, > 0, Vf + V3’ > CT,]. Changing to polar coordinates in the second probability, i.e., V, = R cos 8, V, = R sin 8, 0 G 8 < 2~ and 0 <R < ~0, the above probability becomes

,+:+A:)/2 @(-A,)P[cos8>0,sine-6cose>O, R2>C1] =@(-A,) 2~ ~~~~,s$(Ar,Ci)de

where A, = A, cos 6’ + A, sin 8. The contribution to the power function of if, in this case is

P[V,<O, V~~C2]P[V2>0, v,-SV,>O]

=@(-AI-~)/_q\@[A3-++A2)]q5(x)dx. 2

The contributions to the power functions arising from case (ii) can be obtained from the case (i) above by interchanging (w,, p2) and (wg, p3). The proofs for L = 3 in case (iii) and (iv) are similar.

The case of [L = 21 is also the union of four disjoint cases, namely, (i) El = p2 = F3 <Pa, (ii) ~i~<~~=~~=~~,(iii)~~=~~<~~=~~ and(iv)E,=jZ,<jiL2=ii4. The case (i) is characterized (cf.

264

Volume 14. Number 4 STAl%TICS & PROBABILITY LETTERS 17 July 1992

the minimum lower sets algorithm in Robertson et al., 1988, pp. 24-25) by

min X,, [

WJ, + WJ, WJ, + w3x3 1 w,x, + w*x* + wJ3 w,+w, ’ a WI fw3 w, + w2 + w3 <x,.

In this case, X,“, = Vj and consequently xiz -* = V: + Vz. The contribution to the power function of X,‘, is given by

which is the ninth term in (2.7). The contribution to the power function of Xf, in this case is

P[ V, > O]P[ V* < 0, {w2( wi + w2 + w3)}“*V, + (w,w3)“*1/* < 0,

{Wi( wi + w2 + w3)}1’2Vi - ( w*w3)1’2V2 =G 0, v: + V2’ >, cz] . Changing to polar coordinates, i.e. Vi = R cos 8, V, = R sin 8, 0 G 8 < 21r, 0 < R < m, the above probability reduces to

which is the fourth term in (2.8). The proofs for cases (ii), (iii) and (iv) are similar. If L = 1, then pi = jL2 = jY3 = jZLq = ii, and therefore, X,‘, = 0 and XF2 = V: + L’; + V;?. Now, we see

that [L = 11 if and only if

min X,,

[

WJ, + w*x* WJ, + w,x, w,x, + w*x* + w& w,+w, ’ w,+wj ’ w,+w,+w, 1

WJ, + w,x* + w& + wJT‘$ >

WI + w2 + w3 + w, ’

or equivalently that

w,x, + w,x, + w_J‘j - (w* -i- wj + w,)X, < 0,

(w, + w2)(w3X3 + w&) - (Wj + w4)(wJ, + w2X2) =G 0,

(~~+w,)(w~~~+w~~~)-(w~+w,)(w,~,+w,~~)~O and V3<0.

Writing these inequalities in terms of xi+i -x,, and using the expressions

(w* + w*)

[ 1 1’2 X,-X, = VI,

WlW2

X,-X*= [

(w1+w2+ww3) l’* wdw1+ w2) I [

l/2 v2 - Wl W2(WI + w2) 1 c f

[

W I/* X,-X3= wdw1+ w2 + %) I [ 5 - Wl +w2

w3( WI + w2 + w3) 1 l/2 v2 9

(A.3

265


and simplifying, (A.51 can be expressed as

w4 l/2

1 [ w3 w2 1 l/2

(WI +w,+w,jw v3 +

(WI + W2)(WI + w2 + w3) wdw1+ w2)

v, GO,

w3 l/2

(WI +w2)(w1 +w,+w,) 1 v,<o, (A4

w3 1 l/2

(w,+w2)(wl+w2+w3) w21/2

[

w4 I l/2

+ ( w1+ w2 + w3)W

(WI +w3)v3Goo1

and V, G 0.

Now, make the transformation to spherical coordinates defined by (A.4) above. Then, J = R2 sin 0 and

x12- 1 -’ - V2 + V22 + Vx2 = R2. The limits of integration are: r - a2 G n < $rr -(Ye, 0 < 0 < min(A,, A2) where cy2, cy3, A, and A, are defined in (2.6). The remainder of the proof is like that given for X,‘, with k = I = 4, and this gives the first term in (2.8). •I

Proof of Theorem 2. As in the proof given above, rr$(l_~) is obtained by computing PJS,, 2 C,, L = 11, and summing it over I= 1, 2, 3, 4. Obviously, for [L = 11 this probability is zero. For [L = 41, if2 = 0 and

consequently S,, = vi&/Q. Let h(u) be the density function of u = Q/v. Then, conditioning on U, the first and second terms in (3.5) are obtained from the first and second terms in (2.7). As in the proof of Theorem 1 for the case of [L = 31, there are four subcases to be considered. For the subcase

jZ, = ,i& < j?ij < jY4, we have S,, = v(V,’ + V;‘>/(V,” + Q>, and the required probability is given by P,[V, G 0, V, > 0, V, - SV, > 0, v<V,” + V:)/(Vf + Q> > C,l. Conditioning on V, =x and u = Q/v, and mimicking the argument given for this subcase in the proof of Theorem 1 gives the third term in (3.5). The proof of the other three subcases are similar.

For the case of [L = 21, there are also four subcases to be considered. For the subcase jZI = jIi2 = jIi3 < jZ4, we have S,, = vV;‘/(V12 + Vt + Q), and the contribution to the power function is given by Pp[V2 G 0, V3 > 0, V, - bV2 G 0, vVz/(Vf + Vt + Q) 2 C,]. Conditioning on VI =x, V, = y and u = Q/v, gives the ninth term in (3.5). The proof of the other three subcases are similar.

Note that there is a simple relationship between X& and S,, given by (3.2). Therefore, many -’ properties of the power function of S,, follow from those of xr2. For instance, conditioning on u = Q/v,

~?22(P) = fgs,* >C2]=joxp,[X~2~~C2]du.

The expression for ‘IT; for k = 4 can be obtained by replacing C, by UC, in (2.81, multiplying by h(u) and integrating over (0, m). This completes the proof of the theorem. 0

Acknowledgement

The authors wish to thank Dr. F.T. Wright for use of his one-degree search algorithm and to the reviewers for several helpful suggestions that led to improvements of this paper.

266


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