Testing Non-Linear Ordinal Responses in L2 K Tables

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1 RUHUNA JOURNA OF SCIENCE Vol. 2, September 2007, pp ISSN X 2007 Faculty of Science University of Ruhuna. Testing Non-inear Ordinal Responses in 2 Tables eslie Jayasekara Department of Mathematics, Univerrsity of Ruhuna, Matara 81000, Sri anka. leslie@maths.ruh.ac.lk Takashi Yanagawa Center for Biostatistics, urume University, Japan. yanagawa.takashi@kurume-u.ac.jp Abstract. In comparative studies responses of two populations are often summarized in stratified 2 tables with ordinal categories. A test, called Q Et test, is proposed for testing the homogenuity of the populations against non-linear alternatives in such tables. The asymptotic distributions of proposed test are obtained both under the null and alternative hypothesis. The powers of the Q Et test and extended Mantal test are compared by simulation. ey words: ordinal data, 2 k tables, homogeatiy, nonlinear response, asymptotic distribution, Chi-square tests, confounding variables 1. Introduction Data are often summarized in ordinal 2 tables in comparative medical studies, and statistical tests such as Pearson s chi-squared test (Pearson, 1900), Wilcoxon test(wilcoxon, 1945), Nair s test (Nair, 1986), cumulative chi-squared test (Takeuchi and Hirotsu,1982), and max χ 2 test (Hirotsu,1983) are applied to those data for detecting the difference of two distributions. It is well known that Pearson s chi-squared test has no good powers against ordered alternatives. The Wilcoxon test is specifically designed for testing location difference of two samples; also the tests are asymptotically uniformly most powerful unbiased tests for logistic linear alternatives. Whereas Nair s test is designed for detecting dispersion alternatives. The cumulative chi-squared test and max χ 2 test are ominibus tests developed for a wider class of alternatives including linear and non-linear responses. Here we call the response patterns like A, B and C in Table 1 the linear and the other patterns the non-linear; more specifically, the pattern D, E,, and I respectively called the pattern, pattern,, and pattern. We developed the Q t test (Jayasekara and Yanagawa, 1995; Jayasekara, Nishiyama and Yanagawa, 1999) for non-linear responses in 2 tables. The Q t test is shown to have higher powers than those tests just described when the control and treatment groups show the combination of the patterns of non-linear responses. Now confounding variables such as sex, age, blood pressure and others are involved in medical data and it is important to block their effects on testing. The above statistical tests lack this function and logistic models are conventionally employed. However, as is well known, the result of logistic models depend on the goodness of fit of the models to the data, 18

2 Ruhuna Journal of Science 2, pp , (2007) 19 Table 1 Response probabilities and patterns. Ordered categories Pattern A B C D E F G H I and yet it is not easy to establish the models, in particular, when responses are non-linear and the size of the data is not large. Here we may see the raison d etre of nonparametric tests. As far as we are aware the extended Mantel test (Mantel 1963, indis, Heyman, and och, 1978, Yanagawa 1986)(called EMT test in the sequel) is the only test that has been developed in the sprit. The EMT test adjusts for the effect of the confounding variables by stratification. In this paper we consider the same framework as the EMT test and develop a test for testing the homogenuity against non-linear alternatives. More specifically, considering 2 tables such as those given in Table 2 which have been constructed in the l-th stratum, l = 1, 2,,, to block the effect of confounding variables, we extended the Q t test. It is shown that the extended Q t test has higher power in most cases than EMT test when the alternatives are non-linear. 2. The Test Statistics We suppose in Table 2 that Y l1 = (Y l11,y l12,,y l1k ) and Y l2 = (Y l21,y l22,,y l2k ) are multinomial random vectors independently distributed with parameters n l1,(p l11, p l12,, p l1k ) and n l2,(p l21, p l22,, p l2k ) respectively (l = 1, 2,, ). Suppose that categories B 1, B 2,, B are ordinal (B 1 < B 2 < < B ), and define the odds-ratio of category B k relative to category B 1 by ψ lk = p l11 p l2k /p l21 p l1k (k = 1, 2,, ). The homogenuity of the distributions of the control and treatment groups in the table may be represented by ψ lk = 1 for all k = 1, 2,, and l = 1, 2,,, which we simply denote by ψ 1. Thus the problemma is testing H 0 : ψ 1 against H 1 : ψ lk 1 for some k = 2, and l = 1, 2,,. In particular, considered under the alternatives are the odds ratios derived from the combinations of those linear and non-linear response patterns presented in Table 1. We extend the Q t test(jayasekara and Yanagawa (1995), Jayasekara and Nishiyama(1996) ) for testing H 0 vs. H 1. et c lk be the Wilcoxon scorollarye in the l-th table defined by c l1 = (τ l1 N l )/2 and c lk = Σ k 1 j=1τ l j + (τ li N l )/2 for k = 2, 3,,, where τ lk is the marginal total in Table 2. Note that it is normalized to satisfy Σ k=1τ lk c lk = 0 for l = 1,,. Now for two dimensional vectors a l and b l in l-th stratum we define the inner product of a l and b l by (a l, b l ) = Σ k=1 τ lka lk b lk and the norm of a l by a l = (a l, a l ) 1/2.

3 20 Ruhuna Journal of Science 2, pp , (2007) Table 2 2 table in stratum l, l = 1,,. Ordered Categories Stratum l B 1 B 2 B Total Control Y l11 Y l12 Y l1 n l1 Treatment Y l21 Y l22 Y l2 n l2 Total τ l1 τ l2 τ l N l et c r lk be the r-th power of c lk, and put c lr = (c r l1, c r l2,, c r l), r = 0, 1,, 1. Furthermore let a l0 = c l0 / c lo and a lr = d lr / d lr, where d lr = c lr Σ r 1 j=0(c lr, a l j )a l j, r = 1, 2,, 1. Note that (a lr, a lr ) = { 1 if r = r, 0 if r r, for r, r = 0, 1,, 1. (1) Putting for given tε{1, 2,, 1} A = (a lr ),,2,,;r=1,,t ( t matrix), Y 2 = (Y 12 2 ) ( dimensional vector), and S 2 = n l1n l2 /N l (N l 1), U Et = A Y 2 /S we propose the following Q Et as a test statistic for testing H 0 vs. H 1 : Q Et = U EtU Et for each tε{1, 2,, 1}. et U r be the r-th elemmaent of U Et, then we have U r = and the Q Et may represented as follows: a lry l2 /S, (2) Q Et = U 2 1 +U U2 t. Remark: Q Et is identical to the test statistic of EMT test when t = 1, and to the Wilcoxon test statistic (Wilcoxon, 1945) when t = 1 and = 1. Now under H 0, the conditional distribution of Y l2 given C l = {n l1, n l2, τ l1,, τ l } is multiple hypergeometric with E[Y l2k C l ] = n l2 τ lk /N l Cov[Y l2k,y l2k C l ] = n l1n l2 Nl 2(N l 1) τ lk(δ jk N l τ lk ), for k, k = 1,,, where δ kk = 1 if k = k and 0 otherwise. THEOREM 1. Under H 0, the elemmaents of U Et, i.e., U r, r = 1, 2,,t, are uncorollaryrelated with zero mean and unit variance when conditioned on C = {C l, l = 1,, }. Proof. We first show E[U Et C] = 0. Putting τ l = (τ l1,, τ l ), we have fro a lr τ l = 0. (3)

4 Ruhuna Journal of Science 2, pp , (2007) 21 Thus E[U Et C] = A te[y 2 C]/S = A t(n 12 τ 1/N 1,, n 2 τ /N ) /S = = 0. ( n l2 a l1 τ l/n l,, n l2 a lt τ l/n l ) /S We next compute the conditional covariance matrix of U Et. Since Y l2, l = 1,,, are independent, the conditional covariance matrix V(U Et C) can be expressed as, Since V(U Et C) = A tv(y 2 C)A t /S 2 V(Y l2 C) = A t V(U Et C) = a lr V(Y l2 C)a lr = it follows from (3) that Thus fro Therefore from (4), we have V(Y 2 C) A t/s 2 ( a lr V(Y l2 C)a lr ) /S 2 for r, r = 1,,t. (4) τ l1 n l1 n l2 Nl 2 (N l 1) [N la. 0 lr 0. a lrv(y l2 C)a lr = τ lk n l1 n l2 N l (N l 1) (a lr, a lr ). a lr a lr τ lτ l a lr ], { 1 if r = a lr V(Y l2 C)a r, lr /S2 = 0 if r r, r, r = 0, 1,, 1. V(U Et C) = I t. 3. Asymptotic Distributions Theorem 1 shows that the elemmaents of Q Et are uncorollaryrelated and furthermore from (2) they are linear combinations of Y l2 = (Y l21,,y l2 ). However, their weight vectors, a lr s, depends on N l, which makes the asymptotic theory not straightforward. We assume that when N l the marginal totals n li and τ lk for l = 1,, satisfy: (A1) n li /N l γ li, 0 < γ li < 1, for i = 1, 2, and τ lk /N l ρ lk, 0 < ρ lk < 1, for k = 1, 2,,. To begin with we review the normal approximation of a multiple hypergeometric distribution.

5 22 Ruhuna Journal of Science 2, pp , (2007) 3.1. Normal Approximation of a Multiple Hypergeometric Distribution Plackett (1981) showed that when assumption (A1) is satisfied the asymptotic conditional distribution of X l = (Y l22,,y l2 ) given C l = {n l1, n l2, τ l1,, τ l }, is a 1 dimensional normal with mean m l2 and covariance matrix V l, where m l2 = (m l22,, m l2k ) and V 1 (σ l jk ) with σ lkk = m 1 l11 + m 1 l21 + (m 1 l1k + m 1 l = l2k )δ kk, for k, k = 2,, and l = 1,,. Here the sequence {m lik }, i = 1, 2; k = 1, 2,,, is determined uniquely by equations k=1 m lik = n li, 2 i=1 m lik = τ lk, and m l11 m l2k /m l21 m l1k = ψ lk, for i = 1, 2; k = 1, 2,, and l = 1,,. It is known (Sinkhorn, 1967) that the sequence may be obtained by the following iterative scaling procedure: l1k = n l1, k = 1, 2,, l21 = n l2 [1 + j=2 (ψ l j 1)/] l2k = m (2) lik m (3) lik m (2h) lik m (2h+1) lik n l2 ψ lk [1 + j=2 (ψ l j 1)/], k = 2,, = m(1) lik τ lk l.k = m(2) lik n li m (2) li....,, τ lk = m(2h 1) lik m (2h 1) l.k n li = m(2h) lik m (2h) li.,, h = 1, 2,, and l = 1,, Asymptotic Distributions Under H 0 We first evaluate the weight, a lrk. We write N 1/2 l a lrk = O(1) if and only if N 1/2 l a lrk tends to a constant as N. EMMA 1. If (A1) is satisfied, then (i) N 1 l c lrk = O(1), where c lrk = c r lk, is the r-th power of the k-th Wilcoxon scorollarye in the l-th table, for r = 1, 2,, 1, k = 1, 2,, and l = 1,,. (ii) et a l0k be the k-th elemmaent of a l0. Then N r l (c lr, a l0 )a l0k = O(1), for r = 1, 2,, 1, k = 1, 2,, and l = 1,,. (iii) et d lvk be the k-th component of d lv. If N v l d lvk = O(1), k = 1, 2,,, then for any v = 1, 2,, we have (a) N 2v 1 l d lv 2 = O(1), (b) N r l (c lr, d lv )d lvk / d lv 2 = O(1), l = 1,,. (iv) N r l d lrk = O(1) for r = 1, 2,, 1, k = 1, 2,, and l = 1,,. (v) N 1/2 l a lrk = O(1) for r = 1, 2,, 1, k = 1, 2,, and l = 1,,. Proof. (i) By the definition of c lk, and from (A1), we may get N 1 l c lk = O(1) for l = 1,,. Thus it is obvious that N r l c r lk = O(1). (ii) By the definition of a l0 we have a l0k =

6 Ruhuna Journal of Science 2, pp , (2007) 23 1/N 1/2 l for all k. So from (i) we obtain N (r+1/2) l (c lr, a l0 ) = O(1). Thus we have (ii). (iii) (a) The result may be obtained by the definition of d lv. (b) Expanding the inner product (c lr, d lv ) and applying (i) we may show N (r+l+1) l (c lr, d lv ) = O(1). Now using (a), the result follows. (iv) To prove this result we use induction on r. In case of r = 1, d l1k = c l1k (c l1, a l0 )a l0k, for k = 1, 2,,. Applying (i) and (ii), it follows that N 1 l d l1k = O(1) for k = 1, 2,,. Suppose that the result is true for r = 1, 2,, m 1. Since m 1 d lm = c lm j=0 (c lm, a l j )a l j, = c lm (c lm, a l0 )a l0 m 1 j=1 d l j (c lm, d l j ) d l j 2, it follows that N m l d lmk = O(1) from (i), (ii) and (iii). So the result is true for r = m. Thus by the induction the result follows. (v) From the definition of a lr and also by (iv) the result is straightforward. Next, we consider the asymptotic distribution of the test statistics under H 0. To apply the normal approximation in section 3.1 we represent the t dimensional vector U Et by: U Et = B W/S, (5) where B = (b lr ), b lr = (a lr2 a lr1,, a lr a lr1 ) N 1/2 l, l = 1,, ; r = 1, 2,,t, W = (W 1,W 2,,W ), W l = N 1/2 l (X l n l2 τ l /N l ). THEOREM 2. Under H 0, Q Et is asymptotically distributed as a chi-squared distribution with t degrees of freedom as N, l = 1, 2,,. Proof. From section 3.1 we have m lik = n li τ lk /N l, under H 0. Thus the conditional distribution of W i given C l = {n l1, n l2, τ l1,, τ l } converges in distribution to N 1 (0, l0 ) as N l, where 1 l0 = (σ l jk0), j, k = 2,,, with σ l jk0 = [ρ 1 l1 + δ jkρ 1 lk ]/(γ l1γ l2 ). Furthermore, since N 1/2 l a lrk = O(1) from emmama 1(v), we have b lr = O(1). Thus as N l, l = 1,,, it will be easy to show that U Et = B W/S converges in distribution to a t dimensional normal distribution with mean zero and the covariance matrix 10 V[U Et ] = B B/S2 (6) Now putting M l = 0 ρ l2 (N l ρ l2 ) ρ l2 ρ l3 ρ l2 ρ l ρ l3 ρ l2 ρ l3 (N l ρ l3 ) ρ l3 ρ l... ρ l ρ l2 ρ l ρ l3 ρ l (N l ρ l ) γ l1γ l2,

7 24 Ruhuna Journal of Science 2, pp , (2007) we may show Furthermore, fro B 1 M l l0 M = I 1. M B/S2 I t, where means that the ratio of the both hands side tends to one as N l, l = 1, 2,,. Thus from (6) V[U Et ] I t, and Q Et = U EtU Et follows asymptotically a chi-squared distribution with t degrees of freedom Asymptotic Distribution Under Contiguous Alternatives In this section we obtain the asymptotic distribution of Q Et under alternative hypothesis H 1 : ψ lk = 1 + A lk /N 1/2 l, for k = 2, 3,,, where A lk is a constant. EMMA 2. Under H 1, we may represent m lik = m 0 lik + N1/2 l η lik + O(N 1/2 l ) for i = 1, 2, k = 1, 2,, and l = 1,,, where m 0 lik = n liτ k /N l is the asymptotic mean under H 0, and k = 2, 3,,. η i1 = ( 1) i+1 N 1/2 l γ l1 γ l2 ρ l1 η ik = ( 1) i N 1/2 l j=2 γ l1 γ l2 ρ lk [ψ lk 1 (ψ l j 1)ψ l j j=2 (ψ l j 1)ψ l j ] Proof. Adopting the iterative scaling algoritheorem in section 3.1, we have the following expressions for l1k, l21 l2k, m (2) li1 and m (2) lik, under H 1. l1k = n l1 l21 = n l2 [1 j=2 l2k = n l2 m (2) li1 m (2) lik [ψ lk j=2 (ψ l j 1) (ψ l j 1) = m 0 li1 +( 1) i+1 N l γ l1 γ l2 ρ l1 + o(n 1/2 l )] + o(n 1/2 l )], k = 2, 3,,, j=2 = m 0 lik +( 1) i N l γ l1 γ l2 ρ lk [ψ lk 1 (ψ l j 1) j=2 + o(n 1/2 l ), (ψ l j 1) ]+o(n 1/2 l ).

8 Ruhuna Journal of Science 2, pp , (2007) 25 Using mathematical induction on v, we may show m (v) lik = m0 lik + N1/2 l η lik + o(n 1/2 l ), k = 1, 2,,, and v = 3, 4,. Thus we have the desired results. THEOREM 3. Under H 1, Q Et is asymptotically distributed as a non-central chi-squared distribution with t degrees of freedom. The noncentrality parameter is given by λ = t r=1 δ2 r, where δ r = N lγ l1 γ l2 k=2 a lrkρ lk (ψ lk 1)/S. Proof. From Section 3.1 and emmama 2 it follows that under H 1, the conditional distribution of W l given C l = {n l1, n l2, τ l1,, τ l } converges in distribution to N 1 (η l2, l0 ), where η l2 = (η l22,, η l2 ), and l0 is that given in the proof of Theorem 2. Thus under H 1, U Et = B W/S converges in distribution to t dimentional normal distribution with mean δ Et = B (η 12,, η 2) /S and covariance matrix V[U Et ], which is shown to be I t in the proof of Theorem 2. The r-th elemmaent of δ Et, say δ r, is obtained as: From (7) and ψ l1 = 1, we have δ r = δ r = k=2 N l γ l1 γ l2 k=2 N 1/2 l (a lrk a lr1 )η l2k /S. a lrk ρ lk (ψ lk 1)/S. The theorem is immediately obtained from these results. COROARY 1. The power of U 2 r is approximately maximized when ln ψ lk = β l a lrk, k = 1, 2,,, for some constant β l, l = 1, 2,,. Proof. From the proof of Theorem 3 it follows that Ur 2 follows asymptotically a noncentral chi-squared distribution with one degree of freedom with noncentral parameter δ 2 r. Thus the asymptotic power of Ur 2 for testing H 0 vs. H 1 may be approximated by P(U 2 r χ 2 1(α) H 0 ) Φ(δ r χ(α)), where Φ is the cdf of a standard normal distribution. Since δ r may be represented by δ r = γ l1 γ l2 (a lr, ψ l 1)/S, this power is maximized when ψ 1 = β l a lr, that is when ln ψ lk β l a lrk for some constant β l. From the corollaryollary the statistic Q Et = U1 2 + U U t 2 is viewed as a sum of the statistics that are asymptotically optimum against the alternatives which are expressed as log linearities of the odds ratios with scorollarye a lrk, the standardized r-th power of the Wilcoxon scorollarye.

9 26 Ruhuna Journal of Science 2, pp , (2007) 4. Simulation Studies Simulation was conducted to compare the Q Et, t = 1, 2, 3, 4, test with the EMT test (Mantel 1963, andis, Heyman and och, 1978, Yanagawa 1986). Because the EMT test with Wilcoxon scorollarye is equivalent to the Q E1 test, we herein considered the EMT test with scorollaryes 0, 1, 2,, and 1 assigned to categories B 1, B 2,, and B, respectively. First we assessed Type I error of the Q Et, t = 1, 2, 3, 4 and EMT tests at the significance level α = The response probabilities employed are those listed in Table 1. We considered four strata and combinations of response patterns shown in the first column of Table 3. For example, (,,, ) in the table means that the response probabilities in the 1st stratum are p 111 = p 121 = 0.1, p 112 = p 122 = 0.15, p 113 = p 123 = 0.2, p 114 = p 124 = 0.25, p 115 = p 125 = 0.3; 2nd stratum are p 211 = p 221 = 0.1, p 212 = p 222 = 0.15, p 213 = p 223 = 0.2, p 214 = p 224 = 0.25, p 215 = p 225 = 0.3; and so on. We generated 10,000, four 2 5 tables for each combination of patterns and computed empirical significance levels when n l1 = n l2 = 60, 80, and 100. The results are listed in Table 3. The table shows that Type I error of the Q Et and EMT tests are close to the nominal level for all combinations of patterns. Second we assessed the powers of the Q Et, t = 1, 2, 3, 4, and EMT tests. We conducted similar simulation as above by using again the response probabilities listed in Table 1. Considering the combinations of pattern of distribution of Y 1 from {(,,, ), (,,, ), (,,, ),,(,,, ),(,,, ),(,,, ),(,,, ), (,,, )} we computed the powers of the tests for all combinations of patterns of each distribution, 48 all together, when n l1 = n l2 = 100, l = 1, 2, 3 and 4. The tests which give the largest and second largest powers are listed in Table 4a, 4b, and 4c. For example, the entry of the 2nd row and 3rd column in Table 4a means that when the pattern of Y 1 is and that of Y 2 is the test with the largest power is Q E4 followed by Q E3 ; and the entry of the 2nd row and 4th column in Table 4c means that when the pattern of Y 1 is and that of Y 2 then the test with the largest power is Q E4 followed by Q E3. The tests in the tables show that those tests have equal powers. The tables show that in most combinations, 45 among 48, the powers of the class of the Q Et test are larger or equal to than those of the EMT test. Table 5 lists the maximum, mean and minimum values of the powers of each test for 48 combinations of response patterns considered in Table 4. Inspection of the table shows that the mean and minimum powers of the Q Et test dominates the corollaryresponding values of the other tests, and that the maximum powers of the tests are almost equal. 5. Discussion The Q Et test is proposed for testing the homogeneity against non-linear responses in 2 tables. We took into account the combinations of patterns of linear and non-linear responses summarized in Table 1, and shown that the class of Q Et test is superior to the extended Mantel test (Mantel 1963, andis, Heyman and och 1978, Yanagawa 1986). Those non-linear patterns we considered often appear, for example, in Phase III randomized clinical trials for proving the efficacy of a new drug against the active control,in which the efficacy is sometimes categorized as excellent, effective slightly effective, not effective and aggravation. We emphasize that in such example, the response probabilities like 0.15, 0.25, 0.1, 0.3 and

10 Ruhuna Journal of Science 2, pp , (2007) 27 Table 3 Estimated Type I errors of the Q Et, t = 1, 2, 3, 4, and extended Mantel test (EMT). Pattern Sample size Estimated Type I error levels n l1 = n l2, l = 1, 2, 3, 4 Q E1 Q E2 Q E3 Q E4 EMT (,,, ) (,,, ) (,,, ) (,,, ) (,,, ) (,,, ) (,,, ) (,,, ) (,,, ) (,,, ) (,,, ) (,,, ) (,,, )}

11 28 Ruhuna Journal of Science 2, pp , (2007) Table 4 Tests which give the largest and second largest powers: Y 1 (,,, ) (,,, ) (,,, ) (,,, ) (,,, ) Q E4, Q E3 Q E4, Q E3 Q E3, Q E4 Q E4, Q E2 (,,, ) Q E4, Q E1 (Q E1, Q E2, Q E3, (Q E1, Q E2, Q E3, EMT, Q E2 Q E4, EMT) Q E4, EMT) (,,, ) (Q E1, Q E2, Q E3, (Q E1, Q E2, Q E3, (Q E1, Q E2, Q E3, (Q E1, Q E2, Q E3, Q E4, EMT) Q E4, EMT) Q E4, EMT) Q E4, EMT) (,,, ) Q E2, Q E4 (Q E2, Q E3, Q E2, Q E3 (Q E2, Q E3, Q E4, EMT) Q E2, Q E1 ) (,,, ) (Q E2, Q E3, Q E2, Q E3 (Q E2, Q E3, Q E3, Q E2 Q E4, EMT) Q E4, Q E1 ) (,,, ) Q E3, Q E4 (Q E3, Q E4, Q E2 ) (Q E3, Q E4, Q E2 ) Q E4, Q E3 (,,, (Q E3, Q E4, Q E2 ) Q E4, Q E3 (Q E3, Q E4, EMT) Q E3, Q E4 (,,, ) Q E4, Q E3 Q E4, Q E3 Q E4, Q E3 Q E4, Q E3 (,,, ) Q E4, Q E3 Q E4, Q E3 Q E4, Q E3 Q E4, Q E2 (,,, ) - Q E4, Q E3 Q E4, Q E1 Q E4, Q E3 (,,, ) Q E4, Q E3 - Q E4, Q E3 EMT, Q E3 (,,, ) Q E4, Q E1 Q E1, Q E2 - Q E4, Q E3 (,,, ) Q E4, Q E3 EMT, Q E1 Q E4, Q E3 - Y 2 Table 5 The maximum, mean and the minimum powers of the tests for 48 combinations of the patterns in Table 4. Q E1 Q E2 Q E3 Q E4 EMT Max Mean Min ,i.e. pattern is not unreasonable. It is suggested in the simulation that when all combinations of those response patterns are taken into account the Q E4 test is good choice. The Q Et is shown to be the sum of U 2 r, r = 1, 2,,t, that are asymptotically optimum against the alternatives which are expressed as log linearities of the odds ratios with scorollarye a lrk, the standardized r-th power of the Wilcoxon scorollarye. References Pearson,. (1900), On criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling, Philos. Mag., series 5, 50, (Reprinted 1948 in arl Pearsons Early statistical papers, ed. By E. S. Pearson, Cambridge University Press.) Nair, V. N. (1986), On Testing in Industrial Experiments with Ordered Categorical Data, Journal of American Statistical Association, 58,

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