RJS-2007-jayasekara-yanagawa.dvi RUHUNA JOURNAL OF SCIENCE Vol. 2, September 2007, pp. 18–29 http://www.ruh.ac.lk/rjs/ I S S N 1800-279X © 2007 Faculty of Science University of Ruhuna. Testing Non-Linear Ordinal Responses in L2 × K Tables Leslie Jayasekara Department of Mathematics, Univerrsity of Ruhuna, Matara 81000, Sri Lanka. leslie@maths.ruh.ac.lk Takashi Yanagawa Center for Biostatistics, Kurume University, Japan. yanagawa.takashi@kurume-u.ac.jp Abstract. In comparative studies responses of two populations are often summarized in stratified 2 × K tables with ordinal categories. A test, called QEt test, is proposed for testing the homogenuity of the populations against non-linear alternatives in such tables. The asymptotic distributions of pro- posed test are obtained both under the null and alternative hypothesis. The powers of the QEt test and extended Mantal test are compared by simulation. Key words : ordinal data, L2 × k tables, homogeatiy, nonlinear response, asymptotic distribution, Chi-square tests, confounding variables 1. Introduction Data are often summarized in ordinal 2 × K tables in comparative medical studies, and sta- tistical 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 χ2 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 differ- ence 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 χ2 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 Qt test (Jayasekara and Yanagawa, 1995; Jayasekara, Nishiyama and Yanagawa, 1999) for non-linear responses in 2 × K tables. The Qt 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 Jayasekara and Yanagawa: Testing Non-Linear Ordinal Responses ... Ruhuna Journal of Science 2, pp. 18–29, (2007) 19 Table 1 Response probabilities and patterns. Ordered categories Pattern 1 2 3 4 5 A 0.2 0.2 0.2 0.2 0.2 B � 0.1 0.15 0.2 0.25 0.3 C Q 0.3 0.25 0.2 0.15 0.1 D ∩ 0.15 0.2 0.3 0.25 0.1 E ∪ 0.25 0.2 0.1 0.15 0.3 F �� � � 0.25 0.1 0.2 0.3 0.15 G �� � � 0.1 0.25 0.2 0.15 0.3 H �� � � �� 0.2 0.1 0.3 0.15 0.25 I �� � � �� 0.15 0.25 0.1 0.3 0.2 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, Lindis, Heyman, and Koch, 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 × K tables such as those given in Table 2 which have been constructed in the l-th stratum, l = 1, 2, ··· , L, to block the effect of confounding variables, we extended the Qt test. It is shown that the extended Qt 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 Yl1 = (Yl11,Yl12, ··· ,Yl1k)′ and Yl2 = (Yl21,Yl22, ··· ,Yl2k)′ are multinomial random vectors independently distributed with parameters nl1, (pl11, pl12, ··· , pl1k) ′ and nl2, (pl21, pl22, ··· , pl2k) ′ respectively (l = 1, 2, ··· , L). Suppose that categories B1, B2, ··· , BK are ordinal (B1 < B2 < ··· < BK ), and define the odds-ratio of category Bk relative to category B1 by ψlk = pl11 pl2k/pl21 pl1k (k = 1, 2, ··· , K). The homogenuity of the distributions of the control and treatment groups in the table may be represented by ψlk = 1 for all k = 1, 2, ··· , K and l = 1, 2, ··· , L, which we simply denote by ψ ≡ 1. Thus the problemma is testing H0 : ψ ≡ 1 against H1 : ψlk 6= 1 for some k = 2, ··· K and l = 1, 2, ··· , L. In particular, considered under the alternatives are the odds ratios derived from the combinations of those linear and non-linear response patterns pre- sented in Table 1. We extend the Qt test(Jayasekara and Yanagawa (1995), Jayasekara and Nishiyama(1996) ) for testing H0 vs. H1. Let clk be the Wilcoxon scorollarye in the l-th table defined by cl1 = (τl1 − Nl )/2 and clk = Σk−1j=1τl j + (τli − Nl )/2 for k = 2, 3, ··· , K, where τlk is the marginal total in Table 2. Note that it is normalized to satisfy ΣKk=1τlk clk = 0 for l = 1, ··· , L. Now for two K dimensional vectors al and bl in l-th stratum we define the inner product of al and bl by (al , bl ) = ΣKk=1τlkalkblk and the norm of al by ‖al‖ = (al, al ) 1/2. Jayasekara and Yanagawa: Testing Non-Linear Ordinal Responses ... 20 Ruhuna Journal of Science 2, pp. 18–29, (2007) Table 2 2 × K table in stratum l, l = 1, ··· , L. Ordered Categories Stratum l B1 B2 ··· BK Total Control Yl11 Yl12 ··· Yl1K nl1 Treatment Yl21 Yl22 ··· Yl2K nl2 Total τl1 τl2 ··· τlK Nl Let crlk be the r-th power of clk, and put clr = (c r l1, c r l2, ··· , c r lK ) ′, r = 0, 1, ··· , K − 1. Furthermore let al0 = cl0/‖clo‖ and alr = dlr/‖dlr‖, where dlr = clr − Σr−1j=0(clr, al j)al j, r = 1, 2, ··· , K − 1. Note that (alr, alr′) = { 1 if r = r′, 0 if r 6= r′, for r, r′ = 0, 1, ··· , K − 1. (1) Putting for given tε{1, 2, ··· , K − 1} A = (alr),l=1,2,··· ,L;r=1,··· ,t (KL × t matrix), Y2 = (Y ′12, ··· ,Y ′ L2) ′ (KL dimensional vector), S2 = ∑Ll=1 nl1nl2/Nl(Nl − 1), and UEt = A ′Y2/S we propose the following QEt as a test statistic for testing H0 vs. H1: QEt = U ′ Et UEt for each tε{1, 2, ··· , K − 1}. Let Ur be the r-th elemmaent of UEt , then we have Ur = L ∑ l=1 a′lrYl2/S, (2) and the QEt may represented as follows: QEt = U 2 1 + U 2 2 + ··· + U 2 t . Remark: QEt is identical to the test statistic of EMT test when t = 1, and to the Wilcoxon test statistic (Wilcoxon, 1945) when t = 1 and L = 1. Now under H0, the conditional distribution of Yl2 given Cl = {nl1, nl2, τl1, ··· , τlK} is multiple hypergeometric with E[Yl2k|Cl ] = nl2τlk/Nl Cov[Yl2k,Yl2k′|Cl ] = nl1nl2 N 2l (Nl − 1) τlk(δ jk′ Nl − τlk′ ), for k, k′ = 1, ··· , K, where δkk′ = 1 if k = k′ and 0 otherwise. THEOREM 1. Under H0, the elemmaents of UEt , i.e., Ur, r = 1, 2, ··· , t, are uncorol- laryrelated with zero mean and unit variance when conditioned on C = {Cl , l = 1, ··· , L}. Proof. We first show E[UEt|C] = 0. Putting τl = (τl1, ··· , τlK )′, we have from (1) a′lr τl = 0. (3) Jayasekara and Yanagawa: Testing Non-Linear Ordinal Responses ... Ruhuna Journal of Science 2, pp. 18–29, (2007) 21 Thus E[UEt|C] = A ′ t E[Y2|C]/S = A′t (n12τ ′ 1/N1, ··· , nL2τ ′ L/NL) ′/S = ( L ∑ l=1 nl2a ′ l1τl /Nl , ··· , L ∑ l=1 nl2a ′ lt τl /Nl )′ /S = 0. We next compute the conditional covariance matrix of UEt . Since Yl2, l = 1, ··· , L, are independent, the conditional covariance matrix V (UEt|C) can be expressed as, V (UEt|C) = A ′ tV (Y2|C)At /S 2 = A′t     V (Y′l2|C) . 0 0 . V (Y′L2|C)     At /S 2 V (UEt|C) = ( L ∑ l=1 a′lrV (Y ′ l2|C)alr′ ) /S2 for r, r′ = 1, ··· , t. (4) Since L ∑ l=1 a′lrV (Y ′ l2|C)alr′ = L ∑ l=1 nl1nl2 N 2l (Nl − 1) [Nl a ′ lr     τl1 . 0 0 . τlk     alr′ − a ′ lr τl τ ′ l alr′ ], it follows from (3) that L ∑ l=1 a′lrV (Y ′ l2|C)alr′ = L ∑ l=1 nl1nl2 Nl (Nl − 1) (alr, alr′ ). Thus from (1) L ∑ l=1 a′lrV (Y ′ l2|C)alr′ /S 2 = { 1 if r = r′, 0 if r 6= r′, r, r′ = 0, 1, ··· , K − 1. Therefore from (4), we have V (UEt|C) = It . 3. Asymptotic Distributions Theorem 1 shows that the elemmaents of QEt are uncorollaryrelated and furthermore from (2) they are linear combinations of Yl2 = (Yl21, ··· ,Yl2K ). However, their weight vectors, alr’s, depends on Nl , which makes the asymptotic theory not straightforward. We assume that when Nl → ∞ the marginal totals nli and τlk for l = 1, ··· L, satisfy: (A1) nli/Nl → γli, 0 < γli < 1, for i = 1, 2, and τlk/Nl → ρlk, 0 < ρlk < 1, for k = 1, 2, ··· , K. To begin with we review the normal approximation of a multiple hypergeometric distribu- tion. Jayasekara and Yanagawa: Testing Non-Linear Ordinal Responses ... 22 Ruhuna Journal of Science 2, pp. 18–29, (2007) 3.1. Normal Approximation of a Multiple Hypergeometric Distribution Plackett (1981) showed that when assumption (A1) is satisfied the asymptotic conditional distribution of Xl = (Yl22, ··· ,Yl2K )′ given Cl = {nl1, nl2, τl1, ··· , τlK}, is a K −1 dimensional normal with mean ml2 and covariance matrix Vl , where ml2 = (ml22, ··· , ml2k)′ and V −1 l = (σl jk) with σlkk′ = m−1l11 + m −1 l21 + (m −1 l1k + m −1 l2k)δkk′ , for k, k ′ = 2, ··· , K and l = 1, ··· , L. Here the sequence {mlik}, i = 1, 2; k = 1, 2, ··· , K, is determined uniquely by equations ∑Kk=1 mlik = nli, ∑ 2 i=1 mlik = τlk , and ml11ml2k/ml21ml1k = ψlk, for i = 1, 2; k = 1, 2, ··· , K and l = 1, ··· , L. It is known (Sinkhorn, 1967) that the sequence may be obtained by the following iterative scaling procedure: m (1) l1k = nl1 K , k = 1, 2, ··· , K m (1) l21 = nl2 K[1 + ∑Kj=2(ψl j − 1)/K] m (1) l2k = nl2ψlk K[1 + ∑Kj=2(ψl j − 1)/K] , k = 2, ··· , K m (2) lik = m (1) lik τlk m (1) l.k , m (3) lik = m (2) lik nli m (2) li. , . . . m (2h) lik = m (2h−1) lik τlk m (2h−1) l.k , m (2h+1) lik = m (2h) lik nli m (2h) li. , h = 1, 2, ··· , and l = 1, ··· , L. 3.2. Asymptotic Distributions Under H0 We first evaluate the weight, alrk. We write N 1/2 l alrk = O(1) if and only if N 1/2 l alrk tends to a constant as N → ∞. LEMMA 1. If (A1) is satisfied, then (i) N−1l clrk = O(1), where clrk = c r lk, is the r-th power of the k-th Wilcoxon scorollarye in the l-th table, for r = 1, 2, ··· , K − 1, k = 1, 2, ··· , K and l = 1, ··· , L. (ii) Let al0k be the k-th elemmaent of al0. Then N −r l (clr, al0)al0k = O(1), for r = 1, 2, ··· , K − 1, k = 1, 2, ··· , K and l = 1, ··· , L. (iii) Let dlvk be the k-th component of dlv. If N −v l dlvk = O(1), k = 1, 2, ··· , K, then for any v = 1, 2, ··· , we have (a) N−2v−1l ‖dlv‖ 2 = O(1), (b) N−rl (clr, dlv)dlvk/‖dlv‖ 2 = O(1), l = 1, ··· , L. (iv) N−rl dlrk = O(1) for r = 1, 2, ··· , K − 1, k = 1, 2, ··· , K and l = 1, ··· , L. (v) N 1/2 l alrk = O(1) for r = 1, 2, ··· , K − 1, k = 1, 2, ··· , K and l = 1, ··· , L. Proof. (i) By the definition of clk, and from (A1), we may get N −1 l clk = O(1) for l = 1, ··· , L. Thus it is obvious that N−rl c r lk = O(1). (ii) By the definition of al0 we have al0k = Jayasekara and Yanagawa: Testing Non-Linear Ordinal Responses ... Ruhuna Journal of Science 2, pp. 18–29, (2007) 23 1/N 1/2 l for all k. So from (i) we obtain N −(r+1/2) l (clr, al0) = O(1). Thus we have (ii). (iii) (a) The result may be obtained by the definition of dlv. (b) Expanding the inner product (clr, dlv) and applying (i) we may show N −(r+l+1) l (clr, dlv) = O(1). Now using (a), the result follows. (iv) To prove this result we use induction on r. In case of r = 1, dl1k = cl1k − (cl1, al0)al0k, for k = 1, 2, ··· , K. Applying (i) and (ii), it follows that N−1l dl1k = O(1) for k = 1, 2, ··· , K. Suppose that the result is true for r = 1, 2, ··· , m − 1. Since dlm = clm − m−1 ∑ j=0 (clm, al j)al j, = clm − (clm, al0)al0 − m−1 ∑ j=1 (clm, dl j) dl j ‖dl j‖2 , it follows that N−ml dlmk = 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 alr and also by (iv) the result is straightforward. Next, we consider the asymptotic distribution of the test statistics under H0. To apply the normal approximation in section 3.1 we represent the t dimensional vector UEt by: UEt = B ′W/S, (5) where B = (blr), blr = (alr2 − alr1, ··· , alrK − alr1)′N 1/2 l , l = 1, ··· , L; r = 1, 2, ··· , t, W = (W ′1 ,W ′ 2 , ··· ,W ′ L) ′, Wl = N −1/2 l (Xl − nl2τl /Nl ). THEOREM 2. Under H0, QEt is asymptotically distributed as a chi-squared distribution with t degrees of freedom as N → ∞, l = 1, 2, ··· , L. Proof. From section 3.1 we have mlik = nliτlk/Nl , under H0. Thus the conditional distri- bution of Wi given Cl = {nl1, nl2, τl1, ··· , τlK} converges in distribution to NK−1(0, ∑l0) as Nl → ∞, where ∑−1l0 = (σl jk0), j, k = 2, ··· , K, with σl jk0 = [ρ −1 l1 + δ jkρ −1 lk ]/(γl1γl2). Further- more, since N 1/2 l alrk = O(1) from Lemmama 1(v), we have blr = O(1). Thus as Nl → ∞, l = 1, ··· , L, it will be easy to show that UEt = B ′W/S converges in distribution to a t dimensional normal distribution with mean zero and the covariance matrix V [UEt ]∞ = B ′     ∑10 . 0 0 . ∑L0     B/S2 (6) Now putting Ml =     ρl2(Nl − ρl2) −ρl2ρl3 ··· −ρl2ρlK −ρl3ρl2 ρl3(Nl − ρl3) ··· −ρl3ρlK . . ··· . −ρlK ρl2 −ρlK ρl3 ··· ρlK (Nl − ρlK )     γl1γl2, Jayasekara and Yanagawa: Testing Non-Linear Ordinal Responses ... 24 Ruhuna Journal of Science 2, pp. 18–29, (2007) we may show Ml −1 ∑ l0 = IK−1. Furthermore, from (1) B′     M1 . 0 0 . ML     B/S2 ∼ It , where ∼ means that the ratio of the both hands side tends to one as Nl → ∞, l = 1, 2, ··· , L. Thus from (6) V [UEt ]∞ ∼ It , and QEt = U ′ Et UEt follows asymptotically a chi-squared distribution with t degrees of free- dom. 3.3. Asymptotic Distribution Under Contiguous Alternatives In this section we obtain the asymptotic distribution of QEt under alternative hypothesis H1 : ψlk = 1 + Alk/N 1/2 l , for k = 2, 3, ··· , K, where Alk is a constant. LEMMA 2. Under H1, we may represent mlik = m 0 lik + N 1/2 l ηlik + O(N 1/2 l ) for i = 1, 2, k = 1, 2, ··· , K and l = 1, ··· , L, where m0lik = nliτk/Nl is the asymptotic mean under H0, and ηi1 = (−1)i+1N 1/2 l γl1γl2ρl1 K ∑ j=2 (ψl j − 1)ψl j ηik = (−1)iN 1/2 l γl1γl2ρlk[ψlk − 1 − K ∑ j=2 (ψl j − 1)ψl j] k = 2, 3, ··· , K. Proof. Adopting the iterative scaling algoritheorem in section 3.1, we have the following expressions for m (1) l1k, m (1) l21m (1) l2k, m (2) li1 and m (2) lik , under H1. m (1) l1k = nl1 K m (1) l21 = nl2 K [1 − K ∑ j=2 (ψl j − 1) K + o(N −1/2 l )] m (1) l2k = nl2 K [ψlk − K ∑ j=2 (ψl j − 1) K + o(N −1/2 l )], k = 2, 3, ··· , K, m (2) li1 = m 0 li1 + (−1) i+1Nl γl1γl2ρl1 K ∑ j=2 (ψl j − 1) K + o(N −1/2 l ), m (2) lik = m 0 lik + (−1) iNl γl1γl2ρlk[ψlk − 1 − K ∑ j=2 (ψl j − 1) K ] + o(N −1/2 l ). Jayasekara and Yanagawa: Testing Non-Linear Ordinal Responses ... Ruhuna Journal of Science 2, pp. 18–29, (2007) 25 Using mathematical induction on v, we may show m (v) lik = m 0 lik + N 1/2 l ηlik + o(N 1/2 l ), k = 1, 2, ··· , K, and v = 3, 4, ··· . Thus we have the desired results. THEOREM 3. Under H1, QEt is asymptotically distributed as a non-central chi-squared distribution with t degrees of freedom. The noncentrality parameter is given by λ = ∑tr=1 δ 2 r , where δr = ∑Ll=1 Nl γl1γl2 ∑ K k=2 alrkρlk(ψlk − 1)/S. Proof. From Section 3.1 and Lemmama 2 it follows that under H1, the conditional distri- bution of Wl given Cl = {nl1, nl2, τl1, ··· , τlK} converges in distribution to NK−1(ηl2, ∑l0), where ηl2 = (ηl22, ··· , ηl2K )′, and ∑l0 is that given in the proof of Theorem 2. Thus under H1, UEt = B ′W/S converges in distribution to t dimentional normal distribution with mean δEt = B′(η′12, ··· , η ′ L2) ′/S and covariance matrix V [UEt ]∞, which is shown to be It in the proof of Theorem 2. The r-th elemmaent of δEt , say δr , is obtained as: δr = L ∑ l=1 K ∑ k=2 N 1/2 l (alrk − alr1)ηl2k/S. From (7) and ψl1 = 1, we have δr = L ∑ l=1 Nl γl1γl2 K ∑ k=2 alrkρlk(ψlk − 1)/S. The theorem is immediately obtained from these results. COROLLARY 1. The power of U 2r is approximately maximized when ln ψlk = βl alrk, k = 1, 2, ··· , K, for some constant βl , l = 1, 2, ··· , L. Proof. From the proof of Theorem 3 it follows that U 2r follows asymptotically a noncentral chi-squared distribution with one degree of freedom with noncentral parameter δ2r . Thus the asymptotic power of U 2r for testing H0 vs. H1 may be approximated by P(U 2r ≥ χ 2 1(α)|H0) ≈ Φ(δr − χ(α)), where Φ is the cdf of a standard normal distribution. Since δr may be represented by δr = ∑Ll=1 γl1γl2(alr, ψl − 1)/S, this power is maximized when ψ − 1 = βl alr, that is when ln ψlk ≈ βl alrk for some constant βl . From the corollaryollary the statistic QEt = U 2 1 + U 2 2 + ··· + U 2 t 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 alrk, the standardized r-th power of the Wilcoxon scorollarye. Jayasekara and Yanagawa: Testing Non-Linear Ordinal Responses ... 26 Ruhuna Journal of Science 2, pp. 18–29, (2007) 4. Simulation Studies Simulation was conducted to compare the QEt , t = 1, 2, 3, 4, test with the EMT test (Man- tel 1963, Landis, Heyman and Koch, 1978, Yanagawa 1986). Because the EMT test with Wilcoxon scorollarye is equivalent to the QE 1 test, we herein considered the EMT test with scorollaryes 0, 1, 2, ··· , and K − 1 assigned to categories B1, B2, ··· , and BK , respectively. First we assessed Type I error of the QEt , t = 1, 2, 3, 4 and EMT tests at the significance level α = 0.05. The response probabilities employed are those listed in Table 1. We consid- ered 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 p111 = p121 = 0.1, p112 = p122 = 0.15, p113 = p123 = 0.2, p114 = p124 = 0.25, p115 = p125 = 0.3; 2nd stratum are p211 = p221 = 0.1, p212 = p222 = 0.15, p213 = p223 = 0.2, p214 = p224 = 0.25, p215 = p225 = 0.3; and so on. We generated 10,000, four 2 × 5 tables for each combination of patterns and computed empirical significance levels when nl1 = nl2 = 60, 80, and 100. The results are listed in Table 3. The table shows that Type I error of the QEt and EMT tests are close to the nominal level for all combinations of patterns. Second we assessed the powers of the QEt , 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 Y1 from {( , , , ), (� , � , � , � ), ( Q, Q, Q, Q), ··· ,(�� � � ��, �� � � ��, �� � � ��, �� � � ��),(� , ∩, � � �� , �� � � ��),( Q, ∪, �� � �, �� � � �� ) ,(∩, �� �� , � � � � ��, �� � � �� ), (∪, �� �� , � � � �, � � � � �� )} we computed the powers of the tests for all combinations of patterns of each distribution, 48 all together, when nl1 = nl2 = 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 Y1 is and that of Y2 is the test with the largest power is QE 4 followed by QE 3; and the entry of the 2nd row and 4th column in Table 4c means that when the pattern of Y1 is and that of Y2 then the test with the largest power is QE 4 followed by QE 3. 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 QEt 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 QEt test dominates the corollaryresponding values of the other tests, and that the maximum powers of the tests are almost equal. 5. Discussion The QEt test is proposed for testing the homogeneity against non-linear responses in L2 × K 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 QEt test is superior to the extended Man- tel test (Mantel 1963, Landis, Heyman and Koch 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 some- times 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 Jayasekara and Yanagawa: Testing Non-Linear Ordinal Responses ... Ruhuna Journal of Science 2, pp. 18–29, (2007) 27 Table 3 Estimated Type I errors of the QEt , t = 1, 2, 3, 4, and extended Mantel test (EMT). Pattern Sample size Estimated Type I error levels nl1 = nl2, l = 1, 2, 3, 4 QE 1 QE 2 QE 3 QE 4 EMT ( , , , ) 60 0.052 0.052 0.052 0.052 0.052 80 0.05 0.048 0.048 0.047 0.051 100 0.049 0.049 0.0.05 0.051 0.049 (�, �, � , �) 60 0.054 0.052 0.052 0.049 0.054 80 0.051 0.051 0.048 0.047 0.049 100 0.052 0.052 0.051 0.05 0.053 ( Q, Q, Q, Q) 60 0.053 0.054 0.053 0.05 0.052 80 0.05 0.049 0.05 0.05 0.053 100 0.052 0.051 0.049 0.048 0.05 (∩, ∩, ∩, ∩) 60 0.052 0.051 0.049 0.049 0.051 80 0.049 0.048 0.05 0.05 0.049 100 0.052 0.047 0.05 0.05 0.052 (∪, ∪, ∪, ∪) 60 0.054 0.056 0.052 0.052 0.054 80 0.051 0.051 0.05 0.048 0.051 100 0.052 0.052 0.051 0.051 0.052 (� � �� ,�� � � ,� � �� ,� � �� ) 60 0.053 0.054 0.052 0.053 0.052 80 0.05 0.051 0.051 0.051 0.049 100 0.052 0.051 0.05 0.048 0.052 ( � � ��, � � ��, � � � �, �� � �) 60 0.052 0.05 0.051 0.052 0.053 80 0.049 0.049 0.05 0.047 0.049 100 0.051 0.052 0.051 0.05 0.052 (� � �� � �,� � �� � �,� � �� � �,� � �� � �) 60 0.054 0.054 0.052 0.051 0.053 80 0.052 0.049 0.051 0.05 0.052 100 0.051 0.052 0.051 0.05 0.052 ( � � �� � � , � � �� � � , � � �� � � , � � �� � � ) 60 0.052 0.05 0.051 0.049 0.053 80 0.05 0.05 0.05 0.049 0.05 100 0.05 0.051 0.053 0.054 0.05 (�, ∩, �� � � , � � �� � �) 60 0.053 0.054 0.051 0.05 0.053 80 0.052 0.049 0.049 0.05 0.052 100 0.053 0.052 0.054 0.05 0.055 ( Q, ∪, �� � �, � � �� � � ) 60 0.054 0.052 0.051 0.048 0.053 80 0.05 0.05 0.049 0.047 0.051 100 0.049 0.048 0.049 0.049 0.048 (∩, �� �� , � � � � ��, � � � � �� ) 60 0.053 0.053 0.049 0.051 0.053 80 0.05 0.051 0.051 0.05 0.049 100 0.05 0.049 0.05 0.047 0.05 (∪, �� �� , � � � �, � � � � �� )} 60 0.055 0.051 0.048 0.05 0.054 80 0.05 0.051 0.05 0.05 0.049 100 0.052 0.05 0.049 0.048 0.051 Jayasekara and Yanagawa: Testing Non-Linear Ordinal Responses ... 28 Ruhuna Journal of Science 2, pp. 18–29, (2007) Table 4 Tests which give the largest and second largest powers: Y2 Y1 (�, ∩, �� � � , � � �� � �) ( Q, ∪, �� � �, � � �� � � ) (∩, � � �� , �� � � ��, �� � � �� ) (∪, � � �� , �� � �, � � �� � � ) ( , , , ) QE4, QE3 QE4, QE3 QE3, QE4 QE4, QE2 (�, �, �, �) QE4, QE1 (QE1, QE2, QE3, (QE1, QE2, QE3, EMT, QE2 QE4, EMT) QE4, EMT) ( Q, Q, Q, Q) (QE1, QE2, QE3, (QE1, QE2, QE3, (QE1, QE2, QE3, (QE1, QE2, QE3, QE4, EMT) QE4, EMT) QE4, EMT) QE4, EMT) (∩, ∩, ∩, ∩) QE2, QE4 (QE2, QE3, QE2, QE3 (QE2, QE3, QE4, EMT) QE2, QE1) (∪, ∪, ∪, ∪) (QE2, QE3, QE2, QE3 (QE2, QE3, QE3, QE2 QE4, EMT) QE4, QE1) (� � �� ,� � �� ,� � �� ,� � �� ) QE3, QE4 (QE3, QE4, QE2) (QE3, QE4, QE2) QE4, QE3 ( � � ��, � � ��, � � ��, � � �� (QE3, QE4, QE2) QE4, QE3 (QE3, QE4, EMT) QE3, QE4 (� � �� � �,�� � � ��,� � �� � �,�� � � ��) QE4, QE3 QE4, QE3 QE4, QE3 QE4, QE3 ( � � �� � � , �� � � �� , � � �� � � , �� � � �� ) QE4, QE3 QE4, QE3 QE4, QE3 QE4, QE2 (�, ∩, �� � � , �� �� � �) - QE4, QE3 QE4, QE1 QE4, QE3 ( Q, ∪, �� � �, �� �� � � ) QE4, QE3 - QE4, QE3 EMT, QE3 (∩, �� � � , �� � � ��, �� � � �� ) QE4, QE1 QE1, QE2 - QE4, QE3 (∪, �� � � , �� � �, �� � � �� ) QE4, QE3 EMT, QE1 QE4, QE3 - Table 5 The maximum, mean and the minimum powers of the tests for 48 combinations of the patterns in Table 4. 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