CONDITIONS FOR ROBUSTNESS TO NONNORMALITY ON TEST STATISTICS IN A GMANOVA MODEL

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1 J. Japan Statist. Soc. Vol. 37 No CONDITIONS FOR ROBUSTNESS TO NONNORMALITY ON TEST STATISTICS IN A GMANOVA MODEL Hirokazu Yanagihara* This paper presents the conditions for robustness to the nonnormality on three test statistics for a general multivariate linear hypothesis, which were proposed under the normal assumption in a generalized multivariate analysis of variance (GMANOVA) model. The proposed conditions require the cumulants of an unknown population s distribution to vanish in the second terms of the asymptotic expansions for both the mean and variance of the test statistics. With the proposed conditions, the test statistic can be investigated for robustness to nonnormality of the population s distribution. When the conditions are satisfied, the Bartlett correction and the modified Bartlett correction in the normal case improve the quality of the chi-square approximation even under nonnormality. Key words and phrases: Actual test size, asymptotic expansion, Bartlett correction, chi-square approximation, general multivariate linear hypothesis, modified Bartlett correction. 1. Introduction During the last several decades, there have been many studies on the influences of nonnormality on several statistical test procedures proposed under the normal assumption. For the test size (or the significant level), Hotelling s two-sample test is known to be robust to nonnormality of the population s distribution (i.e., the true distribution), while Hotelling s one-sample test is not. Chase and Bulgren (1971) and Everitt (1979) showed these facts through numerical experiments. Figure 1 shows the results of a similar numerical experiment in the case of sample size n = 20 and the dimension of observation p = 2. Normal, Laplace, uniform, skew-laplace, chi-square and log-normal distributions were used as the true distributions (for details of the setting of true distributions in numerical studies, see Appendix A.1). This figure explains actual test sizes of the two test statistics under the six true distributions when we believe that the data came from the normal distribution, i.e., when we use the F -distribution as the null distribution. From this figure, it is revealed that a variation of the actual test size of the two-sample test, caused by the difference in distributions, is small as opposed to the case of the one-sample test. In the case of the one-sample test, the larger the skewness of the true distributions, the more the difference between the actual and nominal test size grows. If the test size, which we believe to be 10%, becomes 15% or 25%, the corresponding test statistic becomes difficult to use. Received February 27,2006. Revised May 8,2006. Accepted May 15,2006. *Department of Mathematics,Graduate School of Science,Hiroshima University,1-3-1 Kagamiyama, Higashi-Hiroshima,Hiroshima ,Japan.

2 136 HIROKAZU YANAGIHARA 25 Nominal 10% Test 20 Nominal 5% Test Actual Size (%) Actual Size (%) Distributions Nominal Size One Sample Test Two Sample Test Distributions are: 1. Normal 2. Laplace 3. Uniform 4. Skew Laplace 5. Chi Square 6. Log Normal Actual Size (%) Distributions Nominal 1% Test Distributions Figure 1. Actual test sizes of Hotelling s one- and two-sample tests. Although the statistical model in both tests is essentially a multivariate linear model, a large difference is observed in the influence of nonnormality on these two test statistics. A serious difference between these two test statistics appears in the second term of an asymptotic expansion of the mean of the test statistic. Although there are skewnesses of the true distribution in the second term of this expansion of the one-sample test statistic, there are no skewnesses in that of the two-sample test statistic. In this paper, we investigate the difference of the influence theoretically, not numerically. We extend the test statistic for a general multivariate linear hypothesis in the generalized multivariate analysis of variance (GMANOVA) model (Potthoff and Roy (1964)) from Hotelling s one- and two-sample test statistics. Here we use the robustness to nonnormality for the case where the cumulants of the true distribution, i.e., a quantity denoting nonnormality of the true distribution, do not remain in the second terms of asymptotic expansions of the moments for the test statistics. We search for the condition that the cumulants of the true distribution vanish in the second terms of asymptotic expansions for the mean and variance of the test statistic. The condition is clarified through the coefficients of an asymptotic expansion of the null distribution of the test statistic. With the proposed condition, the used test statistic can be investigated for robustness to nonnormality of the true distribution. Furthermore, when the proposed condition is satisfied, the Bartlett correction (Bartlett (1937)) in the normal case, or the modified Bartlett correction (Fujikoshi (2000)), improves the quality of the chi-square approximation under any distributions. Improvement in the quality of the chi-square approximation without estimating higher-order cumulants is important because the bias

3 ROBUSTNESS TO NONNORMALITY ON GMANOVA TESTS 137 of estimators for the cumulants becomes large without having adequate sample size. Especially, the bias of the ordinary estimator of kurtosis proposed by Mardia (1970) becomes serious unless the sample size is huge (see Yanagihara (2007)). The paper is organized as follows: In Section 2, we elaborate on Yanagihara s asymptotic expansions of the null distributions of the three test statistics in a GMANOVA model (Yanagihara (2001)). By using the coefficients of these expansions, the conditions for the test statistics to be robust to nonnormality of the true distribution are obtained in Section 3, followed by the discussion and conclusion in Section 4. Technical details are provided in the Appendix. 2. Asymptotic expansions of null distributions of test statistics in the GMANOVA model The GMANOVA model proposed by Potthoff and Roy (1964) is defined by (2.1) Y = AΞX + EΣ 1/2, where Y =(y 1,...,y n ) is an n p observation matrix of response variables, A =(a 1,...,a n ) is an n k between-individuals design matrix of explanatory variables with the full rank k (< n), X is a p q within-individuals design matrix of explanatory variables with the full rank q ( p), Ξ is a k q unknown parameter matrix, and E =(ε 1,...,ε n ) is an n p unobserved error matrix. It is assumed that ε 1,...,ε n are independent random vectors from ε =(ε 1,...,ε p ), and that the true distribution of ε follows an unknown distribution with a mean E[ε] =0 p and variance-covariance matrix Cov[ε] =I p. Notice that 0 p is a p 1 vector with all elements equal 0. Since this model can be frequently applied to the analysis of growth curve data, it is also called the growth curve model. We consider testing for a general linear hypothesis as (2.2) H 0 : CΞD = O c d, where C is a known c k matrix with the rank c ( k), D is a known q d matrix with the rank d ( q) and O c d is a c d matrix with all elements equal 0. Let S h and S e be the variation matrices due to the hypothesis and the error respectively, i.e., where S h =(C ΞD) (CRC ) 1 (C ΞD), S e = D (X S 1 X) 1 D, Ξ =(A A) 1 A YS 1 X(X S 1 X) 1, R =(A A) 1 A (I n + YS 1 Y )A(A A) 1 Ξ(X S 1 X) Ξ, and S = Y (I n P A )Y. Here, P A is the projection matrix to the linear space R(A) generated by the column vectors of A, i.e., P A = A(A A) 1 A. Then,

4 138 HIROKAZU YANAGIHARA the three criteria proposed for testing (2.2) under normality, in particular, are as follows: (2.3) (i) the likelihood ratio statistic (LR): T LR = {n k (p q)} log( S e / S e + S h ), (ii) the Lawley-Hotelling trace criterion (LH): T LH = {n k (p q)} tr(s h S 1 e ), (iii) the Bartlett-Nanda-Pillai trace criterion (BNP): T BNP = {n k (p q)} tr{s h (S h + S e ) 1 }, (for these results, see e.g., Kariya (1978), von Rosen (1991), Fujikoshi (1993), and Srivastava and von Rosen (1999)). By changing the known matrices C and D, it is possible to express several linear hypotheses. For these examples, see Kshirsagar and Smith (1995, p. 40) and Yanagihara (2001). In order to simplify the formula of asymptotic expansions of the null distributions of the three test statistics, we state some notations of the moments and cumulants of the true distribution, and define some projection matrices. Let µ j1 j m be an mth multivariate moment of ε defined by µ j1 j m = E[ε j1 ε jm ]. Similarly, the corresponding mth multivariate cumulant of ε is denoted by κ j1 j m, e.g., κ abc = µ abc, κ abcd = µ abcd δ ab δ cd δ ac δ bd δ ad δ bc, where δ ab is the Kronecker delta, i.e., δ aa = 1 and δ ab = 0 for a b. Let e 1 and e 2 be independent random vectors from ε, and H, L and M be p p symmetric matrices whose (a, b)th elements are h ab, l ab and m ab, respectively. Then, we define the following multivariate skewness and kurtosis of the transformed ε by H, L and M. κ(h, M) =E e1 [(e 1He 1 )(e 1Me 1 )] {tr(h) tr(m) + 2 tr(hm)} p = κ abcd h ab m cd, a,b,c,d γ1(h, 2 L, M) =E e1 E e2 [(e 1He 2 )(e 1Le 2 )(e 1Me 2 )] p = κ abc κ def h ad l be m cf, a,b,c,d,e,f γ2(h, 2 L, M) =E e1 E e2 [(e 1He 1 )(e 1Le 2 )(e 2Me 2 )] p = κ abc κ def h ab l cd m ef, a,b,c,d,e,f

5 ROBUSTNESS TO NONNORMALITY ON GMANOVA TESTS 139 where the notation p a,b,... means p p a=1 b=1. The ordinary multivariate skewnesses and kurtosis (see e.g., Mardia (1970) and Isogai (1983)) can be expressed as κ (1) 4 = κ(i p, I p ), κ (1) 3,3 = γ2 1(I p, I p, I p ), κ (2) 3,3 = γ2 2(I p, I p, I p ). Let an n n idempotent matrix B whose (i, j)th element is b ij be given by B = A(A A) 1 C {C(A A) 1 C } 1 C(A A) 1 A. Then, the matrices B (d) and B (3) are defined by b ij as b 3 11 b3 1n B (d) = diag(b 11,...,b nn ), B (3) = b 3 n1 b3 nn Furthermore, the projection matrices Ψ and Φ are given by Ψ = Σ 1/2 X(X Σ 1 X) 1 D{D (X Σ 1 X) 1 D} 1 D (X Σ 1 X) 1 X Σ 1/2, Φ = I p Σ 1/2 X(X Σ 1 X) 1 X Σ 1/2. By using these skewnesses, kurtosis and matrices, asymptotic expansions of the null distributions of the test statistics in (2.3) are expressed as in the following lemma (for the proof of the lemma, see Yanagihara (2001)). Lemma 2.1. Suppose that the between-individuals design matrix A and the true distribution of the error matrix E satisfy the following five assumptions: 1 n lim sup a i 4 <, where denotes the Euclidean norm, n n i=1 λ n lim inf n For some constant δ (0, 1/2], n > 0, where λ n is the smallest eigenvalue of A A, max a i = O(n 1/2 δ ), i=1,...,n E[ ε 8 ] <, The Cramér s condition for the joint distribution of ε and εε holds. Then, the null distributions of the three test statistics in (2.3) are expanded as (2.4) P (T x) =G cd (x)+ 1 n 3 β j G cd+2j (x)+o(n 1 ), j=0 where G f (x) is the distribution function of the chi-square distribution with f degrees of freedom and the coefficients β j are given by β 0 = 1 8 m(1) 4 κ(ψ, Ψ) 1 24 {2m(1) 3,3 +3(c 2)(c +1)m(1) 1,1 }γ2 1(Ψ, Ψ, Ψ) 1 8 {m(2) 3,3 2cm(1) 3,1 (c 2)m(1) 1,1 }γ2 2(Ψ, Ψ, Ψ)

6 140 HIROKAZU YANAGIHARA 1 2 {2m(1) 3,1 (2c +1)m(1) 1,1 }γ2 1(Ψ, Ψ, Φ) 1 2 {m(1) 3,1 cm(1) 1,1 }γ2 2(Ψ, Ψ, Φ) 1 2 {m(1) 3,1 (c +1)m(1) 1,1 }γ2 2(Ψ, Φ, Ψ) 1 2 m(1) 1,1 {3γ2 1(Ψ, Φ, Φ)+2γ 2 2(Ψ, Φ, Φ)+γ 2 2(Φ, Ψ, Φ)} + 1 cd(c d 1), 4 β 1 = 1 4 m(1) 4 κ(ψ, Ψ)+1 8 {2m(1) 3,3 4m(1) 3,1 +(3c2 + c 6)m (1) 1,1 }γ2 1(Ψ, Ψ, Ψ) {3m(2) 3,3 2(3c +2)m(1) 3,1 +(c +6)m(1) 1,1 }γ2 2(Ψ, Ψ, Ψ) (2.5) {4m(1) 3,1 (4c +5)m(1) 1,1 }γ2 1(Ψ, Ψ, Φ) + {m (1) 3,1 (c +1)m(1) 1,1 }γ2 2(Ψ, Ψ, Φ) {2m(1) 3,1 (2c +3)m(1) 1,1 }γ2 2(Ψ, Φ, Ψ) m(1) 1,1 {3γ2 1(Ψ, Φ, Φ)+2γ 2 2(Ψ, Φ, Φ)+γ 2 2(Φ, Ψ, Φ)} 1 cd{c + r(c + d +1)}, 2 β 2 = 1 8 m(1) 4 κ(ψ, Ψ) 1 8 {2m(1) 3,3 8m(1) 3,1 +(c + 2)(3c 1)m(1) 1,1 }γ2 1(Ψ, Ψ, Ψ) 1 8 {3m(2) 3,3 2(3c +4)m(1) 3,1 +5(c +2)m(1) 1,1 }γ2 2(Ψ, Ψ, Ψ) 1 2 {m(1) 3,1 (c +2)m(1) 1,1 }{2γ2 1(Ψ, Ψ, Φ)+γ2(Ψ, 2 Ψ, Φ)+γ2(Ψ, 2 Φ, Ψ)} + 1 cd(c + d +1)(1+2r), 4 β 3 = 1 24 {2m(1) 3,3 12m(1) 3,1 +3(c + 1)(c +2)m(1) 1,1 }γ2 1(Ψ, Ψ, Ψ) {m(2) 3,3 2(c +2)m(1) 3,1 +3(c +2)m(1) 1,1 }γ2 2(Ψ, Ψ, Ψ). Here, the constant r is determined by the test statistics as 1/2 (when T is T LR ) r = 0 (when T is T LH ) 1 (when T is T BNP ) and the coefficients m are defined by the between-individuals design matrix A and

7 ROBUSTNESS TO NONNORMALITY ON GMANOVA TESTS 141 the coefficient matrix for hypothesis C as (2.6) m (1) 1,1 = 1 n 1 nb1 n, m (1) 4 = n tr(b 2 (d)) c(c +2), m (1) 3,1 = 1 nb (d) B1 n, m (1) 3,3 = n1 nb (3) 1 n, m (2) 3,3 = n1 nb (d) BB (d) 1 n, where 1 n is an n 1 vector with all elements equal 1. Let u i be a c 1 vector given by (2.7) u i = n{c(a A) 1 C } 1/2 C(A A) 1 a i, (i =1,...,n). Then the coefficients in (2.6) are rewritten as (2.8) m (1) 1,1 = 1 n 2 m (1) 3,1 = 1 n 2 m (2) 3,3 = 1 n 2 n i,j n i,j u iu j, m (1) 4 = 1 n n u i 4 c(c +2), i=1 u i 2 (u iu j ), m (1) 3,3 = 1 n 2 n u i 2 u j 2 (u iu j ). i,j n (u iu j ) 3, i,j Therefore, the coefficients β j (j =0, 1, 2, 3) are determined by the cumulants of ε and sample cumulants of u i (i =1,...,n). Lemma 2.1 points out that the second term in the asymptotic expansion of the null distribution of (2.3) depends on the skewnesses and kurtosis of the true distribution. Therefore, we can generally say that the second terms in asymptotic expansions of the mean and variance of the test statistic depend on the cumulants of ε. However, in the next section, we show conditions that these cumulants vanish in the second terms of the expansions. 3. Conditions for robustness to nonnormality 3.1. Condition for the mean In this subsection, we consider the condition that the cumulants of ε vanish in the second term of an asymptotic expansion for the mean of the test statistic T. Notice that 0 x s dg f (x) =f(f +2) (f +2(s 1)). From the formula (2.4) in Lemma 2.1, we obtain an asymptotic expansion of the mean of T as E[T ]=cd + 1 n 3 β j (cd +2j)+o(n 1 ). j=0

8 142 HIROKAZU YANAGIHARA By using the relation 3 j=0 β j = 0, we can write a concrete form of the expansion of E[T ]as (3.1) E[T ]=cd + 1 n m(1) 1,1 [γ2 1(Ψ, Ψ, Ψ)+γ 2 2(Ψ, Ψ, Ψ) +3{γ1(Ψ, 2 Ψ, Φ)+γ1(Ψ, 2 Φ, Φ)} +2{γ2(Ψ, 2 Ψ, Φ)+γ2(Ψ, 2 Φ, Φ)} + γ2(ψ, 2 Φ, Ψ)+γ2(Φ, 2 Ψ, Φ)] + 1 n cd{r(c + d +1)+d +1} + o(n 1 ) ( = cd 1+ η ) 1 + o(n 1 ). n From (3.1), we can see that η 1 is independent of the cumulants of ε if the coefficient m (1) 1,1 is 0 or all the γ2 j (,, ) are 0 (j =1, 2). It means that a location of the null distribution of the test statistic does not easily change due to the nonnormality of the true distribution. Then, the test statistic becomes robust to nonnormality of the true distribution. Notice that m (1) 1,1 = 0 is equivalent to 1 n 1 na(a A) 1 C {C(A A) 1 C } 1 C(A A) 1 A 1 n =0. Since C(A A) 1 C is a positive definite matrix, the above equation can be rewritten as C(A A) 1 A 1 n = 0 c. Therefore, we obtain the conditions for robustness to nonnormality as in the following theorem. Theorem 3.1. If a considered test satisfies either of the following, Condition 1 or 2: Condition 1: C(A A) 1 A 1 n = 0 c, Condition 2: All the γj 2 (,, ) (j =1, 2) are 0, the second term in the asymptotic expansion of the mean of the test statistic T becomes independent of the cumulants of ε, i.e., E[T ]=cd [1+ 1n ] (3.2) {r(c + d +1)+d +1} + o(n 1 ). Consequently, the location of the null distribution of T does not easily change due to nonnormality of the true distribution. From Appendix A.2, we can see that all the γj 2 (,, ) are 0 if the distribution of ε possesses orthant symmetry defined by Efron (1969). Therefore, we obtain the following corollary (the proof is given in Appendix A.2). Corollary 3.1. Condition 2 holds. If the distribution of ε possesses orthant symmetry, then If the distribution possesses orthant symmetry, this distribution is not skewed. It is well known that the elliptical distribution in which the mean is zero and variance-covariance matrix is diagonal possesses orthant symmetry. Many

9 ROBUSTNESS TO NONNORMALITY ON GMANOVA TESTS 143 authors have reported that a test statistic is robust when the distribution of observation is the elliptical distribution, e.g., Efron (1969), Eaton and Efron (1970), Dawid (1977), Kariya (1981a, 1981b) and Khatri (1988). Robustness to nonnormality, which is caused by Condition 2, corresponds to their results. Although Condition 2 depends on the unknown true distribution, Condition 1 only depends on the setting of the hypothesis testing. By using Condition 1, we can judge whether or not a used test statistic is robust to nonnormality of the true distribution. Here, we can consider two types of the between-individuals design matrix A. Let A be an n (k 1) matrix satisfying P A 1 n 1 n and rank(a )=k 1. When the matrix A has a segment, i.e., A =(1 n A ), then we call such a design matrix the type I between-individuals design matrix. On the other hand, when the matrix A is given by 1 n1 0 n1 (3.3) A =....., 0 nk 1 nk where k i=1 n i = n, then we call such a design matrix the type II betweenindividuals design matrix. Through these matrices, we can obtain settings which satisfy Condition 1 as in the following corollary (the proof is given in Appendix A.3). Corollary 3.2. We consider the two cases of the between-individuals design matrix A, i.e., A is type I or type II. Then, Condition 1 holds if the following conditions of the coefficient matrix for hypothesis C are satisfied. (I) In the case that A is type I, if C =(0 c C ), where C is any c (k 1) matrix with the rank c, Condition 1 holds. (II) In the case that A is type II, if the equation C1 k = 0 c is satisfied, Condition 1 holds. Corollary 3.2 points out that Hotelling s two-sample test satisfies Condition 1 and Hotelling s one-sample test does not. Even if the between-individuals design matrix A satisfying P A 1 n 1 n is not type I or II, the test can be made to satisfy Condition 1 through the transformations A (1 n A), C (0 c C) and Ξ (ξ Ξ ). Furthermore, we can see that testing for the equality, e.g., the equality for the means between k-groups, i.e., H 0 : µ 1 = = µ k, satisfies Condition 1, because matrix A is type II and C1 k = 0 c in this case. Practical tests are testing for the equality of parameters, e.g., the testing for the equality of several means in a multi-way MANOVA model; therefore, it seems that our result is useful for actual applications. If the mean of the test statistic T is expanded as (3.1), then the Bartlett correction can be obtained as ( T = 1 ˆη ) 1 T, n

10 144 HIROKAZU YANAGIHARA where ˆη 1 is a consistent estimator of η 1 in (3.1). If the true distribution is the normal distribution, the Bartlett correction is transformation by the constant coefficient. On the other hand, in general, it is necessary to estimate skewnesses in order to use the Bartlett correction. However, from Theorem 3.1, we can see that the Bartlett correction in the normal case can improve the quality of the chisquare approximation, even if the true distribution is not the normal distribution; this is because η 1 is independent of the cumulants of ε. Therefore, we obtain the following corollary. Corollary 3.3. Let T denote the test statistics adjusted by the Bartlett correction in the normal case as ( 1+ c d 1 ) T LR 2n ( T = 1 d +1 ) T LH. n ( 1+ c ) T BNP n Suppose that either Condition 1 or 2 holds. Then, the asymptotic expansion of the mean of T becomes E[ T ]=cd + o(n 1 ). This naturally implies that the Bartlett correction in the normal case can improve the quality of the chi-square approximation even under nonnormality. Before concluding this subsection, we verify the validity of our theorem through a numerical experiment. The true distributions used are described in Appendix A.1. We dealt with the test for a given matrix in the case of p = 2 and n = 30. The p (p 1) within-individuals design matrix X was generated from the uniform distribution on ( 1, 1) and the coefficient matrix for the hypothesis was D = I q. We prepared two test statistics specified by A and C, and their adjusted versions by the Bartlett correction in the normal case as follows: T 0 : A = A and C = I 3, T 0 : the adjusted version of T 0 by the Bartlett correction in the normal case, T 1 : A =(1 n A ) and C =(0 3 I 3 ), T 1 : the adjusted version of T 1 by the Bartlett correction in the normal case, where an n 3 matrix A was generated from the uniform distribution on (0, 1). It is easy to see that T 1 satisfies Condition 1, and T 0 and T 1 satisfy Condition 2 when the true distributions are 1, 2 and 3. Table 1 shows the actual test sizes of these test statistics when we used the chi-square distribution as the null distribution. The average and standard deviation (S.D.) of actual sizes for every true distribution are also given in the table. From this table, we can see that the S.D. becomes small when either Condition 1 or 2 holds. Therefore, we can say that a variation of the actual test size by the difference in distributions is small in these cases. Then, the Bartlett correction in the normal case improves the quality of the chi-square approximation. From the numerical results, we suggest that the test statistic satisfying Condition 1 should be adjusted by the Bartlett correction in the normal case.

11 ROBUSTNESS TO NONNORMALITY ON GMANOVA TESTS 145 Table 1. Actual test sizes of several test statistics for testing a given matrix. (LR) T 0 T0 T 1 T1 Nominal Sizes Nominal Sizes Nominal Sizes Nominal Sizes Distribution 10% 5% 1% 10% 5% 1% 10% 5% 1% 10% 5% 1% Normal Laplace Uniform Skew-Laplace Chi-Sqare Log-Normal Average S.D (LH) T 0 T0 T 1 T1 Nominal Sizes Nominal Sizes Nominal Sizes Nominal Sizes Distribution 10% 5% 1% 10% 5% 1% 10% 5% 1% 10% 5% 1% Normal Laplace Uniform Skew-Laplace Chi-Square Log-Normal Average S.D (BNP) T 0 T0 T 1 T1 Nominal Sizes Nominal Sizes Nominal Sizes Nominal Sizes Distribution 10% 5% 1% 10% 5% 1% 10% 5% 1% 10% 5% 1% Normal Laplace Uniform Skew-Laplace Chi-Square Log-Normal Average S.D Condition for the variance In this subsection, we consider the condition that the cumulants of ε vanish simultaneously in the second terms of asymptotic expansions for both the mean and variance of the test statistic. From the formula (2.4) in Lemma 2.1, we obtain the asymptotic expansion of the second moment of T as E[T 2 ]=cd(cd +2)+ 1 n 3 β j (cd +2j)(cd +2+2j)+o(n 1 ). j=0 By using the relation 3 j=0 β j = 0, we can write a concrete form of the expansion of E[T 2 ]as (3.4) E[T 2 ]=cd(cd +2) + 1 n [m(1) 4 κ(ψ, Ψ)

12 146 HIROKAZU YANAGIHARA 2{2m (1) 3,1 (cd +2c +6)m(1) 1,1 } {γ1(ψ, 2 Ψ, Ψ)+γ2(Ψ, 2 Ψ, Ψ)} 2{4m (1) 3,1 (cd +4c + 10)m(1) 1,1 }γ2 1(Ψ, Ψ, Φ) 2{2m (1) 3,1 (cd +2c +6)m(1) 1,1 } {γ2(ψ, 2 Ψ, Φ)+γ2(Ψ, 2 Φ, Ψ)} +2(cd +2)m (1) 1,1 {3γ1(Ψ, 2 Φ, Φ)+2γ2(Ψ, 2 Φ, Φ)+γ2(Φ, 2 Ψ, Φ)}] + 2 cd[(cd +2){r(c + d +1)+d +1} +(c + d +1)(1+2r)] n +o(n 1 ) ( = cd(cd +2) 1+ η 2 n ) + o(n 1 ). It is easy to see that the coefficient m (1) 3,1 becomes 0 if m(1) 1,1 from these equations, if either the coefficient m (1) 1,1 = 0 or all γ2 j = 0. Therefore, (,, ) = 0 (j =1, 2) holds and also the equation m (1) 4 = 0 holds simultaneously, η 1 and η 2 become independent of the cumulants of ε. This means that the location and dispersion of the null distribution of the test statistic do not easily change due to nonnormality of the true distribution. Notice that the coefficient m (1) 4 signifies the sample kurtosis of u 1,...,u n, where u i is given by (2.7). Therefore, we obtain the condition for robustness to nonnormality in the following theorem. Theorem 3.2. Suppose that a considered test satisfies either Condition 1 or 2 in Theorem 3.1 and furthermore that the following condition also holds: Condition 3: The sample kurtosis of u 1,...,u n is 0. Then, the second terms in asymptotic expansions of the mean and variance of the test statistic T become independent of the cumulants of ε, i.e., E[T ]=cd [1+ 1n ] {r(c + d +1)+d +1} + o(n 1 ), (3.5) Var[T ]=2cd [1+ 1n ] {4r(c + d +1)+c +3(d +1)} + o(n 1 ). Consequently, the location and dispersion of the null distribution of the test statistic do not easily change due to nonnormality of the true distribution. From Wakaki et al. (2002), if the between-individuals design matrix A is type II and the equation C1 k = 0 c holds, then the coefficient m (1) 4 is rewritten as m (1) 4 = k j=1 n/n j k 2 2k + 2. Therefore, we obtain the following corollary. Corollary 3.4. Suppose that the between-individuals design matrix A is type II and equation C1 k = 0 c holds. Let the sample sizes of k-groups be denoted by n 1,...,n k. If the equation k j=1 n/n j = k 2 +2k 2 holds, then Conditions 1 and 3 hold simultaneously.

13 ROBUSTNESS TO NONNORMALITY ON GMANOVA TESTS 147 Table 2. Robust design to nonnormality in testing the equality. (k =4) 1:1:2:4 1:2:2:5 1:2:4:4 n 1 : n 2 : n 3 : n 4 1:3:4:4 2:5:5:10 (k =5) n 1 : n 2 : n 3 : n 4 : n 5 1:3:3:3:5 3:5:5:7:15 1:1:1:3:3 1:1:3:3:3 3:5:8:12:12 3:7:7:10:15 (k =6) n 1 : n 2 : n 3 : n 4 : n 5 : n 6 2:3:4:5:8:8 2:3:5:5:5:10 3:3:3:4:5:12 3:4:4:4:9:12 3:4:6:6:6:15 3:6:6:8:10:15 3:9:9:9:12:14 4:5:5:5:12:15 4:5:12:12:12:15 4:8:10:15:16:16 4 : 10 : 10 : 10 : 15 : 20 5 : 8 : 12 : 15 : 20 : 20 This result means that the testing equality for the means of k-groups becomes robust to nonnormality by adjusting each sample size. In the ANOVA model, this fact was reported by Box and Watson (1962) and Box and Draper (1975). They called such a design a robust design. Table 2 shows the robust design of our test statistic in the cases of k =4, 5 and 6. If the ratio of sample sizes is satisfied as in Table 2, then the test statistic becomes more robust to nonnormality of the true distribution. The modified Bartlett correction proposed by Fujikoshi (2000) does not only bring the mean close to an asymptotic mean but it also brings the variance close to an asymptotic variance. If the mean and variance of the test statistic T are expanded as (3.1) and (3.4), then the modified Bartlett corrections are given by where ( (1) T =(nˆν 1 +ˆν 2 ) log 1+ T ), ˆν 1 > 0, nˆν 1 +ˆν 2 > 0, nˆν 1 (2) T = T + T ( ˆν2 T ), ˆν 1 < 0, nˆν 1 +ˆν 2 < 0, n ˆν 1 2ˆν 1 (3) T =(nˆν 1 +ˆν 2 ) ˆν 1 = { 1 exp ( T nˆν 1 )}, for any n, ˆν 1 and ˆν 2, 2, ˆν 2 = (cd +2)ˆη 2 2(cd +4)ˆη 1, ˆη 2 2ˆη 1 2(ˆη 2 2ˆη 1 ) and ˆη 1 and ˆη 2 are consistent estimators of η 1 and η 2, respectively. If the true distribution is the normal distribution, the modified Bartlett correction is the transformation by a known monotonic increasing function. On the other hand, in general, it is necessary to estimate skewnesses and kurtosis in order to use the modified Bartlett correction. However, from Theorem 3.2, we can see that

14 148 HIROKAZU YANAGIHARA Table 3. Actual test sizes of several test statistics for testing the equality (LR). Nominal Test Distribution Size m (1) 4 Statistic (i) (ii) (iii) (iv) (v) (vi) Average S.D. 10% 6.0 T T T T T T % 6.0 T T T T T T % 6.0 T T T T T T the modified Bartlett correction in the normal case can improve the quality of the chi-square approximation, even if the true distribution is not the normal distribution; this is because η 1 and η 2 are independent of the cumulants of ε. Therefore, we obtain the following corollary. Corollary 3.5. Let T denote the test statistics adjusted by the modified Bartlett corrections in the normal case as cd +2 {n + 12 } { T c + d +1 (c d 1) log 1+ c + d +1 } n(cd +2) T LH = T BNP + 1 ( c d 1+ c + d +1 ). 2n cd +2 T BNP T BNP Suppose that either Condition 1 or 2 holds, and also Condition 3 holds simultaneously, then the mean and variance of T are expanded as E[ T ]=cd + o(n 1 ) and Var[ T ]=2cd + o(n 1 ). This naturally implies that the modified Bartlett correction in the normal case can improve the quality of the chi-square approximation even under nonnormality. From equations (3.1) and (3.4), when either Condition 1 or 2 holds, and also Condition 3 holds, the mean and the second moment of T LR in the normal case are expanded as { E[T LR ]=cd 1 1 } (c d 1) + o(n 1 ), 2n

15 ROBUSTNESS TO NONNORMALITY ON GMANOVA TESTS 149 Table 4. Actual test sizes of several test statistics for testing the equality (LH). Nominal Test Distribution Size m (1) 4 Statistic (i) (ii) (iii) (iv) (v) (vi) Average S.D. 10% 6.0 T T T T T T T T T % 6.0 T T T T T T T T T % 6.0 T T T T T T T T T E[TLR] 2 =cd(cd +2) {1 1n } (c d 1) + o(n 1 ). These expansions make η 2 = 2η 1. Therefore, we can see that the modified Bartlett correction in the LR test statistic cannot be defined in the normal case. However, the Bartlett correction of the LR test statistic does not only bring a mean close to an asymptotic mean, but it also brings the null distribution close to the chi-square distribution, when the true distribution is also the normal distribution (see e.g., Barndorff-Nielsen and Cox (1984), and Barndorff-Nielsen and Hall (1988)). Therefore, if the true distribution is the normal distribution, the Bartlett correction in the normal case can improve the quality of the chi-square approximation to the same extent as the modified Bartlett correction, although the modified Bartlett correction cannot be defined. However, even under nonnormality, T LR adjusted by the Bartlett correction in the normal case also can

16 150 HIROKAZU YANAGIHARA Table 5. Actual test sizes of several test statistics for testing the equality (BNP). Nominal Test Distribution Size m (1) 4 Statistic (i) (ii) (iii) (iv) (v) (vi) Average S.D. 10% 6.0 T T T T T T T T T % 6.0 T T T T T T T T T % 6.0 T T T T T T T T T improve the quality of the chi-square approximation as in the following corollary (the proof is given in Appendix A.4). Corollary 3.6. Let T LR be the version of T LR adjusted by the Bartlett correction in the normal case. Suppose that either Condition 1 or 2 holds, and also Condition 3 holds simultaneously. Then, T LR can improve the quality of the chi-square approximation even under nonnormality as in the modified Bartlett correction, i.e., E[ T LR ]=cd + o(n 1 ) and Var[ T LR ]=2cd + o(n 1 ). Before concluding this subsection, we verify the validity of our theorem through a numerical experiment. The error distributions used are described in Appendix A.1. We dealt with the test for equality of Ξ in the case of p =2, k = 4 and n = 32. The p (p 1) within-individuals design matrix X was generated from the uniform distribution on ( 1, 1) and the coefficient matrices

17 ROBUSTNESS TO NONNORMALITY ON GMANOVA TESTS 151 for hypothesis were C =(I k 1 1 k 1 ) and D = I q. We prepared the three test statistics specified by the type II design matrix A in (3.3), as follows: Case 1: n 1 = n 2 = n 3 = n 4 =8(m (1) 4 = 6.0), Case 2: n 1 = n 2 =4,n 3 = 8 and n 4 =16(m (1) 4 =0.0), Case 3: n 1 =3,n 2 = n 3 = 4 and n 4 =21(m (1) 4 =6.2). Moreover, we adjusted each test statistic by the Bartlett correction and the modified Bartlett correction. It is easy to see that the three test statistics satisfy Condition 1, and also Condition 3 when A is Case 2. Tables 3, 4 and 5 show actual test sizes of the LR test statistic, HL test statistic and BNP test statistic, respectively. The average and S.D. of actual test sizes for every true distribution are also given in these tables. From these tables, we can see that the S.D. becomes small when Condition 3 holds. Therefore, we can say that a variation of the actual test size by the difference in distributions is small in this case. Then, the modified Bartlett correction in the normal case improves the quality of the chi-square approximation. 4. Discussion and conclusion In this paper, we presented the conditions for robustness to nonnormality. Our conclusion is as follows. First, if either Condition 1 or 2 holds, the cumulants of the true distribution vanish in the second term of the asymptotic expansion for the mean of the test statistic T. This results in the fact that the location of the null distribution of T does not easily change due to nonnormality of the true distribution. In this case, the Bartlett correction without estimating several cumulants improves the quality of the chi-square approximation even under nonnormality. Furthermore, since Condition 1 does not depend on the unknown true distribution, only with Condition 1, it becomes possible to see if the test statistic used is robust to nonnormality of the true distribution. This is an important and useful point of Condition 1. Second, if Condition 3 holds with either Condition 1 or 2, the cumulants of the true distribution vanish in the second terms of the asymptotic expansions for both the mean and variance of the test statistic T. The location and dispersion of the null distribution of T, thus do not easily change due to nonnormality of the true distribution. We use the modified Bartlett correction without estimating several cumulants in this case. By adjusting the ratio of sample sizes in the k- groups, the test statistic for the equality of several means becomes more robust to nonnormality of the true distribution. However, as Box and Watson (1962) suggested, this does not mean that one needs to seek unequal sample sizes to reduce the effect of nonnormality on the test statistic for comparison of means in the k-groups. This is mainly because unequal sample sizes would result in the increase in sensitivity of variance inequalities. Moreover, unequal sample sizes do not result in an optimal design for estimating variance (see Guiard et al. (2000)). All in all, if Condition 1 holds, the test statistic becomes robust to nonnormality of the true distribution. Therefore, it is encouraged to use the betweenindividuals design matrix with the segment and adjust its test statistic by the

18 152 HIROKAZU YANAGIHARA Bartlett correction in the normal case. It is also true that our conditions depend only on the between-individuals design matrix A and the coefficient matrix for hypothesis C. Therefore, the robustness to nonnormality on the test in a GMANOVA model is independent of both the within-individuals design matrix X and the coefficient matrix for hypothesis D. As a result, our conditions essentially coincide with the conditions of testing H 0 : CΞ = O c p in a multivariate linear model as in Wakaki et al. (2002). Appendix Used distributions as true distributions In this subsection, we describe the used distributions as the true distributions for the numerical studies in our paper. The elements of ε are independently and identically distributed and generated from the following six distributions: (i) Normal Distribution: ε j N(0, 1), (κ (1) 3,3 = κ(2) 3,3 = 0 and κ(1) 4 = 0). (ii) Laplace Distribution: ε j is generated from the Laplace distribution with mean 0 and standard deviation 1 (κ (1) 3,3 = κ(2) 3,3 = 0 and κ(1) 4 =3p). (iii) Uniform Distribution: ε j is generated from the uniform distribution on ( 5, 5) divided by the standard deviation 5/ 3(κ (1) 3,3 = κ(2) 3,3 = 0 and κ(1) 4 = 1.2p). (iv) Skew-Laplace Distribution: ε j is generated from the skew-laplace distribution with location parameter 0, dispersion parameter 1 and skew parameter 1 standardized by mean 3/4 and standard deviation 23/4 (κ (1) 3,3 = κ(2) 3,3 1.12p and κ (1) p). (v) Chi-Square Distribution: ε j is generated from the chi-square distribution with 2 degrees of freedom standardized by mean 2 and standard deviation 2(κ (1) 3,3 = κ(2) 3,3 =2p and κ(1) 4 =6p). (vi) Log-Normal Distribution: ε j is generated from the lognormal distribution such that log ε j N(0, 1), standardized by mean e 1/2 and standard deviation e(e 1) (κ (1) 3,3 = κ(2) 3,3 6.18p and κ(1) p). The skew-laplace distribution was proposed by Balakrishnan and Ambagaspitiya (1994) (for the probability density function, see e.g., Yanagihara and Yuan (2005)). It is easy to see that distributions 1, 2 and 3 are symmetric distributions, and distributions 4, 5 and 6 are skewed distributions. Proof of Corollary 3.1. In this subsection, we give the proof of Corollary 3.1. If the distribution of ε possesses orthant symmetry, the third multivariate moment of ε is given by µ abc = E[ε a ε b ε c ]= ε a ε b ε c f(ε 1,...,ε p )dε, where f( ) is a density function of ε. Since the distribution of ε possesses orthant symmetry, the equation f(ε 1,...,ε l,...,ε p )=f(ε 1,..., ε l,...,ε p ) holds for

19 ROBUSTNESS TO NONNORMALITY ON GMANOVA TESTS 153 any ε l s. Let g(ε a,ε b,ε c ) be the marginal distribution of ε a, ε b and ε c. Therefore, µ abc = 0 ε a ε b (ε c ε c )g(ε a,ε b,ε c )dε a dε b dε c =0. Notice that µ abc = κ abc. We obtain κ abc = 0 under an orthant symmetric distribution. Since all γj 2(,, ) consist of κ abc, it implies that all γj 2 (,, ) =0 under an orthant symmetric distribution (j = 1, 2). Proof of Corollary 3.2. In this subsection, we give the proof of Corollary 3.2. For the proof of Corollary 3.2, it is sufficient to confirm C(A A) 1 A 1 n = 0 c. When the between-individuals design matrix A is type I, from the general formula of an inverse matrix (see e.g., Siotani et al. (1985, p. 592)), (A A) 1 is given by ( ) (A A) 1 = na (A A ) 1 w (A A ) 1 A, 1 n W where w = n 1 np A 1 n and W = w(a A ) 1 +(A A ) 1 A 1 n 1 na (A A ) 1. If C =(0 c C ), where C is any c (k 1) matrix with the rank c, then, C(A A) 1 A 1 n = 1 w C {(A A ) 1 A 1 n 1 n WA }1 n = 1 w (n w 1 np A 1 n )C (A A ) 1 A 1 n = 0 c. On the other hand, when the between-individuals design matrix A is type II, (A A) 1 = diag(n 1 1,...,n 1 k ) and A 1 n =(n 1,...,n k ). Notice that C1 k = 0 c. Therefore, C(A A) 1 A 1 n = C1 k = 0 c. From these equations, we obtain the proof of Corollary 3.2. Proof of Corollary 3.6. In this subsection, we give the proof of Corollary 3.6. From the equations (3.5), if either Condition 1 or 2 holds, and also Condition 3 holds, the variance of T LR is expanded as Var[T LR ]=2cd {1 1n } (c d 1) + o(n 1 ). It is known that Var[ T LR ]= ( 1+ c d 1 ) 2 Var[T LR] =2cd + o(n 1 ). 2n Therefore, we obtain the proof of Corollary 3.6.

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