Estimation for parameters of interest in random effects growth curve models

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1 Journal of Multivariate Analysis 98 (2007) Estimation for parameters of interest in random effects growth curve models Wai-Cheung Ip a,, Mi-Xia Wu b, Song-Gui Wang b, Heung Wong a a Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong b College of Applied Sciences, Beiing University of Technology, Beiing 00022, China Received March 2005 Available online 8 September 2006 Abstract In this paper, we consider the general growth curve model with multivariate random effects covariance structure and provide a new simple estimator for the parameters of interest. This estimator is not only convenient for testing the hypothesis on the corresponding parameters, but also has higher efficiency than the least-square estimator and the improved two-stage estimator obtained by Rao under certain conditions. Moreover, we obtain the necessary and sufficient condition for the new estimator to be identical to the best linear unbiased estimator. Examples of its application are given Elsevier Inc. All rights reserved. AMS 99 subect classification: 62H2; 62H5; 62F0 Keywords: Growth curve models; Random-effects; Exact test; Best linear unbiased estimator; Maximum likelihood estimator. Introduction The linear mixed model is a popular choice for the treatment of longitudinal data with random effects, see Laird and Ware [6], Diggle et al. [3], Srivastava and VonRosen [4] and Verbeke and Molenberghs [5]. In this paper, we consider the multivariate model in which m distinct characteristics on each of N individuals taken from r different groups are measured on each of p different occasions. The th characteristic on the ith individual can be assumed to follow the mixed model y i = X β (k) + Z u i + ε i, i =,...,N, =,...,m, i kth group, () Corresponding author. Fax: address: mathipwc@polyu.edu.hk (W.C. Ip) X/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:0.06/.mva

2 38 W.C. Ip et al. / Journal of Multivariate Analysis 98 (2007) where y i = (y i y ip ), y i l is the measurement of the th characteristic at occasion l on the ith individual, X and Z are, respectively, known p q and p c design matrices of full column rank, β (k) is the q vector of regression parameters on the th characteristic in the treatment group k (k =,...,r),u i is c random effect vector, and ε i is p random error vector. If each of the m characteristics that we consider follows a response curve of the same general type over the p occasions, that is, X = X and Z = Z, then for ith individual, where y i = (X I m )β (k) + (Z I m )u i + ε i, y i = Vec((y i,...,y im ) ), β (k) = Vec((β (k),...,β(k) m ) ), ε i = Vec((ε i,...,ε im ) ), u i = Vec((u i,...,u im ) ), denotes the Kronecker product, and Vec( ) operator stacks the columns of a matrix below one another to form a column vector. The individual random effects u i are assumed to be distributed independently as N(0, Σ u ), and independent of the error ε i, with distribution N(0,I p Σ e ), where Σ e is m m positive definite matrix and Σ u is cm cm nonnegative definite matrix, that is, Σ e > 0 and Σ u 0. Thus the covariance matrix of y i is Cov(y i ) = I p Σ e + (Z I m )Σ u (Z I m )=Ω (say). (2) Let Y = (y,...,y N ), U = (u,...,u N ), and E = (ε,...,ε N ), the full multivariate model () can be expressed as Y = (X I m )ξa + (Z I m )U + E, Cov(Vec(Y )) = I N Ω, (3) which is a general growth curve model with multivariate random effects covariance structure, where ξ = (β (),...,β (r) ) is the qm r matrix of the growth curve coefficients, and A = (a,...,a N ) is the r N matrix with full row rank. In particular, if its elements are either or 0 indicating the group from which an observation comes, A is called the group indicator matrix in literature. When m =, model (3) becomes Y = XξA + ZU + E, and Ω = σ 2 e I p + ZΣ u Z, reducing to the single-variable case. Reinsel [,2], Azzalini [], Lange and Laird [7], and Nummi [8,9] considered two special cases: Z = X and Z = X c, where X = (X c : X c ), and showed that the maximum likelihood estimator (MLE) of ξ is identical to its least-squares estimator (LSE) ˆξ = ((X X) X I m )Y A (AA ). (4) However, this is not the case for a general design matrix Z, where the explicit MLE of ξ usually does not exist. Moreover, both the two-stage estimator provided by Khatri [5] and the LSE ˆξ ignore the structure information on the covariance matrix Ω. One alternative is to adopt the improved two-stage estimator suggested by Rao [0] using the structure covariance matrix of the mixed effect model. In many practical situations, only a part of the parameters of model (3) are meaningful to the researcher. For example, in the rat data of Verbeke and Molenberghs [5], of primary interest is the estimation of changes over time and testing whether these changes are treatment dependent. For the mixed linear model with one random effect, Wu and Wang [7] gave a simple estimator

3 W.C. Ip et al. / Journal of Multivariate Analysis 98 (2007) of the part parameter by the reduced model which is often used to remove nuisance parameters. In this paper, we mainly consider this problem under the general growth curve model (3) with random effects. In Section 2, we introduce a new simple estimator for the part parameter of ξ by a reduced model, which can be superior to the LSE under certain conditions. In Section 3, we consider the optimality of the new estimator, and obtain the necessary and sufficient condition the new estimator to be identical to the best linear unbiased estimator (BLUE). Furthermore, we show that the new estimator is superior to the corresponding two-stage estimator under certain conditions. Examples are presented in Section New estimator Without loss of generality, we partition X = (X : X 2 ) in model (3) such that M(X ) M(Z), M(X 2 ) M(Z) ={0}, (5) where X and X 2 are p l and p (q l) matrices, respectively, (0 l q), and M(C) is the range space of any matrix C. Partition ξ = (ξ : ξ 2 ) conformably. Thus model (3) can be rewritten as Y = (X I m )ξ A + (X 2 I m )ξ 2 A + (Z I m )U + E, (6) where ξ 2 is the parameter matrix of primary interest. In what follows, we mainly consider the estimation of ξ 2. Denote Q the p (p c) matrix such that Q Q = I p c and Q Z = 0. Thus Q Q = I p P Z, where P Z = Z(Z Z) Z is the orthogonal proector on the space M(Z). Let M Z = I p P Z = Q Q, premultiplying model (6) by Q I m, we can obtain the reduced model (Q I m)y = (Q X 2 I m )ξ 2 A + ε, Cov(Vec(ε)) = I N(p c) Σ e, (7) which is relevant to the matrix parameters ξ 2 and Σ e. It is readily to see that, under model (7), the MLE of ξ 2 equals to its LSE ] ξ 2 = [(X 2 M ZX 2 ) X 2 M Z I m YA (AA ). (8) Let ξ 2 be partitioned as ξ 2 = (ξ 2,...,ξ 2(q l)), where ξ 2i (i =,...,q l)are m r matrices. Assume that the elements of X are functions of the time variable t, such as X = (f i (t )), then ξ 2i represent the regression coefficients attached to f l+ (t),...,f q l (t), for the m characteristics and r groups. Denote Ψ = (ξ 2,...,ξ 2(q l) ). Applying definition of the restricted maximum likelihood (REML) estimator, we can obtain the REML estimator of Σ e under reduced model (7), that is, Σ e = N i= ( )( ) / Y i Q Ψ(X 2 Q a i ) Y i Q Ψ(X 2 Q a i ) k, (9) where k = N(p c) (q l)r, Y i = (y i,...,y im ), and Ψ = ( ξ 2,..., ξ 2(q l) ). Under the original model (6), estimators ξ 2 and Σ e are also unbiased for ξ 2 and Σ e, respectively.

4 320 W.C. Ip et al. / Journal of Multivariate Analysis 98 (2007) Theorem 2.. (a) ξ 2 N(ξ 2, (AA ) (X 2 M ZX 2 ) Σ e ), (b) k Σ e W m (k, Σ e ), and independent of ξ 2, where W m (k, Σ e ) is the m-dimensional Wishart distribution with k degrees of freedom and parameter matrix Σ e. Proof. (a) is obvious so we only need to prove (b). Noting Vec(AXC) = (C A) Vec(X), thus model (7) can be rewritten as Y i Q = Ψ(X 2 Q a i ) + E i, Cov(Vec(E i )) = I p c Σ e, i =,...,N, (0) where Y i is the same as that in (9), Ψ = (ξ 2,...,ξ 2(q ) ), and E i = (ε i,...,ε im ) Q. Denote Y 0 =(Y Q,...,Y N Q ), X 0 =(X 2 Q a,...,x 2 Q a N ), E 0 =(E,...,E N ), then model (7) can also be rewritten as Y 0 = ΨX 0 + E 0, Cov(Vec(E 0 )) = I N(p c) Σ e. () Thus we get the equivalent forms of Ψ = ( ξ 2,..., ξ 2(q l) ) and Σ e, respectively, Ψ = Y 0 X 0 (X 0X 0 ) = N Y i Q (Q X 2(X 2 M ZX 2 ) a i (AA ) ), i= Σ e = Y 0 (I N(p c) X 0 (X 0X 0 ) X 0 )Y 0 ( /k N )/ = Y i M Z Y i Ψ(X 2 M ZX 2 AA ) Ψ k. i= It follows readily from () that Σ e is independent of Ψ and k Σ e W m (k, Σ e ). The proof of Theorem 2. is completed. Remark 2.. Cov(Vec( ξ 2 )) does not depend on the matrix Σ u. According to Theorem 2., the new estimator ξ 2 can be used to construct an exact test on Ψ, i.e. on ξ 2 for the general linear hypothesis H 0 : LΨG = 0. In fact, the Wilks s Λ is given by Λ = W W + H, where W = k(l Σ e L ) and H = (L ΨG) [ G (X 2 M ZX 2 AA ) G ] (G Ψ L ). On the other hand, from (4), it is easy to obtain that the LSE of ξ 2 under model (6) is ) ˆξ 2 = ((X 2 M X X 2 ) X 2 M X I m YA (AA ), (2) and [ ) Cov(Vec(ˆξ 2 )) = (AA ) (X 2 M X X 2 ) Σ e + ((X 2 M X X 2 ) X 2 M X Z I m Σ u )] (Z M X X 2 (X 2 M X X 2 ) I m. (3)

5 W.C. Ip et al. / Journal of Multivariate Analysis 98 (2007) Combining with Remark 2., it is reasonable to expect that ξ 2 may be superior to ξ ˆ 2 under some conditions. Suppose that X,X 2 and Z in (5) satisfy the following conditions: X X 2 = 0, Z = (X,Z 0 ), M(Z 0 ) M(X i ) ={0}, i =, 2, (4) then M Z X = 0, M X X 2 = X 2 and M Z = M X M X Z 0 (Z 0 M X Z 0 ) Z 0 M X. (5) For the proof of the last equality see [3]. By the use of (4) and the matrix identity (A + BCB ) = A A B(B A B + C ) B A, we can obtain that (X 2 M ZX 2 ) = (X 2 X 2) + (X 2 X 2) X 2 Z 0(Z 0 M XZ 0 ) Z 0 X 2(X 2 X 2). Thus Cov(Vec( ξ 2 )) can be rewritten as [ ) Cov(Vec( ξ 2 )) = (AA ) (X 2 X 2) Σ e + ((X 2 X 2) X 2 Z 0 I m )] ((Z 0 M XZ 0 ) Σ e )(Z 0 X 2(X 2 X 2) I m. (6) Furthermore, under the assumption (4), (3) can be simplified as [ Cov(Vec(ˆξ 2 )) = (AA ) (X 2 X 2) Σ e ) )] + ((X 2 X 2) X 2 Z 0 I m (Σ u ) 22 (Z 0 X 2(X 2 X 2) I m, (7) where (Σ u ) 22 = (0,I m(c l) )Σ u (0,I m(c l) ) is m(c l) m(c l) nonnegative definite matrix. Comparing (6) with (7), we obtain the condition for which the new estimator ξ 2 is superior to the LSE ξ ˆ 2 in the following theorem. Theorem 2.2. If conditions (4) hold, then Cov(Vec( ξ 2 )) < Cov(Vec(ˆξ 2 )) if and only if X 2 Z 0 = 0 and ((Z 0 M XZ 0 ) Σ e )<(Σ u ) 22. (8) For example, X = (,, ), X 2 = (, 0, ), Z 0 = (, 4, 9), and (Σ u ) 22 = a Σ e, then ξ 2 is superior to ξ ˆ 2 only if a> 2 3. Remark 2.2. The condition (8) is irrelevant to the other sub-matrices of Σ u besides (Σ u ) 22. In the following section, we will show that the new estimator ξ 2 can also be superior to the improved two-stage estimator of ξ 2 under some conditions. 3. The optimality of the new estimator In this section, we consider the optimality of the new estimator ξ 2 of ξ 2, the parameter matrix of primary interest, and give the necessary and sufficient condition under which ξ 2 is equal to the BLUE of ξ 2 for the original model (6).

6 322 W.C. Ip et al. / Journal of Multivariate Analysis 98 (2007) Theorem 3.. ξ2 is the BLUE of ξ 2 under model (6), if and only if Z M X X 2 = 0, where M X = I p X (X X ) X. (9) Proof. By Lemma [6], ξ 2 is the BLUE of ξ 2 under model (6) if and only if Cov(Vec( ξ 2 ), (I n P A P X I m ) Vec(Y )) = 0, (20) where n = Npm, that is, (AA ) A ((X 2 M ZX 2 ) X 2 M Z I m )Ω(M X I m ) = 0, which is equivalent to X 2 M ZM X = 0. From (5), we have X M Z = 0, thus (2) X P Z M X = 0 P X P Z M X = 0 P X P Z = P X. Combining Theorem (A0, A) of [2] and the fact that P X = P X + M X X 2 (X 2 M X X 2 ) X 2 M X. (22) Eq. (2) can been simplified to Z M X X 2 = 0. The proof of the Theorem 3. is completed. The BLUE of ξ 2 under model (6) is ξ 2 = C 22. C 2M C Ω /2 YA (AA ), (23) where C i = (X i I m))ω /2,i =, 2, C 22. = C 2 M C C 2,M C = I C (C C ) C. Obviously, ξ 2 is an usually unfeasible estimator since Ω in (23) includes two unknown covariance matrices Σ u and Σ e. However, if condition (9) holds, then by Theorem 3., we have ξ 2 = ξ 2, that means the existence of the explicit ML estimator of the part parameter ξ 2. Theorem 3.2. Under model (6), the following statements are equivalent: (a) Z M X X 2 = 0, (b) ξ 2 = ˆξ 2, (c) ˆξ 2 = ξ 2. Proof. (i) Proof of (a) (c). Similar to the proof of Theorem 3., we have ˆξ 2 = ξ 2 Cov(Vec(ˆξ 2 ), (I n P A P X I m ) Vec(Y )) = 0, which can be simplified as (X 2 M X Z I m )Σ u (Z M X I m ) = 0. (24) Since Σ u 0 is arbitrary, (24) is equivalent to Z M X X 2 = 0orZ M X = 0. From (5), we have Z M X = 0 M(Z) = M(X ) Z M X X 2 = 0, thus (24) is equivalent to Z M X X 2 = 0, that is, (a) (c). (2)

7 W.C. Ip et al. / Journal of Multivariate Analysis 98 (2007) (ii) Proof of (a) (b). Obviously, (a) (b). Now, we consider (b) (a). If (b) holds, then (X 2 M X X 2 ) X 2 M X = (X 2 M ZX 2 ) X 2 M Z. (25) Postmultiplying (25) by Z,wehave(X 2 M X X 2 ) X 2 M X Z=0, which is equivalent to Z M X X 2 = 0. The proof of Theorem 3.2 is completed. Theorem 3.2 shows that the new estimator ξ 2 and the LSE ˆξ 2 can achieve optimality simultaneously, and both necessary and sufficient conditions are Z M X X 2 = 0. Remark 3.. The condition Z M X X 2 = 0 is weaker than the necessary and sufficient condition for ˆξ = ξ under model (6). In fact, by Lemma of [6], we can testify that ˆξ = ξ if and only if Z X = 0or M(Z) = M(X ), which are two special cases for Z M X X 2 = 0. Remark 3. sharpens the intuition that while the explicit MLE of the whole parameter matrix does not exist, it may do so for the part parameter matrix. For the covariance matrix with random effects such as (2), Rao [0] presented an improved two-stage estimator of ξ ˆξ T = ˆξ ((X X) X I m )SQ(Q SQ) Q YA (AA ), where Q = M X Z I m,t = Q Y, and S = Y(I n P A )Y. ˆξ T is derived by covariance adustment in the LSE ˆξ using the concomitant variable T, Rao [0] and Grizzle and Allen [4] proved N r Cov(Vec(ˆξ T )) = N r mk 0 (AA ) {(X I m )Ω (X I m )} N r = N r mk 0 Cov(Vec(ξ )) Cov(Vec(ξ )), (26) where k 0 = rk(m X Z). Clearly, (26) takes the equality if and only if k 0 = 0, which requires M(Z) M(X). In this case the improved two-stage estimator ˆξ T is equal to the LSE ˆξ, and ξ 2 = ˆξ 2 = ˆξ 2T, where ˆξ 2T is the corresponding improved two-stage estimator of ξ 2, ˆξ T = (ˆξ T, ˆξ 2T ). In the following corollary, we will give a set of conditions for the new estimator ξ 2 to be superior to the improved two-stage estimator ˆξ 2T under the case k 0 = 0. Corollary 3.. If M(X) M(Z) = M(X ), M(Z) = M(X ) and Z M X X 2 = 0, then Cov(Vec( ξ ˆ 2T )) > Cov(Vec(ξ 2 )) = Cov(Vec( ξ 2 )) = Cov(Vec(ˆξ 2 )). (27) The conditions M(X) M(Z) = M(X ) and M(Z) = M(X ) ensure k 0 = 0. Upon use of Theorem 3. and (26), we can obtain (27). In order to understand the set of conditions in Corollary 3., without loss of generality, we assume Z = (X : Z 0 ). By (5), we have and M(Z 0 ) M(X) ={0}, Z M X X 2 = 0 Z 0 M X X 2 = 0.

8 324 W.C. Ip et al. / Journal of Multivariate Analysis 98 (2007) According to Theorem (A0, A) of [2], Z 0 M X X 2 = 0 is equivalent to P Z0 P X = 0, thus Z M X X 2 = 0 Z 0 X = 0, (28) and hence the conditions in Corollary 3. are equivalent to Z 0 X = 0, and Z 0 = 0. For example, X = (,,, ),X 2 = (, 2, 3, 4),Z = (X,Z 0 ), where Z 0 = (,,, ), clearly, Z 0 = 0 and Z 0 X = Examples In this section, we shall give two simple examples to illustrate the foregoing results. Example. Consider the model for the rat data (see Verbeke and Molenberghs [4]): β 0 + u 0 + β t i + u z i + ε i if low dose, y i = β 02 + u 0 + β 2 t i + u z i + ε i if high dose, β 03 + u 0 + β 3 t i + u z i + ε i if control, where y i is the observation for the th individual at t i time point, i =,...,p, u = (u 0,u ) is random effect vector with normal distribution N(0, Σ u ), and random error ε i follows the distribution N(0, σ 2 e ). We assume that u,...,u N, ε,...,ε N,...,ε p,...,ε pn are independent each other. Of primary interest is the estimation of the slopes β, β 2 and β 3, and testing whether these slopes are equal to each other. Without loss of generality, we assume that Y = (y,...,y n ), Y 2 = (y n +,..., y n2 and Y 3 = (y n2 +,..., y N ) are the matrices of the observations for the low-dose group, high-dose group and control group, respectively, where y = (y,...,y p ). Denote and Y = (Y : Y 2 : Y 3 ), U = (u,...,u N ), T = (t,...,t p ), Z 0 = (z,...,z p ), X = ( p : T), Z = ( p : Z 0 ) ξ = ( ξ ξ 2 ) ( ) n β0 β = 02 β 0 (n2 n ) 0 (N n2 ) 03, A = 0 n β β 2 β 3, (n 2 n ) 0 (N n2 ) 0 n 0 (n2 n ) (N n 2 ) then model (4.2) can be rewritten as Y = XξA + ZU + E = p ξ A + T ξ 2 A + ZU + E. (30) The covariance matrix of Y is Cov(Vec(Y )) = I N (ZΣ u Z + σ 2 e I p). Denote Q the p (p 2) matrix, such that Q Q = M Z = I p P Z, then the reduced model of (30) may be represented by Q Y = Q T ξ 2A + Q E, Cov(Vec(Q Y)) = I N (σ 2 e I p 2). (3) By (8) and (9), we obtain the new estimators of ξ 2 = (β, β 2, β 3 ) and σ 2 ξ 2 = T M Z T T M Z (Y : Y 2 : Y 3 ),, (29)

9 W.C. Ip et al. / Journal of Multivariate Analysis 98 (2007) or and β r = σ 2 e = T M Z T T M Z Y r, r =, 2, 3, (32) 3 SS r /k, r= (33) where k = N(p 2) 3, Y r = (Y r,...,y pr ), n Y i = y i /n, Y i2 = = n n 2 =n + SS = (y β T) M Z (y β T), SS 2 = SS 3 = = n 2 =n + N =n 2 + y i /(n 2 n ), Y i3 = (y β 2 T) M Z (y β 2 T), and (y β 3 T) M Z (y β 3 T). N =n 2 + y i /(N n 2 ), By Theorem 2., ξ 2 and σ 2 are independent, based on which we can construct a test statistic for testing H 0 : β = β 2 = β 3. Let ( ) H 0 =, 0 then H 0 is equivalent to testing ξ 2 H = 0. The exact test statistic is F = ξ 2 H(H (AA ) H) H ξ 2 /2 (T M Z T), σ 2 e which is an F-statistic and hence F F 2,k if ξ 2 H = 0 holds. By Theorem 2.2, under the case T Z 0 = 0, if the variance of the error σ 2 satisfies σ 2 e <(Z 0 M XZ 0 )(Σ u ) 22 = (Z 0 M XZ 0 ) Var(u ), (34) then ξ 2 is superior to the LSE of ξ 2 ˆξ 2 = (T t p ) (T t p ) (T t p) (Y : Y 2 : Y 3 ). (35) By Theorem 3.2, if T Z 0 = 0 and p Z 0 = 0, then ξ 2 = ˆξ 2, and ξ 2 is the BLUE of ξ 2 under model (29). Furthermore, if Z 0 = 0, by Corollary 3., then ξ 2 is superior to the improved two-stage estimator of ξ 2.

10 326 W.C. Ip et al. / Journal of Multivariate Analysis 98 (2007) Example 2. Consider the model for systolic and diastolic blood pressure data: { () β 0 y i l = + u i + β () x l + v i t l + ε i l if treatment group, β (2) 0 + u i + β (2) x l + v i t l + ε i l if treatment 2 group, i =,...,N, =, 2, l =,...,p, (36) where y il,y i2l are the systolic and diastolic blood pressure for the ith patients (with moderate essential hypertension) at t l time point, respectively, β () 0 and β (2) 0 are fixed intercepts, β () and β (2) are the fixed effects of treatment and 2 on the endpoints, respectively. We assume that random individual effect u i = (u i,u i2,v i,v i2 ) follows distribution N(0, Σ u ), and independent of the error ε i = (ε i,ε i2,...,ε ip,ε i2p ) with distribution N(0,I p Σ e ). Clearly, (36) is the case of model (3) with m = 2, β () 0 β (2) 0 ( ) β () ξ 02 β (2) 02 ( ) ξ = =, A = n 0 (N n ) ξ 2 β () β (2) 0 n, (37) (N n ) β () 2 β (2) 2 and X = ( p, x), Z = ( p, t), U = (u,...,u N ). Here x = (x,...,x p ) and t = (t,...,t p ) are designs vectors on dose and time point. Of primary interest is the testing on ξ 2, the effects of treatment and 2 on systolic and diastolic blood pressure. According to Theorem 2., we can obtain the actual test statistic on ξ 2, Wilks s Λ statistic (2), which is constructed by the new estimator ξ 2 given by (8). Otherwise, the new estimator of ξ 2 is superior to the LSE and two-stage estimator under the conditions of Theorem 2.2 and Corollary 3.. See the following typical designs on dose x and time point t. Take x = (,, 0,, ), t = (, 2, 3, 4, 5), then conditions (8) can be equivalent to Σ e = Cov((ε il,ε i2l ) )<Cov((v i,v i2 ) ). (38) That is, if the covariance matrix of error is lower than the covariance matrix of time effect under Löwner partial ordering, then the new estimator of ξ 2 is superior to the LSE. Take x = (0, 2, 2, 2, 0), t = ( 2,, 0,, 2), it is easy to verify that Z M X x = (0,t x) = (0, 0), so the new estimator of ξ 2 is superior to the two-stage estimator provided by Rao. The above two examples show that Theorem 2.2 and Corollary 3. are helpful for the study on optimal design of experiments too. Acknowledgments This research was supported by a grant from The Hong Kong Polytechnic University Research Committee. Wu s and Wang s research was also partially supported by National Natural Science

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