A multivariate growth curve model with di¨ering numbers of random e¨ects

30 (2000), 525±536

A multivariate growth curve model with di¨ering numbers of random e¨ects

Takahisa Yokoyama

(Received July 22, 1999) (Revised March 13, 2000)

ABSTRACT. In this paper we consider a multivariate growth curve model with covariates and random e¨ects. The model is a mixed MANOVA-GMANOVA model which has multivariate random-e¨ects covariance structures. Test statistics for a general hy- pothesis concerning the adequacy of a family of the covariance structures are proposed.

A modi®ed LR statistic for the hypothesis and its asymptotic expansion are obtained.

The MLE's of unknown mean parameters are obtained under the covariance structures.

The e½ciency of the MLE is discussed. A numerical example is also given.

1. Introduction

Suppose that we obtain repeated measurements of mresponse variables on each of p occasions (or treatments) for each of Nindividuals and that we can use observations of r covariates for each individual. Let x^g†_j be an mp-vector of measurements on the j-th individual in the g-th group arranged as

x^g†_j ˆ …x^g†_11j;. . .;x^g†_1mj;. . . .;x^g†_p1j;. . .;x^g†_pmj†⁰;

and assume that x^g†_j 's are independently distributed as Nmp…m^g†_j ;W†, where W is an unknown mpmp positive de®nite matrix, jˆ1;. . .;N_g, gˆ1;. . .;k.

Further, we assume that mean pro®les of x^g†_j are m-variate growth curves with r covariates, i.e.,

m^g†_j ˆ …B⁰nI_m†x^g†‡Y⁰c^g†_j ;

where B is a qp within-individual design matrix of rank q …Up†, B⁰nIm is the Kronecker product of B⁰ and themmidentity matrix, c^g†_j 's are r-vectors of observations of covariates, x^g†'s aremq-vectors of unknown parameters, Yis an unknown rmp parameter matrix. Let

Xˆ ‰x^1†₁ ;. . .;x^1†_N1;. . . .;x^k†₁ ;. . .;x^k†_NkŠ⁰; NˆN₁‡ ‡N_k:

2000 Mathematics Subject Classi®cation. 62H10, 62H12.

Key words and phrases. multivariate growth curve model, multivariate random-e¨ects covariance structure, likelihood ratio statistic, maximum likelihood estimator.

Then the model of X can be written as

X@N_Nmp…AX…BnI_m† ‡CY;WnI_N†;

…1:1†

where

Aˆ

1_N1 0

...

0 1Nk

0 B@

1 CA

is an Nk between-individual design matrix, 1n is an n-vector of ones, Cˆ ‰c^1†₁ ;. . .;c^1†_N1;. . . .;c^k†₁ ;. . .;c^k†_NkŠ⁰ is a ®xed Nr matrix of covariates, rank ‰A;CŠ ˆk‡r …UNÿp†, Xˆ ‰x^1†;. . .;x^k†Š⁰ is an unknown kmq parameter matrix. Without loss of generality, we may assume that BB⁰ˆIq. The mean structure of (1.1) is a mixed MANOVA-GMANOVA model, and the GMANOVA portion is an extension of Pottho¨ and Roy [5] to the multiple- response case. For the model (1.1) in the single-response case …mˆ1†, see Yokoyama and Fujikoshi [10] and Yokoyama [12]. This type of models has been discussed by Chinchilli and Elswick [2], Verbyla and Venables [9], etc.

For a comprehensive review of the literature on such models, see, e.g., von Rosen [7] and Kshirsagar and Smith [3, p. 85].

In this paper we consider a family of covariance structures W_sˆ …B_s⁰nI_m†D_s…B_snI_m† ‡I_pnS_s; 0UsUq;

…1:2†

which is based on random-coe½cients models with di¨ering numbers of random e¨ects (see Lange and Laird [4]), where D_s and S_s are arbitrary msms positive semi-de®nite andmmpositive de®nite matrices respectively,Bs is the matrix which is composed of the ®rst s rows of B. This family is a gen- eralization of a multivariate random-e¨ects covariance structure proposed by Reinsel [6]. In fact, the covariance structures (1.2) can be introduced by assuming the following model:

x^g†_j ˆm^g†_j ‡ …B_s⁰nI_m†h^g†_j ‡e^g†_j ;

where h^g†_j is an ms-vector of random e¨ects distributed as N_ms…0;D_s†, e^g†_j is an mp-vector of random errors distributed as Nmp…0;IpnSs†, h^g†_j 's and e^g†_j 's are mutually independent. This implies that V…x^g†_j † ˆW_s. In § 2 we derive a canonical form of the model (1.1). A test statistic for testing H0s:WˆWs vs.

H_1s: not H_0s in the model (1.1) has been proposed by Yokoyama [13]. In § 3 we propose test statistics for the hypothesis

H_0s:WˆW_s vs: H_1t:WˆW_t …1:3†

in the model (1.1), where 1Us<tUq. Since the exact likelihood ratio

…ˆLR† statistic for the hypothesis is complicated, it is suggested to use a modi®ed LR statistic, which is the LR statistic for a modi®ed hypothesis. An asymptotic expansion of the null distribution of the statistic is obtained. By making this strong assumption that WˆWs, we can expect to have e½cient estimators. In § 4 we obtain the maximum likelihood estimators (ˆMLE's) of unknown mean parameters under the covariance structures (1.2). In com- parison with the MLE of X when no special assumptions about W are made, we show how much gains can be obtained for the maximum likelihood es- timation of X by assuming that W has the structures (1.2). In § 5 we give a numerical example of the results of § 4.

2. Canonical form of the model

In order to transform (1.1) to a model which is easier to analyze, we use a canonical reduction. We de®ne the submatrices B^ÿ_t and B^ÿ_sVt of B by Bˆ

‰B_t⁰;B^ÿ_t ⁰Š⁰,Btˆ ‰B_s⁰;B^ÿ_sVt⁰Š⁰. LetB be a…pÿq† pmatrix such thatBB⁰ˆIpÿq

and BB⁰ˆ0. Then Gˆ ‰B_s⁰;B^ÿ_sVt⁰;B^ÿ_t ⁰;B⁰Š⁰ˆ ‰G₁⁰;G₂⁰;G₃⁰;G₄⁰Š⁰ is an orthogo- nal matrix of order p, where G₁⁰ ˆ ‰g^1†₁ ;. . .;g^s†₁ Š⁰: ps, G₂⁰ ˆ ‰g^1†₂ ;. . .;g^tÿs†₂ Š⁰: p …tÿs†,G₃⁰ˆ ‰g^1†₃ ;. . .;g^qÿt†₃ Š⁰:p …qÿt†,G₄⁰ˆ ‰g^1†₄ ;. . .;g^pÿq†₄ Š⁰: p …pÿq†.

Therefore, QˆGnImˆ ‰Q₁⁰;Q₂⁰;Q₃⁰;Q₄⁰Š⁰ˆ ‰Q^1†₁ ⁰;. . .;Q^s†₁ ⁰;Q^1†₂ ⁰;. . .;Q^tÿs†₂ ⁰; Q^1†₃ ⁰;. . .;Q^qÿt†₃ ⁰; Q^1†₄ ⁰;. . .;Q^pÿq†₄ ⁰Š⁰ is an orthogonal matrix of order mp. Further, let H ˆ ‰H₁;H₂Š be an orthogonal matrix of order N such that H1 is an orthonormal basis matrix on the space spanned by the column vectors of C. Then, letting Y ˆH₂⁰XQ⁰ˆ ‰Y₁;Y₂;Y₃;Y₄Š ˆ ‰Y₁^1†;. . .;Y₁^s†;Y₂^1†;. . .; Y₂^tÿs†;Y₃^1†;. . .;Y₃^qÿt†;Y₄^1†;. . .;Y₄^pÿq†Š, ‰Y1;Y2;Y3Š ˆY_123† and ‰Y2;Y3Š ˆ Y_23†ˆ ‰Y_23†^1†;. . .;Y_23†^qÿs†Š, the model (1.1) can be reduced to a canonical form

H⁰XQ⁰ˆ Z

Y_123† Y₄

@NNmp m AX~ 0

;CnIN

; …2:1†

where

mˆH₁⁰A‰X;0Š ‡H₁⁰CYQ⁰; A~ˆH₂⁰A;

C ˆQWQ⁰ˆ

C11 C12 C13 C14

C₂₁ C₂₂ C₂₃ C₂₄ C31 C32 C33 C34

C₄₁ C₄₂ C₄₃ C₄₄ 0

BB BB

@

1 CC CC A:

Here we note that …Y;X† is an invertible function of …m;X†. In fact,Ycan be expressed in terms of m and X as

Yˆ …H₁⁰C†^ÿ1mQÿ …H₁⁰C†^ÿ1H₁⁰AX…BnI_m†:

…2:2†

3. Tests for a family of covariance structures

We consider the LR test for the hypothesis (1.3) in the multivariate growth curve model (1.1). This is equivalent to considering the LR test for the hypothesis

H_0s:C ˆ D_s‡I_snS_s 0 0 IpÿsnSs

! vs:

…3:1†

H1t:C ˆ Dt‡ItnSt 0 0 IpÿtnSt

!

in the model (2.1). Since the elements of m in (2.1) are free parameters, for testing the hypothesis (3.1) we may consider the LR test formed by only the density of Y. The model for Y is

Y@N_nmp…‰AX;~ 0Š;CnI_n†;

…3:2†

where nˆNÿr. Let L…X;C† be the likelihood function of Y. It is easy to see that the MLE of X under H0s or H1t is given by X^ˆ …A~⁰A†~^ÿ1A~⁰Y_123†. Then we have

g…C† ˆ ÿ2 logL…X;^ C†

ˆnlogjCj ‡trC^ÿ1‰Y_123†ÿA~X^Y4Š⁰‰Y_123†ÿA~X^Y4Š:

As is seen later on, the minimum of g…C† under H0s or H1t is compli- cated. For simplicity, we consider the LR test for a modi®ed hypothesis

H~_0s:Cˆ C11 0 0 I_pÿsnS_s

vs: H~_1t :C ˆ C_12†…12† 0 0 I_pÿtnS_t

; …3:3†

where C11 and C_12†…12† are assumed to be arbitrary msms and mtmt positive de®nite matrices respectively, and

C_12†…12†ˆ C₁₁ C₁₂ C₂₁ C₂₂

:

We note that the di¨erence between H0s and H~0s is whether or not C11 satis®es a restriction that C₁₁VI_snS_s, and so is the di¨erence between H_1t and H~1t. It is easily seen that

minH~0s

g…C₁₁;S_s† ˆnlog1 nS₁₁

…3:4†

‡n…pÿs†log 1 n…pÿs†

X^qÿs

iˆ1

S_23†…23†^ii† ‡X^pÿq

jˆ1

Y₄^j†0Y₄^j†

!

‡nmp

and

minH~1t

g…C_12†…12†;St† ˆnlog1 nS_12†…12†

…3:5†

‡n…pÿt†log 1 n…pÿt†

X^qÿt

iˆ1

S₃₃^ii†‡X^pÿq

jˆ1

Y₄^j†0Y₄^j†

!

‡nmp;

where

SˆY⁰‰I_nÿA…~A~⁰A†~^ÿ1A~⁰ŠYˆ

S₁₁ S₁₂ S₁₃ S₁₄ S₂₁ S₂₂ S₂₃ S₂₄ S31 S32 S33 S34

S₄₁ S₄₂ S₄₃ S₄₄ 0

BB BB

@

1 CC CC A;

S_12†…12†ˆ S11 S12

S₂₁ S₂₂

!

and Saa^ii†ˆYa^i†0

‰I_nÿA…~A~⁰A†~^ÿ1A~⁰ŠYa^i†. The minimum (3.4) is achieved at C11ˆ1

nS11; Ssˆ 1 n…pÿs†

X^qÿs

iˆ1

S^ii†_23†…23†‡X^pÿq

jˆ1

Y₄^j†0Y₄^j†

! : Therefore, we can obtain the LR test statistic

L~s;tˆ

jS_12†…12†j 1 pÿt

X^qÿt

iˆ1

S₃₃^ii†‡X^pÿq

jˆ1

Y₄^j†0Y₄^j†

!

pÿt

jS11j 1 pÿs

X^qÿs

iˆ1

S_23†…23†^ii† ‡X^pÿq

jˆ1

Y₄^j†0Y₄^j†

!

…3:6† pÿs

for testing H~_0s vs. H~_1t, which may be also used for testing H_0s vs. H_1t. The statistic L~s;t can be expressed in terms of the original observations, using …3:7†

Y₄^j†0Y₄^j†ˆQ₄^j†VxxcQ₄^j†0; Saa ˆQaVxxcaQ_a⁰; S_aa^ii†ˆQ^i†_a VxxcaQ^i†_a ⁰; where V_xxcˆV_xxÿV_xcV_cc^ÿ1V_cx, V_xxcaˆV_xxcÿV_xacV_aac^ÿ1V_axc and

Vˆ ‰X;C;AŠ⁰‰X;C;AŠ ˆ

V_xx V_xc V_xa V_cx V_cc V_ca V_ax V_ac V_aa 0

@

1 A: …3:8†

We can decompose the statistic L~_s;t as L~_s;tˆL~₁L~₂; …3:9†

where

L~₁ˆjS₁₁₂j

jS11j ˆ jS₁₁₂j jS₁₁₂‡S₁₂S^ÿ1₂₂S₂₁j …3:10†

and

L~₂ˆ

jS₂₂j 1 pÿt

X^qÿt

iˆ1

S^ii†₃₃ ‡X^pÿq

jˆ1

Y₄^j†0Y₄^j†

!

pÿt

1 pÿs

X^tÿs

iˆ1

S^ii†₂₂ ‡X^qÿt

iˆ1

S₃₃^ii†‡X^pÿq

jˆ1

Y₄^j†0Y₄^j†

!

pÿs: …3:11†

The statistics L~1 andL~2 are the LR statistics for C12ˆ0 and C22ˆItÿsnSs, respectively.

Lemma 3.1. When the hypothesis H_0s is true, it holds that ( i ) L~₁ and L~₂ are independent,

…ii† E…L~₁^h† ˆGms…¹₂fnÿkÿm…tÿs†g ‡h†Gms…¹₂…nÿk††

Gms…¹₂fnÿkÿm…tÿs†g†Gms…¹₂…nÿk† ‡h†; …iii† E…L~₂^h† ˆ…pÿs†^m…pÿs†h

…pÿt†^m…pÿt†h

G_m…tÿs†…¹₂…nÿk† ‡h†

G_m…tÿs†…¹₂…nÿk††

G_m…¹₂fn…pÿt† ÿk…qÿt†g ‡ …pÿt†h†G_m…¹₂fn…pÿs† ÿk…qÿs†g†

G_m…¹₂fn…pÿt† ÿk…qÿt†g†G_m…¹₂fn…pÿs† ÿk…qÿs†g ‡ …pÿs†h†; where G_m…n=2† ˆp^m…mÿ1†=4Q_m

jˆ1G……nÿj‡1†=2†.

Proof. UnderH_0s, it is easy to verify thatS₁₁₂@W_ms…nÿkÿm…tÿs†;C₁₁†, S12S₂₂^ÿ1S21@Wms…m…tÿs†;C11†, S22@W_m…tÿs†…nÿk;ItÿsnSs†, P_qÿt

iˆ1S₃₃^ii†@ W_m……nÿk†…qÿt†;S_s† and P_pÿq

jˆ1 Y₄^j†0Y₄^j†@W_m…n…pÿq†;S_s†. Further, these statistics are independent. Therefore, L~1 and L~2 are independent. The h-th moment of L~₁ follows from that L~₁ is distributed as a lambda distribution Lms…m…tÿs†;nÿkÿm…tÿs††. The h-th moment of L~2 can be written as

E…L~₂^h† ˆ2^m…tÿs†h…pÿs†^m…pÿs†h …pÿt†^m…pÿt†h

G_m…tÿs†…¹₂…2h‡nÿk††

G_m…tÿs†…¹₂…nÿk†† E jW₂j^pÿt†h jW1‡W2j^pÿs†h

" #

; where W₁ and W₂ are independently distributed, W₁@W_m…n₁;I_m†, W₂@ W_m…n₂;I_m†, n₁ˆ …2h‡nÿk†…tÿs†, n₂ ˆn…pÿt† ÿk…qÿt†. Here, letting Uˆ jW1‡W2j and V ˆ jW2j jW1‡W2j^ÿ1, it is easy to verify that U and V

are independent,V@L_m…n₁;n₂†, Uhas the same distribution as Q_m

iˆ1U_i where the Ui are independent and Ui@w_n²_1‡n2ÿi‡1. The h-th moment of L~2 can be obtained from the above fact.

Using Lemma 3.1, we can obtain an asymptotic expansion of the null distribution of statistic ÿnrlogL~s;t by expanding its characteristic function.

Theorem3.1. When the hypothesis H0s is true, an asymptotic expansion of the distribution function of statistic ÿnrlogL~s;t is

P…ÿnrlogL~s;tUx† ˆP…w²_f Ux† ‡O…M^ÿ2† …3:12†

for large Mˆnr, where f ˆ¹₂m…tÿs†…mt‡ms‡1† and r is de®ned by fn…1ÿr† ˆ 1

12m…tÿs†f6…mt‡ms‡1†k‡6…mt‡1†ms

‡2m²…tÿs†²‡3m…tÿs† ÿ1‡ 1 …pÿt†…pÿs†

f6…pÿq†²k²ÿ6…m‡1†…pÿq†k‡2m²‡3mÿ1gg:

In the single-response case…mˆ1†, the asymptotic expansion (3.12) agrees with the results in Yokoyama [12].

We now consider the exact LR criterion L_s;tⁿ⁼² for H_0s vs. H_1t. Let C^₁₁ˆ1

nS₁₁; S^_sˆ 1 n…pÿs†

X^qÿs

iˆ1

S^ii†_23†…23†‡X^pÿq

jˆ1

Y₄^j†0Y₄^j†

!

; C^_12†…12†ˆ1

nS_12†…12†; S^_tˆ 1 n…pÿt†

X^qÿt

iˆ1

S₃₃^ii†‡X^pÿq

jˆ1

Y₄^j†0Y₄^j†

! : If it holds that

C^11ÿIsnS^sV0 and C^…12†…12†ÿItnS^tV0;

…3:13†

it is easy to show that L_s;tˆL~_s;t. Unless (3.13) holds, we need to solve the problem of minimizing

1

ng…Ds;Ss† ˆlogjDs‡IsnSsj ‡tr…Ds‡IsnSs†^ÿ1C^11

‡ …pÿs†…logjSsj ‡trS^ÿ1_s S^s†

or 1

ng…Dt;St† ˆlogjDt‡ItnStj ‡tr…Dt‡ItnSt†^ÿ1C^_12†…12†

‡ …pÿt†…logjS_tj ‡trS^ÿ1_t S^_t†

under H_0s or H_1t, respectively. However, since the problem is not easy, we consider a lower bound denoted in terms of characteristic roots (Anderson [1]) for the minimum of g…D_s;S_s†=n or g…D_t;S_t†=n. Let l₁V Vl_ms and l₁V Vl_m be the characteristic roots of C^11 and S^s respectively, and let d₁> >d_mt and d₁ > >d_m be ones of C^_12†…12† and S^_t, respectively.

Then, as test statistics for H0s vs. H1t, we obtain Ls;tˆ L~_s;t; if C^₁₁ÿI_snS^_sV0,

R_s; elsewhere,

…3:14†

and

L_s;t ˆ L~s;t; if C^_12†…12†ÿItnS^tV0, Rt; elsewhere,

…3:15†

where

R_sˆ jC^_12†…12†j jS^_tj^pÿt jC^₁₁j Q^m

jˆ1l_jsexp l_j ljsÿ1

( )_pÿs;

R_tˆ

jC^_12†…12†j Q^m

jˆ1djtexp d_j d_jtÿ1

( )_pÿt

jC^11j jS^sj^pÿs :

The statistics L_s;t and L_s;t are approximate LR statistics for H_0s vs. H~_1t and H~0s vs. H1t, respectively. In the single-response case …mˆ1†, we have L_s;tUL_s;tUL_s;t .

4. The MLE's of unknown mean parameters

In this section we obtain the MLE's of unknown mean parameters in the multivariate growth curve model (1.1) withWˆ …B_s⁰nI_m†D_s…B_snI_m† ‡I_pnS_s …ˆWs†and consider the e½ciency of the MLE ofX. This model is reduced to the same canonical form as in (2.1), but the covariance matrix C is given by

C ˆ D_s‡I_snS_s 0 0 I_pÿsnS_s

: It is easily seen that the MLE's of m and X are given by

^

mˆZ and X^ˆ …A~⁰A†~^ÿ1A~⁰Y_123†; …4:1†

respectively. Therefore, from (2.2) the MLE of Y is given by

Y^ˆ …H₁⁰C†^ÿ1ZQÿ …H₁⁰C†^ÿ1H₁⁰A…A~⁰A†~^ÿ1A~⁰Y_123†…BnI_m†:

…4:2†

We now express the MLE's X^ and Y^ in terms of the original observations or the matrix V in (3.8). Noting that

…A~⁰A†~^ÿ1A~⁰Y_123†ˆ …A⁰H2H₂⁰A†^ÿ1A⁰H2H₂⁰X…B⁰nIm†;

…H₁⁰C†^ÿ1H₁⁰ˆ …C⁰C†^ÿ1C⁰; we have the following theorem.

Theorem 4.1. The MLE's of X and Y in the multivariate growth curve model (1.1) with WˆWs are given as follows:

X^ˆ ‰A⁰…INÿPC†AŠ^ÿ1A⁰…INÿPC†X…B⁰nIm†

ˆV_aac^ÿ1V_axc…B⁰nI_m†;

Y^ˆ …C⁰C†^ÿ1C⁰Xÿ …C⁰C†^ÿ1C⁰A‰A⁰…INÿPC†AŠ^ÿ1A⁰…INÿPC†X…B⁰BnIm†

ˆV_cc^ÿ1‰VcxÿVcaV_aac^ÿ1Vaxc…B⁰BnIm†Š;

where P_CˆC…C⁰C†^ÿ1C⁰.

On the other hand, the MLE of X when W has no structures, i.e., is arbitrary positive de®nite is given by

X~ˆ …A~⁰A†~^ÿ1A~⁰H₂⁰XS^ÿ1…B⁰nI_m†‰…BnI_m†S^ÿ1…B⁰nI_m†Š^ÿ1; …4:3†

where SˆX⁰H₂‰I_nÿA…~A~⁰A†~^ÿ1A~⁰ŠH₂⁰X. The result (4.3) is an extension of Chinchilli and Elswick [2] to a multivariate case. It is easily seen that

A~⁰A~ˆVaac; A~⁰H₂⁰XˆVaxc; SˆVxxcÿVxacV_aac^ÿ1VaxcˆVxxac: These imply that

X~ˆV_aac^ÿ1V_axcV_xxac^ÿ1 …B⁰nI_m†‰…BnI_m†V_xxac^ÿ1 …B⁰nI_m†Š^ÿ1: …4:4†

The estimators X^ and X~ have the following properties.

Theorem 4.2. In the multivariate growth curve model(1.1) with WˆW_s it holds that both the estimators X^ and X~ are unbiased, and

V…vec…X†† ˆ^ CsnM;

V…vec…X†† ˆ~ 1‡ m…pÿq†

Nÿ …k‡r† ÿm…pÿq† ÿ1

CsnM;

where M ˆ ‰A⁰…I_NÿP_C†AŠ^ÿ1 and

C_sˆ D_s‡I_snS_s 0 0 I_qÿsnS_s

: Proof. Since X^ˆ …A~⁰A†~^ÿ1A~⁰Y_123†, we have

E…X† ˆ^ X and V…vec…X†† ˆ^ Csn…A~⁰A†~^ÿ1;

which imply the result onX. By an argument similar to the one in Yokoyama^ [12], it can be shown that for any positive de®nite covariance matrix W, E…X† ˆ~ X and

V…vec…X†† ˆ~ 1‡ m…pÿq†

Nÿ …k‡r† ÿm…pÿq† ÿ1

‰…BnI_m†W^ÿ1…B⁰nI_m†Š^ÿ1nM:

Under the assumption that WˆW_s, it holds that ‰…BnI_m†W^ÿ1…B⁰nI_m†Š^ÿ1ˆ Cs, which proves the desired result.

From Theorem 4.2, we obtain

V…vec…X†† ÿ~ V…vec…X†† ˆ^ m…pÿq†

Nÿ …k‡r† ÿm…pÿq† ÿ1CsnM>0;

…4:5†

which implies that X^ is more e½cient than X~ in the model (1.1) with WˆWs. This shows that we can get a more e½cient estimator for X by assuming multivariate random-e¨ects covariance structures. Especially, when p is large relative to N, we can obtain greater gains.

As mentioned in § 3, the MLE's of unknown variance parameters Ss and Ds in the model (1.1) with WˆWs become very complicated. On the other hand, the usual unbiased estimators of S_s and D_s may be de®ned by

S~sˆ 1

n…pÿs† ÿk…qÿs†

X^qÿs

iˆ1

S^ii†_23†…23†‡X^pÿq

jˆ1

Y₄^j†0Y₄^j†

!

and D~sˆ 1

nÿkS11ÿIsnS~s;

respectively. However, there is the possibility that the use of D~_s can lead to a nonpositive semi-de®nite estimate of Ds. These estimators can be expressed in terms of the original observations, again using (3.7).

5. An example

In this section we apply the results of § 4 to the data (see, e.g., Srivastava and Carter [8, p. 227]) of the price indices of hand soaps packaged in four ways, estimated by twelve consumers. Each consumer belongs to one of two groups. It is known (Yokoyama [11]) that the model (1.1) in the case mˆ1,

pˆ4 and Nˆ12 with E…X† ˆ 16

0 !

x1₄⁰‡1₁₂…y₁;y₂;y₃;y₄† and V…vec…X†† ˆ …d²1₄1₄⁰‡s²I₄†nI₁₂

is adequate to the observation matrix X :124. Now we estimate how much gains can be obtained for the maximum likelihood estimation of x by assuming the random-e¨ects covariance structure. Since kˆqˆrˆsˆ1,

‰1₄⁰…d²141₄⁰‡s²I4†^ÿ114Š^ÿ1ˆ …4d²‡s²†=4, Mˆ1=3, d^²ˆ:01353 and s^²ˆ

:00976, it follows from Theorem 4.2 and (4.5) that V…x†=V…^ x† ˆ~ 2=3 and

V…^ x† ÿ~ V…^ x† ˆ …4^ d^²‡s^²†=24ˆ:00266.

Acknowledgements

The author would like to thank the referee and Professor Y. Fujikoshi, Hiroshima University, for many helpful comments and discussions.

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Department of Mathematics and Informatics Faculty of Education

Tokyo Gakugei University Tokyo 184-8501, Japan