Efficiency properties of cell means variance component estimates

Journal of Statistical North-Holland

Planning

and Inference

38 (1994) 159- 178

159

Efficiency properties of cell means variance component estimates Peter

H. Westfall

Department

and

qf’ Irzftirmafion

H. Bremer

Ronald

System

and Quantitatiw

Luhhock.

TX 79409.

Received

29 May 1992; revised manuscript

Sciences,

Te.ya.r Tech University,

USA

received

20 October

1992

Abstract Variance component estimates based on cell means have been shown to have useful diagnostic properties and computational simplicity. Here we establish analytic efficiency properties under conditions involving sample sizes and parameters, in general k-way unbalanced factorial models. The cell means estimates are also established as a special case of the minimum norm quadratic unbiased (or MINQUE) estimates. AMS C/ass$cations Key words:

Numbers:

Primary

Mixed model; MINQUE

62510; secondary estimate;

MIVQUE

62F12. estimate;

relative efficiency

1. Introduction Early attempts at variance component estimation in unbalanced factorial mixed analysis of variance (ANOVA) models relied on sums of squares used for fixed-effects analysis (e.g. Henderson, 1953). The resulting estimates, called the AN0 VA estimates in this article, are inadmissible in certain designs, thereby limiting their appeal (Olsen, et al., 1976). Estimates based on the cell means of the ANOVA table, called the cell means estimates in this article, are useful alternatives to the ANOVA estimates. Several authors have investigated these estimates, including Burdick and Graybill (1984), Tan and Tabatabai (1988), and Khuri (1990). The cell means estimates are like the ANOVA estimates in that certain sums of squares from an ANOVA table are computed, the sums of squares are equated to their expectations, and the resulting system of equations is solved for the unknown variance components. Like the ANOVA method, the cell means method produces simple, unbiased estimates. However, in the cell means method, the first step is to average the

Correspondence

to: Dr. P.H. Westfall,

Texas Tech University,

Department of Information Mail Stop 2101, Lubbock, TX 79409, USA.

0378-3758/94;$07.00 Q 1994-Elsevier SSDI 037%3758(93)E0008-5

Systems

Science B.V. All rights reserved

and Quantitative

Sciences,

P.H. Westfull, R.H. Bremer/Cell

160

means estimates

data for each cell in a k-way factorial design. The resulting factorial design with one observation per cell, implying

averages form a balanced a unique sum-of-squares

decomposition. Along with the usual ‘error sum of squares’ from the unaueraged data, these sums of squares determine the cell means variance components. Because the sum-of-squares mates

decomposition

are uniquely

defined.

is unique In general,

for the balanced the ANOVA

k-way factorial,

estimates

are not

the estiuniquely

defined. The cell means estimates have been shown by Hocking et al. (1989) to have a simple form that allows diagnostic analysis. These diagnostics can point out problems with the data and violations of model assumptions. Since these estimates are unbiased, negative estimates are possible. However, as noted by Hocking et al., negative estimates are not necessarily bad when diagnosing outliers and other possible model defects. Particular observations or groups of observations that are most responsible for the negative estimates can be identified and investigated. Admissibility of the cell means estimates was established for certain classes of models by Klonecki and Zontek (1992). The sufficiency of the cell means and residual sum of squares was established by Khuri (1990), further justifying the use of such estimates. Numerical studies by Bremer (1989) also demonstrated that the cell means estimators are reasonably efficient for particular designs. In this article, we establish analytic efficiency properties for the cell means estimates in general k-way unbalanced mixed models. We find that the cell means estimates possess several efficiency properties. First, the efficiencies approach unity when the design becomes ‘large’ in various ways. Second, they have efficiencies approaching unity when certain variance components become large. In each of these cases, ‘efficiency’ is defined as the ratio of the cell means variance to the variance of the minimum-variance quadratic unbiased (MIVQUE) estimate. Since the MIVQUE variance is the lower bound on the variance of an invariant, quadratic unbiased estimate, it is of interest to see how close the cell means estimate’s variance is to this lower bound. Third, under certain sample-size asymptotics, we show that the cell means estimates are efficient relative to optimal, unbiased, nonnegative estimates based on the random eflects themselves, with or without the normality assumption. Fourth, we show that the vector of cell means estimates is a special case of the minimum norm quadratic unbiased (MINQUE) vector.

2. The mixed factorial model 2.1. Index sets A notation similar to that of Seifert (1979), Khuri (1990) and Hocking (1985) will be used to refer to different terms in the model and corresponding quantities. Let k = number of factors, let the f fixed factors be arbitrarily assigned unique integer

P.H. Westfall, R.H. BremerlCell

labels 1, . . .,f, and the remaining

k -frandom

means estimates

factors be assigned

161

unique integer labels

to a purely random (f+ l), ..., k. Assume that 0
lrl=j},

j=l,...,

k,

(2.1)

j=O

denotes the set of interaction effects of order j with So corresponding to the ‘overall mean’. Note that there are (j”) elements in Sj. The subsets Sj, j= 1, . . ..f. may be partitioned further into subsets representing fixed and random effects: Sj=Ej”~j, where Pj={rES;

r={c 1)...I Cj}, and clbffor

Note Pj = (0) for j = 0 and j >f implying

1=1,...,

j}.

aj = Sj for j >f: Further,

(2.2) let

and fi=u;=raj,

(2.3)

so, s=Fvii. Not all interaction effects need be included in the model; in some nested models certain interaction terms have no physical meaning. In other cases insignificant or unimportant terms might be eliminated for the sake of parsimony. Letting D denote the set of all interaction terms included in the model, define Fj=DnPj, F =D&, Rj= Dnfij, and R=Dnk If D = S, the model is the complete factorial model. For rED, the set D,=(sED;

szr}

(2.4)

P.H. Westjdl, R.H. Bremer/Cel/ means estimates

162

denotes

interactions

the statement

s which contain

of Theorem

interaction

r, and will also be considered

(e.g. in

2 below). Define

D, =Du{e} and

(2.5)

R, =Ru{e}.

2.3. The mixed random-@ticts The mixed random-effects Y’C

EF

model model may be represented

as

u,e,+pJ,r,+E SER

=X$+Ut+&,

(2.6)

where Y is the column vector of observations y(i,, . . . , &, ik+ 1), arranged lexicographitally with rightmost indices varying fastest. The factors 8, and 5, are fixed and random effects associated with the factor combinations r and s, respectively. To explain model (2.6) more completely note that for level combination (i1, . , ik), of the k factor there are n(i,, . . . . ik) observations y(il, . . . . ik,ik+l), where ik+l=l ,..., n(il ,..., ik), and (2.6) means that y(il,...,ik,ik+1)=Cel~~:jsr)+C51~~:jEs)+&~il,...,i~,ir+1) IEF

SSS

For each ~EF the fixed interaction effects B:?:jtr), where ij = 1, . . . , aj and jer, form the a,=nj,,uj-dimensional column vector 0,. Similarly, for each SER the random interform the a,=nj,,aj-dimensional action effects t@!. (,,,,ES,,where ij= 1, . . . . aj and jq random vector 5,. We assume that the <, and E are jointly independent mean zero random vectors, and that the elements within vectors are independent, with common variances & 3 0 and & > 0, respectively. Because our goal is the estimation of variance components, we are not concerned with the estimability of the 0,. Hence, we place no restriction on the fixed effects. We shall restrict the class of allowable models (2.6) as in Khuri (1990) and Seifert (1979). In particular, we allow unbalanced models with both nesting and crossing, but the imbalance may only occur ‘in the last stage’ (see Section 7 for an example). We also assume there are no empty cells, and that certain interaction terms must be present. These conditions are summarized in the following assumptions. Assumption

1. For j = 1, . . . , k, ij runs from 1 to aj, with aj32,

for all j.

per cell, where n(il , . . . , ik) > 1 for all Assumption 2. There are n(il, . . . , ik) observations (iI, . . . . ik), and where n(il, . . . . ik)> 1 for at least one cell (iI, . . . . ik).

P.H. Westfall, R.H. BremeriCell

Assumption

3. If LED and SED, then t=rnsED.

Assumption

4. The effects i; and E are normally

Assumption

5. Model (2.6) includes

163

means estimntes

distributed.

the highest-order

interaction

term <,.

Assumption 3 is given by Seifert (1979) and by Rao and Kleffe (1988, Section 7.4). It insures that a unique sum of squares may be associated with each modelled effect. Assumptions 4 and 5 are required for some, but not all, of the results we shall obtain. Given

Assumption n=

z

1, the dimension ...

il = 1

of U, is (n x a,), where

j!J n(iI,...,i,), ik = 1

and for r=@,

Further

(2.7)

for rED, r#@.

%= i ;Ii~~~i

let a = aI, . . . ,ak be the number

notation:

of cells, and let

b,=a/u,.

(2.8)

Let nh be the harmonic mean of the n(i,, . . . . &), n,,=a/tr[d arithmetic mean of cell sizes, fi = trd /a. Here,

-‘I;

and let fi denote

d=Diag{n(i,,...,&)) is the diagonal

matrix

The matrices

the

(2.9)

of cell sizes

U, may now be defined:

letting

U,=Diag{

lntiI ,,,,, irJ}:

U,=U,

(2.10)

where Uri =

for iGr a, 1 for i&r, (I, i I

and U, = I,. The matrix 1, is the identity matrix size a, 1, is the column vector of size a with all elements equal to one, and Diag{A,} is the block diagonal matrix with diagonal elements Ai. Using the notation [AI; i= 1,2, . ..]=[A. :A,: ...I to concatenate matrices Ai having the same numbers of rows, we have X=[U,;

rEF]

U=[U,;

rER].

and

The vectors

8 and 5 are similarly

(2.11) vertically

stacked

with 0, and 5, subvectors.

P.H. Westfall, R.H. Bremer/Cell

164

means estimates

3. Variance component estimates 3.1. The cell means estimates Consider

where

the cell means

A= A-‘&4

version

for any

of (2.6):

matrix

A having

n rows,

A is given

in (2.9), and

U,=Diag{l,(i,, ....ik)}. As discussed in Scheffk (19.59), there is a unique decomposition of u-dimensional Euclidean space, ‘W, into mutually orthogonal vector subspaces 2r:

where a is the dimension of 7. The orthogonal projection matrix for the subspaces J?~ with respect to the complete factorial model with index set S is given by iIr=bL,i,

(3.1)

i=l

where Lri =

I,,-

l,, lb,/ai,

loi lbi lai,

for iEr, for i$r.

Note that the matrix z, has diagonal the dimension of _%‘ris

elements

identically

equal to nisr(Ui-

1)/a; thus

~=n(Ui-l).

(3.2)

isr

Using subspaces

Assumptions Yp,, ED+,

1-3, the subspaces ps, SES+, can be pooled corresponding to each modelled effect:

uniquely

into

with dimensions dim 5Yr= 1,= CseP, b and D, given in (2.5). To define the ‘pooling’ sets P, define a mapping h from S into D+ as follows. Let s&G. Case 1: There exists an reD such that ssr. Then h(s) is the element of D with minimum cardinality that contains s as a subset. Case 2: s$r for all rsD. Then h(s)=e. Now the ‘pooling’ sets are defined for all rED+ by P,={s&;

h(s)=r}.

(3.3)

165

P.H. Westfull, R.H. Bremer/C’ell means estimates

The projection

of r onto Yr, reD+,

will be denoted

y,, and is given by (3.4)

Note that

The following

cell means

qrm=

II rr

%vn=

y’(z-yD)

‘mean squares’

l12/WL

are defined:

PER>

(3.5)

Y/ne,

where pn denotes the projection matrix for the column space of [X: U], and where n, is the number of degrees of freedom for error in the model with all factors fixed. Note that n,=n-a+l,,

(3.6)

where le=ClsP,?r. Let %‘(A) denote the column space of a real matrix A, and let q(A) projection matrix. Following Seifert (1979), we have for SED

denote

its

so that .9’(U,)=b;‘U,U;=

_ -

1 L,, 1LS

(3.7)

tsD

where L, is given in (3.4). This allows simple calculation quadratic forms (3.5): &,,)=trL,

c Au,G+&-~-’

of the expected

values of the

l(O,)

SER

=A+ C ~,b,lb,+~,tr(L,d-‘)l(l,b,) s3r

SER (3.8)

and (3.9)

166

P.H. Westjdl,

R.H. BremerlCell

means estimates

where A is given in (2.9), b, is given in (23, nh is the harmonic 1, is the dimension of _Yr. The cell means variance

component

estimates

ratic forms qr,,,, rER+ , to their expectations

are obtained

mean of the cell sizes and by equating

(3.8) and (3.9), then solving

the quad-

the resulting

system of equations for & and c#J~.Specifically, define Qm=(qrm; reR+)‘. Letting 4=(~#+; TER+)‘, we have E(Q,)= r,,,4, and &= T; ’ Qm is the vector of cell means estimates. Where i(q,,) denotes the row of Q,,, occupied elements within the vector satisfy i(q,,) < i(q,) triangular matrix with unit diagonals. 3.2. MINQUE

and MIVQUE

For rgR, let pr = &/de,

v,=

by qs,,,, we require that the ordering of s c t. This makes T, an upper

whenever

estimates

and let yr denote an a priori guess at the value of pI. Define

c y,u,u:+1 rsR

and w,=

v,‘-

v)?x(x’v,-lx)-x’v,-‘.

Then the MINQUE quadratics (denoted MINQE(U,I) emphasize unbiasedness invariance) and are

by Rao and Kleffe (1988) to Q,=(qrr; rER+!‘, where qry= Y’W’,U,U~W, Y, rER, and qey= Y’W,W,Y. Thus, E(Q,)= T,$J, and &,=T{‘Qy is the vector of MINQUE estimates. Individual variance component estimates are &. (Assumptions l-3 guarantee unbiased estimability of all variance components, hence Ty is invertible.) When the values y are replaced by the actual values p, the MIVQUE estimates &,, result. These estimates have minimum variance in the class of invariant quadratic unbiased estimators (Rao, 1971), but are not usable in practice since the parameters p are unknown. We use these estimates as benchmarks to evaluate the performance

of the cell means estimates.

4. Efficiency of cell means estimates relative to MIVQUE In this section we describe various conditions under which the efficiency of a cell means estimate relative to the corresponding MIVQUE approaches unity. We consider sequences of models of the form (2.6), indexed by z, z = 1,2, . . . , with fixed number of factors k and fixed model, i.e. fixed index sets F,R, and D, but with the ai, the n(il, . . . . ik), and the 4 possibly depending on Z. Efficiencies are implicitly considered for T+ co, but for notational simplicity, the dependence of the various quantities upon T will not always be indicated explicitly. Since all estimates are unbiased, ‘relative efficiency’ of the cell means estimator &,, of c#+ relative to the MIVQUE &, is reasonably defined as the ratio Var(&,,)/Var(&,).

P.H. Westfull, R.H. Bremer/Cell

Asymptotic

efficiency of the &, for increasing

167

means estimates

cell sizes and/or

increasing

numbers

of cells is given in Theorem 1. Consider and

with

q$>O.

model,

TER+, (i) ylh--tX#, or (ii) Ui+x,

Then

Var(&,)/Var(&,)+l: from

with Assumptions

(2.6)

either

fbr

of

l-4, the

with all parameters

following

g$jxed,

conditions

insures

,for some iEc, i$r, and ti is bounded

awuy

1.

Theorem 1 considers aspects of the design controlled by the researcher. Under condition (i), we may claim that &,, is efficient when the harmonic mean of the cell sizes nh is large. If the number of levels ai of the factors can be increased, the cell sizes need not be large, as indicated by condition (ii). The condition that the arithmetic average of the cell sizes ii is bounded away from 1 can be relaxed if Assumption 5 is added, as given in corollary 1. Corollary 1. Consider and with &>O.

Efficiency given in

model

Thenjbr

(2.6)

bvith Assumptions

rER, r#c,

Var(&,)/Var(&,)+l

of &, in terms of the true parameters

l-5,

with all parameters

if ui-tr;,,for

4,fixed,

some i~c, i$r.

4, for fixed a, and n(i,, . . . , ik), is

Theorem 2. Consider model (2.6) with Assumptions l-4, with all ai and n(i, , . . , &)jxed and D, given in (224). Consider rER. lf ps+x ,for some SED,, then Var(&,)/Var(&,)+l. Unlike Theorem 1, Theorem 2 considers aspects of the model that are not controlled by the researcher. The theorem states that &,, is efficient when & is large relative to & for any s 2 r. Naturally, the precision of the estimate will be better if the cell sizes and/or the levels of the factors are also large, as given by Theorem 1. Note that efficiency of the estimate &,, is not covered by Theorem 2. This estimate represents a special case and requires different assumptions. Theorem 3. Consider model (2.6) with Assumptions 1-4, with ull at and n(il, . . . . ik) Let H = Max,..(lsl) so that Sn represents the set ofall highest-order interactions in the model. lf

fixed.

5-lps+ then Var(&,)/Var($,,)-+

6,6(0, x)

.fbr SE!&,

i 6,~[0, co) jbr

SER, s$Sf,,

1.

The theorem states that Jem is efficient when the highest-order ance(s) is(are) large relative to the error variance.

interaction

vari-

168

P.H. Westfall, R.H. Bremer/Cell

means estimates

The proofs of Theorems l-3 and Corollary 1 are deferred. Because the form of Var(&,) can be extremely complicated, evaluation of efficiency is greatly simplified by augmenting the vector Y with data Yaus so every (iI, . . . , ik) cell in the design has exactly M observations,

where M = max [n(il, . . . , &)I. The augmented

model is then

or more simply, r,=x,e+

U,5+s,.

(4.1)

Let the cell means estimates from model (4.1) be denoted &,, PER+. The augmented model is a balanced factorial design. Under Assumptions 1-4, the cell means estimates are uniformly minimum variance quadratic unbiased estimators assuming invariance (Rao and Kleffe, 1988, p. 174). Because the MIVQUE estimates from the original model (2.6) are also invariant quadratic unbiased estimates under model (4.1), their variances can be no smaller than the variances of the cell means estimates from model (4.1). This proves Proposition 1. For all TER+,

Thus we may establish efficiency of the cell means estimates relative to MIVQUE they are efficient relative to the cell means estimates from the augmented model. Since T,,, is upper triangular with unit diagonals, we have

if

where h, = 1. Using the fact that Var( Y’ A Y) = 2 tr (A Cov( Y) A Cov( Y)> for square symmetric A with AX=O, and using (3.7) repeatedly, we obtain

(4.2) where

P.H. Westfall, R.H. BremerlCell

means estimates

169

We have D, in (2.4) b,, A, and U, in (2.8)-(2.10) respectively, 9n in (3.9, n, in (3.6) and L, is the projection of F onto _Yr which has dimension 1,. Writing (4.2) as Var(&,,)=2A+44,B+24,2C1 we see that the term A does not depend on A, and is therefore identical for the original model (2.6) and the augmented model (4.1). Thus, if we subscript the quantities B and C with m or CIto indicate an original

or augmented

V4LJ Var(L) Theorems conditions.

model, we have _ I = 4&(B,-B,)+2~%(C,-C,) 24+44,B,+2@,2C,

1 and 2 are proven

by showing

(4.3)

’

(4.3) tends to zero under the appropriate

Proof of Theorem 1. Define the usual 0 and o notation: for real sequences {xr} and {y,}>x,=O(y,)‘fI 1 x J y 71is b ounded uniformly in z; and xZ= o(y,) if xr/yZ-+O as z+ co. w h ere P, are the pooling sets given in (3.3); Note Var(~~,,)~2~,2/1,~2~,2/(IP,Ia,), hence, it will suffice to show that all terms in the numerator of (4.3) have order o(a; ‘). We now consider the orders of the various terms in the variance expression (4.2). Because @s=CtsD, $,b,b &&LED,

b, d bdslRl,

we have

@,=O(b,). From (3.8) all elements Ti 1 have order O(1).

of T,,, are less than or equal to 1, implying

all elements

h,

of

Consider ls=CteP. ItdCteP, a, < a, I P, 1; hence, I, = O(a,). Using ai - 12 ai/‘2, we also obtain 1/1,=0(1/u,). Further, note that n,an---a, implying l/n,< l/{u@-- 1)) where n, is given in (3.6). By the Cauchy-Schwarz inequality Jtr ABJ
11121112 tr(L,A-‘L,

A-‘)<=-,

(4.4) nh2

where nh2 is the harmonic

mean of the squared

cell sizes, nh2 =u/tr

Am2. We also have

tr(U,A-‘&A-‘U:(I-B,,)}
formula

(4.2). Using

the preceding

inequalities

and bounds,

obtain

P.H. Westfall, R.H. Bremer/Cell

170

means estimates

and

Note that these bounds Hence,

also apply for the variance

of the augmented

estimate

&.

Var(&J ^ Var(&,)

-l=O(&)+O(&) +O(

(4.5)

b:(fiil)nh)+o(&)-

Since hr=nigrui, we have h, +a if ai_fOl: for any i$r. Noting also that fi >nh and nh2 3 n,,, we see that all terms in (4.5) become negligible under either condition (i) or (ii) of the statement of the theorem, proving the result for PER. To prove the result for J,,, use the bound Var($,,)a2&/n. This is true since the MIVQUE

for q& based on the random 1 < Var(&J ’ Var(&,)

n ’ < ’

efSects themselves

is nmlc’~.

Hence,

afi a&- 1)’

which implies the result since fi+cc cannot apply to the error variance.)

under Cl

condition

(i). (Note

that condition

(ii)

Proof of Corollary 1. If the highest order interaction parameter, c#I,,is in the model, the terms h,, of (4.2) are identically zero for all rER, r fc. Thus, the last two terms of (4.5) are identically zero in this case, proving the result. 0 Proof of Theorem 2. Let P nlax=max{p,}=p,($ tED,

where D, is given in (2.4). Because h,,= 1, Var (4%.) 2 2 b&, (1, h,2)- 1 d& 2 4e’ P&,.

(4.6)

After factoring out c#I~,the numerator of (4.3) is a linear function of the pt, for tED,, and has order O(p,,,). The result follows from (4.6) since pmax>ps+~. 0 To prove Theorem

3 we use the following

Proposition 2. Under the conditions in (3.5),

of Theorem

propositions. 3, W, as in Section

3.2 and gD given

P.H. Westfall, R.H. BremerlCell

Proof. Note that

means estimates

171

V, = I + z UD,, U’, where U and X (used below) are given in (2.1 l),

D,, = Diag j (p,/~)1,~},

and where

lim D,, = Dd = Diag {6, Znrj . *API

(4.7)

Note that for large r, %(U D:!‘)=%(U), lim V;‘x=x. T”I)

for x&?(U),

Using (4.7), (4.Q and some matrix

(4.8)

algebra,

we have further

lim r V;‘x=(UD6U’)+x,

for x&Z(U), where A+ denotes

implying

(4.9)

r-+z

the Moore-Penrose

inverse.

Results

(4.8) and (4.9) imply

lim V;‘=Z-P(U)

z+ x

Consider

now V;‘x(x’V;‘x)-x’V;‘=

,;Wq

,;1’2x1

,;I’?

Letting E, denote a matrix whose columns form an orthonormal an expression (valid for large z) for V; I” is

(o.n.) basis for V(U),

V,-“2=E,(I+Zdpr)-1’2E:+z-~(U), where A,, is the diagonal

matrix of positive eigenvalues

X1 and X2 whose columns respectively, note that S[ V;r’2x] Since %‘(UDj!2)=%?(U)

form

o.n. bases

=6?[ v;l’2x11 for large z, lim,,,

of UD,,U’.

of ‘??(X)n%(U)

Defining and

matrices

%(X)n%‘(U),

+Y[XJ.

(4.10)

V;1/2Y[V;1’2X1]

Vi1/2=0.

Thus,

lim V~112P[V;1/2X]V~1/2=(Z-P[U])P[X2](Z-Y[U]),

c--rcc

implying lim W,=Z-P[X:U]=Z-PD.

0

7-m

Proposition

3. Under the conditions

of Theorem

3, 1)z W, U, II = O(l), for

all SER.

P.H. Westfall, R.H. Bremer/Ce//

172

Proof. From

(4.9) obtain z v;’

SO

that

means estimates

U,=( u&u’)+

u,

11z Vi ’ U, /(= O(1). Using (4.10), ~~,-‘x(x’v,-‘x)-x’v,-‘u,=zv,-“~~[v~~’~x~]~~~’~u~ =E,(l/z

“2+A,,)-“2E:9y

fp2X1]

x E,(llz 1’2+ A,,)- 1’2E; U,. The result follows from the inequalities 0 and from the convergence of A,,. Proof of Theorem 3. Consider E(Y’W,W,Y)=

llAB/I d I(All IlBll and IIA+BI( < lIA(I + IlBll,

that

1 qkz+dver, ssR

where c,, = II W, U,j12 and c,, = II Wpl12. Hence,

Ap= Y’ w,wpy/c,,- ~(4c,,) 4,. SER

By Propositions 2 and 3, ~,~=n,+o(l) and c,,=O(l/~~), respectively, where n, is given in (3.6). Noting that Var(&)dVar(&J, and observing from (4.2) that Var(&,,)= @O(r2), we have Var{(c,,/c,,)&)

(4.11)

=deZO(l/z2).

Now consider Var(YW,W,Y)=2

/I I/

2

C&W,USU:W,+Awpw~ seR

=2&e’

c

II 2

II

(P,l~)(~w,u,)(u:w,)+w,w,

SER

(4.12)

=2&{n,+o(l)}. Using (4.1 l), (4.12), Propositions

2 and 3, and the covariance

Var(&,)=2&{1ln,+o(l)}, which implies the result since Var(&,,)=

24%/n,.

0

inequality,

we have

P.H. Westfall, R.H. BremerlCell

5. Efficiency of cell means estimates

means estimates

relative to estimates

173

based

on the random effects In this section we drop the normality assumption which the cell means estimates become efficient MIVQUE (without Replace Assumption

Assumption

Under

assuming normality) based 4 with Assumption 4’.

4’. For rER+, the components

this condition, &,=

the MIVQUE

:,;;p

;:

and identify conditions under relative to estimates that are

on the random

of C, have finite fourth

of & based on the random

eficts

themselves.

moments

pL,.

effects t,, uR+,

is

:‘=“,

i (The o subscript denotes ‘optimal’). Note that the &,, are nonnegative intuitively appealing estimates, that would be used if the random effects were observable. Note also that these optimal estimates are used as ‘targets’ in the development of the MINQUE estimators (Rao, 1972). The following theorem identifies conditions under which the cell means estimates are asymptotically efficient relative to these ideal estimates.

Theorem 4. Consider model (2.6), with Assumptions moments q$ and pL, arejxed, and that pr-$f>O i=l , . . . . k, and q+;o, then Var(&,)/Var(&,)+

l-3 and Assumption 4’. Assume all for some rER+. If ai+E, for all

1.

a,, rER, and Var(&,)=(p,-4,2)/n; it will Proof. Noting that Var(&,)=(~L,-q5~)/ suffice to show a,Var(&,,)+p,-4: for rER and n Var(&,)+p,--q5: to prove the result. Using (3.8), the upper triangular matrix T,,, has all nonzero off-diagonal elements tending to zero. Hence,

(5.1)

174

P.H.

Wes$all,

R.H. Bremer/Cell

means estimates

Now, for any SER,

+ 1 (~~-3~:)tr(U;L,U,diag{U;L,U,})/(1,2b,2) eel& +(~e-3~,2)tr(Ued-1L,d-1

VA

xdiag{U,d-‘L,d-‘U:.})/(1,2hS), where diag {A} is the diagonal matrix A. Also

matrix containing

(5.2) the diagonal

elements

Var(q,,)=24,2/ne+(pL,-34,2)tr((I--D)diag(l-%J)/n,2. Using us/is= 1 +o(l), as well as inequality result for the ‘normal’ portion of (5.2):

(4.4), we obtain

a,(2~s2/(l,b,2)+4~,~,/(1,h,Znh)+2~,2 For the ‘nonnormal’ portion metric. Hence, for tED,, t#s, a,(tr(UILsU,diag{

of the square

(5.3) the following

tr(d_lL,d-‘L,)/(1,2b,2))~2~~

of (5.2), note that tr(A diag{A})d

uiL,U,})/(1,2@))+0.

convergence

(5.4)

tr(A*) for A sym-

(5.5)

Further, a,(tr(U,d-‘L,d-‘U:.diag{

U,d-‘L,A-‘U~})/(1~b~))+O.

(5.6)

For t = s, note that tr(U:~,U,diag{~i:Z,U,S)dtr(UIL,U,diag{U:L,U,))~h,21,. Using (3.1) we have

a,(tr(UiL,U,diag{ Combining

u:L,D,})/(lfb,2))-+1.

(5.7)

(5.3)-(5.7) yields a,Var(q,,)+p,-&,

(5.8)

for all SER, implying u,Var {o(l)qs,,,} =o(l) for all SED,. Because u,/n,
P.H.

Westjdl,

R.H. Bremer/Cel/

means estimcctes

175

so that

Because

the lower

n Var($,,)

and

upper

bounds

tend

to unity

= pL,- &, which proves the result for r = e.

6. Cell means estimates

in this expression,

we have

0

as a special case of MINQUE

When the highest-order interaction term 5, is in the model, the cell means estimates are limits of the MINQUE estimates described in Section 3. This result establishes the cell means estimates as a special case of the MINQUE(a) estimates described by Westfall (1987). Theorem 5. Consider model (2.6), with Assumptions parameters

4 and the number of levels for the factors,

per cell, n(i, ,

z-l

l-3 and Assumption 5, and with all a, and the number ofobservations

, ik), fixed. Suppose yI = y,(7) are sequences 6,~(0, x) yr+ i &E[O,~;C)

Then jbr all YE%“, lim,,,

qf a

priori guesses@

which

for r=c, for rER, rfc.

&= 4,.

Proof. From Proposition 2, we have lim,,, qey = neqem. Because %?(U,) G%‘( U,), the remaining quadratics converge to zero unless suitably normalized. From (4.9) we have lim zV;‘x=(UD,U’)+ X+00

for x&(U,),

x.

(6.1)

Note that U = U, 0, since w(U) = %?(U,). Hence, (UDgU’)+

Using the identity

=(U,UD,U’U;)+.

(AA’)+

= A { (A' A)+}’ A’, we have

Note that

hence, for any matrices

A and B having

A’(UD,U’)+B=,?(UD,fl’)-‘B.

n rows,

(6.2)

P.H. Westfall, R.H. BremerlCell

176

From (3.4) we have Y=Clsn+ A,=~,,6,b,>O.

means estimates

Y,. Note that (UO, 6’) Y, = CssRbs U, VAY,= ;1, Y,, where

Hence, (u&u’)-’

r,=n;r

Using (3.4) (3.7), (6.1)-(6.3)

r,.

(6.3)

and the orthogonality

of Y, to x for KR, obtain

for PER,

Denote the limiting vector of quadratic forms Qm (using the normalizing constant 22 for KR), and order the components of Qa, and Q,,, as in Section 4, then

Qrn=T~Qrn, where the matrix T, is lower triangular with positive diagonal elements. Because the limiting quadratics for the MINQUE estimation scheme are related to the quadratics for the cell means estimates via a one-to-one transformation, the result follows. 0

7. Example and concluding remarks To illustrate

the concepts,

S=D={1,2,12}, Sr={l,2}, R={2,12}. Also, Dr={l,12},

consider

a complete

two-way

mixed

design.

S2=R2=R2={12}, F1=pl={l}, F={O,l), Dz={2,12}, and D12={12}. Model (2.6) is

Here, and

or

where p is generally used instead of 0 (0),il runs from 1 to aI, i2 runs from 1 to a2 and i3 runs from 1 to n(ir , i2). The set notation is usually suppressed in referring to the index sets, thus tj,!,*il would be more succinctly given as li,Tiz. The form of the cell means estimates for this example can be found in Hocking et al. (1989). A numerical study of the efficiencies of the cell means estimates discussed in this paper for the model discussed above is summarized in Bremer (1989). The result of Theorem 2 is very clearly demonstrated in the efficiency plots. The efficiency of the cell means estimates of 42 and 4r2 monotonically increased in all models considered as p12 was increased from 0 to 5. The efficiency of the cell mean estimates of C#J~ is nearly 1 when p12 = 2 in all the cases examined. The efficiency of the cell means estimate of (IJ is 0.9 or better when p12 = 2 in all cases considered. The effect on the efficiencies of ingeasing II,, can be seen when comparing the efficiency plots for model 5 (rr,,= 5.7)

P.H. Westfall, R.H. Bremer/Cell

means estimates

and model 6 (q, = 1.5) or model 9 (q, = 6.7) and model For example,

the efficiency of the cell means estimate

171

10 (q, = 2.0) of Bremer (1989). of q& for model 5 versus model

6 with plz=O.l and pz=O.l is 0.997 versus 0.898. Theorem 1 guarantees efficiency of the cell means estimates of & and 4i2 will approach 1 if the become large (q, + 00). Consider next the above example without 2 in D. This corresponds to a nested design with factor 2 nested in factor 1. The pooling sets of (3.3) are

that the cell sizes two-fold PI = {l},

Pi2 = {2,12} and P,=@ I n a nested design the number of levels of factor 2 nested in a particular level of factor 1 may vary from level to level. This is not allowed by Assumptions 1 and 2 of Section 2.3 and this is what is meant when we say the imbalance may occur only in the last stage (of nesting). We have established that the cell estimates are efficient estimators in various situations. Their utility for diagnostic purposes has already been established; this paper provides justifications for using them further as point estimates. Researchers are justified in using these estimates when the design is ‘large’, as indicated in Theorems 1, 4 or Corollary 1, or when they think that the variance components themselves are ‘large’, as indicated in Theorems 2 and 3. Further comments Comment 1. As noted by Klonecki and Zontek (1992), the cell means estimates may not be admissible, for example, when the highest-order interaction term is excluded from the model. If not, then Theorems l-3 and Corollary 1 also describe the behavior of the estimates that uniformly dominate the cell means estimates. Comment 2. Since the conditions of Theorem 4 are so stringent, it is reasonable to ask, “Do all of the usual variance component estimates have the same efficiency property?” The answer is ‘no’. Westfall (1987) shows that the ANOVA estimates do not possess this property in the unbalanced one-way model.

References Bremer, R.H. (1989). Numerical study of small sample variances of estimators of variance components in the two-way factorial model. Comm. Statist. B 18, 985-1009. Burdick, R.K. and Graybill, F.A. (1984). Confidence intervals on linear combinations of variance components in the unbalanced one-way classification. Technometrics 26, 131-136. Henderson, C.R. (1953). Estimation of variance and covariance components. Biometrics 9, 226-252. Hocking, R.R. (1985). The Analysis qfknear Models. Brooks-Cole, Monterey, CA. Hocking, R.R., Green, J.W. and Bremer, R.H. (1989). Variance-component estimation with model-based diagnostics. Technometrics 31, 227-239. Khuri, A.I. (1990). Exact tests for random models with unequal cell frequencies in the last stage. J. Statist. Plan. If: 24, 177-193. Klonecki, W. and Zontek, S. (1992). Admissible estimators of variance components obtained via submodels Ann Statist. 20, 1454-1467. Olsen, A., Seely, J. and Birkes, D. (1976). Invariant quadratic estimation for two variance components, Ann. Statist. 4. 878-890.

P.H.

178

Wesrfall,

R.H.

Bremer/Cell

means

estimates

Rao, C.R. (1971). Minimum variance quadratic unbiased estimation of variance components. J. Multicariate Anal. 1, 445-456. Rao, CR. (1972). Estimation of variance and covariance components in linear models. J. Amer. Statist. Assoc. 67, 112-115. Rao, C.R. and Kleffe, J. (1988). Estimation qf Variance Components and Applications. North-Holland, Amsterdam. Schefft, H. (1959). The Analysis of Variance. Wiley, New York. Seifert. B. (1979). Optimal testing for fixed effects in general balanced mixed classification models. Math. Operationsforsch. Statist. Ser. Statistics 10, 237-255. Tan, W.Y. and Tabatabai, M.A. (1988). Harmonic mean approach to unbalanced random effects models under heteroscedasticity. Comm. Statist. A 17, 1261-1286. Westfall, P.H. (1987). Computable MINQUE-type estimates of variance components. J. Amer. Statist. Assoc.

82. 586-589.