analysis model exactly. This method received virtually no attention in the litera ture until it was rediscovered by Ihara and Kano (1986). They adapted it to the.
NONITERATIVE
ESTIMATION
BATTERY Michael
Noniterative multiple battery is employed
FACTOR
W. Browne*
FOR THE
ANALYSIS and
Krishna
MODEL Tateneni*
methods for estimating the block diagonal factor analysis are provided. A partitioning
and inter-battery
and factor
extension
unique variance matrix in of the covariance matrix
procedures
manner. The resulting methods involve the computation nondiagonal block of the covariance matrix. An example
MULTIPLE
are applied
in a stagewise
of a conditional is provided.
inverse
of a
1. Introduction Inter-battery factor analysis was introduced by Tucker (1958) as a method for investigating relationships between two batteries of tests. Ordinary least squares estimates (Tucker, 1958) and maximum likelihood estimates (Suzuki, 1976; Browne, 1979) can be expressed in closed form. The inter-battery factor analysis model was subsequently generalized to the multiple battery factor analysis model that treats several batteries (McDonald, 1970). Unlike inter-battery factor analysis, multiple battery factor analysis requires an iterative procedure (Browne, 1980) for obtaining minimum discrepancy estimates. In order to provide a noniterative procedure for multiple battery factor analysis the present article will make use of an approach that has been employed in standard exploratory factor analysis. This is the partitioning method for estimating unique variances. Each unique variance estimate is expressed as a function of a subset of the elements of the covariance matrix. It was first proposed by Albert (1944) who considered the errorless situation where a covariance matrix satisfies the factor analysis model exactly. This method received virtually no attention in the litera ture until it was rediscovered by Ihara and Kano (1986). They adapted it to the analysis of data subject to sampling fluctuation and suggested criteria for choosing variables involved in the partitioning. The partitioning method works well in practice and has received a substantial amount of subsequent attention (Kano, 1990a, 1990b, 1991 ; Cudeck, 1991). One of the subsequent developments was due to Kano (1989) who proposed an approach that has the advantage of retaining as many elements of the covariance matrix as possible when a partitioning is carried out. This method makes use of a conditional inverse, and has had a strong influence on the methodology of the
Key
Words
stagewise
* The
and
Phrases
estimation
Ohio
43210-1222,
State
USA.
; noniterative
University
, Department
method,
multiple
of Psychology,
battery
factor
1885 Neil
analysis,
Avenue
Mall,
unique
variance,
Columbus,
OH
present
paper.
procedure factor
Here,
involving extension
a single
inverse
formula depends
on
Wishart
of the procedure
2.
applicable
the
special
stages
least
factor
of standard
stage, The
in the squares
multiple
a two and
are
choice
of into
of condi step
considered. factor
stage
a pair
inter-battery
battery
factor
as
can be synthesized
inverse.
procedure
ordinary
case
derived
at the first
two
a conditional
and
be
The
is on noniterative
in the
will
analysis
estimation
likelihood
The multiple battery
method
factor
involving
main
it is also
the
at the second.
maximum focus
of
an inter-battery
procedures
algebraic
tional
a variant
;
The
analysis
but
analysis.
analysis model
Consider g batteries, each consisting of pq tests, q =1, g. The multiple battery factor analysis structure for the covariance matrix of the p= pl + ••• + pg tests is (e.g., Browne, 1980, Section 2) I =AA'+
T
(1)
where A is a p x k factor matrix and ?' is a p x p block diagonal unique variance matrix partitioned as Al A=
(2)
A2 Ag
and !F11
0
•.
0
0
W22
...
0
(3) 0
Each
diagonal
negative
unique
•••
W
variance
matrix,
2F22,is assumed
to be non
definite.
The standard battery
within-battery
0
factory
exploratory analysis
factor
model
analysis
model is a special
with one test per battery
q =1, •• •, p. Suppose that S represents the usual unbiased a sample of size N and that S is partitioned as S11
S12
S19
S21
S22
S2g
Sgl
Sgt
case of the multiple
; so that g=p estimator
and p,=1,
of I based
on
S= ...
Sgg
We shall provide consistent noniterative estimators of the Y'qq. The derivation will make use of inter-battery factor analysis (Tucker, 1958; Browne, 1979) and the
factor extension procedure (Dwyer, 1937). review the necessary aspects of inter-battery the factor extension procedure.
3.
Inter-battery
factor
The inter-battery multiple
battery
batteries
involved
a and form
b so that
factor
The following section will therefore factor analysis and its relationship to
analysis
factor
analysis
analysis
is g=2.
model (Tucker,
model considered
The two batteries
the inter-battery
Xaa AbAa
2 where
will be represented
covariance
_
1958) is a special
in Section
structure
case of the
the number
of
by the subscripts
will be represented
AaAb `bb
in the
(4)
where Aa is a pa x k factor matrix with k:< Pa and Ab is a Pb x k factor matrix with k pb. Equation (4) differs from equation (1) in that the reparametrization Iqq= AqAq+ Pqq, q = a, b, has been employed. The sample covariance matrix, S, is partitioned correspondingly as _
Saa Sba
Sab Sbb
Two types of estimator will be considered. Ordinary least squares estimators are obtained by minimizing the discrepancy function
FOLS(S, 1)=+tr[{D1(S_1)}2], S
(OLS)
(5)
where DS=Diag[S]. In many practical applications of this discrepancy function, S is replaced by a correlation matrix, R, so that DS is replaced by I. The term Ds is included in (5) to define OLS estimators that are equivariant (Lehman, 1983, Section 3.2) under changes of scale ; that is, scale changes in the original variables are reflected by corresponding scale changes in the estimators. Maximum Wishart Likelihood (MWL) estimators are obtained by minimizing the discrepancy function FMWL(S, 27)=1n I S I -In 1271+tr[S27-']-P.
(6)
There is greater indeterminacy (Tucker, 1958) in the inter-battery factor analysis model than in the standard factor analysis model or the multiple battery factor analysis model. If Aa and Ab satisfy (4) then so will Aa =AaT and Ab = AbT-1", where T is any k x k nonsingular matrix. It is therefore necessary to impose kz identification conditions on the factor matrices instead of k(k-1)/2 as in standard factor analysis. Let Dy be a k x k diagonal matrix that is not specified a priori. Two alternative sets of k2 identification conditions that may be used are
AaD6aAa=AbD6bAb=D7,
(7)
that are convenient for use in conjunction with the OLS discrepancy
function in (5),
or
AaI asAa= AbI bbAb= D7,
(8)
that are convenient for use in conjunction with the MWL discrepancy function in (6). The factor extension procedure (Dwyer, 1937)will be regarded here as a special case of inter-battery factor analysis where one of the two factor matrices is fixed. The inter-battery covariance structure in (4) is assumed but either Aa or Ab is prespecified and the remaining factor matrix is estimated. Known results are summarized for subsequent use in two lemmas. Parameter estimates are summarized in Lemma 1 using ordinary least squares and Lemma 2 using maximum Wishart likelihood. Lemma 1. (OLS estimates) : Let the diagonal elements of the k x k diagonal matrix Da represent the k largest singular values of the pa x pb matrix Rab= Ds112SabDSb/2 and let the columns of pa x k matrix Ua and Pb x k matrix Ub be the corresponding left and right singular vectors so that U¢Rab-DaUb, RabUb= UaDa, UaUa= UbUb-Ik.
(9)
In inter-battery factor analysis, the OLS estimates of Aa and Ab, subject to the identification conditions (7), are (Tucker, 1958) Aa=Dsa i2UaDa "2 11b=DSb2Ubba~2
(10)
and the OLS estimates of Iaa and `'bb are Iaa-Saa
Ibb=Sbb.
(11)
In factor extension where Ab is assigned a prescribed value, Ab, the conditional minimizer of FOLS(S, 1) with respect to Aa is (Dwyer, 1937) Aalb S¢bDlbAb(AbDsbAb) 1.
(12)
Similarly, if Aa is assigned a prescribed value, Aa, the conditional minimizer of FoLS(S,I) with respect to A'b is Abla-(AaDSQAa)-IAaDSQSb¢.
(13)
In both conditional minimizations of FoLS(S,1) assigning a prescribed value to one of the factor matrices, the conditional minimizers ±aa and £bb are still given by (11). o Lemma 2. (MWL estimates)
: Let Saa2 represent
a square root of Saa :
Saa =Saa Saa2' Saa = Saa2rSa112
Let the diagonal elements of the k x k diagonal matrix Da represent the k largest singular values of the pa x pb matrix Saa'2SabSb6'2i and let the columns of the pa x k matrix of the Ua and the Pb x k matrix Ub be the corresponding left and right singular vectors so that
Ua(Saa12SabSbb/2~)-DaUb, (Saa /2SabSbb/2') Ub UaDa, UaUa= Ub Ub= lk. The MWL estimates of Aa and Ab, subject to the identification are (Browne, 1979, equation (3.19)) Aa=Saa2UaDa/2 and
the
MWL
estimates
of 'aa
and
conditions
Ab-Sbb UbDa'2
Ibb
Iaa=Saa
(15) (8),
(16)
are
Ibb=Sbb•
(17)
The choice of matrix square root satisfying (14) does not influence the estimates in (16). For computational purposes it is usually convenient to choose the Cholesky (lower triangular) square root. The maximum likelihood inter-battery factor matrix estimates, Aa and Ab satisfy the equations (Browne, 1979, equations (3.10), (3.11)) Aa = SabSbbAb(11bSbbAb)-1 Ab=(A¢SaaAa)-lAaSaa Sab• This formulae
suggests
that
that
compatible
are
if Ab and
Aa are
with
MWL
prespecified, inter-battery
noniterative factor
Aalb=SabSbbAb(AbSbbAb)-1• Abla = (AaS-1Aa)-lAaSaa Sab•
4.
Parameter
estimation
in multiple battery
factor
factor analysis
extension are
(18) (19)
analysis
An adaptation to multiple battery factor analysis of the assumption made for the use of the partitioning method in standard factor analysis will first be specified. Assumption
1.
If any battery factor matrix,
Aq, q =1, •••, g, is deleted from A in
(2), the remaining battery factor matrices can be stacked to form two new disjoint factor matrices, I'2q and I'sq, each of rank k. No battery factor matrix, A,-, may have rows in both r2q and T3q. This assumption is an adaptation of the assumption stated in Theorem 5.1 of Anderson and Rubin (1955) and is a sufficient condition for A to be identified up to post multiplication by an orthogonal matrix.. It implies that g >_3. For simplicity of notation we shall treat the situation where q =1 and drop the subscript, q, from F2, and Psq. We shall separate the tests into three sets, S1, S2
and S3. S1 consists of tests from the first battery. Each of the remaining tests is assigned to either S2 or S3 with the restriction that no battery may have some tests in S2 and others in S3. After a possible reordering of batteries, the covariance matrix, 1=AA'+ q', in (1) may then be partitioned as 111
112
113
E21
122
E23 =
A1A1 + I'll r2A1
131
132
`33
r3A1
A1r2
Alr2
r2 r2 + r22
r2 F3"
r3r2
(20)
F3 FY + T33
where Al A=
1-2 F3
The submatrices,
?F11 and
q =
0
0
0
r22
0
0
0
r33
(21)
r2 and r3, of A in (21) are formed from one or more of the Aq in
(2). Similarly the submatrices, r22 and 1' , Of T are formed from one or more W', in (3) and may be block diagonal. The requirement of Assumption 1 that no battery should have tests in both S2 and S3 is made to ensure that 123 in (20) should not depend on any nonzero elements of W. Another requirement (Assumption 1) that has to be met in the assignment of tests into S2 and S3 is that 123 should be of rank k, or equivalently that r2 and r3 should be of full column rank, k. 4.1 Stagewise estimation of unique variance matrices Let the sample covariance matrix S, the population correlation matrix P and the sample correlation matrix R be partitioned conformably with I' in (20). The generalized
communality
matrix will be defined as H11=A1Ai.
(22)
The estimation of H11 will be considered first. Thereafter estimates of f'i1 will be obtained. In order to show consistency of estimators we shall require the following assumption about S. Assumption
2.
S converges
in probability
to I as N *c.
A stagewise approach is employed. Firstly estimates . 2 and t are obtained from an inter-battery factor analysis of S23. Then an estimate Al is obtained from a factor extension analysis of S13 treating r3 as specified, and a separate estimate Al is obtained from S21 treating r2 as specified. The required estimates are given in Lemma 1 and Lemma 2. Finally an estimate of H11 is given by : f11=A1Ai. This estimate need not be symmetric but appropriate taken subsequently.
(23) symmetrizing
steps will be
Proposition 1. (Stagewise OLS estimation). Let the k x k diagonal matrix Da represent the first k singular values of R23=DS2112S23DS312 and let U2 and U3 repre sent the corresponding left and right singular vectors respectively. Then the stagewise OLS estimate of H11 is H11= S13DS31/2P23DS21/2S21 =DS;2R13P23R21Ds;2
(24)
where P23=
is the (Moore-Penrose)
generalized
U3Da ' U2
inverse of the OLS estimate
P23= Da212I'2.V3 DC2'2. H11 is a consistent
Proof
estimator
of H11.
Use of (10) with a = 2 and b = 3 shows that r2 = DS22U2Dar2T
I'3 = DS32U3Da/2T`
where T is an arbitrary k x k nonsingular matrix, introduced to allow for any possible transformation of the inter-battery factor matrices, without relying on any specific identification conditions. Use of (12) with a=1, b=3 and Ab= fl shows that A1-S13Ds311 3(l 3D33l 3)-1 =DSI2R13U3Da1/2T and use of (13) with a=1,
(25)
b=2 and Ab=F2 gives
Ai = (r2DSZI'2)-1h2DSZS21 = T-1Da1/2U2R21Ds,2 Substitution of (25) and (26) into (23) yields (24). Since 1Y11is a continuous function of S on a neighborhood Assumption 2 that H11 is consistent. Proposition values of
2. (Stagewise
MWL estimation).
(26)
of 1, it follows from El
Let Da represent the first k singular
5221/2523S331/2t_R221/2R23R331'2~ and let stagewise
U2 and
U3 represent
MWL
estimate
the corresponding
left and right
(27) singular
vectors.
The
of H11 is
H11= S13I+23S21 = DS,2R13P23R21 DS;2
(28)
where
I23
5331/21 U3Da 1U25221/2 P23= R331/2, U3Da-'U2 R221/2
(29)
are reflexive conditional
inverses of the MWL estimates ±23= 2r3
H11 is a consistent Proof
estimator
1523=D6212X23D63/2
(30)
of H11.
We now use Lemma 2 writing SaQ2=5222as 5222= DS22R222
Application
of (16) with a=2 and b=3 shows that I'2 = DS22R222 U2Da12T
I3 = DS22R332 U3Da'2T -1'
where T has the same function as in the proof of Proposition with a=1, b=3 and Ab=I3 to give
1. Now (18) is applied
Al = S13S3311 3(r3 S331r3)-1 = DSi2R13R331'2r U3Da1l2i1.
(31)
Similarly (19) with a=1, b=2 and Ab=F2 gives A =(I2 DS-21 r2)-lv2 DSZS21 =T-1Da112UzR22112R21Ds~2
(32)
Substitution of (31) and (32) into (23) yields (28). It is straightforward the matrix P23 in (29) is a reflexive conditional inverse of
to verify that
P23=R21/22 U2DaU3R33zr in (30). i.e.
P23P23P23= P23 P23P23P23=1523 O
The estimators in (24) and (28) are related to a corresponding estimator for standard factor analysis given by Kano (1989, equation (8)) but differ in the choice of generalized inverse and the rationale employed in the derivation. Alternatively 123 in (20) may be chosen to be a nonsingular k x k matrix as in standard factor analysis (Albert, 1944; Ihara & Kano, 1986). In general this means that not all tests are employed in the estimation of H11. Consequently the sets S1, S2 and S3 are supplemented by a fourth set, S4, of tests that are not used for the estimation of H11. The multiple battery situation differs from the standard situa tion only in that the requirement of Assumption 1 that no battery factor matrix, A, may have rows in both I'2 and F3 is made and 113 and 121 are square matrices instead of vectors. This approach to the estimation of communalities was referred to as PACE (PArtitioned Covariance Estimation) by Cudeck (1991). Proposition 3. (PACE). Suppose that L.23is a nonsingular consistent estimator of H11 is
k x k matrix.
Then a
H11-S13S231S21 =DS12R13R231R21DS;2
(33)
Once a nonsymmetric estimate H11 of H11 has been obtained from Proposition 1, Proposition 2 or Proposition 3, it is replaced by the symmetric matrix HS11that yields a best least squares fit to fill :
Hg11= 2(fill+Hll) A corresponding
measure of lack of symmetry of Hll is the skew symmetric matrix
E= 2(fill-Ail) which will be null if Al is symmetric. A possibly indefinite estimate of the unique variance is obtained from X11=S11-Hsll.
(34)
Since x'11is a continuous function of the consistent estimator Hl,, it follows that 4ril is also consistent. If an estimate !'ll in (34) is indefinite, its spectral decomposition J.il= 1 ui/liuz i=1
is obtained and ?'11 is replaced by the positive semidefinite matrix that is best fitting in a least squares sense : Pi
4i1=ZuiAiu'. A >o
(35)
Proposition 3 is equivalent to the special cases of Proposition 1 and Proposition 2 where S2 and S3 both contain only k tests. This has disadvantages. There are zero degrees of freedom involved in the calculation of the estimates I1, P2, Al and A2 so that the residual matrices S23 r2l 3, S13 A11 3 and S21 I'2Ai are always null. It would therefore be surprising if the estimator of Proposition 3, which uses less information, were not less precise than the estimators of Propositions 1 and 2. On the other hand, Proposition 3 does not involve a singular value decomposition and therefore requires less computation. Similar methods are used to obtain the remaining within battery unique vari ance matrix estimates x'22,• ••, 4r99 In stagewise MWL estimation, the singular values, a;, of the P2 X fi3 matrix in equation (27) of Proposition 2 are sample canonical correlation coefficients. If the multiple battery factor analysis model holds and if Assumption 1 is valid, k of the corresponding population canonical correlation coefficients are nonzero and the rest are zero. The sample canonical correlations, a;, j =1, •••, Min(fi 2, P 3), may there fore be inspected to see if the assumption of k inter-battery factors in reasonable.
A likelihood ratio test (Browne, 1979,equation (3.17))could be carried out but would strictly be valid only if the partitioning employed had been prechosen and not if the methods of Section 4.2 were used. This process may be repeated each of the g times a Hzzis calculated. The singular values of Proposition 1 may be inspected in a similar manner to check the assumption of k inter-battery factors. No associated test is available. 4.2 Partition choice In practice a prior partitioning of tests into sets S2 and S3 is not available and the choice of partition must be made making use of the data. In the case of standard factor analysis, Ihara and Kano (1986)suggested that S2 and S3be selected so as to maximize the absolute value of the determinant of the k x k matrix R23. Stepwise algorithms that maximize the determinant at each step are available (Cudeck, 1991; Kano, 1990b). The Gauss-Jordan pivots involved in the computa tion of I R23~are chosen one at a time so as to be as large as possible subject to no battery being involved in both S2 and S3. The procedure is terminated when k pivots have been chosen. While this approach does not guarantee the maximum for the absolute value of I R231,it does appear to result in a R23that is sufficiently well conditioned for practical purposes. This approach is easily modified for the multiple battery estimator of Proposition 3. The only modification required is that no battery should have tests in both S2 and S3. Additional modifications are required for use in conjunction with Propositions 1 and 2. We no longer require a k x k nonsingular R23,but rather a R23of as large an order as possible, of rank at least k and that can be approximated as well as possible by a matrix of rank k. The sweep procedure described by Cudeck (1991, pp. 40-42) may be modified for this purpose. The largest k pivots are selected as described by Cudeck, but ensuring that no battery has tests in both S2 and S3. After this has been done, if any test from a battery has been assigned to one of the sets, the remaining tests are assigned to the same set. Remaining batteries are assigned to S2 or S3 according to the following two criteria : (i) After sweeping on the first k pivots, the remaining potential pivots in R23should be as close to zero as possible. (ii) The number of tests in S2 should be as close to the number of tests in S3 as possible.
• • • •
No further sweeps are carried out after the first k. The general aim is to select a R23 with the following properties : It involves all batteries except the battery that forms Sl . It contains a k x k submatrix with a large determinant . Sweeping out this k x k submatrix will result in small residual elements in R23. The number of rows and number of columns of R23 should be as close as possible .
4.3 Estimation of factor loadings After the estimate, ? , of the block diagonal unique variance matrix has been obtained, an estimate, A, of the factor matrix is required. This may be obtained as a further stage obtaining the conditional minimizer of either F,,,, (S, 1) or FMWL(S, ~) with respect to A given ?P. The result for FOLSis a simple adaptation of a well-known result in standard factor analysis. Proposition 4. Let Al> A2 Apbe the eigenvalues of DS 112(S ¶) Ds 1'2,let DA be a k x k diagonal matrix with the k largest eigenvalues as diagonal elements and let V be a p x k matrix with the corresponding eigenvectors as columns. The conditional minimizer of ,,,,(S, 1) with respect to A given ?P, subject to the identification conditions, AD. 'A=Dx, is A= DS/2VD1/2 and the corresponding minimum value is p '2 /`i •
FoLS= Z
i=k+1
7
Some care must be taken to allow for the possibility of ?I1'being singular when computing the conditional minimizer of FMWLwith respect to A. An adaptation (Browne, 1980) of an approach
due to Jennrich
and Robinson (1969) is used.
Proposition 5. Let Al< A2< •• • < /lp be the eigenvalues of S-1'2 'S-1/2', let A be a k x k diagonal matrix with the k smallest eigenvalues as diagonal elements, let D,= I A and let V be a p x k matrix with the corresponding eigenvectors as columns. The conditional minimizer of FMWL(S,I) with respect to A given ?If, subject to the identification conditions, AS-'A=D,, is A=S112V(I-DA)112 and the corresponding
minimum value is p
FMWL L (A '+ln li-1).
(36)
i=k+1
4.4 Applicability of methods to sample correlation matrices The multiple battery factor analysis model (1) was presented as a covariance structure and estimators of the parameters based on the sample covariance matrix, S, were obtained. This was done primarily to justify the use of the maximum Wishart likelihood discrepancy function FMWLin (6). It is common practice, how ever, to regard both standard factor analysis and inter-battery factor analysis as correlation structures and to apply them to the sample correlation matrix, R, rather
than to the covariance matrix, S. The correlation the inter-battery covariance structure (1) is P=ApA'p+
structure
that corresponds
to
Y'p
where Ap=D61r2A and Y'p=Dd112tfD61'2, so that estimators multiple battery correlation structure are
of parameters
in the
(37)
AP =Da-1/2A and
To = AT 112WD 6 1/2 Since specified
all estimators in Propositions
that 1-5
we have may
considered
be carried
(38) are
out
equivariant,
replacing
S by
all computations R and
DS by
I
without affecting the values of the correlation structure parameter estimates A and ?ui obtained from (37) and (38).
5.
An example
As an example we shall make use of data reported in Jackson (1975, Table 1), and employed in Browne (1980) to illustrate the maximum likelihood solution in multiple battery factor analysis. Four personality traits (1, •••, 4) were measured under each of five judgmental sets (A, •••, E) on a sample of size 480. The 20 x 20 correlation matrix was employed to obtain the unique variance matrix estimates using the methods based on Proposition 1 (SOLS), Proposition 2 (SMWL) and Proposition 3 (PACE). The iterative Gauss-Seidel procedure (Browne, 1980) for obtaining full maximum Wishart likelihood (FMWL) estimates was also applied for comparative purposes. Estimates of each of the five unique variance matrices were obtained by each of the four estimation procedures and the rescaling of (38) applied. Results are shown in Table 1. The SMWL and SOLS estimates are very close to each other and are possibly a little closer to the iterative FMWL estimates than are the simpler PACE estimates, although this trend is not clear cut. Since the aim of the example is to compare alternative noniterative methods with the iterative maximum Wishart likelihood estimates, the same conditional minimizer, A, of FMWLprovided in Proposition 5 was employed in conjunction with each of the three noniterative unique variance estimates of ?1'. The rescaling of (37) was carried out in each case. In order to facilitate comparison of the noniter ative estimates with the FMWL estimate of A, a Varimax rotation (Kaiser, 1958) was applied to each of the four factor matrices. Results are shown in Table 2. Again differences between methods are small and will not lead to any differences in interpretation. It is of interest to investigate the closeness of the three noniterative estimates
Table Unique
Variance
1 Matrices
of the factor matrix to the iterative FMWL estimate. This can be done by compar ing values of FM,L in (36). These are shown in Table 3. As the FMWL solution minimizes FMWLwith respect to W and A simultaneous ly, it yields the smallest (best) discrepancy function value. The SMWL solution is next. This is not surprising since it uses FMWLin all stages. Differences between the SOLS and SMWL solution are negligible, however. The PACE solution is furthest away from FMWL. These differences are of theoretical interest only since the results of different solutions shown in Tables 1 and 2 are equivalent for practical purposes. Calculation times using a 486 DX2 66 microcomputer are also shown. Although the SMWL solution involves a little more computation than the SOLS solution the difference in computation time was too small to be detectable in the present example. There is a trend for solutions that are closer to the FMWL solution to require more computation time. Neither differences in results of alternative analyses, nor in the time taken are of any real consequence, however. When comparing results, one should bear in mind that the results of the
Table 2 Rotated Factor
Varimax
Table 3 Function Values
MWL Discrepancy
noniterative
methods
differences
in estimates
assigning give will
batteries
slightly depend
was
required.
4 seconds, function
to
different on the
communalities were
depend
that
between S2 and
different
S3.
employed a limit
Thus
was
correct
used,
of ten
reported
and that
there of the
implementations
may
presented was in Table
decimal
here.
imposed
places
3 and was
Fifty the
procedure
for may
two
procedure
a minimum obtained.
be minor
of PACE
taken for the iterative FMWL A convergence criterion of
iterations
to two
Times
implementations different
for the example
of the 5.6 seconds
and Computing
partitionings
results. Also the time convergence criterion.
When
instead value
on the
Loadings
solution .0001 on iterations took
discrepancy
2.
6.
Concluding
comments
The feasibility of noniterative solutions in multiple battery factor analysis has been demonstrated. All three noniterative solutions presented can also be applied in standard factor analysis. The PACE solution then is equivalent to the solutions given by Cudeck (1991) and by Kano (1990b). The other two solutions, SOLS and SMWL, when applied in standard factor analysis, are related to Kano's (1989) method but make use of different conditional inverses. Initial experimentation suggests that the methods given here are promising but more experience will be required before any firm conclusions can be reached. Multiple battery factor analysis is employed in situations where relationships between tests within batteries are not of interest and relationships between tests in different batteries alone are investigated. The number of tests in different batt eries need not be the same. One application of this model is in the analysis of multitrait-multimethod data of the type employed in the present example. Methods are treated as batteries and the same number of traits is investigated with each method. Some authors advocate a restricted multiple battery factor analysis model with some factor loadings specified to be zero for the analysis of data of this type (c.f., Marsh, Byrne, & Craven, 1992 and references therein). A related approach would be to use one of the multiple battery factor analysis methods described here in conjunction with an oblique rotation to a partially specified target (Browne, 1972). Misspecifications of zero loadings could then be detected by the inspection of loadings of the rotated matrix that correspond to zero target elements.
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