The association structure between manifest variables arising from the single-factor model is investigated using partial correlations. The additional insights to the practitioner provided by partial correlations for detecting a single-factor model are discussed. The parameter space for the partial correlations is presented, as are the patterns of signs in a matrix containing the partial correlations that are not compatible with a single-factor model

G. Hadley

H. Yanai

H.F. Kaiser

J. Whittaker

John W. McDonald

L. Guttman

M.F. Salgueiro

Maria de Fátima Salgueiro

P.J. Rousseeuw

Peter W. F. Smith

T.W. Anderson

English

The association structure between manifest variables arising from the single-factor model is investigated using partial correlations. The additional insights to the practitioner provided by partial correlations for detecting a single-factor model are discussed. The parameter space for the partial correlations is presented, as are the patterns of signs in a matrix containing the partial correlations that are not compatible with a single-factor model. Key words: anti-image correlation matrix, elliptical tetrahedron, factor analysis, factor partial correlation, manifest partial correlation

McDonald, John

Smith, Peter W F

Salgueiro, Maria

Institute of Education EPrints

THE MANIFEST ASSOCIATION STRUCTURE OF THE
SINGLE-FACTOR MODEL: INSIGHTS FROM PARTIAL
CORRELATIONS
Maria de Fa´tima Salgueiro
departamento me´todos quantitativos, iscte business school
Peter W.F. Smith
southampton statistical sciences research institute,
university of southampton
John W. McDonald
southampton statistical sciences research institute,
university of southampton
July 8, 2007
The research of the first author was supported by the Fundac¸a¯o para a Cieˆncia e a
Tecnologia and by the Southampton Statistical Sciences Research Institute.
The authors thank the Editor, the Associate Editor and three Reviewers for their
constructive comments which helped to improve the paper.
Correspondence should be sent to Maria de Fa´tima Salgueiro
E-Mail: fatima.salgueiro@iscte.pt
Phone: +351-963077654
Fax: +351-217903941
Psychometrika Submission July 8, 2007 2
THE MANIFEST ASSOCIATION STRUCTURE OF THE SINGLE-FACTOR
MODEL: INSIGHTS FROM PARTIAL CORRELATIONS
Abstract
The association structure between manifest variables arising from the
single-factor model is investigated using partial correlations. The additional
insights to the practitioner provided by partial correlations for detecting a
single-factor model are discussed. The parameter space for the partial
correlations is presented, as are the patterns of signs in a matrix containing
the partial correlations that are not compatible with a single-factor model.
Key words: anti-image correlation matrix, elliptical tetrahedron, factor
analysis, factor partial correlation, manifest partial correlation.
Psychometrika Submission July 8, 2007 3
1. Introduction
Factor analysis is a classical approach to modeling multivariate data where all
variables are treated on an equal footing. Traditionally, these are thought of as models
for covariances or correlations, but partial correlations can be used.
Guttman (1953) advocated that all off-diagonal elements of the inverse correlation
matrix close to zero was a criteria for ruling out a common-factor space for the
population. Kaiser (1970), in a Presidential address to the Psychometric Society, stated
that “among Guttman’s landmark papers . . . one particular theme appears repeatedly:
given a correlation matrix, we should always look carefully at its inverse, in order to
assess the sampling adequacy of the data for factor-analytic purposes”. The scaled, to
have ones on the diagonal, inverse variance matrix has off-diagonal elements (i, j) equal
to the negatives of the partial correlation between variables i and j, after conditioning
on the remaining variables (Whittaker, 1990, page 143). This matrix is known as the
anti-image correlation matrix and is available in standard statistical packages such as
SPSS, SAS and STATA. For further details on the relationship between correlation
matrices and anti-image correlation matrices, see Yanai and Mukherjee (1987).
This paper presents three main results and two corollaries that provide extra
insights into single-factor models, based on partial correlations and on the pattern of
their signs. Section 2 presents necessary, but not sufficient, constraints on the
parameter space of the population partial correlations implied by a single-factor model.
Section 3 defines factor partial correlations and derives the association structure
between manifest variables after marginalizing over the latent variable in a single-factor
model. Also presented in Section 3 are patterns of signs of partial correlations that are
not compatible with a single-factor model. Section 4 re-expresses the classical tetrad
conditions as a function of the manifest partial correlations. Section 5 contains a
discussion.
Psychometrika Submission July 8, 2007 4
2. The Parameter Space of the Single-Factor Model
The classical single-factor model can be written as XM = λL+ δ, where XM is
the vector of the p manifest variables X1, X2, . . . , Xp, L is the factor or latent variable,
λ is a p× 1 vector of factor loadings, and δ is a vector of p variables representing
random measurement error and indicator specificity. Variables are considered to be
measured as deviations from their means, so E[XM] = 0 and E[L] = 0. The classical
single-factor model assumes that E[Lδ] = 0, E[δ] = 0, var[δ] is diagonal and that L
and δ are multivariate normal, although the results presented in this paper do not
require these normality assumptions. To identify the model, the latent variable L is
scaled to have unit variance. The variance matrix for XM, with elements denoted by
σij , is ΣM = λλ
T +Θ, where Θ is the p× p diagonal variance matrix of δ, with
non-negative elements. A necessary and sufficient condition for ΣM to be a variance
matrix of a single-factor model is that p(p− 1)/2− p independent tetrad conditions are
satisfied and 0 ≤
σkiσij
σkj
≤ σii for one pair (j 6= k) for each i (Anderson and Rubin, 1956).
The tetrad conditions are σkiσlj − σliσkj = 0, for all distinct i, j, k and l, from 1 to p.
Results in this paper are presented under the assumption that variances are
positive and that all variance matrices are positive definite. Therefore, a necessary but,
as will be seen later, not sufficient condition on the scaled inverse variance matrix is
that it is positive definite.
Result 1. The p× p anti-image correlation matrix is positive definite if and only if its
determinant is strictly positive and, for p ≥ 4, all its symmetric (p− 1)× (p− 1)
submatrices are positive definite. Hence, when p = 3, the positive definiteness
constraint on the anti-image correlation matrix implies
1− 2 ρ12.3 ρ13.2 ρ23.1 − ρ
2
12.3 − ρ
2
13.2 − ρ
2
23.1 > 0, (1)
where ρij.k is the partial correlation coefficient between Xi and Xj given Xk.
The proof of Result 1 follows from Hadley (1961), Equation 7-69. The different
combinations of the three ρij.k form a convex set which is symmetric with respect to
rotations corresponding to permutations of the components of (ρ12.3, ρ13.2, ρ23.1).
Psychometrika Submission July 8, 2007 5
Figure 1 displays the boundary of the convex set in a three-dimensional plot. These
plots should be contrasted to those in Figure 1 of Rousseeuw and Molenberghs (1994):
both are elliptical tetrahedrons, but with different orientations.
=========================
Insert Figure 1 about here
=========================
Ignoring for the moment the shaded area, Figure 2 supports the statement that any
horizontal cross section of the surface defined by Figure 1 is an ellipse.
=========================
Insert Figure 2 about here
=========================
3. The Relationship Between the Manifest and the Factor Partial
Correlations
Let X be the vector of the p+ 1 random variables, partitioned as
[
XM
T L
]T
,
with positive definite scaled inverse variance matrix partitioned as
T =

 I TML
TLM 1

 ,
where I is a p× p identity matrix since the single-factor model assumes that the
manifest variables are conditionally independent, given the latent variable. The p× 1
vector TML = TLM
T contains the non-zero elements −τiL.(iL), the negative of the
partial correlation between manifest variable Xi and latent variable L, where (iL)
denotes the remaining p− 1 variables in X, after removing Xi and L. Henceforth, the
τiL.(iL) are called factor partial correlations.
The positive definiteness constraint on the matrix T implies that its determinant is
positive, i.e., |1− TLMTML| > 0, and that the determinants of all its symmetric
submatrices are also positive. Therefore,
p∑
i=1
τ 2iL.(iL) < 1. (2)
Psychometrika Submission July 8, 2007 6
Also, because the τ are partial correlations, −1 < τiL.(iL) < 1. The parameter space for
the factor partial correlations is defined by Equation 2, with the additional constraint
that τiL.(iL) 6= 0. Indeed, for the single-factor model to be of interest all factor partial
correlations need to be of substantive interest, and not only non-zero mathematically.
Note that Equation 2 defines an open unit hyper-sphere.
Result 2. The manifest partial correlations can be written as a function of the factor
partial correlations as
ρij.(ij) =
τiL.(iL) τjL.(jL)
{(1− τ 2iL.(iL))(1− τ
2
jL.(jL))}
1/2
, i 6= j ∈ {1, 2, . . . , p}, (3)
where ρij.(ij) is the manifest partial correlation between Xi and Xj, given the remaining
p− 2 manifest variables.
The proof follows. Inverting T , the scaled inverse variance matrix of X, gives
T−1 =

 I TML
TLM 1


−1
=

 (I − TMLTLM)
−1 −(I − TMLTLM)
−1TML
−TLM(I − TMLTLM)
−1 (1− TLMTML)
−1

 .
(4)
The correlation matrix of XM is given by scaling (I − TMLTLM)
−1. Hence, the
anti-image correlation matrix is given by scaling I − TMLTLM to have ones on the
diagonal. The off-diagonal element (i, j) is given by −
τiL.(iL) τjL.(jL)
{(1−τ2
iL.(iL)
)(1−τ2
jL.(jL)
)}1/2
. Since the
off-diagonal elements of the population anti-image correlation matrix are the negatives
of the population manifest partial correlations, Equation 3 follows.
Two corollaries of Result 2 follow.
Result 2a. Marginalizing the single-factor model over the latent variable L yields a
joint distribution for the manifest variables with no zero manifest partial correlations,
that is, with no zero entries in the anti-image correlation matrix.
Since all factor partial correlations are assumed non-zero, and the manifest partial
correlations are given by Equation 3, all τiL.(iL) 6= 0 imply all ρij.(ij) 6= 0. Therefore,
there are no zero entries in the anti-image correlation matrix and the proof is complete.
Psychometrika Submission July 8, 2007 7
From Equation 3, the sign of ρij.(ij) equals the product of the signs of τiL.(iL) and
τjL.(jL). Let τ = −TML be a p× 1 column vector of the factor partial correlations and
let ρ be a p× p symmetric matrix containing the manifest partial correlations as
off-diagonal elements. Recall that λ is the vector of factor loadings.
Result 2b. The off-diagonal elements of sign(ρ) represent a pattern of signs for the
manifest partial correlations that is compatible with a single-factor model if
sign(ρ) = sign(τ τ T ) = sign(λλT ) (5)
(for the off-diagonal elements). This result refers to population parameters and assumes
the tetrad conditions are satisfied. The second equality follows from Salgueiro (2003,
Equation 5.16): λ⋆i =
τiL.(iL)
1−
∑p
k=1 τ
2
kL.(kL)
. Indeed, up to scaling, each factor loading can be
written as a function of the factor partial correlations.
Note that, when p = 3, Equation 3 defines a system of three equations of the form
τ 2iL.jk =
1
1 +
ρjk.i
ρij.kρik.j
. (6)
Since 0 < τ 2iL.jk < 1, then
ρjk.i
ρij.kρik.j
> 0. Hence, three manifest variables can only define a
single-factor model if their anti-image correlation matrix is positive definite and the
product of the three manifest partial correlation coefficients is positive. Revisiting
Figure 2, the shaded area represents the admissible region for the three manifest partial
correlation coefficients in a single-factor model. The ellipse represents the positive
definiteness constraint given by Equation 1. Additionally, there is now the constraint of
a positive product of the three partial correlations. Indeed, it is not possible to have a
single-factor model with three manifest variables in which just one or all three ρij.k are
negative. Also, if τiL.jk is negative and the other two factor partial correlations are
positive (or the opposite), then all manifest partial correlations in column (row) i of the
anti-image correlation matrix will be negative, and all the remaining manifest partial
correlations will be positive.
When p ≥ 4, Equation 5 can be used to derive possible patterns of signs for the
manifest partial correlations, given the hypothesized structure of signs for the factor
Psychometrika Submission July 8, 2007 8
partial correlations. Alternatively, by considering all possible patterns of signs for the
factor partial correlations, Equation 5 can be used to obtain all possible patterns of
signs for the manifest partial correlations.
4. An Extension of the Tetrad Conditions
The classical tetrad conditions for the single-factor model can be re-expressed in
terms of partial correlations.
Result 3. When p ≥ 4, the following p(p− 1)/2− p tetrad conditions for the partial
correlations have to be satisfied in a single-factor model:
ρki.(ki)ρlj.(lj) − ρli.(li)ρkj.(kj) = 0, (7)
with i, j, k and l distinct, from 1 to p, where ρki.(ki) is the partial correlation coefficient
between Xk and Xi, given the remaining p− 2 manifest variables.
The proof follows by repeated substitution of Equation 3 into the left hand side of
Equation 7.
5. Discussion
This paper has provided some useful results for the practitioner. Inspection of
sample partial correlations between manifest variables can provide evidence for and
against the single-factor model. If the pattern of signs of the sample partial correlations
do not obey Result 2b, then, provided this is not due to sampling error, the
single-factor model can be rejected. Indeed, population partial correlations between
manifest variables in a single-factor model should be non-zero. Therefore, provided
there is adequate power for the tests that these are zero, from Result 2a, sample partial
correlations not significantly different from zero also rule out the single-factor model.
However, if power for these tests is small, the single-factor model should still be
considered, even if there are non-significant sample partial correlations. Methods for
estimating overall power for graphical Gaussian models were investigated by Salgueiro
et al. (2005). As the single-factor model can be parameterized as a graphical Gaussian
Psychometrika Submission July 8, 2007 9
model, the results presented by Salgueiro et al. (2005) can be used for the single-factor
model.
Psychometrika Submission July 8, 2007 10
References
Anderson, T.W. and Rubin, H. (1956). Statistical inference in factor analysis. In
Proceedings of the third Berkeley Symposium on Mathematical Statistics and
Probability, Vol. V, 111-150, Berkeley: University of California Press.
Guttman, L. (1953). Image theory for the structure of quantitative variates.
Psychometrika, 18, 277-296.
Hadley, G. (1961). Linear Algebra. Massachusetts: Addison-Wesley Publishing
Company, Inc.
Kaiser, H.F. (1970). A second generation Little Jiffy. Psychometrika, 35, 401-415.
Rousseeuw, P.J. and Molenberghs, G. (1994). The shape of correlation matrices. The
American Statistician, 48 (4), 276-279.
Salgueiro, M.F. (2003). Distributions of Test Statistics for Edge Exclusion for Graphical
Models. PhD thesis, University of Southampton.
Salgueiro, M.F., Smith, P.W.F. and McDonald, J.W. (2005). Power of edge exclusion
tests in graphical Gaussian models. Biometrika, 92 (1), 173-182.
Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. Chichester:
John Wiley & Sons.
Yanai, H. and Mukherjee, B.N. (1987). A generalized method of image analysis from an
intercorrelation matrix which may be singular. Psychometrika, 52 (4), 555-564.
Psychometrika Submission July 8, 2007 11
Figures
Figure 1.
Set of all possible partial correlation coefficients (ρ12.3, ρ13.2 and ρ23.1) between three variables.
-1.0 -0.5 0.0 0.5 1.0
 
-
1.
0
-
0.
5
0.
0
0.
5
1.
0
 
a)
-1.0 -0.5 0.0 0.5 1.0
 
-
1.
0
-
0.
5
0.
0
0.
5
1.
0
 
b)
-1.0 -0.5 0.0 0.5 1.0
 
-
1.
0
-
0.
5
0.
0
0.
5
1.
0
 
c)
-1.0 -0.5 0.0 0.5 1.0
 
-
1.
0
-
0.
5
0.
0
0.
5
1.
0
 
-1.0 -0.5 0.0 0.5 1.0
 
-
1.
0
-
0.
5
0.
0
0.
5
1.
0
 
e)
-1.0 -0.5 0.0 0.5 1.0
 
-
1.
0
-
0.
5
0.
0
0.
5
1.
0
 
f)d)
Figure 2.
The admissible region (shaded) for the three partial correlation coefficients compatible with a single-
factor model: ρ13.2 on the horizontal axis, ρ23.1 on the vertical axis and ρ12.3 taking values of a) 0.1,
b) 0.5, c) 0.9, d) −0.1, e) −0.5, f) −0.9. The ellipse represents the positive definiteness constraint for
partial correlations.


The Manifest Association Structure of the Single-Factor Model: Insights from Partial Correlations

Salgueiro, Maria de Fátima

Smith, Peter W.F.

McDonald, John W.

Southampton (e-Prints Soton)

  THE MANIFEST ASSOCIATION STRUCTURE OF THE SINGLE-FACTOR MODEL: INSIGHTS FROM PARTIAL CORRELATIONS  MARIA DE FÁTIMA SALGUEIRO, PETER W. F. SMITH,  JOHN W. MCDONALD  ABSTRACT  The association structure between manifest variables arising from the single-factor model is investigated using partial correlations. The additional insights to the practitioner provided by partial correlations for detecting a single-factor model are discussed. The parameter space for the partial correlations is presented, as are the patterns of signs in a matrix containing the partial correlations that are not compatible with a single-factor model.   Southampton Statistical Sciences Research Institute Methodology Working Paper M07/07 THE MANIFEST ASSOCIATION STRUCTURE OF THESINGLE-FACTOR MODEL: INSIGHTS FROM PARTIALCORRELATIONSMaria de F´ atima Salgueirodepartamento m´ etodos quantitativos, iscte business schoolPeter W.F. Smithsouthampton statistical sciences research institute,university of southamptonJohn W. McDonaldsouthampton statistical sciences research institute,university of southamptonMarch 30, 2007The research of the ﬁrst author was supported by the Funda¸ c¯ ao para a Ciˆ encia e aTecnologia and by the Southampton Statistical Sciences Research Institute.The authors thank the Editor, the Associate Editor and three Reviewers for theirconstructive comments which helped to improve the paper.Correspondence should be sent to Maria de F´ atima SalgueiroE-Mail: fatima.salgueiro@iscte.ptPhone: +351-963077654Fax: +351-217903941Psychometrika Submission March 30, 2007 2THE MANIFEST ASSOCIATION STRUCTURE OF THE SINGLE-FACTORMODEL: INSIGHTS FROM PARTIAL CORRELATIONSAbstractThe association structure between manifest variables arising from thesingle-factor model is investigated using partial correlations. The additionalinsights to the practitioner provided by partial correlations for detecting asingle-factor model are discussed. The parameter space for the partialcorrelations is presented, as are the patterns of signs in a matrix containingthe partial correlations that are not compatible with a single-factor model.Key words: anti-image correlation matrix, elliptical tetrahedron, factoranalysis, factor partial correlation, manifest partial correlation.Psychometrika Submission March 30, 2007 31. IntroductionFactor analysis is a classical approach to modeling multivariate data where allvariables are treated on an equal footing. Traditionally, these are thought of as modelsfor covariances or correlations, but partial correlations can be used.Guttman (1953) advocated that all oﬀ-diagonal elements of the inverse correlationmatrix close to zero was a criteria for ruling out a common-factor space for thepopulation. Kaiser (1970), in a Presidential address to the Psychometric Society, statedthat “among Guttman’s landmark papers ...one particular theme appears repeatedly:given a correlation matrix, we should always look carefully at its inverse, in order toassess the sampling adequacy of the data for factor-analytic purposes”. The scaled, tohave ones on the diagonal, inverse variance matrix has oﬀ-diagonal elements (i,j) equalto the negatives of the partial correlation between variables i and j, after conditioningon the remaining variables (Whittaker, 1990, page 143). This matrix is known as theanti-image correlation matrix and is available in standard statistical packages such asSPSS, SAS and STATA. For further details on the relationship between correlationmatrices and anti-image correlation matrices, see Yanai and Mukherjee (1987).This paper presents three main results and two corollaries that provide extrainsights into single-factor models, based on partial correlations and on the pattern oftheir signs. Section 2 presents necessary, but not suﬃcient, constraints on theparameter space of the population partial correlations implied by a single-factor model.Section 3 deﬁnes factor partial correlations and derives the association structurebetween manifest variables after marginalizing over the latent variable in a single-factormodel. Also presented in Section 3 are patterns of signs of partial correlations that arenot compatible with a single-factor model. Section 4 re-expresses the classical tetradconditions as a function of the manifest partial correlations. Section 5 contains adiscussion.Psychometrika Submission March 30, 2007 42. The Parameter Space of the Single-Factor ModelThe classical single-factor model can be written as XM = λL + δ, where XM isthe vector of the p manifest variables X1, X2, ..., Xp, L is the factor or latent variable,λ is a p × 1 vector of factor loadings, and δ is a vector of p variables representingrandom measurement error and indicator speciﬁcity. Variables are considered to bemeasured as deviations from their means, so E[XM] = 0 and E[L] = 0. The classicalsingle-factor model assumes that E[Lδ] = 0, E[δ] = 0, var[δ] is diagonal and that Land δ are multivariate normal, although the results presented in this paper do notrequire these normality assumptions. To identify the model, the latent variable L isscaled to have unit variance. The variance matrix for XM, with elements denoted byσij, is ΣM = λλT + Θ, where Θ is the p × p diagonal variance matrix of δ, withnon-negative elements. A necessary and suﬃcient condition for ΣM to be a variancematrix of a single-factor model is that p(p − 1)/2 − p independent tetrad conditions aresatisﬁed and 0 ≤σkiσijσkj ≤ σii for one pair (j  = k) for each i (Anderson and Rubin, 1956).The tetrad conditions are σkiσlj − σliσkj = 0, for all distinct i, j, k and l, from 1 to p.Results in this paper are presented under the assumption that variances arepositive and that all variance matrices are positive deﬁnite. Therefore, a necessary but,as will be seen later, not suﬃcient condition on the scaled inverse variance matrix isthat it is positive deﬁnite.Result 1. The p × p anti-image correlation matrix is positive deﬁnite if and only if itsdeterminant is strictly positive and, for p ≥ 4, all its symmetric (p − 1) × (p − 1)submatrices are positive deﬁnite. Hence, when p = 3, the positive deﬁnitenessconstraint on the anti-image correlation matrix implies1 − 2ρ12.3 ρ13.2 ρ23.1 − ρ212.3 − ρ213.2 − ρ223.1 > 0, (1)where ρij.k is the partial correlation coeﬃcient between Xi and Xj given Xk.The proof of Result 1 follows from Hadley (1961), Equation 7-69. The diﬀerentcombinations of the three ρij.k form a convex set which is symmetric with respect torotations corresponding to permutations of the components of (ρ12.3, ρ13.2, ρ23.1).Psychometrika Submission March 30, 2007 5Figure 1 displays the boundary of the convex set in a three-dimensional plot. Theseplots should be contrasted to those in Figure 1 of Rousseeuw and Molenberghs (1994):both are elliptical tetrahedrons, but with diﬀerent orientations.=========================Insert Figure 1 about here=========================Ignoring for the moment the shaded area, Figure 2 supports the statement that anyhorizontal cross section of the surface deﬁned by Figure 1 is an ellipse.=========================Insert Figure 2 about here=========================3. The Relationship Between the Manifest and the Factor PartialCorrelationsLet X be the vector of the p + 1 random variables, partitioned as XMT L T,with positive deﬁnite scaled inverse variance matrix partitioned asT =I TMLTLM 1,where I is a p × p identity matrix since the single-factor model assumes that themanifest variables are conditionally independent, given the latent variable. The p × 1vector TML = TLMT contains the non-zero elements −τiL.(iL), the negative of thepartial correlation between manifest variable Xi and latent variable L, where (iL)denotes the remaining p − 1 variables in X, after removing Xi and L. Henceforth, theτiL.(iL) are called factor partial correlations.The positive deﬁniteness constraint on the matrix T implies that its determinant ispositive, i.e., |1 − TLMTML| > 0, and that the determinants of all its symmetricsubmatrices are also positive. Therefore,p  i=1τ2iL.(iL) < 1. (2)Psychometrika Submission March 30, 2007 6Also, because the τ are partial correlations, −1 < τiL.(iL) < 1. The parameter space forthe factor partial correlations is deﬁned by Equation 2, with the additional constraintthat τiL.(iL)  = 0. Indeed, for the single-factor model to be of interest all factor partialcorrelations need to be of substantive interest, and not only non-zero mathematically.Note that Equation 2 deﬁnes an open unit hyper-sphere.Result 2. The manifest partial correlations can be written as a function of the factorpartial correlations asρij.(ij) =τiL.(iL) τjL.(jL){(1 − τ2iL.(iL))(1 − τ2jL.(jL))}1/2, i  = j ∈ {1,2,...,p}, (3)where ρij.(ij) is the manifest partial correlation between Xi and Xj, given the remainingp − 2 manifest variables.The proof follows. Inverting T, the scaled inverse variance matrix of X, givesT−1 =I TMLTLM 1−1= (I − TMLTLM)−1 −(I − TMLTLM)−1TML−TLM(I − TMLTLM)−1 (1 − TLMTML)−1.(4)The correlation matrix of XM is given by scaling (I − TMLTLM)−1. Hence, theanti-image correlation matrix is given by scaling I − TMLTLM to have ones on thediagonal. The oﬀ-diagonal element (i,j) is given by −τiL.(iL) τjL.(jL){(1−τ2iL.(iL))(1−τ2jL.(jL))}1/2. Since theoﬀ-diagonal elements of the population anti-image correlation matrix are the negativesof the population manifest partial correlations, Equation 3 follows.Two corollaries of Result 2 follow.Result 2a. Marginalizing the single-factor model over the latent variable L yields ajoint distribution for the manifest variables with no zero manifest partial correlations,that is, with no zero entries in the anti-image correlation matrix.Since all factor partial correlations are assumed non-zero, and the manifest partialcorrelations are given by Equation 3, all τiL.(iL)  = 0 imply all ρij.(ij)  = 0. Therefore,there are no zero entries in the anti-image correlation matrix and the proof is complete.Psychometrika Submission March 30, 2007 7From Equation 3, the sign of ρij.(ij) equals the product of the signs of τiL.(iL) andτjL.(jL). Let τ = −TML be a p × 1 column vector of the factor partial correlations andlet ρ be a p × p symmetric matrix containing the manifest partial correlations asoﬀ-diagonal elements. Recall that λ is the vector of factor loadings.Result 2b. The oﬀ-diagonal elements of sign(ρ) represent a pattern of signs for themanifest partial correlations that is compatible with a single-factor model ifsign(ρ) = sign(τ τT) = sign(λλT) (5)(for the oﬀ-diagonal elements). This result refers to population parameters and assumesthe tetrad conditions are satisﬁed. The second equality follows from Salgueiro (2003,Equation 5.16): λi =τiL.(iL)1−Ppk=1 τ2kL.(kL).Note that, when p = 3, Equation 3 deﬁnes a system of three equations of the formτ2iL.jk =11 +ρjk.iρij.kρik.j. (6)Since 0 < τ2iL.jk < 1, thenρjk.iρij.kρik.j > 0. Hence, three manifest variables can only deﬁne asingle-factor model if their anti-image correlation matrix is positive deﬁnite and theproduct of the three manifest partial correlation coeﬃcients is positive. RevisitingFigure 2, the shaded area represents the admissible region for the three manifest partialcorrelation coeﬃcients in a single-factor model. The ellipse represents the positivedeﬁniteness constraint given by Equation 1. Additionally, there is now the constraint ofa positive product of the three partial correlations. Indeed, it is not possible to have asingle-factor model with three manifest variables in which just one or all three ρij.k arenegative. Also, if τiL.jk is negative and the other two factor partial correlations arepositive (or the opposite), then all manifest partial correlations in column (row) i of theanti-image correlation matrix will be negative, and all the remaining manifest partialcorrelations will be positive.When p ≥ 4, Equation 5 can be used to derive possible patterns of signs for themanifest partial correlations, given the hypothesized structure of signs for the factorpartial correlations. Alternatively, by considering all possible patterns of signs for thePsychometrika Submission March 30, 2007 8factor partial correlations, Equation 5 can be used to obtain all possible patterns ofsigns for the manifest partial correlations.4. An Extension of the Tetrad ConditionsThe classical tetrad conditions for the single-factor model can be re-expressed interms of partial correlations.Result 3. When p ≥ 4, the following p(p − 1)/2 − p tetrad conditions for the partialcorrelations have to be satisﬁed in a single-factor model:ρki.(ki)ρlj.(lj) − ρli.(li)ρkj.(kj) = 0, (7)with i, j, k and l distinct, from 1 to p, where ρki.(ki) is the partial correlation coeﬃcientbetween Xk and Xi, given the remaining p − 2 manifest variables.The proof follows by repeated substitution of Equation 3 into the left hand side ofEquation 7.5. DiscussionThis paper has provided some useful results for the practitioner. Inspection ofsample partial correlations between manifest variables can provide evidence for andagainst the single-factor model. If the pattern of signs of the sample partial correlationsdo not obey Result 2b, then, provided this is not due to sampling error, thesingle-factor model can be rejected. Indeed, population partial correlations betweenmanifest variables in a single-factor model should be non-zero. Therefore, providedthere is adequate power for the tests that these are zero, from Result 2a, sample partialcorrelations not signiﬁcantly diﬀerent from zero also rule out the single-factor model.However, if power for these tests is small, the single-factor model should still beconsidered, even if there are non-signiﬁcant sample partial correlations. Methods forestimating overall power for graphical Gaussian models were investigated by Salgueiroet al. (2005). As the single-factor model can be parameterized as a graphical Gaussianmodel, the results presented by Salgueiro et al. (2005) can be used for the single-factormodel.Psychometrika Submission March 30, 2007 9ReferencesAnderson, T.W. and Rubin, H. (1956). Statistical inference in factor analysis. InProceedings of the third Berkeley Symposium on Mathematical Statistics andProbability, Vol. V, 111-150, Berkeley: University of California Press.Guttman, L. (1953). Image theory for the structure of quantitative variates.Psychometrika, 18, 277-296.Hadley, G. (1961). Linear Algebra. Massachusetts: Addison-Wesley PublishingCompany, Inc.Kaiser, H.F. (1970). A second generation Little Jiﬀy. Psychometrika, 35, 401-415.Rousseeuw, P.J. and Molenberghs, G. (1994). The shape of correlation matrices. TheAmerican Statistician, 48 (4), 276-279.Salgueiro, M.F. (2003). Distributions of Test Statistics for Edge Exclusion for GraphicalModels. PhD thesis, University of Southampton.Salgueiro, M.F., Smith, P.W.F. and McDonald, J.W. (2005). Power of edge exclusiontests in graphical Gaussian models. Biometrika, 92 (1), 173-182.Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. Chichester:John Wiley & Sons.Yanai, H. and Mukherjee, B.N. (1987). A generalized method of image analysis from anintercorrelation matrix which may be singular. Psychometrika, 52 (4), 555-564.Psychometrika Submission March 30, 2007 10FiguresFigure 1.Set of all possible partial correlation coeﬃcients (ρ12.3, ρ13.2 and ρ23.1) between three variables.-1.0 -0.5 0.0 0.5 1.0 -1.0-0.50.00.51.0 a)-1.0 -0.5 0.0 0.5 1.0 -1.0-0.50.00.51.0 b)-1.0 -0.5 0.0 0.5 1.0 -1.0-0.50.00.51.0 c)-1.0 -0.5 0.0 0.5 1.0 -1.0-0.50.00.51.0 -1.0 -0.5 0.0 0.5 1.0 -1.0-0.50.00.51.0 e)-1.0 -0.5 0.0 0.5 1.0 -1.0-0.50.00.51.0 f) d)Figure 2.The admissible region (shaded) for the three partial correlation coeﬃcients compatible with a single-factor model: ρ13.2 on the horizontal axis, ρ23.1 on the vertical axis and ρ12.3 taking values of a) 0.1,b) 0.5, c) 0.9, d) −0.1, e) −0.5, f) −0.9. The ellipse represents the positive deﬁniteness constraint forpartial correlations.

A generalized method of image analysis from an intercorrelation matrix which may be singular.

A second generation Little Jiﬀy.

Distributions of Test Statistics for Edge Exclusion for Graphical Models.

Graphical Models in Applied Multivariate Statistics.

Image theory for the structure of quantitative variates.

Linear Algebra.

Power of edge exclusion tests in graphical Gaussian models.

Statistical inference in factor analysis.

The shape of correlation matrices.

The manifest association structure of the single-factor model: insights from partial correlations

anti-image correlation matrix, elliptical tetrahedron, factor analysis, factor partial correlation, manifest partial correlation,

Maria Salgueiro

Peter Smith

John McDonald

Research Papers in Economics

UCL Discovery

THE MANIFEST ASSOCIATION STRUCTURE OF THESINGLE-FACTOR MODEL: INSIGHTS FROM PARTIALCORRELATIONSMaria de Fa´tima Salgueirodepartamento me´todos quantitativos, iscte business schoolPeter W.F. Smithsouthampton statistical sciences research institute,university of southamptonJohn W. McDonaldsouthampton statistical sciences research institute,university of southamptonJuly 8, 2007The research of the first author was supported by the Fundac¸a¯o para a Cieˆncia e aTecnologia and by the Southampton Statistical Sciences Research Institute.The authors thank the Editor, the Associate Editor and three Reviewers for theirconstructive comments which helped to improve the paper.Correspondence should be sent to Maria de Fa´tima SalgueiroE-Mail: fatima.salgueiro@iscte.ptPhone: +351-963077654Fax: +351-217903941Psychometrika Submission July 8, 2007 2THE MANIFEST ASSOCIATION STRUCTURE OF THE SINGLE-FACTORMODEL: INSIGHTS FROM PARTIAL CORRELATIONSAbstractThe association structure between manifest variables arising from thesingle-factor model is investigated using partial correlations. The additionalinsights to the practitioner provided by partial correlations for detecting asingle-factor model are discussed. The parameter space for the partialcorrelations is presented, as are the patterns of signs in a matrix containingthe partial correlations that are not compatible with a single-factor model.Key words: anti-image correlation matrix, elliptical tetrahedron, factoranalysis, factor partial correlation, manifest partial correlation.Psychometrika Submission July 8, 2007 31. IntroductionFactor analysis is a classical approach to modeling multivariate data where allvariables are treated on an equal footing. Traditionally, these are thought of as modelsfor covariances or correlations, but partial correlations can be used.Guttman (1953) advocated that all off-diagonal elements of the inverse correlationmatrix close to zero was a criteria for ruling out a common-factor space for thepopulation. Kaiser (1970), in a Presidential address to the Psychometric Society, statedthat “among Guttman’s landmark papers . . . one particular theme appears repeatedly:given a correlation matrix, we should always look carefully at its inverse, in order toassess the sampling adequacy of the data for factor-analytic purposes”. The scaled, tohave ones on the diagonal, inverse variance matrix has off-diagonal elements (i, j) equalto the negatives of the partial correlation between variables i and j, after conditioningon the remaining variables (Whittaker, 1990, page 143). This matrix is known as theanti-image correlation matrix and is available in standard statistical packages such asSPSS, SAS and STATA. For further details on the relationship between correlationmatrices and anti-image correlation matrices, see Yanai and Mukherjee (1987).This paper presents three main results and two corollaries that provide extrainsights into single-factor models, based on partial correlations and on the pattern oftheir signs. Section 2 presents necessary, but not sufficient, constraints on theparameter space of the population partial correlations implied by a single-factor model.Section 3 defines factor partial correlations and derives the association structurebetween manifest variables after marginalizing over the latent variable in a single-factormodel. Also presented in Section 3 are patterns of signs of partial correlations that arenot compatible with a single-factor model. Section 4 re-expresses the classical tetradconditions as a function of the manifest partial correlations. Section 5 contains adiscussion.Psychometrika Submission July 8, 2007 42. The Parameter Space of the Single-Factor ModelThe classical single-factor model can be written as XM = λL+ δ, where XM isthe vector of the p manifest variables X1, X2, . . . , Xp, L is the factor or latent variable,λ is a p× 1 vector of factor loadings, and δ is a vector of p variables representingrandom measurement error and indicator specificity. Variables are considered to bemeasured as deviations from their means, so E[XM] = 0 and E[L] = 0. The classicalsingle-factor model assumes that E[Lδ] = 0, E[δ] = 0, var[δ] is diagonal and that Land δ are multivariate normal, although the results presented in this paper do notrequire these normality assumptions. To identify the model, the latent variable L isscaled to have unit variance. The variance matrix for XM, with elements denoted byσij , is ΣM = λλT +Θ, where Θ is the p× p diagonal variance matrix of δ, withnon-negative elements. A necessary and sufficient condition for ΣM to be a variancematrix of a single-factor model is that p(p− 1)/2− p independent tetrad conditions aresatisfied and 0 ≤σkiσijσkj≤ σii for one pair (j 6= k) for each i (Anderson and Rubin, 1956).The tetrad conditions are σkiσlj − σliσkj = 0, for all distinct i, j, k and l, from 1 to p.Results in this paper are presented under the assumption that variances arepositive and that all variance matrices are positive definite. Therefore, a necessary but,as will be seen later, not sufficient condition on the scaled inverse variance matrix isthat it is positive definite.Result 1. The p× p anti-image correlation matrix is positive definite if and only if itsdeterminant is strictly positive and, for p ≥ 4, all its symmetric (p− 1)× (p− 1)submatrices are positive definite. Hence, when p = 3, the positive definitenessconstraint on the anti-image correlation matrix implies1− 2 ρ12.3 ρ13.2 ρ23.1 − ρ212.3 − ρ213.2 − ρ223.1 > 0, (1)where ρij.k is the partial correlation coefficient between Xi and Xj given Xk.The proof of Result 1 follows from Hadley (1961), Equation 7-69. The differentcombinations of the three ρij.k form a convex set which is symmetric with respect torotations corresponding to permutations of the components of (ρ12.3, ρ13.2, ρ23.1).Psychometrika Submission July 8, 2007 5Figure 1 displays the boundary of the convex set in a three-dimensional plot. Theseplots should be contrasted to those in Figure 1 of Rousseeuw and Molenberghs (1994):both are elliptical tetrahedrons, but with different orientations.=========================Insert Figure 1 about here=========================Ignoring for the moment the shaded area, Figure 2 supports the statement that anyhorizontal cross section of the surface defined by Figure 1 is an ellipse.=========================Insert Figure 2 about here=========================3. The Relationship Between the Manifest and the Factor PartialCorrelationsLet X be the vector of the p+ 1 random variables, partitioned as[XMT L]T,with positive definite scaled inverse variance matrix partitioned asT = I TMLTLM 1 ,where I is a p× p identity matrix since the single-factor model assumes that themanifest variables are conditionally independent, given the latent variable. The p× 1vector TML = TLMT contains the non-zero elements −τiL.(iL), the negative of thepartial correlation between manifest variable Xi and latent variable L, where (iL)denotes the remaining p− 1 variables in X, after removing Xi and L. Henceforth, theτiL.(iL) are called factor partial correlations.The positive definiteness constraint on the matrix T implies that its determinant ispositive, i.e., |1− TLMTML| > 0, and that the determinants of all its symmetricsubmatrices are also positive. Therefore,p∑i=1τ 2iL.(iL) < 1. (2)Psychometrika Submission July 8, 2007 6Also, because the τ are partial correlations, −1 < τiL.(iL) < 1. The parameter space forthe factor partial correlations is defined by Equation 2, with the additional constraintthat τiL.(iL) 6= 0. Indeed, for the single-factor model to be of interest all factor partialcorrelations need to be of substantive interest, and not only non-zero mathematically.Note that Equation 2 defines an open unit hyper-sphere.Result 2. The manifest partial correlations can be written as a function of the factorpartial correlations asρij.(ij) =τiL.(iL) τjL.(jL){(1− τ 2iL.(iL))(1− τ2jL.(jL))}1/2, i 6= j ∈ {1, 2, . . . , p}, (3)where ρij.(ij) is the manifest partial correlation between Xi and Xj, given the remainingp− 2 manifest variables.The proof follows. Inverting T , the scaled inverse variance matrix of X, givesT−1 = I TMLTLM 1−1= (I − TMLTLM)−1 −(I − TMLTLM)−1TML−TLM(I − TMLTLM)−1 (1− TLMTML)−1 .(4)The correlation matrix of XM is given by scaling (I − TMLTLM)−1. Hence, theanti-image correlation matrix is given by scaling I − TMLTLM to have ones on thediagonal. The off-diagonal element (i, j) is given by −τiL.(iL) τjL.(jL){(1−τ2iL.(iL))(1−τ2jL.(jL))}1/2. Since theoff-diagonal elements of the population anti-image correlation matrix are the negativesof the population manifest partial correlations, Equation 3 follows.Two corollaries of Result 2 follow.Result 2a. Marginalizing the single-factor model over the latent variable L yields ajoint distribution for the manifest variables with no zero manifest partial correlations,that is, with no zero entries in the anti-image correlation matrix.Since all factor partial correlations are assumed non-zero, and the manifest partialcorrelations are given by Equation 3, all τiL.(iL) 6= 0 imply all ρij.(ij) 6= 0. Therefore,there are no zero entries in the anti-image correlation matrix and the proof is complete.Psychometrika Submission July 8, 2007 7From Equation 3, the sign of ρij.(ij) equals the product of the signs of τiL.(iL) andτjL.(jL). Let τ = −TML be a p× 1 column vector of the factor partial correlations andlet ρ be a p× p symmetric matrix containing the manifest partial correlations asoff-diagonal elements. Recall that λ is the vector of factor loadings.Result 2b. The off-diagonal elements of sign(ρ) represent a pattern of signs for themanifest partial correlations that is compatible with a single-factor model ifsign(ρ) = sign(τ τ T ) = sign(λλT ) (5)(for the off-diagonal elements). This result refers to population parameters and assumesthe tetrad conditions are satisfied. The second equality follows from Salgueiro (2003,Equation 5.16): λ⋆i =τiL.(iL)1−∑pk=1 τ2kL.(kL). Indeed, up to scaling, each factor loading can bewritten as a function of the factor partial correlations.Note that, when p = 3, Equation 3 defines a system of three equations of the formτ 2iL.jk =11 +ρjk.iρij.kρik.j. (6)Since 0 < τ 2iL.jk < 1, thenρjk.iρij.kρik.j> 0. Hence, three manifest variables can only define asingle-factor model if their anti-image correlation matrix is positive definite and theproduct of the three manifest partial correlation coefficients is positive. RevisitingFigure 2, the shaded area represents the admissible region for the three manifest partialcorrelation coefficients in a single-factor model. The ellipse represents the positivedefiniteness constraint given by Equation 1. Additionally, there is now the constraint ofa positive product of the three partial correlations. Indeed, it is not possible to have asingle-factor model with three manifest variables in which just one or all three ρij.k arenegative. Also, if τiL.jk is negative and the other two factor partial correlations arepositive (or the opposite), then all manifest partial correlations in column (row) i of theanti-image correlation matrix will be negative, and all the remaining manifest partialcorrelations will be positive.When p ≥ 4, Equation 5 can be used to derive possible patterns of signs for themanifest partial correlations, given the hypothesized structure of signs for the factorPsychometrika Submission July 8, 2007 8partial correlations. Alternatively, by considering all possible patterns of signs for thefactor partial correlations, Equation 5 can be used to obtain all possible patterns ofsigns for the manifest partial correlations.4. An Extension of the Tetrad ConditionsThe classical tetrad conditions for the single-factor model can be re-expressed interms of partial correlations.Result 3. When p ≥ 4, the following p(p− 1)/2− p tetrad conditions for the partialcorrelations have to be satisfied in a single-factor model:ρki.(ki)ρlj.(lj) − ρli.(li)ρkj.(kj) = 0, (7)with i, j, k and l distinct, from 1 to p, where ρki.(ki) is the partial correlation coefficientbetween Xk and Xi, given the remaining p− 2 manifest variables.The proof follows by repeated substitution of Equation 3 into the left hand side ofEquation 7.5. DiscussionThis paper has provided some useful results for the practitioner. Inspection ofsample partial correlations between manifest variables can provide evidence for andagainst the single-factor model. If the pattern of signs of the sample partial correlationsdo not obey Result 2b, then, provided this is not due to sampling error, thesingle-factor model can be rejected. Indeed, population partial correlations betweenmanifest variables in a single-factor model should be non-zero. Therefore, providedthere is adequate power for the tests that these are zero, from Result 2a, sample partialcorrelations not significantly different from zero also rule out the single-factor model.However, if power for these tests is small, the single-factor model should still beconsidered, even if there are non-significant sample partial correlations. Methods forestimating overall power for graphical Gaussian models were investigated by Salgueiroet al. (2005). As the single-factor model can be parameterized as a graphical GaussianPsychometrika Submission July 8, 2007 9model, the results presented by Salgueiro et al. (2005) can be used for the single-factormodel.Psychometrika Submission July 8, 2007 10ReferencesAnderson, T.W. and Rubin, H. (1956). Statistical inference in factor analysis. InProceedings of the third Berkeley Symposium on Mathematical Statistics andProbability, Vol. V, 111-150, Berkeley: University of California Press.Guttman, L. (1953). Image theory for the structure of quantitative variates.Psychometrika, 18, 277-296.Hadley, G. (1961). Linear Algebra. Massachusetts: Addison-Wesley PublishingCompany, Inc.Kaiser, H.F. (1970). A second generation Little Jiffy. Psychometrika, 35, 401-415.Rousseeuw, P.J. and Molenberghs, G. (1994). The shape of correlation matrices. TheAmerican Statistician, 48 (4), 276-279.Salgueiro, M.F. (2003). Distributions of Test Statistics for Edge Exclusion for GraphicalModels. PhD thesis, University of Southampton.Salgueiro, M.F., Smith, P.W.F. and McDonald, J.W. (2005). Power of edge exclusiontests in graphical Gaussian models. Biometrika, 92 (1), 173-182.Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. Chichester:John Wiley & Sons.Yanai, H. and Mukherjee, B.N. (1987). A generalized method of image analysis from anintercorrelation matrix which may be singular. Psychometrika, 52 (4), 555-564.Psychometrika Submission July 8, 2007 11FiguresFigure 1.Set of all possible partial correlation coefficients (ρ12.3, ρ13.2 and ρ23.1) between three variables.-1.0 -0.5 0.0 0.5 1.0 -1.0-0.50.00.51.0 a)-1.0 -0.5 0.0 0.5 1.0 -1.0-0.50.00.51.0 b)-1.0 -0.5 0.0 0.5 1.0 -1.0-0.50.00.51.0 c)-1.0 -0.5 0.0 0.5 1.0 -1.0-0.50.00.51.0 -1.0 -0.5 0.0 0.5 1.0 -1.0-0.50.00.51.0 e)-1.0 -0.5 0.0 0.5 1.0 -1.0-0.50.00.51.0 f)d)Figure 2.The admissible region (shaded) for the three partial correlation coefficients compatible with a single-factor model: ρ13.2 on the horizontal axis, ρ23.1 on the vertical axis and ρ12.3 taking values of a) 0.1,b) 0.5, c) 0.9, d) −0.1, e) −0.5, f) −0.9. The ellipse represents the positive definiteness constraint forpartial correlations.

Crossref

WOS:000261959200007 (Nº de Acesso Web of Science)The association structure between manifest variables arising from the single-factor model is investigated using partial correlations. The additional insights to the practitioner provided by partial correlations for detecting a single-factor model are discussed. The parameter space for the partial correlations is presented, as are the patterns of signs in a matrix containing the partial correlations that are not compatible with a single-factor model

Salgueiro, M. F.

Smith, P. W. F.

McDonald, J. W.

Repositório Institucional do ISCTE-IUL

  THE MANIFEST ASSOCIATION STRUCTURE OF THE SINGLE-FACTOR MODEL: INSIGHTS FROM PARTIAL CORRELATIONS  MARIA DE FÁTIMA SALGUEIRO, PETER W. F. SMITH,  JOHN W. MCDONALD  ABSTRACT  The association structure between manifest variables arising from the single-factor model is investigated using partial correlations. The additional insights to the practitioner provided by partial correlations for detecting a single-factor model are discussed. The parameter space for the partial correlations is presented, as are the patterns of signs in a matrix containing the partial correlations that are not compatible with a single-factor model.   Southampton Statistical Sciences Research Institute Methodology Working Paper M07/07 THE MANIFEST ASSOCIATION STRUCTURE OF THESINGLE-FACTOR MODEL: INSIGHTS FROM PARTIALCORRELATIONSMaria de Fátima Salgueirodepartamento métodos quantitativos, iscte business schoolPeter W.F. Smithsouthampton statistical sciences research institute,university of southamptonJohn W. McDonaldsouthampton statistical sciences research institute,university of southamptonMarch 30, 2007The research of the first author was supported by the Fundaçāo para a Ciência e aTecnologia and by the Southampton Statistical Sciences Research Institute.The authors thank the Editor, the Associate Editor and three Reviewers for theirconstructive comments which helped to improve the paper.Correspondence should be sent to Maria de Fátima SalgueiroE-Mail: fatima.salgueiro@iscte.ptPhone: +351-963077654Fax: +351-217903941Psychometrika Submission March 30, 2007 2THE MANIFEST ASSOCIATION STRUCTURE OF THE SINGLE-FACTORMODEL: INSIGHTS FROM PARTIAL CORRELATIONSAbstractThe association structure between manifest variables arising from thesingle-factor model is investigated using partial correlations. The additionalinsights to the practitioner provided by partial correlations for detecting asingle-factor model are discussed. The parameter space for the partialcorrelations is presented, as are the patterns of signs in a matrix containingthe partial correlations that are not compatible with a single-factor model.Key words: anti-image correlation matrix, elliptical tetrahedron, factoranalysis, factor partial correlation, manifest partial correlation.Psychometrika Submission March 30, 2007 31. IntroductionFactor analysis is a classical approach to modeling multivariate data where allvariables are treated on an equal footing. Traditionally, these are thought of as modelsfor covariances or correlations, but partial correlations can be used.Guttman (1953) advocated that all off-diagonal elements of the inverse correlationmatrix close to zero was a criteria for ruling out a common-factor space for thepopulation. Kaiser (1970), in a Presidential address to the Psychometric Society, statedthat “among Guttman’s landmark papers . . . one particular theme appears repeatedly:given a correlation matrix, we should always look carefully at its inverse, in order toassess the sampling adequacy of the data for factor-analytic purposes”. The scaled, tohave ones on the diagonal, inverse variance matrix has off-diagonal elements (i, j) equalto the negatives of the partial correlation between variables i and j, after conditioningon the remaining variables (Whittaker, 1990, page 143). This matrix is known as theanti-image correlation matrix and is available in standard statistical packages such asSPSS, SAS and STATA. For further details on the relationship between correlationmatrices and anti-image correlation matrices, see Yanai and Mukherjee (1987).This paper presents three main results and two corollaries that provide extrainsights into single-factor models, based on partial correlations and on the pattern oftheir signs. Section 2 presents necessary, but not sufficient, constraints on theparameter space of the population partial correlations implied by a single-factor model.Section 3 defines factor partial correlations and derives the association structurebetween manifest variables after marginalizing over the latent variable in a single-factormodel. Also presented in Section 3 are patterns of signs of partial correlations that arenot compatible with a single-factor model. Section 4 re-expresses the classical tetradconditions as a function of the manifest partial correlations. Section 5 contains adiscussion.Psychometrika Submission March 30, 2007 42. The Parameter Space of the Single-Factor ModelThe classical single-factor model can be written as XM = λL + δ, where XM isthe vector of the p manifest variables X1, X2, . . . , Xp, L is the factor or latent variable,λ is a p × 1 vector of factor loadings, and δ is a vector of p variables representingrandom measurement error and indicator specificity. Variables are considered to bemeasured as deviations from their means, so E[XM] = 0 and E[L] = 0. The classicalsingle-factor model assumes that E[Lδ] = 0, E[δ] = 0, var[δ] is diagonal and that Land δ are multivariate normal, although the results presented in this paper do notrequire these normality assumptions. To identify the model, the latent variable L isscaled to have unit variance. The variance matrix for XM, with elements denoted byσij , is ΣM = λλT + Θ, where Θ is the p × p diagonal variance matrix of δ, withnon-negative elements. A necessary and sufficient condition for ΣM to be a variancematrix of a single-factor model is that p(p − 1)/2 − p independent tetrad conditions aresatisfied and 0 ≤σkiσijσkj≤ σii for one pair (j 6= k) for each i (Anderson and Rubin, 1956).The tetrad conditions are σkiσlj − σliσkj = 0, for all distinct i, j, k and l, from 1 to p.Results in this paper are presented under the assumption that variances arepositive and that all variance matrices are positive definite. Therefore, a necessary but,as will be seen later, not sufficient condition on the scaled inverse variance matrix isthat it is positive definite.Result 1. The p × p anti-image correlation matrix is positive definite if and only if itsdeterminant is strictly positive and, for p ≥ 4, all its symmetric (p − 1) × (p − 1)submatrices are positive definite. Hence, when p = 3, the positive definitenessconstraint on the anti-image correlation matrix implies1 − 2 ρ12.3 ρ13.2 ρ23.1 − ρ212.3 − ρ213.2 − ρ223.1 > 0, (1)where ρij.k is the partial correlation coefficient between Xi and Xj given Xk.The proof of Result 1 follows from Hadley (1961), Equation 7-69. The differentcombinations of the three ρij.k form a convex set which is symmetric with respect torotations corresponding to permutations of the components of (ρ12.3, ρ13.2, ρ23.1).Psychometrika Submission March 30, 2007 5Figure 1 displays the boundary of the convex set in a three-dimensional plot. Theseplots should be contrasted to those in Figure 1 of Rousseeuw and Molenberghs (1994):both are elliptical tetrahedrons, but with different orientations.=========================Insert Figure 1 about here=========================Ignoring for the moment the shaded area, Figure 2 supports the statement that anyhorizontal cross section of the surface defined by Figure 1 is an ellipse.=========================Insert Figure 2 about here=========================3. The Relationship Between the Manifest and the Factor PartialCorrelationsLet X be the vector of the p + 1 random variables, partitioned as[XMT L]T,with positive definite scaled inverse variance matrix partitioned asT =I TMLTLM 1 ,where I is a p × p identity matrix since the single-factor model assumes that themanifest variables are conditionally independent, given the latent variable. The p × 1vector TML = TLMT contains the non-zero elements −τiL.(iL), the negative of thepartial correlation between manifest variable Xi and latent variable L, where (iL)denotes the remaining p − 1 variables in X, after removing Xi and L. Henceforth, theτiL.(iL) are called factor partial correlations.The positive definiteness constraint on the matrix T implies that its determinant ispositive, i.e., |1 − TLMTML| > 0, and that the determinants of all its symmetricsubmatrices are also positive. Therefore,p∑i=1τ 2iL.(iL) < 1. (2)Psychometrika Submission March 30, 2007 6Also, because the τ are partial correlations, −1 < τiL.(iL) < 1. The parameter space forthe factor partial correlations is defined by Equation 2, with the additional constraintthat τiL.(iL) 6= 0. Indeed, for the single-factor model to be of interest all factor partialcorrelations need to be of substantive interest, and not only non-zero mathematically.Note that Equation 2 defines an open unit hyper-sphere.Result 2. The manifest partial correlations can be written as a function of the factorpartial correlations asρij.(ij) =τiL.(iL) τjL.(jL){(1 − τ 2iL.(iL))(1 − τ2jL.(jL))}1/2, i 6= j ∈ {1, 2, . . . , p}, (3)where ρij.(ij) is the manifest partial correlation between Xi and Xj, given the remainingp − 2 manifest variables.The proof follows. Inverting T , the scaled inverse variance matrix of X, givesT−1 =I TMLTLM 1−1=(I − TMLTLM)−1 −(I − TMLTLM)−1TML−TLM(I − TMLTLM)−1 (1 − TLMTML)−1 .(4)The correlation matrix of XM is given by scaling (I − TMLTLM)−1. Hence, theanti-image correlation matrix is given by scaling I − TMLTLM to have ones on thediagonal. The off-diagonal element (i, j) is given by −τiL.(iL) τjL.(jL){(1−τ2iL.(iL))(1−τ2jL.(jL))}1/2. Since theoff-diagonal elements of the population anti-image correlation matrix are the negativesof the population manifest partial correlations, Equation 3 follows.Two corollaries of Result 2 follow.Result 2a. Marginalizing the single-factor model over the latent variable L yields ajoint distribution for the manifest variables with no zero manifest partial correlations,that is, with no zero entries in the anti-image correlation matrix.Since all factor partial correlations are assumed non-zero, and the manifest partialcorrelations are given by Equation 3, all τiL.(iL) 6= 0 imply all ρij.(ij) 6= 0. Therefore,there are no zero entries in the anti-image correlation matrix and the proof is complete.Psychometrika Submission March 30, 2007 7From Equation 3, the sign of ρij.(ij) equals the product of the signs of τiL.(iL) andτjL.(jL). Let τ = −TML be a p × 1 column vector of the factor partial correlations andlet ρ be a p × p symmetric matrix containing the manifest partial correlations asoff-diagonal elements. Recall that λ is the vector of factor loadings.Result 2b. The off-diagonal elements of sign(ρ) represent a pattern of signs for themanifest partial correlations that is compatible with a single-factor model ifsign(ρ) = sign(τ τ T ) = sign(λλT ) (5)(for the off-diagonal elements). This result refers to population parameters and assumesthe tetrad conditions are satisfied. The second equality follows from Salgueiro (2003,Equation 5.16): λi =τiL.(iL)1−Ppk=1 τ2kL.(kL).Note that, when p = 3, Equation 3 defines a system of three equations of the formτ 2iL.jk =11 +ρjk.iρij.kρik.j. (6)Since 0 < τ 2iL.jk < 1, thenρjk.iρij.kρik.j> 0. Hence, three manifest variables can only define asingle-factor model if their anti-image correlation matrix is positive definite and theproduct of the three manifest partial correlation coefficients is positive. RevisitingFigure 2, the shaded area represents the admissible region for the three manifest partialcorrelation coefficients in a single-factor model. The ellipse represents the positivedefiniteness constraint given by Equation 1. Additionally, there is now the constraint ofa positive product of the three partial correlations. Indeed, it is not possible to have asingle-factor model with three manifest variables in which just one or all three ρij.k arenegative. Also, if τiL.jk is negative and the other two factor partial correlations arepositive (or the opposite), then all manifest partial correlations in column (row) i of theanti-image correlation matrix will be negative, and all the remaining manifest partialcorrelations will be positive.When p ≥ 4, Equation 5 can be used to derive possible patterns of signs for themanifest partial correlations, given the hypothesized structure of signs for the factorpartial correlations. Alternatively, by considering all possible patterns of signs for thePsychometrika Submission March 30, 2007 8factor partial correlations, Equation 5 can be used to obtain all possible patterns ofsigns for the manifest partial correlations.4. An Extension of the Tetrad ConditionsThe classical tetrad conditions for the single-factor model can be re-expressed interms of partial correlations.Result 3. When p ≥ 4, the following p(p − 1)/2 − p tetrad conditions for the partialcorrelations have to be satisfied in a single-factor model:ρki.(ki)ρlj.(lj) − ρli.(li)ρkj.(kj) = 0, (7)with i, j, k and l distinct, from 1 to p, where ρki.(ki) is the partial correlation coefficientbetween Xk and Xi, given the remaining p − 2 manifest variables.The proof follows by repeated substitution of Equation 3 into the left hand side ofEquation 7.5. DiscussionThis paper has provided some useful results for the practitioner. Inspection ofsample partial correlations between manifest variables can provide evidence for andagainst the single-factor model. If the pattern of signs of the sample partial correlationsdo not obey Result 2b, then, provided this is not due to sampling error, thesingle-factor model can be rejected. Indeed, population partial correlations betweenmanifest variables in a single-factor model should be non-zero. Therefore, providedthere is adequate power for the tests that these are zero, from Result 2a, sample partialcorrelations not significantly different from zero also rule out the single-factor model.However, if power for these tests is small, the single-factor model should still beconsidered, even if there are non-significant sample partial correlations. Methods forestimating overall power for graphical Gaussian models were investigated by Salgueiroet al. (2005). As the single-factor model can be parameterized as a graphical Gaussianmodel, the results presented by Salgueiro et al. (2005) can be used for the single-factormodel.Psychometrika Submission March 30, 2007 9ReferencesAnderson, T.W. and Rubin, H. (1956). Statistical inference in factor analysis. InProceedings of the third Berkeley Symposium on Mathematical Statistics andProbability, Vol. V, 111-150, Berkeley: University of California Press.Guttman, L. (1953). Image theory for the structure of quantitative variates.Psychometrika, 18, 277-296.Hadley, G. (1961). Linear Algebra. Massachusetts: Addison-Wesley PublishingCompany, Inc.Kaiser, H.F. (1970). A second generation Little Jiffy. Psychometrika, 35, 401-415.Rousseeuw, P.J. and Molenberghs, G. (1994). The shape of correlation matrices. TheAmerican Statistician, 48 (4), 276-279.Salgueiro, M.F. (2003). Distributions of Test Statistics for Edge Exclusion for GraphicalModels. PhD thesis, University of Southampton.Salgueiro, M.F., Smith, P.W.F. and McDonald, J.W. (2005). Power of edge exclusiontests in graphical Gaussian models. Biometrika, 92 (1), 173-182.Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. Chichester:John Wiley & Sons.Yanai, H. and Mukherjee, B.N. (1987). A generalized method of image analysis from anintercorrelation matrix which may be singular. Psychometrika, 52 (4), 555-564.Psychometrika Submission March 30, 2007 10FiguresFigure 1.Set of all possible partial correlation coefficients (ρ12.3, ρ13.2 and ρ23.1) between three variables.-1.0 -0.5 0.0 0.5 1.0 -1.0-0.50.00.51.0 a)-1.0 -0.5 0.0 0.5 1.0 -1.0-0.50.00.51.0 b)-1.0 -0.5 0.0 0.5 1.0 -1.0-0.50.00.51.0 c)-1.0 -0.5 0.0 0.5 1.0 -1.0-0.50.00.51.0 -1.0 -0.5 0.0 0.5 1.0 -1.0-0.50.00.51.0 e)-1.0 -0.5 0.0 0.5 1.0 -1.0-0.50.00.51.0 f)d)Figure 2.The admissible region (shaded) for the three partial correlation coefficients compatible with a single-factor model: ρ13.2 on the horizontal axis, ρ23.1 on the vertical axis and ρ12.3 taking values of a) 0.1,b) 0.5, c) 0.9, d) −0.1, e) −0.5, f) −0.9. The ellipse represents the positive definiteness constraint forpartial correlations.

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The manifest association structure of the single-factor model: insights from partial correlations

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