Table Of ContentLecture Notes in Statistics 120
Edited by P. Bickel, P. Diggle, S. Fienberg, K. Krickeberg,
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Maia Berkane
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Latent Variable Modeling and
Applications to Causality
, Springer
Maia Berkane
Department of Mathematics
Harvard University
Cambridge, MA 02138
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Preface
This volume gathers refereed papers presented at the 1994 UCLA conference on "La
tent Variable Modeling and Application to Causality." The meeting was organized by the
UCLA Interdivisional Program in Statistics with the purpose of bringing together a group
of people who have done recent advanced work in this field. The papers in this volume are
representative of a wide variety of disciplines in which the use of latent variable models is
rapidly growing. The volume is divided into two broad sections. The first section covers
Path Models and Causal Reasoning and the papers are innovations from contributors in
disciplines not traditionally associated with behavioural sciences, (e.g. computer science
with Judea Pearl and public health with James Robins). Also in this section are contri
butions by Rod McDonald and Michael Sobel who have a more traditional approach to
causal inference, generating from problems in behavioural sciences.
The second section encompasses new approaches to questions of model selection with
emphasis on factor analysis and time varying systems. Amemiya uses nonlinear factor
analysis which has a higher order of complexity associated with the identifiability condi
tions. Muthen studies longitudinal hierarchichal models with latent variables and treats
the time vector as a variable rather than a level of hierarchy. Deleeuw extends exploratory
factor analysis models by including time as a variable and allowing for discrete and ordi
nal latent variables. Arminger looks at autoregressive structures and Bock treats factor
analysis models for categorical data. Berkane brings differential geometric approaches to
covariance structure models and shows how estimation can be improved by use of the
different measures of curvatures of of the model. Bentler considers the problem of esti
mating factor scores and proposes equivariant estimators as an alternative to conventional
methods such as maximum likelihood. Jamshidian and Browne offer new algorithms and
computational techniques for parameter estimation.
All these contributions present new results not published elsewhere. I believe this volume
will provide the readers with opportunities to capture the various connections between
techniques, viewpoints and insights that would not be possible otherwise.
I would like to thank the following sponsors who made the conference happen through
their support and help: The Drug Abuse Research Center, the Cognitive System Labo
ratory, the National Institute on Alcohol Abuse and Alcoholism, the National Research
on Evaluation, Standards and Student Testing, the National Research Center for Asian
American Mental Health, and the National Science Foundation.
Maia Berkane
Lexington, MA
1996
Contents
Preface
Causality and Path Models
Embedding Common factors in a Path Model 1
Roderick P. McDonald
Measurement, Causation and Local Independence
in Latent Variable Models 11
Michael E. Sobel
On the Identifiability of Nonparametric Structural Models 29
Judea Pearl
Estimating the Causal effects of Time Varying Endogeneous Treatments
by G-Estimation of Structural Nested Models 69
James Robins
Latent Variables
Model as Instruments, with Applications to Moment Structure Analysis 119
Jan de Leeuw
Bias and Mean Square Error of the Maximum Likelihood Estimators
of the Parameters of the Intraclass Correlation Model 133
Maia Berkane and Peter M. Bentler
Latent Variable Growth Modeling with Multilevel Data 149
Bengt Muthen
High-Dimensional Full-Information Item Factor Analysis 163
Darrell R. Bock and Steven Shilling
Dynamic Factor Models for the Analysis of Ordered Categorical Panel data 177
Gerhard Arminger
Model Fitting Procedures for Nonlinear Factor Analysis Using
the Errors-in-Variables Parameterization 195
Yasuo Amemiya and Ilker Yalcin
Multivariate Regression with Errors in Variables: Issues
on Asymptotic Robustness 211
Albert SatorTa
Non-Iterative fitting of the Direct Product Model for Multitrait-Multimethod
Correlation Matrices 229
Michael E. Browne and H. F. Strydom
An EM Algorithm for ML Factor Analysis with Missing Data 247
Mortaza Jamshidian
Optimal Conditionally Unbiased Equivariant Factor Score Estimators 259
Peter M. Bentler and Ke-Hai Yuan
Embedding common factors in a path model
Roderick P. McDonald
University of Illinois
1. Introduction
In models containing unobservable variables with mUltiple
indicators--common factors, true scores--it seems to be common
practice not to allow, except in special cases, nonzero
covariances of residuals of indicators of the common factors and
other variables in the model. Indeed, those models, such as
LISREL and LISCOMP, that separate a "measurement" model from a
"structural" model, exclude these by their defining matrix
structure. Yet it is not at all obvious that such covariances
should not be allowed, and it seems desirable to look for some
explicit principles to apply to such cases, rather than settle
the matter by default. This question will be examined in the
context of the reticular action model. The following section
sets out some necessary preliminaries from the case of path
models without common factors, then section 3 treats the problem
of embedding a block of common factors in a path model. Section 4
gives a numerical example.
2. Path models without common factors
Let v be a qx1 random vector of observable variables, that
is, of variables whose values can be realized in a sampling
experiment. We suppose v is so scaled that its expected value is
null. A special case of the Reticular Action Model (RAM--see
McArdle and McDonald, 1984) may be written as
v = Bv+e (1)
with
Cov{v} = L, (2)
Cov{e} = \f, (3)
where the elements ~jk' \lfjk' of B, \f are functions of a tx1 vector
of parameters. We will refer to the components of e neutrally as
error terms. From (1) without further assumptions we have
e (I-B)v, (4)
and
v (I-B) -le. (5)
We define
. . A [ujkl = (I B) -1,
and ~mmed~ately have, further,
L (I - B)-l\f(I - B) ,-1 (6)
or
L A\fA' , (7)
and
\f A-1LA ,-1. (8)
We write
<I> Cov{v, e},
2
and by (3) and (5) we have
<I> =A'¥ (9)
and
<I> =L (I-B) , (10)
With the variables
(11)
or
y = BOy + rx + d, (12)
following a widely accepted though unnecessary convention that
separates variables corresponding to null rows of B in (1) into
an mx1 vector x of exogenous variables and the remainder as a p x
1 vector y of endogenous variables.
It is easily seen that there is a correspondence between a
specification of the structural model (1) and the specification
of a path diagram, such that (a) the nodes (vertices) of the
diagram correspond to the variables v,," .,v (b) a possibly
q,
nonzero Bjk corresponds to a directed arc (an arrow) from the
node vk to the node Vj (c) a possibly nonzero ajk corresponds to a
directed path (a chain of unidirectional arcs) from the node Vk
to the node Vj' (d) a possibly nonzero \j!jk corresponds to a
nondirected arc (a line) between the nodes Vj and Vk' including a
nondirected arc from Vj to itself corresponding to a variance.
We will always suppose that Diag{B} is null; thus, there cannot
be a directed arc from a variable to itself.
We consider only recursive models here. A recursive model is
one in which (possibly after permuting components of v) B is a
(lower) triangular matrix. (In the corresponding path diagram,
there are no closed loops formed by unidirectional arcs.) It is
nonrecursive if and only if it is not recursive as just defined.
(Some writers--e.g. Fox, 1984; Bollen, 1989--require also that in
a recursive model '¥ is diagonal. Others--e.g. Wright, 1934;
Duncan, 1975; Kang and Seneta, 1980--do not.) It is well-known
that if B is lower triangular then A is lower triangular with
unities on the diagonal (and conversely). We rewrite (1) as
Vj B(j)'v(j)+ej, (13)
and (5) as
Vj a(j)'e(j) +ej, j=l, ... , q, (14)
where B(j)" a(j)' contain the j-1 elements of the jth row of Band
of A except the diagonal unity, and v(j)' e(j) contain the
corresponding components of v, e.
We note that while certainly at most j-1 components of B(j)',
a(j)' are nonzero, in applications some number less than j-1 may
be nonzero because of the pattern of B and the patterning of A
that is in general induced by the pattern of B. We therefore
rewrite (13) as
(15)
and (14) as
3
Vj = a's(j)es(j) + ej' (16)
in which I3r(j) contains only the rj:S:j-1 nonzero elements of l3(j) and
as(j) contains only the sj:S:j -1 nonzero elements of a(j) and vr(j)'
es(j) the corresponding components of v(j) and e(j).
We will say that Vk directly precedes Vj if in the path
diagram there is a directed arc from Vk to Vj. Correspondingly,
I3jk is then possibly nonzero. We will say that Vk precedes Vj if
there exists at least one directed path from vk to vj.
Correspondingly, ujk is then possibly nonzero. We will say that
two variables are ordered/unordered if one precedes/does not
precede the other.
We are now in a position to list a number of possible
"rules" suggested in the literature governing the specification
of nondirected paths. The word "rule" is used here deliberately
to avoid pre-judging the status of these as assumptions or
definitional statements. They are at least commonly offered as
rules for the guidance of a user of path models. We consider the
following:
1. The Direct Precedence Rule (DPR) states that
if I3jk *0, then <l>kj=O.
That is, the error-term of variable Vj is orthogonal to each
variable in its own equation ( Land, 1969; Kang and Seneta, 1980;
Wright, 1960; Schmidt, 1982; Duncan, 1975; James, Mulaik and
Brett, 1982).
2. The General Precedence Rule for Variables (GPRVl states
that
if ujk *0, then <l>kj=O
That is, if Vk precedes Vj' directly or indirectly, vk and ej are
orthogonal. (See McDonald, 1985. Duncan, 1975, pp. 27-28 states
that in a recursive model "each dependent variable is
predetermined with respect to any other dependent variable that
occurs later in the causal ordering.")
3. The General Dependence Rule for Errors (GPREl states that
if ujk * 0, then IjIjk = 0
(Kang and Seneta, 1980; McDonald, 1985). That is, the errors ek,
ej of ordered variables are orthogonal.
4. The Exogenous Variable Rule (EVR) states that
if V is exogenous and Vj is endogenous, then <l>kj = 0 or,
k
equivalently, IjIjk = O. (See, for example, Fox, 1984; Bollen,
1989. Note that we do not here assume that the variables are
ordered. Note also that the components of e are not interpreted
as exogenous variables in the application of this rule.)
5. The Orthogonal Errors Rule (OERl states that
IjIjk = 0
unless both Vj and v are exogenous ( Fox, 1984; Land, 1969;
k
Bollen, 1989).
This list of six rules appears to cover the main
recommendations offered, authol~itatively, in the literature. In
principle one or more of these "rules" may be used by an
investigator to determine the location of the nondirected arcs
when a substantive hypothesis has been represented by a
specification of directed arcs. The problem is that it does not
seem obvious how to choose between them. McDonald (in prep.)
proves the following mathematical propositions, intended to
4
assist in an evaluation of these rules. The proofs follow fairly
readily from the simple algebra of model (1).
Proposition 1: The direct precedence rule is a sufficient
condition that a (recursive or nonrecursive) path model (without
unobservable variables with mUltiple indicators) is just
identified, if all nonzero ~jk' \l!jk are unconstrained.
Corollary: Under any stronger rule the model is
overidentified.
Proposition 2: In a recursive model, the general precedence
rules for variables and for errors imply each other.
Proposition 3: In a recursive model the orthogonal errors
rule implies the general precedence rules which imply the direct
precedence rule. The orthogonal errors rule implies the exogenous
variables rule. (Here "implies" can be read as "includes" and "is
at least as strong as".)
Note that we do not have a relationship of implication
between the general precedence rules and the exogenous variables
rule, or, more importantly, between the latter and the direct
precedence rule. By Proposition 1 the direct precedence rule
gives a weak sufficient condition for identifiability. In
contrast, it is well-known that the exogenous variables rule does
not.
3. Path models containing common factors
In the more recent literature the term "latent variable" has
been used, commonly without explicit definition, in a number of
distinct senses, including but not confined to its original sense
as a synonym for a latent trait or common factor. To avoid
confusion, we will avoid the term "latent variable" and confine
attention to cases where the random variables in the model are
(i) observable variables, (ii) error-terms, and (iii) latent
traits or common factors (including in particular "true scores")
defined by the weak principle of local independence (McDonald,
1981) and hence necessarily possessing more than one indicator.
We define a selection matrix J as a p x q matrix of zeros and
unities that selects a proper subset v from the components of v*,
q x 1, by
v = Jv*. (17)
We then have the general form of the Reticular Action Model as
v = J(I - B)-ie, (18)
yielding
Cov{v} = Lv = J(I-B)-l'¥(I-B) ,-lJ', (19)
where now B, '¥ are path coefficient matrices and error covariance
matrices of v*.
The mathematical statement that
y=All +E (20)
with
COV{E} Ily, diagonal, (21)
so that
Cov{y} AL~~A ' + Ily , (22)
where
