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Parameter Estimation and Model Construction
for Recursive Causal Models with

Unidimensional Measurement

presented by

David Wayne Gerbing

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Date July 27, 1979

 

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1“-

PARAMETER ESTIMATION AND MODEL CONSTRUCTION
FOR RECURSIVE CAUSAL MODELS WITH

UNIDIMENSIONAL MEASUREMENT

By

David Wayne Gerbing

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Psychology

1979

ABSTRACT
PARAMETER ESTIMATION AND MODEL CONSTRUCTION

FOR RECURSIVE CAUSAL MODELS WITH
UNIDIMENSIONAL MEASUREMENT

BY

David Wayne Gerbing

Two kinds of linear models often used in the social sciences are
measurement models and causal models. Measurement models specify the
relation of latent variables to observed variables and causal models
relate latent variables to each other so that their ordering in the
model specifies the outcome of an underlying causal process. The
purpose of this paper is to compare the utility of two distinct
estimation procedures which are embodied in two separate computer
programs: PACKAGE and LISREL. The comparison is limited to recursive
models with unidimensional measurement models.

LISREL is based on the more recently developed procedures of "full
information maximum likelihood". LISREL simultaneously estimates the
parameters of both the measurement and causal models. PACKAGE is based
on least squares techniques. The parameters of the measurement model
are estimated with centroid factor analysis with communalities in the
diagonal, and then, in a separate step, the estimated correlations
among the latent variables are subjected to an ordinary least squares
(OLS) path analysis.

The results of this paper show PACKAGE to be a superior technique.

David Wayne Gerbing

If the model is correctly specified, then both PACKAGE and LISREL
recover the underlying structure, although LISREL does so at a much
greater cost in computer time than does PACKAGE. If the model is
misspecified, then LISREL spreads the errors related to the misspeci-
fied equations throughout the entire system so that all of the
parameter estimates tend to be affected by the misspecification. In
particular, even if the measurement model is correctly specified, a
misspecification of the causal model precludes the correct recovery of
the correlations among the latent variables. 0n the other hand,
PACKAGE not only separates the estimation of the parameters of the two
models, it is based on single equation techniques which localize the
errors.

Some authors have claimed that the use of the LISREL first deri-
vatives for detecting misspecification is superior to the use of
residuals. Instead, the use of derivatives was shown to be misleading
in certain cases, and under none of the circumstances investigated in
this paper did the derivatives provide more information than the
residuals.

One of the claimed advantages for LISREL was the capability of
allowing for correlated disturbance terms and correlated measurement
errors. Yet counter-examples were constructed in which the use of
correlated errors was misleading. In particular, these examples
demonstrate that, contrary to the current literature, omitted variables
do not lead to correlated errors. False indication of correlated
errors can be produced by (a) ad-hoc composites and (b) missing paths.
In certain cases the covariance of "correlated disturbances" is exactly
the OLS residual for a misspecified model defined by the deletion of a

path.

Dedicated to my best friend

Monica Gerbing

ii

ACKNOWLEDGMENTS

I would first like to thank my parents, Gordon and Marilyn
Gerbing, and my grandparents, Alfred and Edna Bottorf, who have
encouraged me from the beginning to be intellectually curious. Not
only did they spend so many hours tutoring me from an early age, I
was always treated with respect and trusted with responsibility. I
would also like to thank those few teachers in high school and college
who were able to inspire, teach, and at the same time provide encour-
agement and understanding. For the past five years as a graduate
student, I would especially like to thank my committee members not
only for serving on the committee, but also for the additional support
and knowledge they have provided throughout graduate school

Charles Wrigley provided some helpful criticisms of the
dissertation--especially regarding the historical development of
factor analysis.

Neal Schmitt provided suggestions for revising particular
passages and also contributed to the overall organization of the
dissertation. He has also answered many of my questions and provided
encouragement and guidance over the last five years.

Raymond Frankmann, in addition to contributing to the disserta-
tion, has been an invaluable resource person since I arrived at
Michigan State. He has been the one person I could turn to when, as

was often the case, I needed help in negotiating through the maze of

iii

paperwork and bureaucratic procedures--or just learning my way around
campus. His help was always appreciated, but especially so when it
was most needed—-for this final certification.

Bill Schmidt has contributed greatly to my graduate education.
He has provided instruction over a wide variety of material that would
not otherwise have been available. Moreover, he was the source of
much of the material in this dissertation.

Jack Hunter was my chairman and the guiding force throughout
graduate school. He, more than anyone else, has contributed to my
career and he is the person who encouraged me to major in quantitative
psychology. I am grateful for the opportunity to work with him. Jack
is unique in that he is both a mathematician and at the same time a
substantive social scientist. He is also a friend who could provide
guidance and moral support whenever it was needed. It was his emphasis
on meaning and his ability to challenge the prevailing traditions that
formed the theme of this work. I look forward to continuing our
friendship and our professional relationship well beyond graduate
school.

My wife Monica is not only my best friend, but she has contri-
buted hundreds of hours in directly helping me survive graduate school.
She did all of the typing for this dissertation and she did it
flawlessly-~including the mathematical terminology and Greek letters.
She has also helped me in all other phases of life. I couldn't ask

for a better dea1--and I hope she couldn't either.

iv

LIST OF
LIST OF
‘ LIST OF
Chapter

I.

TABLE OF CONTENTS

TABLES O O O O O O O O O O O O O O C O
FIGURES O O O O O O O O O O O O O O O

APPENDICES . . . . . . .

INTRODUCTION . . . . . . . . . . . . . .

Causal Models . . . . . . . . . . . . . .
An example of a causal model . . . . . . . . .
Structural equations . . . . . . . . . . . .
The covariance structure of a causal model

Measurement Models . . . . . . . . . . . .
Latent variables . . . . . . . . . . . . .
The covariance structure of unidimensional measurement

madels O O O O O O O O O O O I O O 0

Evaluation of the Full Causal Model . . . . . . .
OLS and centroid factor multiple groups analysis . .
LISREL . . . . . . . . . . . . . . . .
The LISREL model . . . . . . . . . . . .
The LISREL covariance structure . . . . . . . .
The metric of the latent variables in a simultaneous
LISREL analysis . . . . . . . . . . . .
LISREL parameter estimation . . . . . . . . .
LISREL two-step analysis . . . . . . . .

Detecting Misspecification . . . . . . . . . .
The actual vs. the specified model . . . . . . .
The residual matrix (OLS and LISREL) . . . . . .
Confidence intervals about parameter estimates (OLS
and LISREL) . . . . . . . . . . . . . .
First derivatives (LISREL) . . . . . . . . . .
Likelihood ratio test (LISREL) . . . . . . . .
"Reliability" of the model (LISREL) . . . . . .
Miscellaneous oddities (LISREL) . . . . . . .

Page
viii

ix

U'lboNv—i

00‘

12
12
16
17
19

21
25
26

27
27
27

28
28
30
32
33

Chapter

II.

III.

IV.

OVERVIEW AND PROCEDURE . . . . . . . . . .
PACKAGE vs. LISREL . . . . . . . . . . . .

The Equivalence of MGRP and OLS with LISREL Given
Correct Models . . . . . . . . . . . .

Procedure . . . . . . . . . . . . .
TWO-STEP vs. ONE-STEP ESTIMATION: PACKAGE vs. LISREL .
Single Equation vs. Full Information Methods . .

An Example of a Misspecified Causal Model . . .
Analysis of the measurement model . . . . . . .
OLS path analysis . . . . . . . . . . . .
LISREL simultaneous analysis . . . . . . . . .
A comparison of OLS and LISREL . . . . . . . .
Analysis of a variety of models and specifications .

SPECIFICATION ERROR IN BOTH CAUSAL AND MEASUREMENT
MODELS : AD-HOC COWOSITES o o o o o o o o

The Pattern of Residuals from an Ad-hoc Composite
Misspecified as a Construct . . . . . .

A Re-interpretation of the Costner and Schoenberg
AtlaIYSis O O O O O O O O O O O O O O O

A More Complicated Model . . . . . . . . . .

The Use of Derivatives in Detecting Misspecification in

Ad-hoc Composites . . . . . . . . . . . .
A re-analysis of previous studies . . . . . . .
An extrapolation of the Sorbom procedure . . .

Summary . . . . . . . . . . . . . . . .

MISSING VARIABLES, MISSING PATHS, AND CORRELATED
ERROR S O O O O C O O I O O O O O I

The Effect of a Missing Variable on the Fit of the
Model . . . . . . . . . . . . . . . .

Previous assumptions . . . . . . . . . . . .

A test of the assumptions: Deletion of an endogenous
variable . . . . . . . . . . . . . .

A test of the assumptions: Deletion of an exogenous
variable . . . . . . . . . . . . . .

Correlated Errors in Practice: The Use of Derivatives
to Respecify a Model . . . . . . . . . .

vi

Page
34

34

36
38
42
42
45
45
46
47

51
51

54

56

61
64
68
68
7O

72
74
74
74

75

80

82

Chapter

Correlated Errors:

APPENDICES . .

LIST OF REFERENCE NOTES

LIST OF REFERENCES

Summary

vii

Page
86
88

101

102

LIST OF TABLES

Table Page
1 LISREL Notation . . . . . . . . . . . . . l8

2 Correct and Estimated Factor Correlations from OLS
and LISREL and the Residuals . . . . . . . . 48

3 Parameter Values and Estimates . . . . . . . . 49

and n 50

2 3 O O O
5 Comparison of PACKAGE vs. LISREL . . . . . . . 52

4 The Residuals of the Indicators of n

6 Residuals from the Endogenous Indicators in Figure 7 . 58
7 MGRP Residuals and Partial Correlations . . . . . 62

8 Correlations of the Endogenous Indicators with E
Partialled Out . . . . . . . . . . . . 63

9 The Second Order Correlations of the Endogenous
Indicators . . . . . . . . . . . . . . 64

10 LISREL Residuals and Derivatives . . . . . . . 66

11 Partial Correlations . . . . . . . . . . . 67

12 Second Order Correlations . . . . . . . . . . 67

13 Derivatives of the Measurement Errors of the Endogenous
Indicators . . . . . . . . . . . . . . 69

14 Residuals of the Misspecified Model of Figure 10 . . 79

15 Derivatives for the Model in Figure 6 Computed by
LISREL O O O O 0 O O O O O O O O O O 83

viii

Figure

10

11

12

13

14

LIST OF FIGURES

A path diagram . . . . . . . . . . . . .
A two-factor multiple groups measurement model

A flow chart of the model building process .

The LISREL partitioning of Z . . . . . .

Four actual causal models used in this study . . .

Actual model (a) from Figure 5 and an accompanying
misspecified model . . . . . . . . . . .

The actual and misspecified models from Costner and
Schoenberg (1973) with parameter estimates computed
by LISREL O I O C O O O O O O O O O 0

Actual model (a) from Figure 5 and a second
misspecified model . . . . . . . . . .

The actual and respecified models presented by
Duncan (1975) . . . . . . . . . . . .

Actual model (d) from Figure 5 and an accompanying
misspecified model . . . . . . . . .

Deletion of a block of exogenous variables which
affects only a single endogenous variable . . .

Actual model (d) from Figure 5 and misspecified and
respecified versions of this model . . . . .

Actual model (a) from Figure 5 and a third
misspecified model with correlated disturbance
terms 0 O O O O O O O O O O O I 0 0

An example of MGRP iterations . . . . . . . .

ix

Page

13
19

40

45

57

65

76

78

81

81

85

90

Appendix
A

B

LIST OF APPENDICES

COMMUNALITIES IN A MULTIPLE GROUPS ANALYSIS

ANALYTIC SOLUTION FOR THE PARAMETER ESTIMATES OF
MISSPECIFIED MODELS . . . . . . . .

COMPUTATIONAL DETAILS OF A SIMULTANEOUS LISREL
ANALYSIS 0 C 0 C O I C O C O O O O

RELATIVE SIZES OF THE INDICATOR RESIDUALS WITHIN AND

BETWEEN CONSTRUCTS IN AN AD-HOC COMPOSITE IF ALL
THE INDICATORS HAVE EQUAL FACTOR LOADINGS .

Page

88

91

95

100

CHAPTER I

INTRODUCTION

Causal Models

 

The primary goal of science is theory construction. Most theories
are descriptions of the causal processes among a set of variables. The
direct study of the underlying causal processes, i.e., the study of the
system dynamics, requires the construction of mathematical models of
these processes expressed in terms of differential or difference equa-
tions (e.g., Hunter & Cohen, Note 1; Hunter, Nicol & Gerbing, Note 2).
A model of dynamics allows one to construct a developmental sequence of
the variables in the system--a trajectory--which can be compared to the
actual behavior of the system.

Within the social sciences, however, the causal processes are
usually not studied directly. Instead a causal theory, most often
expressed qualitatively, is used to predict the relations among the
variables. The model is tested by observing these relations §f£g£_the
operation of the causal processes. Thus the distinction is drawn
between (a) the underlying causal processes and (b) the effect these
processes have on the relations between the system variables. And the
outcome of the process, not the process per se, is usually the object
of study within the social sciences.

The statistical methodology employed to test a causal theory by

studying the outcome of the causal processes is, in part, provided by

regression analysis. For example, if a model identifies variable X as
a causal antecedent of variable Y, then it follows that variables X and
Y are related. If this relation is linear, then (a) the variables
should be correlated, and (b) the slope parameter of the regression of
Y on X, which indicates the linear relation between Y and X with the
other predictor variables in the equation held constant, is interpreted
as the indicator of the causal impact X has on Y.

A "complete" system involves a set of variables such that many of
the variables are consequent to some variables and antecedent to other
variables. A causal theory which predicts such a network of relations
can be tested by the analysis of a set of simultaneous regression
equations. But the distinction should always be maintained between
this model of relations among the variables following the outcome of
the causal processes and the model of dynamics which is tested by the
outcome model, even if the process model exists only in the form of
verbal relations.

An example of a causal model. Consider a theory which predicts

 

that X1 causes X2 and that X2 causes X3 but X1 does not directly

influence X3. The implications of this theory for the relations among

the variables can be represented in either diagrammatic or equation
form. If the relations among the variables are linear, the equations

which define the model are:
X2 ’ lex1 + U2

X + U

X3 7 p32 2 3

The equivalent representation of the model in diagrammatic form appears

in Figure 1. X1 is the single exogenous variable of the system (i.e.,

no antecedents Specified) and X2 and X3 are the two endogenous

variables of the system. No constant term appears in the equations
since the variables are mean deviates. The error terms, also called

residuals or disturbances and denoted by U indicate that the speci-

i,
fication of the causal influences of the variables X2 and X3 is
incomplete. Each disturbance is also specified to be uncorrelated
with the predictor variables in the corresponding equation and with

the other disturbances. That is, the disturbances are specified as

random influences.

1’21

Figure l. A path diagram.

p32

The model illustrated above is called a recursive model. For some
authors (e.g., Duncan, 1975) a model is recursive if all of the causal
linkages or paths form a hierarchy, i.e., the paths flow so that a path
never leads back to the same variable from which the path began. Other
authors (e.g., Heise, 1975) require that a recursive model be both
hierarchical and that the disturbance terms across equations be inde-
pendent. In this paper, the latter definition will be used to aid the
distinction between hierarchical models with correlated disturbances
and recursive models.

Structural equations. The regression equations are called
structural equations and the regression weights are called structural
parameters or path coefficients. The diagram is called a path diagram.

The structural equations and/or the path diagram define a causal model

which is a specific interpretation of the underlying causal processes
described by the theory. Since the relations between the variables are
linear, the expected differences (denoted by "A") would be given by:

AX AX and AX

2 a p21 1 3 = p32sz
For example, if manipulation or natural processes changed the value of
X1 by one unit for all of the individuals in the population, then the

theory specifies that the mean difference in X2 would be:

sz = 921(1) = p21
and predicted mean difference in X3 would be:
Ax3 = p32AX2 = p32921
Thus the path coefficients indicated the amount of change that can be
expected in the system variables given a change in antecedent variables
following the operation of the underlying causal processes.

The distinction between structural equations and regression
equations which are not structural equations is the distinction between
theory and blind prediction. If the variables can be measured, any
variable can be regressed on any other variable or set of linearly
independent variables without consideration of causality. For example,
consider the situation in which two variables, Y2 and Y3, share a
common antecedent Y1 but are not causally related in any other way.

The variable Y3 can be mathematically expressed as a function of Y2
even though this relationship does not mirror an existing causal
relationship. From a purely mathematical perspective, the equation

Y3 = YY + U implies AY = YAYZ, but this derived relationship among

2 3

the changes in the two variables has no counterpart in physical

processes. That is, a manipulation of a unit change in Y2 independent

of Y will not cause a change in Y

1 3'

The covariance structure of a causal model: An example. The key

 

to the evaluation of causal models is the calculation of the covari-
ances among the variables predicted by the model. The central idea is
that the model imposes constraints on the covariances among the
variables and it is these constraints on which the test of the model
is based. For example, consider the causal model presented at the

beginning of this paper in which X1 causes X2 and X2 causes X3.

Multiplying both sides of the equation for X by X yields:

3 1

x3X1 = p32X2X1 + x1U3

Taking expected values,
E(X3X1) = p32E(X2X1) + E(X1U3)
Since the variables are mean deviates,
0(X3X1) = p320(X2X1) + 0(X1U3)
But 0(X1U3) = 0. And if the variables are standardized,

r31 = p32‘21

Finally, r since there is only a single predictor in the

p32 = 32

equation for X3, so

r31 = r32’21

Thus the model imposes a structure upon the correlations among the
variables. In this simple example it is this single constraint which
provides a test of the model by comparing the observed value of r31

with the predicted value E obtained by multiplying the observed

31

values of r32 and r21.

This test of the model that r

A

= r is equivalent to tests based

31 31

on partial correlation coefficients and multiple regression

coefficients. Since

r = r31 ' r32r21 r31 ' r31

31-2 2 a 2 a = 2 3 2 k
(1 ‘ r32) (1 ‘ r21) (1 ' r321) (1 ' r21)

the model implies r

 

 

31.2 = 0. If X3 is regressed on X1 and X2 and 831.2

is the corresponding multiple regression weight, then:

A

 

B = r31 ' r32r21 = r31 ’ r31
31-2 1 _ r 2 1 _ 2
21 r21
Thus, the model implies 831.2 = O. The variables X3 and X1 are related
since r31 # 0, but only through the intervening variable X2.

The general principle of constructing a model defined by regres-
aion equations and then using this model to derive the constraints
imposed by the model on the covariances among the variables has been
known for over 50 years. In 1921, Wright introduced "path analysis" as
a test of recursive causal models, though it was not until Simon's
(1957) work, followed by Duncan (1966) and Blalock (1964) that path
analysis began to be used by social scientists other than econometri-
cians. The computation of the predicted correlations of a causal model
is covered extensively in the literature on causal models. Standard
references are the texts by Heise (1975), Duncan (1975), Asher (1976),
and the article by Lewis-Beck (1974).

Measurement Models

Latent variables. Before the variables can be related by a set of

 

regression equations, they must be measured. The variables of
interest, the variables in the causal model, are called the latent
variables since they are not directly observed. The observed variables

are the indicators of the latent variables. The relation between the

indicators and the latent variables is specified by the measurement or
factor model. The measurement model is a set of simultaneous regres-
sion equations of the indicators regressed on the latent variables.

Each measure is an imperfect indicator of a latent variable
because of random response error and invalidity. The problem of
measurement error for the analysis of causal models is more serious
than simply the lack of precise measures of the latent variables. The
analysis of a causal model based on fallible data yields biased
parameter estimates (e.g., Wiley, 1973). For a regression equation
with a single predictor, measurement error in the independent variable
attenuates the slope coefficient in proportion to the reliability of
the independent variable. However, for multiple regression equations
the effect of measurement error of the independent variables on the
estimates of the regression coefficients is usually unpredictable.
Wiley (1973) presented an example of a two-predictor multiple
regression equation in which the presence of measurement error reversed
the magnitude of the sample parameter estimates from the values of the
population parameters. The problem of measurement error can be
countered by providing multiple indicators of each latent variable.
The use of multiple indicators improves the precision of the estimates
of the latent variables and allows reliability estimates of the
composite scores to be computed. Traditionally, these reliability
estimates are used to correct for attenuation the correlations among
the latent variables computed from the observed composites.

The covariance structure of unidimensional measurement models.
The testing of measurement models in which the measures are partitioned

into clusters such that the measures in each cluster are postulated to

be alternate indicators of only a single common latent variable is
called Spearman factor analysis or oblique multiple groups analysis
(Tryon, 1939; Holzinger, 1944). The original conceptualizations were
provided by Spearman in 1904. The computations of a multiple groups
analysis involve any factor analytic method by which a single factor
is extracted from each group. The computations provide estimates of
the factor loadings of each indicator on the group factors.

An extensive discussion of the construction and evaluation of
unidimensional measurement models, an example of a procedure called
confirmatory factor analysis, is provided by Hunter (Note 3) and Hunter
and Gerbing (Note 4). Measurement models are evaluated according to
the same principles which underlie the evaluations of causal models.
The basic idea is to derive the covariances among the observed
variables predicted by the model. The covariance structure of
unidimensional measurement models is given below.

As presented in Figure 2, let the observed variables X1 and X2 be
indicators of latent variable (or true score or factor) F and let Y be
an indicator of G. The errors of measurement are specified as random.

The curved double-headed arrow in Figure 2 represents a correlation

without the specification of causality.

9% O
6% a

Figure 2. A two-factor multiple groups measurement model.

In equation form,

X = 81F + e

l 1
X2 = 82F + e2
and Y = BYG + eY

where r(F,ei) = r(G,ei) = r(ei,ej) = 0.

The covariance structure of the indicators can be described by
considering (a) the covariances among indicators of different factors,
and (b) the covariances among indicators of the same factor. Consider

first the relation between X1 and Y. The model implies:

XlY = BIBYFG + BlFe1 + BYGeY + eleY

E(X1Y) = BIBYE(FG) + BlE(Fe1) + BYE(GeY) + E(e1eY)

Standardizing the observed and latent variables,

8 B

r11! 2 1 YrFG

where 81 and 82 are standardized regressions weights. Since

B=r

1 and BY = r

1F YG

by substitution,

rlY = rlrrrcrcr

Thus one test of the measurement model is the comparison of the

observed value of rl with the value predicted by the model rlY’ which

Y

is the product of the observed values of r rFG’ and r

1F, GY'

Hunter and Gerbing (Note 4) call the expression r1Y = rlFrFGrGY
the product rule for external consistency of items. By similar logic,

they also derive the product rule for external consistency of items and

10

factors, which is,

r19 = rlrrrc

If X1 and X2 are both indicators of the game factor F, an
equivalent test based on these covariance restrictions can be used to
test the unidimensional measurement model. The product rules imply
that the correlations of X1 and X2 with other items or other factors

are proportional to their respective factor loadings on their own group

factors. That is, for some other factor C,

These proportionality rules are equivalent to the tetrad difference
criterion, which is usually written as

r1Gr2F = rZGrlF 0r rerZF = rZleF
They were presented in the context of factor analysis by Spearman in
1907 and first illustrated in the context of data analysis by Burt
(1909) and Spearman (1914).

The stability of this proportionality of the correlations of two
indicators of the same factor across other indicators and/or factors is
indexed by the following formula, originally used as a measure of the
similarity of two factors.
k§1r(xixk) 11(ij k)

(2 n {r(XiXk)} 2)L5 (:12 {r<xjxk)} 2)

k=l

11

Hunter (1973) called this index a "similarity coefficient".
Conceptually, the "intercolumnar criterion" is an "unadjusted correla-
tion . . . a product-moment coefficient based on absolute deviations,
instead of on the deviations about the means" (Burt, 1940, p. 343). If
communalities are placed in the diagonal of the correlation matrix, the
value of ¢ ranges from -1 to 1 with ¢ equaling 1 or -1 if the two

items, X and Xj’ have correlations that are perfectly proportional.

i

The product rules for external consistency and the implied propor-
tionality rules specify the relations between indicators of a factor
with variables external to the factor. The constraint imposed on the
correlations between indicators of the same factor is called the rule
for internal consistency. The internal consistency product rule is a
special case of the external consistency product rule in which rFG = 1.
That is,

1:12 = rlFrFZ

where X1 and X2 are both indicators of F. This product rule is
Spearman's (1904) general factor equation and is a special case of
Thurstone's (1931) "fundamental theorem of factor analysis" for the
single factor solution.

Other authors have independently discovered these properties. For
example, Tryon called indicators which meet these constraints
"collinear" (Tryon & Bailey, 1970). J6reskog (1971) called the latent
variables "congeneric measures" and Burt (1976) called the latent
variables "point variables" if the indicators of the latent variables
satisfy these constraints. Kenny (1979) called the product rules for

internal and external consistency the rules for "homogeneity within

constructs" and "homogeneity between constructs".

12

Evaluation of the Full Causal Model

The analyses of the regression equations which define the measure-
ment and causal models are similar. The analysis begins with the
construction of the model, either a measurement model or a causal model
or both. Estimates of the parameters of the regression equations which
define the model are computed from the observed sample correlation
matrix. Given the model and the parameter estimates, the covariance
matrix among the variables implied by the model can be computed. The
comparison of implied and actual covariance matrices provides an
evaluation of the fit of the model to the data. Since the fit of the
model can usually be improved by constructing a revised model, the
previous steps are repeated until the investigator has obtained a good
fitting model or abandons the project. The steps involved in such an
analysis are illustrated in the flow chart in Figure 3.

Competing statistical procedures exist for computing parameter
estimates. Should the measurement and causal models be analyzed
separately or simultaneously? Should least squares or maximum
likelihood methods be used? Should single equation or full information
methods be used? The central issue of this paper is a comparison of
these competing techniques.

OLS and centroid factor multiple groups analysis. Hunter has
argued that testing the full causal model can be accomplished in two
separate steps: test the multiple indicator measurement model using a
multiple groups analysis, and then submit the estimated correlation
matrix between latent variables from the multiple groups analysis to a
path analysis. Examples of this procedure are found in Hunter, Hunter

and Lopis (in press) and Hunter, Gerbing, and Boster (Note 5).

13

 

CONSTRUCT
MODEL

 

 

 

 

 

ESTIMATE MODIFY
PARAMETERS MODEL

 

1+

 

 

 

 

 

 

J

 

EVALUATE
FIT

 

 

 

NO

INTERPRET YES SATls- NO ANDON
PARAMETER ‘7 FACTORY PROJECT?
ESTIMATES FIT?

YES

 

 

 

 

 

 

 

 

 

 

__.C.... )i

Figure 3. A flow chart of the model building process.

14

The initial computations of a multiple groups analysis are
presented in Holzinger (1944), Hunter (Note 3), and Hunter and Gerbing
(Note 4). The method consists of extracting a first centroid factor
from each cluster of indicators. The extraction of this group factor is
accomplished by computing the correlations of the indicators with the
factors and the correlations among the factors. If communalities are
not used in the analysis, then the first centroid factor is simply the
sum of the indicators which define the group, i.e.,

F=X1+X2+X3+...+Xn

where X X

1’ 2’ '°" Xn are the indicators of factor F. The multiple
groups analysis is based on the repeated application of the rule "the

covariance of a sum is the sum of the covariances" to the following

correlational formulas:

 

 

o(X ,F.)

_ ii
r<x1’Fj) ' o(Xi)o(Fj)
o(F ,F )

= 11
r(Fi’Fj) o(Fi)o(Fj>

The remaining step in the analysis is the computation of the communality
estimates which is accomplished with an iterative procedure outlined in
Hunter (Note 3) and Hunter and Gerbing (Note 4). The communality of
interest for each observed variable is the communality of the indicator
across the remaining indicators in the corresponding group. As Hunter
(Note 3) demonstrated, if these communalities are inserted in the
diagonal of the correlation matrix, then the observed correlation matrix
among the indicators is transformed to the covariance matrix of true

scores. Hunter (Note 3) also demonstrated, by example, that factoring

15

a covariance matrix of true scores implies that the computed factor
loadings and factor-factor covariances are corrected for attenuation
(Spearman, 1907) due to the measurement error in the composite (i.e.,
factor) scores. At the same time, the use of communalities also
eliminates the Spuriously large "part-whole" correlations between
indicator and factor, i.e., communalities eliminate the upward bias of
correlating an indicator with a composite of which it is a part.

However, if the indicators can not be measured without error,
communalities must be used in the multiple groups analysis if parameter
values are to be correctly estimated. The convergence of this iterative
process for these data is discussed in Appendix A.

The input to the multiple groups analysis is the correlation
matrix of the observed variables. The input to the path analysis is the
correlation matrix among the latent variables. Parameter estimates of
recursive models have traditionally been accomplished with ordinary
least squares (OLS). That is, the estimated parameter values are those
values that minimize the sum of the squared errors of the sample data
points about the sample regression surface for each equation in the
model.

Both centroid factor multiple groups analysis and OLS path

analysis are called single equation estimation procedures. That is, the

 

computations of these analyses are accomplished "equation by equation";
only the variances and covariances of the variables which appear in a
particular equation are used in the computations of the estimated para-
meters of that equation. The general equation for a multiple indicator
measurement model is

Xi = 11F + ei

16

since each indicator or observed variable is a function of only a single

factor plus measurement error. In a recursive model, the factors can be
ordered so that all variables causally antecedent to a given factor are
listed ahead of it. If the variables are so listed, then the general
equation for a recursive path analysis is

i—l

F=£PF+U
1 k=1 ik k 1

The computation of both the centroid factor multiple groups
analysis and the OLS path analysis may be accomplished with PACKAGE
(Hunter & Cohen, 1969). The use of the multiple groups subprogram is
explained by Hunter and Cohen (Note 6) and Hunter and Gerbing (Note 4).
The path analysis subprogram is explained by Hunter (Note 7).

LISREL. Based on the work of Lawley (1943) and Bock and Bargmann
(1966), J6reskog (e.g., 1967, 1978) has developed an alternative
analysis of causal and measurement models which differs from the OLS
with correction for attenuation strategy along two different dimensions.
J6reskog has also developed a computer program, LISREL, which contains
the computational algorithms (J6reskog & SSrbom, 1978). Both analytic
methods, OLS with correction for attenuation and LISREL were developed
to account for the biasing effects of measurement error. However,

LISREL simultaneously estimates the parameters of the measurement and

 

causal models. LISREL also is not restricted to testing multiple
indicator measurement models in which each indicator is an indicator of
only a single factor, i.e., multiple groups measurement models. Nor
does LISREL require the measurement errors to be uncorrelated. In terms
of the causal model, LISREL is not restricted to the analysis of

recursive path models. The causal model may be nonrecursive in that the

17

disturbances may be correlated and/or "feedback loops" may appear in
the model.

The complexity of the general LISREL model requires the use of
matrix algebra in its representation. Since J6reskog's parameteriza-
tion of the model includes an explicit distinction between exogenous
and endogenous variables, there is also a much greater notational
complexity. Many of these symbols are defined in Table 1. Other
symbols include ny for the covariance matrix of the exogenous and
endogenous observed variables and Z for the covariance matrix of all of
the indicators.

The LISREL model. Joreskog has parameterized the general measure-

 

ment model in separate equations for the exogenous latent variables and

a model for the endogenous latent variables.

y Ayn +_e and E(fl_€')‘-"O

pxl pxm mxl pxl

X

Ax; +6 and E(§§') = 0

qx1 qxn nxl qxl
The 1th row of the factor pattern matriceslly andllx corresponds to the
ith endogenous or exogenous indicator. The jth column of each of the
matrices refers to the jth endogenous or exogenous factor. Thus if the
scalar equations take the form of

y1 = Ky n + Ei or xi = Ax g + 61

i i

as is the case for the multiple indicator models with one indicator per
factor, then each rowofAyandAx will contain a single A with the
remainder of the elements in the row equal to zero.

The parameterization of the causal model is best understood by

beginning closer to the more traditional path analytic parameterization,

18

Table 1

LISREL Notation

 

Variables

Number
Variable of such Covariance
psymbol variables matrix

 

 

 

 

Observed x q 2
xx
Exogenous Unobserved 5 n ¢
Measurement error 6 q 96
Observed 2
y p yy
Unobserved n m C
Endogenous
Measurement error e p 08
Disturbance c m w
Parameters
Scalar Matrix Function
Ax
Ax A x 54x1
Measurement
A
model Y
A A i
y y E ——v-yi
F Y
Y
Causal g n
model Bji

19

N

F.=£P F +U,
1 k=1 ik k 1

where there are N latent variables, designated by F. In matrix form,

£= PE +9
le NxN le le

If a distinction is made between exogenous and endogenous factors, then
the model can be rewritten as

n=An+I§+£

mxl mxm mxl mxn nxl mxl
For recursive models, A is a lower triangular matrix with 0's down the
main diagonal and U) E E(5__c_') is a diagonal matrix with the disturbance
variances down the main diagonal.
However, J6reskog's parameterization of the causal model is based

on the following transformation of the above equation.

(I - A)fl = P §_+ C

B3=F§+£ andEgg')=0
where B E I - A. Thus B has 1's down the main diagonal and the path
coefficients are reversed in sign. For recursive models, B is still

lower triangular.

The LISREL covariance structure. Given the three equations which

 

define the model, it is now possible to derive the covariances among
the indicators predicted by the model. The derivation is based on the

partitioning of 2 shown in Figure 4.

 

I
I

Z = - - - -I -----
I
I

 

Figure 4. The LISREL partitioning of Z.

20

For example, to compute the implied covariances among the indicators of

the exogenous variables,

Xxx; E(xx') = E{(l\x_€_+_6_)(Ax§_+_6_)'}

MAX; g'AX') + meg 53') + E(§§'Ax') + E(§_6_')

=AXE(§§_')AX' +AXE(_§') + E(§§_')Ax' + E(_<_5_§)'

__ '
AX¢AX + 96

which is the usual expression for the correlation of the observed

variables expressed as a function of the factor pattern matrix, the

correlation matrix among the oblique factors, and the communalities.

Similar derivations lead to

X E Egyxf) =

yx E{(Ayn+_€></\x§+ 9'}

= AyE(fl_€_') A)"

and

yy: E(Xy) = E{(Ay_r_1_+_e_)(l\y3_+_§)}

A (3A ' + O
Y Y "5

But, the covariances among the endogenous factors, C55 E(fllf) , can be

decomposed according to the causal model. Since

B11 = P §_+'£,

So
, . —1 -1 -1 -1 ,
Egg) =E{(B r§_+3 3;)(3 £§_+B 5)}
= 3'1 r ¢B"1 + B.1 ws"1
Similarly,
E(3§') = B’lrcp
So

2 = AB'1r¢A'
yx y x

21

1

-1 - -1 -1
=11 B I" B' +3 3' A'+0
yc <1 1 )y _E

2
YY
And, from before,

2: =A¢A'+o
XX x X —

5
Thus the predicted covariance matrix of the indicators, 2, has been

parameterized in terms of the estimators of the four covariance

A A A A

matrices, ¢, w, GE, 96’ and the four parameter matrices, P, B, Ay,

A

A .
x

In the two-step strategy of OLS preceded by centroid factor
analysis, (a) the correlations among the indicators are parameterized
only in terms of the factor pattern, the factor correlations, and the
communalities, and (b) the covariances among the factor correlations
are parameterized only in terms of the causal model. In contrast,
J6reskog simultaneously parameterizes the covariances among the
indicators in terms of both the measurement and causal models. The
complexity of the algebra varies greatly across the two strategies, but
the key ideas remain the same: (a) construct a model, (b) derive the
implications of this model for the covariance structure of the
variables, i.e., compute E, and (c) test the model by comparing E to
the observed sample covariance matrix S.

The metric of the latent variables in a simultaneous LISREL
analysis. A potential source of confusion in the interpretation of the
simultaneous LISREL analysis is the metric of the latent variables. If
the variance of each latent variable is not set by the user, the model
is underidentified. The classic solution to this problem is to set the
variance of the factor to 1, i.e., to standardize the factors. Instead,

Jbreskog and Sfirbom (1978) chose to fix the factor loading of one of

22

the indicators of each latent variable at 1.00. Jfireskog does not
clearly indicate the effect of this specification on the parameter
estimates other than that "the scales for [the latent variables] have
been chosen to be the same as for [the corresponding indicators whose
factor loadings were set at 1.0]" (1978, p. 468). The phrase "the
scales [areJ the same as" is often interpreted to imply that the metric
of the latent variable is set at the metric of the corresponding indi-
cator with a fixed factor loading of 1.0, though this is not true.

The implications of this practice can be derived by beginning with

the basic relationship
5
Y=An+e or o§=A20§+02€ or 02=-—L——

It follows that, for the specific case a: = oi = 1 in which the true

value of A is used,

_ 2 2 2_ _ 2
l-ATRUE+OE 0" Ge’l ATRUE

That is, the value of 0% computed by LISREL is determined by the actual
value of A if the value of 0% is not fixed a priori by the user. But
if A is fixed at 1.0 while oi remains free,

2 2
o - 08 1 - (l - A )
_JL_____ TRUE = A2

2 1 TRUE

 

Q
II
II

Fixing the value of a factor loading at 1.0 determines the
variance of the corresponding latent variable for computational
purposes, but this variance is an arbitrary and meaningless metric
which is neither the variance of the observed variable nor is it the
actual variance of the latent variable. Consequently, the computed
covariance matrices of the latent variables and of the latent and

observed variables are expressed in a metric which does not correspond

23

to the parameters of the model used to generate the data.

Moreover, the maximum likelihood estimates (and the corresponding
standard errors) of the regression parameters of the measurement model
do not correspond to the parameters of the model used to generate the

data because the relation

2
o = A302 + 06
Y1 1

must be satisfied. If the factor loading A of the first indicator of

n is fixed at 1.0, then obviously A1(TRUE) # A1(LISREL)° But consider

the ith indicator. The value of o; = 1 is given. LISREL computes the
2 2
values of on and 0E as
i

2 = A2
on 1(TRUE)

2
Oei ’ 1 " A1(TRUE)
Thus, 2 2
Cy ' OE
A1(LISREL) = i 2 1

2
1 ‘ (1 " Ai(TRUE))
12
1(TRUE)

 

2
iigTRUE)

A2

1(TRUE)
For example, if the true or actual factor loadings of the indicators of
a given latent variable were .80, .60, and .40, and if the factor
loading of the first variable was set at 1.0, then the computed factor
loadings would be 1.00, .75, and .50. The sets of factor loadings are

equivalent except for the constant of proportionality, .80.

24

If the variance of the latent variables is 1.0, the covariance
matrices and the measurement model parameters happen to correspond to
the correct values in the standardized solution. Unfortunately, LISREL
prints only the factor pattern matrix and the covariance matrix of the
endogenous variables (in addition to the causal parameter matrices) in
the standardized solution. If the full factor loading matrix, the full
matrix of covariances among latent variables, or the covariances among
latent and observed variances is desired in which the latent variables
are expressed in variances chosen by the user, then the following
procedure is suggested.

Use an arbitrary metric such as that obtained with fixing an
indicator of each latent variable at 1.0. Take the value of 0:

provided by LISREL and substitute it into the following formula in

which a; is given by the data and the value of oi is chosen by the
user.
2 2
o - OE
A=l___2_
0‘n

The LISREL program can then be rerun with a single factor loading for
each latent variable set at the corresponding value of A as determined
by the formula. If the metric of the observed variable is to match the

metric of the latent variable, the formula simplifies to

If the observed variables have variance 1.0 and the latent variables

are chosen to have variance 1.0 also, then

A = 1 - oe

25

which is just the value of A computed by LISREL in the standardized
solution.

LISRELgparameter estimation. LISREL estimation is based on a

 

method called "full information maximum likelihood" (FIML). Instead of
estimating the value of each parameter from a different subset of the
covariances, LISREL uses all of the data in the computation of each
parameter estimate (unless particular restrictions are placed on the
model).

LISREL assumes that the latent variables and the measurement
errors are generated by a multivariate normal distribution with a mean
vector_g and covariance matrix X. Wishart (1928) has shown that the
observed sample covariance matrix S has what is now called a Wishart
density, which is expressed in terms of the covariance matrix 2 and the
sample size n. The parameter estimates of LISREL are given by the set
of values which maximizes the value of the corresponding multivariate
likelihood function. The effective log likelihood function of the
Wishart density, i.e., the logarithm of the density function with
constant terms deleted, is

log L(Z) = -1/2 n [log IZI + tr(Z-IS)]
In practice, however, LISREL minimizes a transformation of log L called
F where

F(Z) = log IX] + tr(SZ-1) - log ISI - (p + q)
Since log IS] and p and q are constants for a given 8, and since the
sign of F is the opposite of the sign of log L, maximizing log L is
equivalent to minimizing F. In terms of computations, J6reskog has
chosen to minimize F since this transformation of log L is directly

related to the likelihood ratio test of the fit of the model--a topic

26

which will be examined in more detail later.

F is minimized by computing the partial first derivatives of each
of the free parameters with respect to F, setting the resulting set of
simultaneous equations to zero, and solving for the values of the para-
meters. Unfortunately, an analytic solution is not available for most
models, so the function is minimized with an iterative procedure from
numerical analysis. The method of Fletcher and Powell (1963) is "a
rapidly converging iterative procedure for minimizing a function of
several variables when analytical expressions for the first-order
derivatives are available" (Jfireskog, 1969, p. 187). The procedure is
based on an approximation of the inverse of the matrix of second
derivatives. As the program iterates, the approximation improves until
it converges to the best approximation of the inverse at the minimum of
F. Details of the method as applied to LISREL are provided by Gruvaeus
and Jareskog (Note 8). As with most iterative procedures, there is the
problem of local minima: "If there are several minima of F there is
no guarantee that the method will converge to the absolute minimum"
(Jareskog & sorbom, 1978).

LISREL two-step analysis. Although most studies in the literature

 

have used LISREL to simultaneously estimate the parameters of the
causal and measurement models and provide the test of fit for the model
as a whole, LISREL could also be used in a two-step analysis (e.g.,
Kohn & Schooler, 1978). LISREL could be used instead of centroid
factor multiple groups analysis to perform a FIML multiple groups
analysis to test the measurement model and estimate the factor corre-
lations. LISREL could then be used instead of OLS to do the path

analysis on the estimated factor correlations. However, for a

27

recursive model with independent disturbances, Land (1973) has demon—
strated that "FIML reduces to equation by equation least squares
regression provided that there are no cross-equation constraints on the
coefficients" (p. 42).

Detecting,Misspecification

 

The actual vs. the specified model. The goal of the investigator

 

is to build a properly specified model. Ordinarily, the interpretation
of the parameter estimates would not be of interest unless the model
was a valid representation of the outcome of the underlying causal
processes. The problem is that typically the initial model specified
by the investigator does not fit the data as well as a competing model
composed from the same set of variables. That is, the initial

specified model is usually not the actual model of the relations

 

 

between the variables which exist following the operation of the under-
lying causal processes.

The most important task facing the researcher after a model has
been constructed is the detecting and subsequent correction of misspec-
ification. Thus the first two questions the researcher usually asks
after the first computer run are (a) how badly did the model fit? and
(b) how can the model be revised to improve the fit? Both OLS and
LISREL offer several indices of the fit of both_the model as a whole
and for specific equations of the model.

The residual matrix (OLS and LISREL). The residual matrix is
defined by the difference of the predicted covariances implied by the
equations of the model and the observed covariances, i.e., E - S in the
notation of LISREL. The OLS program contained within PACKAGE (i.e.,

PATHPAC) prints the residual matrix and the sum of the squared

28

residuals which indicates the fit of the model as a whole. If desired,
this sum may be tested for significance. Although LISREL does not
print the sum of the squared residuals, the residual matrix is printed
from which the sum of squared residuals could be computed.

The residual matrix is also used to pinpoint specification errors.
Residuals which exceed the boundaries of sampling error indicate lack
of fit. In terms of a causal model, large residuals may indicate the
addition or deletion of a path or a reordering of the variables. The
residuals of a unidimensional measurement model may indicate the
necessity of repartitioning the variables into a new set of clusters.
The residuals from a LISREL simultaneous analysis of measurement and
causal models are less interpretable since a large residual may imply
misspecification in either or both models. An assessment of LISREL in
this regard is one of the goals of this study.

Confidence intervals about parameter estimates (OLS and LISREL).
In path models, a proposed path may be deleted if the corresponding
structural parameter is small. The usual standard error for a regres-
sion weight underestimates the correct standard error for an OLS
estimated parameter since the factor correlations have been implicitly
corrected for attenuation. The usual test for the significance of a
correlation coefficient may be applied to the factor loadings
uncorrected for attenuation. LISREL prints the asymptotic standard
error of each parameter estimate which can be used to construct a
confidence interval about the parameter estimate. Since these standard
errors are based on the large sample properties of maximum likelihood
estimators, their validity for small samples is in question.

First derivatives (LISREL). LISREL prints the values of the first

 

29

partial derivatives of F with respect to all the parameters--fixed,
free or constrained. The purported usefulness of these derivatives for
detecting a specific source of misspecification is based on their
relation to the iterative procedure used in the search for the minimum
of F(E). At convergence, the values of the partial derivatives for the
free parameters will be zero by construction. The derivatives of the
free parameters should be approximately zero according to the model
assumptions. If LISREL does not converge, the non-zero derivatives
should indicate the misspecified free and constrained parameters.

The test for locating a potential misspecification for a solution
which has converged is based on the values of the derivatives of the
fixed parameters. If the model does not fit the data well, some of the
derivatives of the parameters fixed at constant values should differ
from zero. LISREL prints the first derivatives of all the elements in
the eight matrices which define the parameterization of 2. Even if the
model is specified as a recursive model with uncorrelated measurement
errors, LISREL still prints the values of the complete B matrix, i.e.,
the derivatives of all the possible paths including those in the
"opposite" direction, and the values of the complete w matrix and the
O8 and 06 matrices including the covariances of the disturbances and
the covariances of the measurement errors respectively.

SBrbom (1975) recommends the use of the first derivatives of the
parameters over the use of the residual matrix (i.e., E - S) in
detecting misspecification. He advises that "we should relax the . . .
restriction for that element which gives the largest decrease in F(E)"
(p. 143). That element is the element with the largest partial deriva—

tive. However, the use of the partial derivatives for detecting

30

misspecification has not been adequately studied nor has the usefulness
of this information been compared to the usefulness of the information
supplied by the residual matrix.

Likelihood ratio test (LISREL). The likelihood ratio test can be

 

used as a test of the fit of the model as a whole. The null hypothesis
H0 is that the observed covariances S conform to the constraints
imposed on the covariances of the observed variables by the model,
i.e., that S = E to within sampling error. The alternative hypothesis
H1 is that Z is not described by E, i.e., that "Z is any positive
definite matrix" (Jareskog & Sfirbom, 1978, p. 14) with no restrictions
placed on its structure.

The test is based on the comparison of the value of the likelihood
function. Since 3 is the unconstrained maximum likelihood estimator of

log L, log L is obtained by the substitution of S for 2. The value of

1

Log L becomes the value of log L with Z substituted for 2. The values

0
of log L evaluated at S and E are compared by the statistical test

based on the likelihood ratio or its logarithm, Ll — L0. That is,
under H0,
2 Lo
x ~—-2 log'i—- with df = %(p + q)(p + q + l) - t
1

where t is the number of independent parameters to be estimated.
Since

< log L unless E - S = 0

log L 1

0
the question addressed by the likelihood ratio test is, how large a
discrepancy between the observed S and the predicted E is required for
a given sample size to indicate that the model which was used to

generate Z is false?

31

The x2 statistic (Long, 1976) can be expressed as

Lo

-2 log —-
L
1

-2 log L + 2 log L

0 1

n[log IS] + tr(SE—1) - log‘SI- (p + q)]

n F(E)

In order to directly obtain the X2 value, Jareskog has chosen to find
the value of the parameters which maximize the value of the likelihood
function by minimizing F (since F(§) and log L differ only by a
constant).

Jareskog warns that "the values of x2 should be interpreted very
cautiously because of the sensitivity of x2 to various model assump-
tions such as linearity, additivity, multinormality, etc., but also for
other reasons" (1978, p. 448). The primary "other reason" is the
sensitivity of any significance test to sample size. No model is
perfect, so any model will be rejected by the x2 test given sufficient
sample size.

A potentially more useful application of the x2 statistic is the
test of specific parameters by comparing the x2 values for nested
models. A model M1 is nested within the more general model M2 if M1 is
defined by "constraining one or more of the free parameters in M2 to be
fixed" (Long, 1976, p. 170). Under the null hypothesis that the para-
meter is zero, the difference in the x2 values of the two models is
also distributed approximately as x2 with one degree of freedom for
large sample sizes. J6reskog (1978) does not discuss the formal
comparison of values in terms of significance tests. He suggests that
a model may be "relaxed" by introducing more constraints. "If the drop

in x2 is large compared to the difference in degrees of freedom, this

32

is an indication that the change made in the model represents a real
improvement. If . . . the drop in x2 is close to the difference in
number of degrees of freedom, this is an indication that the improve-
ment in fit is obtained by capitalizing on chance and the added para-
meters may not have any real significance or meaning" (p. 448).

"Reliability" of the model (LISREL). Tucker and Lewis (1973) have

 

noted some of the problems with the x2 statistic as a test of fit of
the model and have proposed an alternative indicator of global fit
which was originally developed for maximum likelihood confirmatory

factor analysis.

"amount of covariation explained by
C0 - C9 the proposed model"

0 — C ' E(C ) _ "total amount of covariation available
0 q n
to be explained

 

C0 is the sum of squares of the off-diagonal elements of the sample
covariance matrix divided by the degrees of freedome-the number of
unique elements in the matrix, (n)(n + 1)/2 where n is the number of
variables. Cq is the sum of squared covariances not explained by the
model and is approximately equal to the ratio of sum of squares of

the partial correlations of the variables with the group factors
partialed out to the degrees of freedom of the model--the same degrees
of freedom for the x2 test of fit. E(Cq) is the expected value of the
sum of squared covariances not explained by the model.

This "reliability" coefficient indicates the extent that a
measurement model explains or accounts for the covariation among the
observed variables. Burt (1973) notes that "Tucker and Lewis's
statistic is sensitive to sample size in its computation of the

expected value of explained covariance (sampling variability), but it

is much less subject to large sample sizes than is the likelihood ratio

33

since it focuses on covariation rather than on total variation

(p. 148). The sampling distribution of the coefficient is unknown, but
Tucker and Lewis (1973) indicate that "any accepted solution should
have a high coefficient of reliability" (p. 9) which, from their
examples, apparently means at least somewhere about .9.

Tucker and Lewis (1973) also noted that the sum of the correla-
tions among the observed variables with the factors partialed out is
approximately equal to the value of the function F minimized by LISREL.
Since the value of F(2) at its minimum given the model and the observed
covariances is simply the obtained X2 statistic multiplied by the
sample size, Burt (1973) recommends using this "reliability" coeffi-
cient for the more general simultaneous analysis of the causal and
measurement models.

Miscellaneous oddities (LISREL). Anecdotal evidence gathered from
previous experience with LISREL indicates that certain misspecified
models never converge, even after hundreds of (expensive) iterations.
Concomitant with the lack of convergence is the existence of some
"strange" parameter values which continue to become stranger as the
program continues to iterate. Also, as the program iterates, the
residual matrix can become almost completely flat at zero even though
the program never converges and some of the parameter estimates are

nonsense values.

CHAPTER II

OVERVIEW AND PROCEDURE

PACKAGE vs. LISREL

 

There are at least three competing strategies for the analysis of
fully recursive causal models: (a) the PACKAGE analysis, i.e., a
centroid factor multiple groups analysis with communalities followed by
an OLS path analysis, (b) the LISREL analysis, a FIML simultaneous
analysis of the measurement and causal models, and (c) a separate FIML
confirmatory factor analysis (Jareskog, 1966, 1967, 1979), followed by
a FIML analysis of the factor correlations to test the causal model.
However, Land (1973) has noted that in a recursive model, the FIML
analysis of the factor correlations is identical to OLS. Thus the only
difference between methods (a) and (c) is in the method of factor
analysis for the measurement model. The contrast of greatest interest
is between (b), the one—step LISREL procedure, and (a), the two-step
analysis. That is, one main objective of this paper is to assess the
utility of analyzing the measurement and causal models separately.

Hunter's strategy of separately analyzing the measurement and
causal models has not received much attention. Some social scientists
still seem content with the more traditional OLS estimation procedure
without considering the need for good measurement. Those who recognize
the biasing effect of measurement error on the parameter estimates of

the causal model have used the one-step LISREL analysis. Jareskog has

34

35

never suggested the possibility of a separate analysis in his writings
even though he has used LISREL for confirmatory factor analyses (e.g.,
J6reskog & Sfirbom, 1978) in other contexts. Duncan (1975) has written
chapters on specification error, measurement error, and multiple indi-
cator models (which include the biasing effects of measurement error on
OLS estimators), but the possibility of preceding OLS with correction
for attenuation is not discussed. Duncan even concludes that "the
contribution of multiple indicators should not be exaggerated" (1975,
p. 137) in reference to some of the problems involved in estimating the
parameters of both models simultaneously. Burt (1976) identified some
of the undesirable properties of the effects of misspecification of the
measurement model on the estimation of the parameters of the causal
model, but he never considered the possibility of a separate analysis.

Does the additional cost, the potential problems with local minima
characteristic of such iterative procedures, and the potential sensi—
tivity to misspecification justify the use of LISREL over the strate-
gies embodied in PACKAGE? In what situations do both approaches do
equally well and in what situations do the approaches give poor
results, or worse yet, results which look good but are, in fact, wrong?
Which combination of strategies best leads the investigator to uncover
the original structure with the most accurate set of parameter esti-
mates given the inevitable sampling error and initial misspecification
of the causal and measurement models?

The analysis strategies are evaluated in terms of both the
validity of the statistical estimates and the usefulness of the program
in providing information for the construction of revised, better-

fitting models. Parameter estimates are usually not interpretable

36

unless the model fits the data. But if the model is misspecified, what
is important is not the validity of the parameter estimates, but the
information provided on how to correctly specify the model. It is
precisely this kind of information which rarely appears in the
literature.

What patterns of errors are associated with various specification
errors such as variable deletion? Which set of misspecification
indices allow one to most accurately pinpoint specification error: OLS
and centroid factor multiple group residuals, LISREL simultaneous
analysis residuals, or LISREL first derivatives? Which is the better
indicator of the overall fit of the model: sum of the squared OLS
residuals, the x2 test of fit, or the "reliability" of the model?

The issue is this: One reason for the recent popularity of LISREL
is its ability to account for the influence of measurement error, to
analyze nonrecursive models, and to allow for the possibility of
specifying constraints on the equality of parameter estimates.

However, if_measurement error is accounted for in an independent
confirmatory factor analysis, if_only recursive models are considered,
and if_no constraints are placed on the parameter equalities, is there
still an advantage to using LISREL? And how can either approach,
LISREL or OLS, best be used to recover underlying structure?

The Equivalence of MGRP and OLS with LISREL Given Correct Models

If the specified model is correct, then the OLS preceded by
correction for attenuation strategy, the one-step LISREL analysis, and
the two-step LISREL analysis all provide correct parameter eatimates
and indicate a perfect fit of the model. The primary difference

between the approaches is cost. The total execution time in seconds is

37

approximately one second for the PACKAGE subprograms MGRP and PARTIAL
for the analysis of the measurement model and PATH for the OLS
parameter estimates of the causal model. For the simultaneous LISREL
analyses, all free parameters were initialized at 0.6 except the

causal parameters relating the latent endogenous variables which were
initialized at -0.5. For the LISREL confirmatory factor analyses, the
start values for the factor loadings, error variances and factor corre—
lations were arbitrarily set at 0.5, 0.8, and 0.3 respectively. Either
of the LISREL analyses used 64 seconds of execution time for a 18 x 18
correlation matrix. Thus for models of the size considered in this
paper, LISREL is 64 times as expensive as PACKAGE. For larger models,
the difference increases more than proportionately.

The MGRP analysis is equivalent to a LISREL confirmatory factor
analysis when the measurement model is defined by the partitioning of
the observed variables into mutually exclusive clusters. This is some-
what surprising in view of the attention the maximum likelihood
approach has recently received. For example, a chapter on confirmatory
factor analysis in a factor analysis text by Mulaik (1972) is a chapter
on Jdreskog's work. The centroid factor analysis multiple groups
alternative is not mentioned. Mbreover, in an early section of the
text on the history of factor analysis, Spearman (1904) is given credit
for developing "the first common-factor-analysis model" (p. 6), but the
section closes with "more recently Bock and Bargmann (1966) and
J5reskog (1969) considered hypothesis testing from the point of view of
fitting a hypothetical model to the data . . . . The author expects, as

a consequence, factor analysis will be more fruitfully used in the

future in the development of structural theories in psychology and in

 

38

other areas of study as well" (p. 10). What Mulaik (1972) has
perceived as linear development is in actuality circular! Finally,
Jareskog (1966, 1967, 1969, 1970, 1971, 1978) has never mentioned the
multiple groups method. He has even introduced the terms "congeneric
measurement" and "the hypothesis of congenerism" (1971, p. 132) to
describe what traditionally has been called "unidimensional measure-
ment" or "the hypothesis of unidimensionality" which has been tested
empirically with Spearman (1904) factor analysis using the centroid
method. There is no necessary reason for tying the conceptual approach
of a confirmatory factor analysis to the computational method of FIML.

The critical differences between OLS and LISREL emerge if the
model is not correctly specified. The two methods will be shown to
diverge widely in response to such errors. In summary, the OLS and
multiple groups method localizes errors whereas LISREL spreads the
error over the entire matrix. Thus LISREL is much harder to use in
generating a more appropriate revised model.

The focus of this paper is on specification error, so sampling
error is ignored throughout. However computations done along these
lines suggest that sampling error adds further problems for LISREL
since it tends to capitalize more on chance than do the OLS methods.
Procedure

It seems reasonable to expect that some consistent patterns can be
captured and described if either PACKAGE or LISREL are valid approaches
for data analysis and theory construction. The chief restrictions in
this paper are that only recursive, causal models and unidimensional
measurement models free from correlated errors of measurement are

considered. All variables, latent and observed, are standardized.

39

(1) "Arbitrarily" construct a model. Begin with a full model
which is complete with all parameter values in both the causal and
measurement models. Four causal models are illustrated in Figure 5.
If necessary, the regression coefficients can be varied in either the
causal or the measurement models, but in this paper, all latent
variables will have three indicators with the following factor
loadings: .8, .6, and .4.

(2) Compute 2. Given the rules of path analysis, the causal and
measurement models and the parameter values of both models imply the
correlations among the observed variables. The correlations among the
latent variables are also computed as a prerequisite to computing the
correlations among the observed variables.

(3) Model misspecification for observed data. Three kinds of
misspecifications are possible, depending on what part of the model, if
any, is misspecified.

(a) Misspecify causal model only.
(b) Misspecify measurement model only.
(c) Misspecify both measurement and causal models.

(4) Estimategparameters and evaluate fit. Estimate parameters

 

with:
(a) a LISREL simultaneous measurement and causal analysis;
(b) separate measurement and causal analyses:
(i) a PACKAGE analysis with centroid factor multiple
groups followed by OLS path analysis.
(ii) a LISREL analysis with FIML multiple groups
analysis followed by a OLS path analysis with

LISREL and/or a FIML path analysis.

40

 

 

 

 

 

 

 

 

 

Figure 5. Four actual causal models used in this study.

41

The fit of the model can be evaluated with Z - S, the x2 test of fit,
and the "reliability" coefficient.

(5) Revise the model. Describe how the model would be modified

 

given the output of the programs and compare the revision to the

correct model.

CHAPTER III

TWO-STEP vs. ONE-STEP ESTIMATION: PACKAGE vs. LISREL

Single Equation vs. Full Information Methods

 

Since a single equation method such as MGRP or OLS estimates the
parameters of each equation separately, the effect of the misspecifi—
cation of a single equation or just a few equations would be confined
to those equations. The source of the misspecification should thus be
relatively easy to locate. Errors in estimating one parameter are
somewhat independent of errors in estimating parameters in other
equations. However, the use of a full information procedure such as
LISREL implies that the effects of a misspecification in one part of
the model will be absorbed by the entire set of computations so that
the errors in different parameters may be intimately interwoven. A
full information procedure assumes that the entire specified model is
the correctly specified model and simultaneously computes all the para-
meter estimates gizgp_this assumption. If the model is misspecified,
the diffusion of the specification error throughout the model may
hinder the location of the source of misspecification. WOrse yet, a
detected misspecification may appear in the wrong place. The use of
LISREL for the simultaneous estimation of both measurement and causal
models would presumably exacerbate these potential problems since
misspecification in the measurement model would influence the compu-

tations of the causal model and vice versa.

42

43

Burt (1976) is one of the few authors who has studied the effects
of misspecification and the resulting consequences for the interpre-
tation of the parameter estimates computed by LISREL. He is particu-
larly concerned with the effects of misspecified measurement models on
the estimated causal parameters. He cites problems in interpretation
that result from the possible confusion of two potential sources of
"empirical meaning" of a latent variable or factor. The empirical
meaning of a factor is the meaning assigned to that variable in terms
of its relations to the observed variables. Since each parameter
estimate in a full information technique is based on all of the
observed covariances, empirical meaning is potentially assigned to an
unobserved variable both in terms of its own indicators ("epistemic
criteria") gpd the indicators of the remaining unobserved variables
("structural criteria"). "Interpretational confounding occurs when an
individual assumes that an unobserved variable is assigned empirical
meaning in terms of epistemic criteria when in fact it is assigned
empirical meaning in terms of structural criteria" (Burt, 1976, p. 14).

In computational language, what Burt has stated is that the
correlations of a given factor with other variables are not only based
on the correlations of its indicators with those other variables (as in
multiple groups), but depend also on the correlations between the other
variables. In multiple groups analysis, the estimated correlation
between factors A and B depends only on the observed correlations
between the indicators of A and B. But in LISREL, the estimated corre-
lation between A and B also depends on the correlations between
indicators of A and C, of B and C, and even C and D! This problem is

compounded in simultaneous LISREL by dependence on the postulated path

44

model.

The problem results from the characteristic of a FIML method to
force the parameter estimates to maximize the fit of the entire model
to the data. If a unidimensional measurement model is specified, then
the program will attempt to satisfy the product rules of internal and
external consistency in the computation of the parameter estimates.

But if the measurement model is incorrectly specified, the lack of

 

external consistency in the observed correlations implies that the
misspecified indicators will differentially correlate with other
factors and the indicators of those factors (e.g., Hunter, Gerbing &
Boster, Note 5). The estimated correlations of the misspecified indi-
cators vary as a function of the strength and type of causal relations
in the particular model.

Consider the problem opposite to that considered by Burt (1976),
i.e., a perfectly specified measurement model but a misspecified path
model. How might errors in the causal model affect the parameter
estimates of the measurement model? Two analytic strategies will be
compared, a one-step and a two-step analysis:

(a) a LISREL simultaneous analysis of the indicator correlations.

(b) an OLS path analysis of the estimated factor correlations

from a multiple groups factor analysis or from a LISREL
confirmatory factor analysis.
The comparison is evaluated in terms of the accuracy of the estimated
parameters and the usefulness of the information provided for detecting
the misspecification in the causal model. The strategy is to examine a
particular model in detail and then to determine how well the results

from the comparison applied to a particular model generalize to a

45

variety of models and misspecifications.
An Example of a Misspecified Causal Model

Consider the following causal model and the associated misspeci-
fied causal model defined by the deletion of the path from n2 to n3.
These models appear in Figure 6. The misspecified model was subjected

to MGRP, LISREL CFA, OLS, and a simultaneous LISREL analysis.

 

 

Misspecified model

Figure 6. Actual model (a) from Figure 5 and an accompanying
misspecified model.
The parameters of the causal model in Figure 6 were set at

Y = .30, 821 = B = .35, and B .40. The four variables of the

31 32 =
model were given three indicators apiece with factor loadings of .80,
.60, and .40. Using the product rules for internal and external
consistency and the implied correlations among the factors, the 2
matrix was generated for the actual model and analyzed under the

constraints of the misspecified model.

Analysis of the measurement model. Since the measurement model

 

46

was perfectly specified, both the multiple groups confirmatory factor
analysis with the PACKAGE subprogram MGRP and the full information
maximum likelihood confirmatory factor analysis with LISREL recovered
the factor loadings and factor correlations perfectly. Thus the OLS
path analysis was performed on the actual factor correlations.

OLS path analysis. As calculated in Appendix B, the OLS estimates

 

of the parameters of the misspecified model are:

A

Y = Y
821 = 821
831 = 831 + 821832

Since Y = r(n1, g) and 821 = r(nl, n2), the values of Y and 821 are
equal to the corresponding parameter values in the properly specified
model. The value of €31 differs from 831 since the actual correlation

between n1 and n3 is determined by the direct effect DE and the

31 ' B31

indirect effect IE (see Lewis-Beck, 1974). In the misspeci-

31 = 821332
fied model, the indirect influence of n1 on n3 has been eliminated by
the deletion of the path, i.e., by fixing 832 = 0. Thus the parameter
estimate for 831 must "absorb" this indirect effect. Although the
value of B31 is biased since B31 # 831, OLS made the "rational" adjust-
ment to the misspecification since 831 is the total effect of n1 on n3
831 = TE31. Given this "adjustment",
although the paths from g to n3 and from n1 to n3 are different in the

in the actual model. That is,

actual and misspecified models, the total influence of g on n3 and n1
on n3 remains unchanged.
The OLS residuals of the misspecified model are calculated in

Appendix B. Since the decomposition of r(€, n1), r(£, n2), and

47

r(nl,n2) according to the misspecified model involved only terms

consisting of Y and B the predicted values of these correlations

31’
equaled their actual values. Since the total effect of g on n3 and n1

on n3 remained unchanged, their residuals were also zero. Thus all the

residuals were zero except for:

ReS(n2,n3)
= B32(1 ‘ 821)

The residual matrix of the OLS solution provided useful information
regarding the correct specification of the model. The positive
residual between Hz and n3 correctly indicated that the relation
between n2 and n3 had been underpredicted by the model, although it
does not indicate the direction of the missing path. Since the remain-
ing residuals were equal to zero, they correctly indicated that the

rest of the model had been correctly specified.

LISREL simultaneous analysis. The factor loadings of the measure-

 

ment model were almost perfectly recovered despite misspecification of
the causal model. The only error larger than .003 was for the first
indicator of n1 where an estimate of .76 was obtained instead of the
correct .80.

However, the simultaneous LISREL analysis has a very damaging
problem as shown in Table 2: the estimated factor correlations are
wrong. Even though this run was made without sampling error, the
estimated correlation between n2 and n3 is off by .29. Thus even
though the measurement model was correctly specified, the simultaneous
LISREL analysis generated large errors in the estimated correlations of

the latent variables. Moreover, there is nothing in the LISREL

48

Table 2

Correct and Estimated Factor Correlations
from OLS and LISREL and the Residuals

n1 n2 n3 E

 

n1 1.00 .35 .49 .30

 

n2 .35 1.00 .52 .11
n3 .49 .52 1.00 .15
E .30 .11 .15 1.00

a) The correct factor correlations.

 

 

 

 

 

 

 

 

n1 n2 n3 E n1 n2 n3 E
ml 1.00 .43 .55 .30 n1 .00 -.08 -.06 .00
Hz .43 1.00 .23 .13 n2 -.08 .00 .29 -.02
n3 .55 .23 1.00 .17 H3 -.06 .29 .00 -.02
E .30 .13 .17 1.00 E .00 -.02 -.02 .00
b) The factor correlations c) The error in the estimated
computed from a LISREL factor correlations from the
simultaneous analysis. simultaneous analysis.
n1 n2 n3 E n1 n2 n3 E
n1 1.00 .35 .49 .30 ml .00 .00 .00 .00
n2 .35 1.00 .17 .11 n2 .00 .00 .35 .00
n3 .49 .17 1.00 .15 n3 .00 .35 .00 .00
E .30 .11 .15 1.00 E .00 .00 .00 .00
d) The factor correlations e) The error in the estimated
computed from an OLS factor correlations from an

analysis. OLS analysis.

49

printout to suggest that these estimates are wrong.

The estimated factor correlations from LISREL have values such
that an OLS path analysis of the estimated correlations would result in
an indication of perfect fit. That is, LISREL placed all of the error
into the estimated factor correlations and maintained perfect consis-
tency between the factor correlations and the path coefficients. Since
the misspecification occurred in the causal model only, this is the
exact reverse of reality. Thus the residual correlations in LISREL
suggest that the location of error is in the measurement model even
though the measurement model is perfect. The residual correlations are
zero for the causal model even though it is quite wrong.

Thus most of the estimates of the causal parameters of the mis-
specified causal model were different from the corresponding OLS
estimates. The parameter estimates for the OLS and the simultaneous
LISREL analyses are presented in Table 3 along with the actual values
of the parameters from the properly specified model. Computational
details are presented in Appendix C. The OLS estimates are closer than
the LISREL estimates; the LISREL estimate of 821 deviates from the
correct value by .08 whereas OLS deviates by 0, and the LISREL devia—

tion of 831 is .20 while the OLS deviation is only .14.

Table 3

Parameter Values and Estimates

 

 

Actual

value OLS LISREL
Y .30 .30 .30
821 .35 .35 .43
B .35 .49 .55

31

50

In terms of pinpointing misspecification, the information provided
by the LISREL analysis added nothing to the information inferred from
the OLS residuals. Analysis of the full causal model yielded nine
times as many residuals, i.e., a 12 x 12 matrix of indicator residuals
instead of a 4 x 4 matrix of factor residuals. Each factor correlation
in the 4 x 4 matrix is replaced by a 3 x 3 block of nine residuals in
the 12 x 12 matrix. The six blocks of residuals formed approximately
the same pattern as the six OLS factor residuals except that the
pattern in the LISREL analysis was more blurred than the pattern of the
OLS residuals. All of the residuals between the indicators of the
single exogenous factor and the indicators of the endogenous variables,
except between n2 and n3, were only approximately zero since they
varied in magnitude from .002 to .037. The residuals between the indi-
cators of n2 and n3, which replaced the single OLS factor residual of

.352, are presented in Table 4.

Table 4

The Residuals of the Indicators of n2 and n3

 

Indicators of n2

 

l 2 3
Indicators 1 .18 .14 .09
2 .14 .10 .07
of n3
3 .09 .07 .05

In Appendix C, it is demonstrated that the indicator residuals
from the simultaneous analysis are a function of the estimated factor

correlations. The sizes of the indicator residuals in each block of

51

nine residuals correspond to the size of the corresponding factor
residual except that each indicator residual is proportionately
diminished by the measurement error in the corresponding indicators.

A comparison of OLS and LISREL. The similarities and differences

 

between OLS and LISREL observed in the analyses of the model presented
in Figure 6 are presented in Table 5. The crucial difference between
OLS and LISREL is that, for the OLS path analysis, the factor correla-
tions were estimated at their correct values by the confirmatory factor
analysis. However, for the LISREL simultaneous analysis, the factor
correlations were adjusted to match the estimated path coefficients and
yield a model of apparent perfect fit--regardless of the amount of
error in the estimated path coefficients.

Analysis of a variegy of models and specifications. The conclu-

 

sions above were supported in many other models. These additional
examples misspecified the given model in at least four distinct ways:

(a) deletion of a causal path;

(b) deletion of a latent variable;

(c) addition of a causal path;

(d) reversal of the direction of a causal path.
The correct models were those of Figure 5 and Figure 6. The factor
correlations and the factor loadings were perfectly reproduced by
multiple groups analysis in each misspecified model. The information
provided by the derivatives was redundant with the information provided
by the residuals in either the LISREL analysis of the path model or the
full simultaneous analysis. The full LISREL analysis always correctly
recovered the factor loadings of the measurement model to within a

reasonable degree of error, but the parameter estimates of the causal

52

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53

model were farther off than the estimates provided by the OLS analysis.
In all cases, the one-step LISREL procedure matched its estimated
factor correlations to the erroneous path coefficients. Thus whereas
the multiple groups analysis always correctly reproduced the factor
correlations, the LISREL one-step procedure was always way off. This
is a crucial difference since it is discrepancies between observed and
reproduced correlation coefficients which provide the best guide to the
detection and remediation of errors in the causal model. LISREL always
falsely suppressed these discrepancies.

In summary, if the measurement model is correct, then the two-step
procedure will correctly estimate the correlations between the latent
variables. However the LISREL one-step analysis will be correct only
if the causal model is algp correct. Thus for LISREL, a correct

measurement model provides no clues as to the correct causal model.

CHAPTER IV

SPECIFICATION ERROR IN BOTH CAUSAL AND MEASUREMENT MODELS:
AD-HOC COMPOSITES

A composite score may be defined as the sum over 3px set of
component measures, irrespective of the content or statistical behavior
of the components. If all of the component measures are indicators of
the same latent variable, though perhaps with varying degrees of
reliability, then the observed composite score is an estimate of the
score for that construct. Whether or not the component measures are
indicators of a construct is determined by the procedures of construct
validity. The correlations of the indicators of a construct satisfy
one product rule for internal consistency in terms of their correla-
tions with each other, and a second product rule for external consis-
tency in terms of their relations with other variables.

A composite defined by a set of measures which are not indicators
of a construct is called an "ad-hoc composite", i.e., an ad-hoc
composite is an aggregation of related but conceptually and statis-
tically distinct constructs. Such composites are usually constructed
according to the procedures of content validity and have been useful in
applications in education and industrial psychology where empirical
prediction is the primary objective. As Cronbach (1970, p. 44) wrote,
"Nothing in the logic of content validation requires that the universe
or test be homogenous in content." However, a misspecification occurs
when the indicators of distinct constructs are collapsed together and

54

55

specified as alternate indicators of a single latent variable.

The embedding of such an amorphous measure in a causal model
insures that both the measurement and causal models are misspecified.
The lack of external consistency of the indicators of the misspecified
construct implies that the component constructs of the ad-hoc composite
tend to differentially relate with the other latent variables in the
model. Thus the component constructs would occupy different positions
in a properly specified model. This is illustrated in the Hunter,
Gerbing and Boster (Note 5) analysis of the Christie and Geis (1970)
Mach IV measure of Machiavellianism.

Unfortunately, the most common statistical analysis for the con-
struction and evaluation of measurement scales is not a confirmatory
factor analysis. Instead, the usual analysis, such as that provided by
the SPSS subprogram RELIABILITY, is based primarily on (a) itemrtotal
correlations, and (b) coefficient alpha or some other index of
"internal consistency" reliability. The use of item-total and relia-
bility statistics as indices of unidimensionality can lead to crude,
nebulous measures. Following Hunter (Note 3), consider a scale defined
by 10 items which breaks into two clusters of five items each. Items
within each cluster intercorrelate .7 with each other, but all corre-
lations between items in different clusters are only .2. Although the
scale is clearly two dimensional, coefficient alpha is .88. The
uncorrected item-total correlations are .69, and itemrtotal correla—
tions calculated with communalities are .67. Thus an examination of
only item-total correlations or coefficient alpha can lead to the
acceptance of a scale as unidimensional when it is actually an ad-hoc

composite.

56

The problem with coefficient alpha as an indicator of dimension-
ality was noted by Green, Lissitz and Mulaik (1977) and by Hunter
(Note 3). The amount of measurement error in a composite score is a
concept which is logically independent of the dimensionality of the
scale. If each component is perfectly measured, then the reliability
of the composite score is also 1.0 no matter what the relations between
the components might be. The problem with itemrtotal correlations
(even if corrected for upward bias by the use of communalities) is that
although an indicator will correlate most highly with its own properly
specified construct, it can have a substantial correlation with any
other correlated factor. Thus the item-total correlation is only

meaningful on a comparative basis; there is no absolute size criterion

 

for unidimensionality.

The Pattern of Residuals from an Ad-hoc Composite Misspecified as a
Construct

Consider the actual and misspecified models illustrated in Figure
7. The misspecification was defined by the pooling of the indicators
of n1 and n2 into a composite which is misspecified as the "construct"
a; The purpose of this section is to identify a characteristic pattern
of the residuals of such a misspecification. The residuals among the
four indicators as computed by LISREL are presented in Table 6. The
dotted lines in Table 6 indicate the partitioning of the indicators
into their respective sub-clusters based on the actual model.

The first consideration is the relative sizes of the factor
loadings in the actual and misspecified models. The actual construct
of an indicator is causally antecedent to the indicator, so the corre-
lation of the indicator with other variables is mediated by the

construct. That is, according to the product rule for external

57

.60 e

.30
.50 .70

.80
o e

Actual model

 

 

Misspecified model

Figure 7. The actual and misspecified models from Costner and
Schoenberg (1973) with parameter estimates computed by LISREL.

58

Table 6

Residuals from the Endogenous Indicators in Figure 7

 

 

Y1 y, y3 y4
I
y1 .oo .17 . -.04 -.04
I
y2 .17 .00 . - 04 -.04
_________ 1 - _ _ _ _ - _ _ _
y3 — 04 -.04 . oo 05
I
- 04 -.04 . .05 .oo

59

consistency,

r(i.2) = r(i.n) r(n.2)
where n is the actual construct of indicator 1 and z is some other
variable. In classical reliability theory, this result is expressed as
"the maximum value of the validity coefficient . . . is equal to the
reliability index . . ." (Magnusson, 1967, p. 150) where the relia-
bility index is the correlation between the indicator and the

construct. If 2 E E, i.e., if

r(i,?1’) = r(i.n) r013)
then
r(i,n) > r(i,'fi) or A > L
n. D
1 i
For example, A11 = .45 and Afi. = .30, as shown in Figure 7.

1

In terms of predicting the pattern of residuals for the misspeci-
fied model, consider first the correlations among the indicators of the
same properly specified construct. The correlation among these indi-
cators adheres to the product rule for internal consistency, i.e., for

h

the it and jth indicators of n,

r(i.j) = r(i.n) r(n,j)
But for the misspecified model,

£ua>=flnmflmp
And, from above,

I). A |>|A~A~|
Di nj Hi nj

If the residual is defined as
Res(1.j) s r(1.j) - £(1.j)
then the expected residuals of an ad-hoc composite misspecified as a

construct should be positive for the indicators of the same construct

60

in the correctly specified model, as they are in this example in which
Res(y1,y2) = .09 and Res(y3,y4) = .11.

Now consider the residuals among the indicators of different
constructs in the ad-hoc composite. The observed correlations for the
indicators were computed with the product rule for external consistency.

For example,
r(y1.y3) = r(y1.nl)r(n1.n2) r(nZJZ)

(.45) (.48) (.55) = .119

But the predicted correlations among these indicators which have been
falsely placed in the same cluster will again be computed using the

product rule for internal consistency. For example,

(.30) (.53) = .159
So the residual is .119 - .159 = .040.

Thus there are two competing influences on the relative sizes of
r(i,j) and r(i,j), where i and j are indicators of different constructs
in the actual model. As shown before, the actual factor loadings are
larger than the computed factor loadings. This discrepancy is large
and the residual is positive, to the extent r(nl,n2) is small. What
is new for this situation is the presence of r(nl,n2) in the expres-
sion for the actual correlation r(i,j). The smaller r(n1,n2), the
smaller the product r(i,rh)‘r(nl,n2):r(n2,j). Let the value of
r(n1,n2) decrease, but let the remainder of the actual model be
unchanged. If the value of this product decreases faster than the
discrepancies between the actual and computed factor loadings, the

residuals could even become negative, as they are in this example. But

61

these residuals should at least tend to be smaller than the residuals
of indicators of the same construct. This result is proved in Appendix
D for the special case of equal factor loadings in the actual model.

A Re-interpretation of the Costner and Schoenberg Analysis

 

Costner and Schoenberg (1973) investigated the effects of misspec-
ification with the strategy utilized in this paper. One of their
models, which is presented in Figure 7, was misspecified to form what
is here labelled an ad-hoc composite. Costner and Schoenberg (1973)
predicted the pattern of residuals resulting from the analysis of this
model without any formal justification. They simply indicated that
"our intuition is that . . . we might expect large residuals between
. . . the indicators of [E] on the one hand and the indicators of [n1]
on the other, or between indicators of [E] and [n2]" (p. 175).
Following an analysis of the misspecified model with LISREL, the
authors noted that "the actual pattern of residuals . . . does not
conform to this pattern at all" (p. 176).

How should the model be respecified given the residual matrix?
Costner and Schoenberg (1973) believed that the respecification of a
model when confronted with a matrix of nonzero residuals among the
indicators should be to allow the error variables for indicators with
the largest residual to be correlated. They respecified the model by
relaxing the r(sl,sz) = 0 constraint and noted that (a) the respeci-
fied model fits relatively well, and (b) this result was misleading
since the respecified model was not the correct model. The authors
then concluded on the basis of this example and others that "the
respecification suggested by an intuitive appraisal of the pattern of

residuals may be grossly misleading" (p. 177). Rejecting the

62

information provided by the residuals, the authors go on to devise a
laborious procedure for detecting misspecification which involves
separately testing all possible combinations of two-indicator models
and then testing specified combinations of three-indicator models.

However, the obtained pattern of residuals does conform to the
predictions of the previous section. The residuals between indicators
of the same construct in the correctly specified model are positive.
The residuals between indicators from different constructs are not only
smaller but negative in this example. Contrary to the conclusion of
Costner and Schoenberg (1973), the residuals do appear to provide
useful information for the respecification of this misspecified model.
The incorrect assumption is that positive residuals imply correlated
errors.

Does the use of PACKAGE provide information not provided by
LISREL? The application of the multiple groups analysis to the Costner
and Schoenberg (1973) example generated the residuals and partial

correlations presented in Table 7.

Table 7

MGRP Residuals and Partial Correlations

 

 

 

 

Residuals Partial Correlations
Y1 Y2 Y3 Y4 Y1 Y2 Y3 Y4
y1 .00 .09 ' —.06 -.07 y1 1.00 .14 ' -.07 -.06
I I
y 09 000 I -006 7.001 y .14 1000 ' -008 -007
2 _______ , _______ 2 ______ L _______
y3 -.06 -.06 ' .00 .11 y3 -.07 -.08 ' 1.00 .17
I I
y4 -.07 -.01 ' .11 .00 y4 -.06 -.07 ' .17 1.00
I I

63

The residuals derived from PACKAGE follow exactly the same pattern as
the LISREL residuals. However, since the specified model contains only
two factors, the LISREL simultaneous analysis is equivalent to a
confirmatory factor analysis.

Alternate residuals can be obtained by partialling out a single
factor at a time-—a straightforward operation with the PACKAGE
subprogram PARTIAL following the use of the MGRP subprogram. In the
actual model r(n1,n2) can be decomposed entirely into the spurious
influence of an exogenous variable E, i.e., r(n1,n2-E) = 0. This
same relationship among the factors is mirrored by the indicators of
the factors, as illustrated in Table 8. Again, contrary to the con-
clusion of Costner and Schoenberg (1973), the residuals provide

directly usable information for locating specification errors.

Table 8

Correlations of the Endogenous Indicators with E Partialled Out

 

 

Y1 y2 Y3 Y4
yl 1.00 .21 : .oo .oo
y2 .21 1.00 : .oo oo
y3 .oo .oo : 1.00 .21
y4 .oo .00 z .21 1.00

Finally, Costner and Schoenberg (1973) could have applied a test
suggested by Spearman in 1914 that would also have unambiguously deter-
mined that the misspecified model contained an ad-hoc composite.
Spearman noted that the proportionality constraint which follows from

the product rule for external consistency implies that the correlations

64

of two indicators of the same construct across other variables are
"perfectly correlated" (1914, p. 109). These "intercolumnar" or
"second order" correlations may be computed by correlating the corre-
lations of each pair of indicators with the diagonal value of 1.00
defined as missing data. The resulting matrix of second order corre-
lations is presented in Table 9. Actually, similarity coefficients
(e.g., Hunter, 1973) should be used instead of the second order corre-
lation coefficients since proportionality is a stricter criterion than
linearity. Second order correlations were computed because of the

availability of the computer program with a missing data provision.

Table 9

The Second Order Correlations of the Endogenous Indicators

 

 

I
I
yz 1.00 1.00 1 .26 08
__________ '_ _ _ — ._ _ _ _ _
y3 .05 .26 . 1.00 1.00
I
y4 —.02 .09 , 1.00 1.00

A More Complicated Model

 

Since the misspecified model analyzed by Costner and Schoenberg
(1973) contained only two latent variables, a simultaneous LISREL
analysis is equivalent to a LISREL CFA analysis. Consider the actual
and misspecified models in Figure 8. The factor loadings of the
correctly specified factors were correctly recovered while the factor
loadings of the ad-hoc composite were attenuated from their true values.

The simultaneous and CFA solutions in terms of factor loadings,

65

residuals, and derivatives were identical. The LISREL residuals among
the six indicators and the derivatives for the measurement error corre-
lations between the six indicators of the ad-hoc composite are presen-
ted in Table 10. The pattern is redundant between the two matrices.
Contrary to the assertion of Costner and Schoenberg (1973), the
residuals clearly indicate subdivision of the indicators into two sets.
Contrary to S6rbom (1975), the derivatives falsely suggest correlated

errors .

 

 

 

Actual model

 

Misspecified model

Figure 8. Actual model (a) from Figure 5 and a second misspecified
model.

The partial correlations of the indicators of the ad-hoc composite

from the PACKAGE analysis are presented in Table 11. This pattern

66

Table 10

LISREL Residuals and Derivatives

 

 

 

 

Residuals
Y1 y2 y3 Y4 y5 Y6
.00 .23 .14 : -.11 -.11 .08
23 .00 .09 : -.11 -.11 .08
14 .09 .00 : -.08 —.08 .06
____________ 1“ - _ _ _ _ _ - _ _ _ _
-.11 —.11 -.08 1 .00 .17 .11
—.11 -.11 -.08 : .17 .00 .06
I
-.08 -.O8 -.06 I .11 .06 .00
Derivatives
VI 22 Y3 Y4 ys Y6
I
.00 -.41 -.23 , .24 .21 .13
I
-.41 .00 -.13 I .22 .18 11
I
-.23 —.13 .00 , .14 .12 .07
____________ '- _ _ _ _ _ _ _ _ _ _ _
.24 .22 .14 , .00 -.37 .20
I
.21 .18 .12 , —.37 .00 .09
I
13 .11 .07 , -.20 -.09 .00

67

Table 11

Partial Correlations

 

 

 

V1 2, Y3 Y4 ys Y6
y1 .00 .28 .16 : -.19 -.15’ -.11
y2 .28 .00 .10 : - 15 -.12 -.08
y3 .16 .10 .00 : -.11 -.08 -.06
y4 -.19 -.15 -.11 : .00 .28 .16
y5 -.15 - 12 —.08 : .28 .00 .10
y6 -.11 -.08 -.06 : .16 .10 .00
Table 12
Second Order Correlations

y1 y2 y3 y4 y5 y6
yl 1.00 1.00 1.00 : .04 .11 .08
y2 1.00 1.00 1.00 : .11 .14 .10
y3 1.00 1.00 1.00 : .10 11 .07
y4 .04 .11 .10 : 1.00 1.00 1.00
y5 11 .14 .11 : 1.00 1.00 1.00

.08 .10 .07 : 1.00 1.00 1.00

 

68

the partial correlations was again similar to the pattern of the
residuals and/or derivatives from the LISREL analysis. The matrix of
second order correlations is presented in Table 12.

In practice, the indicators of the ad-hoc composite might not be
listed in subclusters as they are in the examples presented here. Use
of the PACKAGE subprogram ORDER on the matrix of partial or second
order correlations would reorder the variables so that the indicators
which define the component constructs are listed consecutively.

The Use of Derivatives in Detecting Misspecification in Ad-hoc
Composites

 

 

A re-analysis of previous studies. SSrbom (1975) was interested

 

in models with correlated errors. He accepted Costner and Schoenberg's
(1973) conclusion regarding the problems with the residual matrix for
detecting misspecification, but he sought a less troublesome alterna-
tive than the procedure outlined by Costner and Schoenberg. SBrbom
(1975) advocated the use of the first derivatives as an alternative to
the residuals for locating misspecification. "We should relax the
zero-restriction for that element which gives the largest decrease in
F" (p. 143), i.e., the element with the largest first derivative. He
demonstrated that the procedure worked for an actual model with corre-
lated errors which was misspecified as a model with uncorrelated
errors.

However, Sfirbom's (1975) procedure would lead to false conclusions
in the case of the collapsed indicator model or ad-hoc composite. For
the Costner and Schoenberg (1973) example, the matrix of first deriva-
tives for the covariance matrix of measurement errors is given in

Table 13. The pattern of first derivatives is redundant with the

69

Table 13

Derivatives of the Measurement Errors of the Endggenous Indicators

 

y1 y2 Y3 Y4
y1 .00 -.23 I .07 .09
l
y2 - 23 .00 I .07 10
________ 4_-______.
y3 .07 07 I .00 - 12
I
y4 .09 10 I -.12 .00
I

information supplied by the matrix of residuals. The only difference
between the patterning of the derivatives and the residuals is that the
derivatives have opposite algebraic signs. If the model were respeci-
fied by falsely assuming correlated measurement errors for Y1 and y2
and for Y3 and y4, then the actual structure would not be recovered, as
Costner and Schoenberg (1973) have already noted.

Saris, Pijper and Zegwaart (1978) also noted that the Costner and
Schoenberg (1973) procedure for detecting misspecification was "quite
time-consuming and not completely clear as to how one should proceed in
all circumstances" (p. 152). Following Sbrbom (1975), they were
concerned only with first derivatives, but they sought to improve
SBrbom's procedure by considering the correlations among the deriva-

tives. For each of the m fixed parameters, they computed the function

In. A
W1 7 jflrij 1
where 91 is the estimated first derivative of the ith parameter, and
$11 is the estimated correlation between the derivatives of the 1th and
th

j parameters. They proposed "a stepwise procedure . . . where at

70

each step the restriction is dropped with the highest 3_in absolute
value" (Saris et al., 1978, p. 158).

Saris et al. (1978) began with two-factor models which contained
(a) correlated measurement errors and/or (b) observed variables which
were indicators of both factors. The misspecified model was always the
corresponding two-factor model with unidimensional measurement, i.e.,
no correlated error variances and all indicators were indicators of
only a single factor. They concluded that "the Saris procedure per-
forms better than Sfirbom's" (p. 163). However, like SBrbom, these
authors never began with a recursive model with unidimensional measure-
ment. Like Sorbom, they never considered residuals.

Thus the "improvement" suggested by Saris et al. (1978) leads to
the same error of the Sarbom (1975) method when applied to the Costner
and Schoenberg (1973) example. Since all derivatives except the
derivatives of the measurement errors of the endogenous indicators
shown in Table 13 were zero, the application of the Saris et al. method
to these data would also lead to a false respecified model which con-
tains correlated measurement errors. The only potential difference
between the procedures is that the correlated measurement errors might
be added in a different order.

An extrapolation of the Sarbom_procedure. As constraints of the

 

model are relaxed, the apparent fit of the model continually improves.
In the ad-hoc composite example, the suggestion of SBrbom (1975) to
assume correlated measurement errors does improve the apparent fit of
the model though the respecified model is false. The question
addressed in this section is, what is the end result of allowing

increasingly more measurement error covariances?

71

The largest residual from the LISREL analysis of the misspecified
model in Costner and Schoenberg (1973) was between the indicators of
y1 and y2. The largest derivative was between the measurement error
covariances of the corresponding error terms. If this error covariance
is freed, it assumes the value of .198, the fit of the model improves
since F(§) decreases from .588 to .142, and the factor loadings of the
two indicators decrease about .04. The largest residual is now between
y3 and y4 and the largest derivative is between the corresponding
covariance between their error terms. If the model is respecified
again by adding this second measurement error covariance, the model
fits perfectly, i.e., F(§) = 0.000. The estimated regression para-
meters continue to change with each respecification since the factor
loadings of y3 and y4 decrease about .09 and, more interestinly, the
coefficient relating the latent variables becomes equal to 1.00.

The generality of this result can be checked by examining the more
complicated model which appears in Figure 8. The misspecified model
was successively respecified by freeing a new measurement error covar-
iance on each respecification. The chosen covariance on each round
corresponded to the largest derivative or, equivalently, the largest
residual. The model was respecified until perfect fit was obtained.

In general, as more measurement error covariances are added, the
factor loadings decrease in magnitude while the estimated regression
parameters of the causal model increase in magnitude. The fit of the
model continues to improve until six covariances have been included,
at which point F(§) - .014. Yet in this "perfect" model, B is equal to
the nonsense value of .911. And this "perfect" model is, in actuality,

a misspecified ad-hoc composite.

72

The pattern of correlated errors reveals the nature of the
misspecification. The misspecified model fit the data perfectly if all
the covariances between measurement errors of indicators of the same
construct were unconstrained. For the Costner and Schoenberg (1973)
model this criterion was achieved after only two measurement error
covariances were freed since the ad-hoc composite contained the indi-
cators from only two factors and each factor had only two indicators.
The present model required six free error covariances since there were
three indicators for each factor.

The apparent fit of a model with correlated measurement errors
means only that the model is misspecified. If the model fits poorly
without correlated errors, but fits very well with correlated errors,
and if the indicators can be partitioned into clusters with positively
correlated errors within clusters and zero or negative covariances
between clusters, then at least one of the latent variables is an
ad-hoc composite. The indicators should be partitioned accordingly.
However as was shown earlier, the same analysis can be performed
directly on the original model residuals. There is no need to obtain
an intermediate solution with "correlated errors".

Summary

In contemporary path models, a construct is defined as a latent
variable whose indicators form a unidimensional set in the sense of
Spearman (1904). To use such models, one must "reduce" composites
which are measured by a conglomerate of constructs into component
variables. Such an ad-hoc composite can be regarded as a model which
has been misspecified by collapsing the indicators of distinct

constructs into a single latent variable. Moreover, the recognition of

73

such a misspecification can be accomplished by the examination of the
residuals--although previous work has claimed that the residuals were
not helpful in detecting misspecification. Contrary to previous
claims, the derivatives from a LISREL analysis were shown not to add
any more information than the residuals. Indeed the recommendations
of Sgrbom (1975) and Saris et al. (1978) were shown to lead to the

false assertion of correlated errors.

CHAPTER V

MISSING VARIABLES, MISSING PATHS, AND CORRELATED ERRORS

No existing path model contains all the relevant causal factors.
If only complete models were capable of analysis using OLS, then there
would be no situations in which OLS could be used. This is precisely
the claim of a number of contemporary authors. They argue that missing
variables always result in correlated disturbances and hence the use of
OLS is never justified in real data sets. The following examples show
this claim to be false. On the contrary, these examples suggest that
most recommendations for correlated disturbances are misguided
responses to data sets which call for the addition of missing paths in
the misspecified models.
The Effect of 3 Missing Variable on the Fit of the Model

Previous assumptions. The traditional estimation procedure used

 

in path analysis is OLS, which assumes that the disturbance terms
across equations are uncorrelated. However, several authors have
argued that correlated disturbances could be produced by missing
variables. These arguments are important since the complete specifi-
cation of all of the variables in a causal process is impossible.
Presumably, if a common antecedent of some of the endogenous
variables is omitted, the omitted variable will "appear" in the corre-
sponding disturbance terms since the disturbance represents the

influences which operate on the "dependent variable" of the equation

74

75

which are not described in the set of predictor variables in the
equation. For example,

"[The realism of] the assumption that the disturbance
terms are mutually uncorrelated in any given instance will
depend upon the completeness of the causal system . . . .
Whenever common causes of the disturbance terms for two or
more equations can be located or measured, they should be
explicitly introduced into the equations as additional
variables" (Namboodiri, Carter, & Blalock, 1975, pp. 446
and 448).

"If the same explanatory factor is excluded from.more
than one equation, the effect of that factor will be present
in more than one error term and will cause the error terms
to be somewhat correlated . . ." (Hanushek & Jackson, 1977,
pp. 230 and 231).

"The uncorrelated residuals assumption is basically
equivalent to the assertion that there is no confounding

variable impinging upon both X1 and X2 where by a confounding

variable is meant any unmeasured factor that directly

influences two or more of the measured variables" (Asher,
1976, p. 16).

"Correlated error terms arise when omitted variables
simultaneously influence different observed variables"
(Saris, Pijper, & Zegwaart, 1978, p. 161).

A test of the assumptions: Deletion of an endogenous variable.

 

The authors cited above believe that a missing variable results in
correlated errors. The example in Figure 9, which was presented by
Duncan (1975), appears to contradict this principle, although this
example did not appear in a discussion of correlated errors. Duncan

(1975) noted that "the OLS estimator of Y in [the respecified model]

31

estimates Y without bias. The principle is that insertion of an

21832
"intervening variable" into one path of an initial model does not
inValidate that model, but merely elaborates it" (p. 109). The

following discussion is an elaboration and extension of these

principles.

76

 

Actual model

 

 

Respecified model

Figure 9. The actual and respecified models presented by Duncan
(1975).

77

Consider the more complex model presented in Figure 10 where the

misspecification is defined by the deletion of n The parameter

3.
estimates presented in the misspecified model were computed with OLS.
The residuals of the misspecified model are presented in Table 14.
The pattern of residuals is that which would be produced by the
deletion of a path. If the misspecified model is respecified by the

inclusion of a path from n to n4 with 841 = .09, then (a) the other

1
parameter estimates remain unchanged and (b) the respecified model fits
perfectly, i.e., all residuals are zero. The difference between the
two models is that in the misspecified model, the elimination of the
path from n1 to n3 to n4 was not replaced by a direct path from n1 to

n4. Thus the causal impact of n on 114 was not accounted for in the

1
misspecified model.

Any model is incomplete in the sense that there are missing
variables. Otherwise all multiple correlations in the model would be
1.00. What is crucial is not that variables are missing, but that the
causal effects of the missing variables are represented by paths
connecting the variables which are observed. If paths corresponding to
indirect causation are present, then any subset of a recursive model
can also be fit by a recursive model. Deleted endogenous variables
will not affect the fit of the model if the direct antecedents of the
deleted variable directly influence the direct consequents of the
deleted variable. If this conditiOn is met, the total effects of the
antecedent variables on the consequent variables are the same in both
models. That is, deletion of an endogenous variable does not lead to

a misspecification unless paths are incorrectly deleted in the reduced

model. Thus the deletion of an endogenous variable does not imply that

78

 

 

Actual model

.30 0 .30

 

.30 a .30

Misspecified model

Figure 10. Actual model (d) from Figure 5 and an accompanying
misspecified model.

79

Table 14

Residuals of the Misspecified Model of Figure 10

 

 

E 01 02 04 05
E .00 .00 .00 .03 .01
01 .00 .00 .00 .09 .03
n2 .00 .00 .00 .00 .00
n4 .03 .09 .00 .00 .03

.01 .03 .00 .03 .00

80

the disturbance terms of any of the variables remaining in the model
should be correlated.

A test of the assumptions: Deletion of an exggenous variable. If

 

a single variable is the only consequent of a block of variables and
this variable is antecedent to another block of variables, then the
deletion of the first block of variables implies that the specified
model will fit the data perfectly. This general situation is illus-
trated in Figure 11. The variables in the deleted block of variables
do not directly influence the correlations of the variables in the con-
sequent block of variables. Thus, the result of deleting these varia-
bles is similar to the result from deleting an endogenous variable.

The deletion of the variable(s) does not affect the fit of the model
and so does not lead to correlated disturbance terms between any of the
remaining variables in the model.

The deletion of an antecedent variable or block of antecedent
variables becomes salient, however, if the deleted variable(s) directly
influence variables within the consequent block of variables. For
example, the misspecified model presented in Figure 12 which is defined
by the deletion of E does not fit the data generated by the correct

model because E directly influences n2 and n as well as n1. The

3
misspecified model could be "fixed" by adding a path from n1 to n3, but

this respecification amounts to replacing the spurious effect of E on

and n by a direct effect. However, even though the respecified

n1 3
model is conceptually incorrect, it fits the data perfectly, so there

are no correlated disturbance terms in the respecified model.

81

 

 

ANTECEDENT
BLOCK OF
VARIABLES

 

40+

CONSEQUENT
IBIKNSK CM?

 

VARIABLES

 

 

 

Figure 11.

Actual model

 

 

@+

 

CONSEQUENT
BLOCK OF
VARIABLES

 

Specified model

Deletion of a block of exogenous variables which affects
only a single endogenous variable.

 

Misspecified model

Figure 12.

 

 

Actual model

 

Respecified model

Actual model (d) from Figure 5 and misspecified and
respecified versions of this model.

 

82

Correlated Errors in Practice: The Use of Derivatives to Respecify a
Model

 

Given a maximum likelihood estimation procedure such as that used
by LISREL, a test for locating a potential misspecification for a
solution which has converged is based on the values of the first
partial derivatives of the likelihood function with respect to each of
the fixed parameters. If the model fits the data poorly, some of the
derivatives of the parameters fixed at constant values should differ
from zero. S5rbom (1975) recommends that a misspecified model may be
respecified by using these derivatives. He advises that "we should
relax the . . . restriction for the element which gives the largest
decrease in F(E)" (p. 143). That element is the element with the
largest partial derivative.

Consider the actual and misspecified model in Figure 6. Since 832
was falsely fixed at zero in the misspecified model, the derivative
with respect to 832 should not be zero in a LISREL solution of the
misspecified model given the data generated by the actual model.
However there were three nonzero first derivatives in the LISREL
analysis of the path model in Figure 6 and these derivatives appeared
in both the B and 0 matrices. The corresponding derivatives were also
nonzero in the LISREL simultaneous analysis of the model in Figure 6 in
which each of the latent variables was measured with three indicators
with factor loadings of .80, .60, and .40. These derivatives appear in
Table 15.

In both the two-step and simultaneous LISREL solutions, the
largest derivative was for o(E2,C3). In the simultaneous analysis the
value of this derivative was almost twice the size of the next largest

derivative over all the parameters, which is the derivative for 832.

83

Table 15

Derivatives for the Model in Figure 6 Computed by LISREL

 

 

 

—— 6.-....
0(E2,E3) ‘-53 -.34
B32 ‘-46 -.18
8 ‘-40 -.15

23

84

If the model were respecified by relaxing the constraint which yielded
the largest derivative, the fit of the respecified model would be
improved, but the respecified model would not be the model which
generated the data. Thus for this model, SSrbom's (1975) advice is
false. Moreover, it was the deletion of a path--not the deletion of a
variable--that led to the generation of correlated disturbances. The
rules for models with correlated disturbances are just the rules for
models with missing paths.

The equivalence between correlated disturbances and a misspecifi-
cation defined by a missing path is even clearer in the following
example. The actual and misspecified models in Figure 13 are the same
models presented in Figure 6 except that the three disturbance covar-
iances are ppp_constrained at zero in the misspecified model. These
disturbance terms are included in Figure 13 for heuristic purposes
only. The parameter estimates of the misspecified model are computed
in Appendix B.

The estimated regression parameters of the two misspecified models
are equal. Of particular interest, however, is the comparison between
the single nonzero OLS residual and the single nonzero disturbance
covariance: they are equal. That is,

The correlated disturbance term in the nonrecursive misspecified model
is simply the lack of fit in the corresponding recursive misspecified
model. The use of correlated disturbances provides no more information
than was available from the information provided by the traditional OLS
solution. If the disturbance covariances are fixed at zero as they are

for the OLS solution, the model does not fit. If the disturbance

85

 

 

Actual model

8<c1.c,_>

 

    

Misspecified model

Figure 13. Actual model (a) from Figure 5 and a third misspecified
model with correlated disturbance terms.

86

covariances are free parameters whose values are to be computed from
the data, then the model fits the data perfectly. But to say that this
model with free disturbance covariances fits the data is to say nothing
at all given that the OLS solution does not fit the data.

Correlated Errors: Summary

 

There are situations such as the analysis of longitudinal data
(Hunter, Coggin, & Gerbing, Note 9) in which an analysis with corre-
lated errors is appropriate. But the results of this paper suggest
that there are at least two situations in which the use of correlated
errors is inappropriate or at best misleading: (a) correlated measure—
ment errors as a substitute for decomposing ad-hoc composites, and
(b) correlated disturbances as a substitute for addition of a path.

To propose a model with correlated errors in either of these two
circumstances is to do nothing more than propose a misspecified non-
recursive, unidimensional model. To claim that "OLS is biased if there
are correlated disturbances in the presence of a deleted path" is to
say nothing more than that the parameter estimates from OLS are wrong
because the model is false. If LISREL with correlated disturbances is
used instead of OLS to estimate the parameters, the model is still
false. The parameter estimates are different but equally misleading.
Although the apparent fit of the respecified model with correlated
errors may be dramatically improved, the model is still wrong.

Most of the simulation studies which examine the properties of the
statistical procedures for simultaneous equation models begin with a
model or models which contain correlated errors, e.g., Hanushek and
Jackson (1977) or Cragg (1968). Hanushek and Jackson (1977) concluded

that "there is a noticeable increase in the bias in the ordinary least

87

squares estimator as the correlation between the error terms in the two
equations increases" (p. 237), which is only to say that as the model
becomes more and more false, the OLS estimators become less and less
useful. 0r consider the studies by Costner and Schoenberg (1973),
SBrbom (1975), or Saris et al. (1978) in which they attempted to deter-
mine useful indices for locating misspecification. These authors, with
but the one exception in Figure 7, used a given model with correlated
errors to generate the data and then attempted to fit a recursive,
unidimensional model to this data. Under these conditions it is not
surprising that all of these authors have failed to realize that the
residuals may contain some very useful information for detecting
misspecification.

Contemporary use of correlated errors and correlated disturbances
has led to erroneous models in most cases. Perhaps the fact that
LISREL permits correlated errors should be regarded as a flaw in the

program instead of an advantage.

APPENDICES

APPENDIX A

COMMUNALITIES IN A MULTIPLE GROUPS ANALYSIS

Unlike most contemporary authors, Kenny (1979) recognizes that the
classical multiple groups analysis is a confirmatory factor analysis.
However he completely abandons multiple groups in favor of LISREL.

A "factor analytic solution to [a multiple indicator model] . . . is
the multiple group solution . . . which is rarely used. Like most
factor analytic solutions, it . . . suffers from the communality
problem; that is, communalities must be specified in advance" (p. 138).
This is a serious misunderstanding of the factor analysis literature
since communalities, like any other parameter, must be estimated. This
is true of LISREL as well. The LISREL estimate of the communality of

2

the ith observed variable is A1.

Nunnally (1978) is one of the strongest contemporary advocates of
the classical multiple groups approach, but he argues that "the
diagonal unities in a centroid factor multiple groups analysis can
make the loadings seem spuriously high . . . . In a sense, there is
nothing wrong with this, because it is the correct mathematical
solution. The illusory appearance of large loadings could be reduced
by . . . the use of SMCs as communalities . . . . However, this is not
really necessary" (p. 420). Nunnally (1978) not only fails to

recognize the possibility of iterating for the communality values, he

88

89

does not recognize the need.

However, Hunter (Note 7) has shown that except in certain unusual
cases, the communalities in a multiple groups analysis will converge to
their correct values on successive iterations. The convergence for
each indicator of the measurement model used in this study is graphed
in Figure 14.

The 0th iteration for each indicator is the initial computation
obtained before the first iteration, i.e., the value Nunnally (1978)
recommends. If "the correct mathematical solution" implies that the
underlying structure is recovered with perfect accuracy in the absence
of sampling or specification error, and if the indicators are measured
with error, then communalities must be inserted into the diagonal of

the correlation matrix.

90

1.000-

I

 

.80

 

 

 

 

 

Cb \
2
H
O C
a: 60
CD
_J
c: I
CD
I-—
L)
a: .40
U.
0200'"
0 4.000 8.000 121000 ' 16:000

NUMBER OF ITERRTIONS

Figure 14. An example of MGRP iterations.

APPENDIX B

ANALYTIC SOLUTION FOR THE PARAMETER
ESTIMATES OF MISSPECIFIED MODELS

The parameter estimates and consequently the residuals of the
misspecified causal models can be solved analytically by the following
algorithm developed for this paper:

(1) Generate the predicted correlations of the correctly

specified model, 2.

(2) Generate the predicted correlations of the misspecified
model, 2.

(3) Solve for the parameters of the misspecified model in
terms of E.

(4) Express the solution of the parameters of the misspecified
model in terms of the parameters of the correctly specified
model.

This procedure is illustrated for the misspecified models which appear

in Figure 6 and Figure 13 and the corresponding actual model, which is

the same in both figures. For the models in Figure 6,

r(E.nl) = Y £(a.n1) = I

91

92

r(n1,n2) = 821 r(nl’nz) = 821
r(nl.n3) = 831 + 821832 r(nl,n3) = 831
r("2"”3) = 832 + 831821 r("2’”3) = 31821

Given these correlations, the OLS parameter estimates and the
residuals of the misspecified model may be computed. The OLS estimates
of the parameters of the misspecified model are

A

Y = r(nl.E) 821 = r(02.nl) 831 = r(n3.nl)

which can be expressed in terms of the parameters of the actual model.

A

Y = r(n19g) = Y
831 = r("3’“1) = 831 + 821832
The OLS residuals of the misspecified model are defined by S - 8.

Since theilmatrix is the observed matrix for these examples, the

residuals are defined by Z - 2.

I
.4
I
4)
ll
.4
l
.4
II
C

r(€,nl) - £(53n1) —

r(E.02) - £(a.n2) = 1321 — 1821 = 1321 - 1321 = o

I
4
'm

r(E.n3) - £(E.n3) -

YB B B 0

+

.4

'm
u:
h)

'm
h)
[—0

I

31 ’ Y 21 32 =

831 + 821832 " B31 ’ 821832 = 0

r(02.n3) - r(nz.n3) = 832 + B31821 - 831821

2
’ 832 + 831321 ‘ B31521 ’ 821832

I
‘m

93

Consider the misspecified model in Figure 13 which is identical to
the misspecified model in Figure 6 except that the disturbance covar-
iances are unconstrained. The covariance structure of the misspecified

model is presented below.

r(E.nl) = i

A

r(nl.n3> = 331 + 0(cl.c3)

r(n2.n3) = 831821 + 8318(61.62) + 8210(61.c3) + 0(E2:E3)
The parameters of the misspecified model appear to be just identified
since there are six equations in six unknowns, Y, 821, 831, 6(E1,C2),
0(cl,c3), and 6(c2,c3). (The disturbance variances are functions of
these six parameters.)

Since there are as many equations as there are unknowns, the
parameters of the misspecified model can be expressed in terms of the
predicted correlations. Moreover, a just identified model fits any
data set perfectly since there are no overidentifying restrictions or
degrees of freedom to test the fit of the model. So each predicted

correlation r equals the corresponding actual correlation r Since

ij ij '
the actual correlations can be expressed in terms of the parameters of
the actual model, solving for the parameters of the misspecified model
in terms of the correlations implied by the misspecified model is
equivalent to expressing the parameters of the misspecified model in

terms of the parameters of the actual model for a just identified

model. That is, the FIML solution computed by LISREL can be obtained

94

heuristically for the just identified model. These computations for
this example are presented below.

Y = E(§.n> = r(E.nl) = Y

r(nz.E) r(02.E) B

 

A 21

B =———-—— =—————=—-——=B

21 Y Y Y 21

g = r(n3.E) = r(n3.E) = Y831 + Y832821 = B + B B
31 y Y Y 31 32 21

A

= r(nlinz) - 821

= B - B

21 21

= 0

= (831832821) (8 8+31 832 821)

= 0

= (832+ 831821) (8 31+ 832 821)(821)
2
’ 1332'l ’ 821)

APPENDIX C

COMPUTATIONAL DETAILS OF A SIMULTANEOUS LISREL ANALYSIS

The Relation Between the Indicator Residuals and Factor Residuals

 

Consider the model in Figure 6 such that each latent variable has
three indicators with factor loadings .80, .60, and .40, and Y = .30,
821 = 831 = .35, and 832 = .40. Also consider the residuals from the
simultaneous LISREL analysis between the indicators of n2 and 03 which
appear in Table 4. Since the residuals are defined as the difference
of the corresponding observed and predicted correlations, these
residuals may be computed by first computing the observed and predicted
correlations.

The observed indicator correlations were generated by the product

rule for external consistency. For example, since the actual correla-

tion between NZ and n3 reported in Table 2 is .523,

(.8) (.523) (.8)

= .334
LISREL computed the corresponding predicted correlations according to
the same product rule. The difference is that the estimated values of
the factor correlations were substituted for the actual values. That
is,

r(3'21’3’31) = A21r("2’”3) A31

95

96

Since the factor loadings were correctly recovered by LISREL,

A

A21 = A31 = .80

The LISREL estimate of r(n2,n3), reported in Table 2, was .234.

£(y21.y31> = (.8) (.234) (.8)
= .150

Thus the residual of the indicators y21 and y31 is

Res(y21’Y31) E r(Y219Y31) — r(y21!Y3l)

.334 - .150

= .184

So,

And .184 is the same value computed by LISREL as listed in Table 4.

The keys to reconstructing the residuals between indicator corre-

lations are the residuals among the factor correlations and the

respective factor loadings. This can be more explicitly stated by

first recomputing the residual between y21 and y31.
RES(y21,y31) E r(Y213y31) - r(y21’y31)
= A21"”2’”3)A31 ' A21"“2’”3)A31

[Res(n2,n3)] (A21A31)

That is,

Res(y21,y3l) = (.523 - .234) (.8) (.8)

II

(.289) (.8) (.8)

= .184

This result can be generalized to any Factors F and G with indicators

1 and j respectively,

Res (yFi’yGi) = Res (F,G) AFi AGJ.

97

The entire block of residuals for the indicators of factors F and

G can be described as

_ I
(‘11 rFG) AFAG
where;F is the vector of factor loadings for the NF indicators of F

and A is the vector of factor loadings for the N factor loadings of

-G G

G. The resulting "block" of residuals is a matrix of order NF x NG'

For example, let n23; F, n3 3 G, A% = Aé E [,8 .6 .4]. Then

I- 1 p

.18 .14 .09 .64 .43 .32.1

.14 .10 .07 (.289) .48 .36 .24

.04 .07 .05 .32 .24 .16
L J - L a

 

 

 

 

For this example in which each factor in the model had the same
number of indicators with an identical pattern of factor loadings, the
entire residual matrix of indicator correlations can be expressed as a
Kronecker product of (a) the entire residual matrix of the factor
correlations and (b) the factor loadings. That is,

(BIND " 2311111)) = (ZFAC ' iF.II.(:)® (AA')
12x12 4x4 3x3
The Kronecker product, designated by(3> is defined as the matrix formed
by justaposing the scalar products of every element of the first matrix
with the entire second matrix. That is, every element of (ZFAC - EFAC)
is "replaced" by the multiple of that element with the matrix (AAf).

Thus the complete 12 x 12 matrix of residuals defined by Z - E
where 2 is computed in a LISREL simultaneous analysis can be expressed
as a function of the actual and estimated factor loadings and factor

correlations. The actual factor loadings and factor correlations are

given and the estimated factor loadings are approximately equal to the

98

actual factor loadings. The problem of reconstructing the residuals of
the indicator correlations has been reduced to reconstructing the
residuals of the factor correlations.

Computation of the Parameter Estimates of the Causal Model

 

LISREL selects those parameter estimates which minimize a function
F(E) of the residuals among the indicator correlations. But minimizing
the residuals among the indicator correlations is just minimizing the
residuals among the factor correlations. Since LISREL is a full infor-
mation technique, the parameter estimates are computed to

simultaneously minimize the factor correlation residuals across the

 

entire model.

Given the present example in which the misspecification is defined
by the deletion of the path from n2 to n3, there are two conflicting
"pressures" which must be simultaneously resolved. First, the only

source of the correlation between and in the misspecified model

T12 03

is the common antecedent n1, i.e.,
but the actual correlation between n2 and n3 is decomposed as
r(nz.n3) = 832 + 821831

To the extent 832 is large, B and 832 must be biased upward to

31
minimize Res(n2,n3). That is, LISREL "would like" to increase the

and 8 until

values of B 32

31

A

But, as a second consideration, LISREL cannot adjust both 831 and 832

without affecting the residuals Res(nl,n2) and Res(nl,n3). To the

99

extent that

8>B and 8

31 31 > B

32 32

the values of Res(n1,n2) and Res(n ) will increase.

2’"3
The resulting parameter values computed by LISREL are a compro-

mise. The values of 831 and 832 are increased over their OLS counter-

parts, which reduces Res(n2,n3) for the LISREL solution compared to the

A

corresponding OLS residual. However, this increase in B and 832

31
increases Res(nl,n3) and Res(n2,n3) over their OLS counterparts, which
are equal to zero. Thus the simultaneous LISREL solution provided

incorrect factor correlations and obscured the detection of the

misspecification in terms of the residuals.

APPENDIX D

RELATIVE SIZES OF THE INDICATOR RESIDUALS WITHIN
AND BETWEEN CONSTRUCTS IN AN AD-HOC COMPOSITE
IF ALL THE INDICATORS HAVE EQUAL FACTOR LOADINGS

The examination of the residuals from a misspecified ad-hoc
composite should reveal a pattern in which the largest residuals were
between the indicators of the same construct. The residuals of the
indicators of different constructs may be negative, but they should at
least tend to be smaller than the residuals of indicators of the same
nconstruct. For example, let yi and yj be indicators of nF and yk be
an indicator of nG. Let all of the actual factor loadings be equal.

If yi, yj, and yk are pooled together and misspecified as a construct

n, then Resij - Res1k = (rij - rij) - (rik - rik)

= rij ' r13 ' r1k + rik

= riFer ' rifirjfi'- rirrrcrck + rifirkfi
If riF = er = rkG’
R s - Res = r2 - r r - r2 r + r r
8 ij ik 12 ffi jfi iF FG ffi kfi'
2 2

r1F ' riFrFG
So if rFG < 1 and if the actual factor loadings are equal, then the
residual for y1 and yj, indicators of the same construct, will always

be larger than the residual for yi and yk, indicators of different

constructs.

100

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