TESTING LATENT VARIABLE MODELS WITH SURVEY DATA
(2nd Edition)
FOREWORD
This monograph is concerned with the Social Science process of testing, or validating, as it is
sometimes called, theoretical models involving unobserved or latent variables with survey data (an
unobserved or latent variable can still be "measured"--it has observed variables as "indicators" or
instances of that latent variable). It is motivated partly by several years of ad hoc reviewing for a
variety of journals in the Social Sciences and a qualitative review of journal articles to determine
current practices in survey model testing. As a result, the book critically reviews current practices
and selectively suggests improvements in those practices based primarily on the social science
methods literature and my own research. Because I also inject anecdotal material such as my own
personal experience, the result is more than a recitation of approaches and techniques available
elsewhere. Thus, the primary aim of the monograph is to be of assistance to researchers in the Social
Sciences in their testing of theoretical models involving latent variables1 and survey data.
Because I am a captive of my discipline, a branch of social science research that investigates
socio-economic exchanges between firms (also known as marketing channels research--e.g.,
exchanges between business firms), the book's examples and some of its comments about theoretical
model testing practices using survey data involve research in Marketing. Nevertheless, because
empirical research in Marketing follows the same practices and conventions for theoretical model
testing as the other branches of the Social Sciences, this book and its suggestions have application
across the Social Sciences. Parenthetically, one of the early monographs on latent variables and
structural equation analysis was from an author in Marketing, Richard Bagozzi [Bagozzi 1980b], and
a theoretical model from his substantive research (Bagozzi 1980a) appears as a pedagogical example
in many monographs on latent variables and structural equation analysis (e.g., Bollen, 1989; Dillon
and Goldstein, 1984; Jöreskog and Sörbom, 1996b; etc.).
My first experience with latent variable was while testing a theoretical model with multiple
dependent or endogenous variables using survey data. After trying Ordinary Least Squares (OLS)
regression then Canonical Correlation to estimate the path coefficients in the hypothesized structural
model, I decided to use structural equation analysis, specifically LISREL, because of its ability to
jointly estimate simultaneous linear equations, which OLS regression can not do, and its ability to
model the effects of measurement error, which neither OLS regression nor Canonical Correlation can
do. I was subsequently surprised by how difficult structural equation analysis was to use.
While are other structural equation analysis programs besides LISREL (e.g., EQS, AMOS,
2
etc.), when compared to OLS regression for example structural equation analysis using any of these
1. The book is restricted to models that have ordinal, continuous, etc. variables rather than
classification (e.g., categorical) variables. Categorical variables are briefly discussed when they
are used as independent or exogenous variable, but models involving them as dependent or
endogenous variables are beyond the scope of this research.
 2004 Robert A. Ping, Jr. 1
programs seems to be as much an "art" as it does an estimation technique. Thus, this book is also
intended to help make analyzing survey data using structural equation analysis a little easier.
The monograph is organized around the process of testing theoretical models involving latent
variables and survey data. It brings together what is known about this process, and it selectively adds
to this body of knowledge. The book begins with a discussion of theoretical model testing involving
survey data that highlights the six steps in this process. Then it discusses each of these steps,
providing suggestions for improving this process and several pedagogical examples involving realworld survey data. It concludes with unresolved issues in this process and needed research in this
area.
Along the way it suggests an additional procedure for achieving model-to-data fit using
structural equation analysis, and it provides a suggestion for easily executed pretests using scenario
analyses. A contribution of the monograph is its accessible discussion of the problem of
"inadmissible solutions" in structural equation analysis (i.e., parameter estimates that do not make
sense) and model-to-data fit difficulties, with suggested remedies. The book discusses matters that
may be well-known to methodologists but may not be as well known to substantive researchers, such
as "the puzzle of about six indicators" in structural equation analysis, a discussion of error-adjusted
regression, several alternatives to address the persistence of about six indicators or less in structural
equation analysis including the use of single summed indicators in structural equation analysis, and
the use of a nonrecursive model to investigate directionality or causality. It provides accessible
discussions of several overlooked but valuable statistics such as Average Variance Extracted (AVE)
and Root Mean Squared Error of Approximation (RMSEA), and it suggests an estimator of AVE that
does not rely on structural equation analysis. The monograph selectively discusses estimating latent
variable interactions, and provides a rationale for the more frequent inclusion of interactions and
quadratics in survey model tests. It calls for additional attention to measure consistency, and thus
model-to-data fit, in structural equation analysis, and argues for higher thresholds for acceptable
reliability based on average extracted variance. The book also renews calls for caution in
generalizing from a single study because of the unavoidable risks from violations of methodological
assumptions and the use of inter-subject research designs to test intra-subject hypotheses. It also
(re)emphasizes the implications of reliability and facets of validity as sampling statistics that have
unknown sampling distributions, and it suggests bootstrapping for reliabilities and facets of validity.
In addition, it suggests alternatives to omitting items in structural equation analysis to improve
model-to-data fit, which should be especially useful for older measures established before structural
equation analysis became popular.
The book assumes the reader is familiar with the terminology and logic of structural equation
analysis, and a software program for the analysis of structural equations (e.g., LISREL, EQS, AMOS,
etc.) (the interested reader is directed to excellent introductory treatments of structural equation
analysis such as Hayduk 1987). Nevertheless, I have tried to make the book as accessible as possible.
The list of those I should thank in this book is long and certainly incomplete. My first
exposures to empiricism and the scientific method occurred before my terminal degree. Chris Allen
2. Other available structural equation analysis software includes AMOS (Arbuckle, 1988),
COSAN (Fraser, 1988), EQS (Bentler, 1985), LINCS (Schoenberger and Arminger, 1988), and
LISCOMP (Muthen, 1984).
 2004 Robert A. Ping, Jr. 2
and Ralph Katerberg, among others, enriched those earlier exposures. My first encounter with
structural equation analysis was while working with Bob Dwyer. Neil Ritchey helped refine those
first encounters. My thinking about empiricism, survey model testing and structural equations has
been influenced by the writings of Jum Nunnally; James Anderson, Richard Bagozzi, David Gerbing
and John Hunter; Peter Bentler; Kenneth Bollen; Michael Browne and Robert Cudeck; John Fox and
Michael Sobel; Leslie Hayduk; Karl Jöreskog and Dag Sörbom; and John Kenny, to name a few.
This book is on my academic web site for several reasons. A web version seems to be more
useful than a printed version. It allows me to direct e-mail inquiries about survey model testing to
detailed material in this book that the inquirer can immediately access. An online monograph also
enables me to "publish" this material comparatively rapidly, although I suspect that potential readers
are less aware of it than they would be if it were in a "hard copy" book. I can also use my recent
research and experiences to periodically extend and revise the book without the rigors of formally
publishing a revised edition. Because the book is searchable using the "Find" function available in
Microsoft WORD, a web version also may be more easily searched and thus more useful than a
printed version. The book is in Microsoft WORD, rather than HTML or Adobe Acrobat, as a
compromise between formatting and download times. However, it still may seem to download rather
slowly. It was also auto-formatted from WordPerfect to WORD, so in addition to my own errors of
omission and commission there may be reformatting errors as well.
Although the book is copyrighted, you are welcome to print parts or all of it for your personal
use. My only request is that you remember to cite the monograph when that is appropriate (the APA
citation format for the book is, Ping, R.A. (2003b), Testing Latent Variable Models with Survey
Data, [on-line monograph], www.wright.edu/~robert.ping/lv/toc1.htm).
Finally, if you see anything you like or dislike, any errors or things you would like to see
included or explained better, etc., please e-mail me with the details. Thank you in advance for your
comments, and Bon Appetit.
Robert A. Ping, Jr.
rping@wright.edu
 2004 Robert A. Ping, Jr. 3
TESTING LATENT VARIABLE MODELS WITH SURVEY DATA
INTRODUCTION
This monograph concerns the process of testing theoretical models (i.e., hypothesis testing)
involving unobserved or latent variables3 and survey data. These unobserved or latent variables have
multiple item (observed) measures, and diagrammatically an unobserved or latent variable is
'indicated' by (i.e., 'points to') its items (e.g., in Figure A the unobserved or latent variable U has the
observed variables or items u1, u2, u3, u4 and u5). The suggestions in the monograph for the process
of testing theoretical models containing these variables are based on a review of substantive articles
in the Social Sciences that qualitatively judged compliance with generally accepted procedures for
testing these models using survey data, and a review of the recent social science methods literature.4
The book selectively extends recent results in the methods literature, and proposes novel applications
of several others. For example, it suggests several alternatives to omitting items in structural
equation analysis to improve model-to-data fit, that could be especially useful for older measures
established before structural equation analysis became popular. The monograph contributes an
accessible discussion of the well-known problems of "inadmissible solutions" in structural equation
analysis (i.e., parameter estimates that do not make sense) and model-to-data fit, with suggested
remedies, including an additional procedure for achieving model-to-data fit using structural equation
analysis, and it provides a suggestion for easily executed pretests using scenario analyses.
It also discusses matters that may be well-known to methodologists but may not be as well
known to substantive researchers, such as a discussion of error-adjusted regression, the use of single
summed indicators in structural equation analysis, and the use of a nonrecursive model to investigate
directionality or causality. The monograph provides accessible discussions of several overlooked but
3. The book assumes criterion (i.e., ordinal, continuous. etc.) dependent or endogenous variables
rather than classification (e.g., categorical) variables (i.e., latent class models are not discussed).
4. I reviewed articles reporting theoretical model tests involving survey data in the Journal of
Consumer Research, the Journal of Marketing Research, the Journal of Marketing, Marketing
Science, the Journal of the Academy of Marketing Science, the Journal of Retailing, the Journal
of Personal Selling and Sales Management, and the Journal of Business Research from 1980 to
the present. I also selectively reviewed methodological articles in the Psychological
Bulletin/Psychological Methods, Psychometrika, Multivariate Behavioral Research, Sociological
Methodology, Sociological Methods and Research, the Journal of Marketing Research, and
Quality and Quantity, for this same period. I had planned to complete similar reviews of the
major journals in Psychology, Education, Political Science, Sociology, etc., but shortly after
starting those reviews it became apparent that research in Marketing could be argued to be
representative of the good and bad practices involved in testing theoretical model involving
survey data in the Social Sciences. The resulting sub-sample of social science articles also may
restrict any raised hackles resulting from my critical remarks about model testing practices in the
sample to raised hackles in Marketing.
 2004 Robert A. Ping, Jr. 4
valuable statistics such as Average Variance Extracted (AVE) and Root Mean Squared Error of
Approximation (RMSEA), and it suggests an estimator of AVE that does not rely on structural
equation analysis. It also selectively discusses recent advances in latent variable interactions and
quadratics, and provides a rationale for the more frequent inclusion of interactions and quadratics in
survey model tests. The book calls for additional attention to measure consistency, and thus modelto-data fit, in structural equation analysis, and argues for higher thresholds for acceptable reliability
based on average extracted variance. It also (re)emphasizes the implications of reliability and facets
of validity as sampling statistics with unknown sampling distributions.
Finally, it provides numerous explanations and examples,5 and overall it is intended as a
contribution to continuous improvement in the use of generally accepted procedures for theoretical
model tests involving unobserved variables and survey data in the Social Sciences.
THEORETICAL MODEL TESTING STUDIES
While not everyone would agree, Bollen (1989:268) states the objective of theoretical model
testing studies that involve unobserved variables and survey data: "In virtually all cases we do not
expect to have a completely accurate description of reality. The goal is more modest. If the model...
helps us to understand the relations between variables and does a reasonable job of matching
(fitting) the data, we may judge it (the model) as partially validated. The assumption that we have
identified the exact process generating the data would not be accepted" (the underling and
parenthesized words are mine). My own experience and the qualitative review mentioned earlier
seem to suggest that the objective of theoretical model testing is to attempt to disqualify (i.e.,
disconfirm) antecedents (i.e., independent or exogenous variables) of the focal constructs in a
theoretical model. Anecdotally, theoretical model testers usually approach this task with the hope
that few antecedents will be disconfirmed. Perhaps as a result, one characterization of theoretical
model testing might be that it is a theory-driven search for the important antecedents of one or more
focal constructs.
Nevertheless, social science researchers appear to agree that the process of testing or validating
theoretical models with survey data is addressed by first determining the adequacy of the measures of
the unobserved variables in the model, then determining the reasonableness or adequacy of the
hypothesized model. The first of these, measure adequacy, is typically determined using conceptual
definitions of the unobserved or latent variables, along with observed variables or items that "tap
into" or measure these unobserved or latent variables, and, increasingly, model-to-data fit and
parameter estimates from measurement models that utilize structural equation analysis. Model
adequacy is determined using hypotheses, and model-to-data fit and parameter estimates from
structural models that also utilize structural equation analysis.
5. Because my Social Science research involves theoretical research in Marketing, the book's
examples are from studies conducted in that venue. Nevertheless, because the processes and
conventions of theoretical model testing using survey data are generally the same across the
Social Sciences, the examples should be understandable and useful throughout the Social
Sciences.
 2004 Robert A. Ping, Jr. 5
SIX STEPS IN THEORETICAL MODEL TESTING These researchers also appear to agree that
specifying and testing models using unobserved variables with multiple item measures of these
unobserved variables and survey data involves
i) defining model constructs,
ii) stating relationships among these constructs,
iii) developing appropriate measures of these constructs,
iv) gathering data using these measures,
v) validating these measures, and
vi) (in)validating the model (i.e., testing the stated relationships among the constructs).
However based on articles I reviewed (see Footnote 4), there also appears to be considerable latitude
in some cases, and confusion in others, regarding how these steps are carried out in model tests using
latent variables and survey data.
For example, in response to calls for increased psychometric attention to measures in theoretical
model tests (e.g., in Marketing, Churchill, 1979; Churchill and Peter, 1984; Cote and Buckley, 1987,
1988; Heeler and Ray, 1972; Peter, 1979, 1981; Peter and Churchill, 1986; among others), reliability
and validity now receive more attention in these tests, when compared to the results of the study.
However, the reviewed articles exhibited significant variation in what constitutes an adequate
demonstration of valid and reliable measures when unobserved variables and survey data were
involved. For example in some articles, steps v) (measure validation) and vi) (model validation)
involved separate data sets. In other articles a single data set was used to validate both the measures
and the model. In some articles the reliabilities of measures used in previous studies were reassessed.
In other articles, measures that were reliable in a previous study were simply assumed to be reliable
in all subsequent studies, including the present study (i.e., without "confirmation"6 of reliability in
the present study). Similarly, in some articles many facets of validity for each measure were
examined, even for previously used measures. In other articles few facets of measure validity were
examined, and validities for existing measures were also assumed to be constants (i.e., once judged
to be valid a measure is assumed to be valid in all subsequent studies).
Further, methodologists in the Social Sciences have long warned about regression's potential for
coefficient bias and sample-to-sample coefficient variation (inefficiency) because of measurement
error in the independent and dependent variables (Bohrnstedt and Carter, 1971; see Rock, Werts,
Linn and Jöreskog, 1977; Warren, White and Fuller, 1974; and demonstrations in Cohen and Cohen,
1983). Nevertheless based on the articles I reviewed, regression still appears to be acceptable in some
venues as an estimation technique for survey data with variables that contain measurement error. In
addition, although many of the studies I reviewed acknowledged the risk of generalizing from a
single study,7 in some cases there was little subsequent concern about the appropriateness or amount
6. In this case "confirmation" is failure to disconfirm, not necessarily that what is confirmed is
always true (see Popper, 1959).
7. Generalizing from a single study involves recommending interventions based on a single
study, which ignores the possibility that "confirmed"/not disconfirmed hypotheses could be
disconfirmed in a subsequent study, and disconfirmed associations could be "confirmed" (i.e., not
 2004 Robert A. Ping, Jr. 6
of generalizing from a single study. Because there are other examples, major and minor, such as little
apparent concern in some cases about violations of the assumptions underlying the estimation
techniques used in latent variable model tests using survey data (e.g., the use or ordinal data with
covariant structure analysis, which assumes continuous data), it seems fair to say there appear to be
fewer generally accepted (i.e., widely accepted) principles of model validation using unobserved
variables and survey data than there could be.8
Fortunately there have been important advances in (in)validating latent variable models involving
survey data.9 These include new results in developing, testing and evaluating multiple item measures,
and estimating models employing these measures. However, some of these developments have
appeared in literatures not widely read or easily understood by all substantive researchers in the
Social Sciences.
Thus, this monograph is intended for these researchers, and one of its objectives is to selectively
identify areas for continuous improvement in the process of testing latent variable models involving
survey data.
The first step in model testing using latent variables and survey data is discussed next.
STEP I IN LATENT VARIABLE-SURVEY MODEL TESTING-DEFINING MODEL CONCEPTS
Models with unobserved or latent variables have multiple item (observed) measures of the
unobserved variables because we observe only indirect evidence or indications of these unobserved
or latent variables. For example, we can directly observe or measure a concept or construct 10 such as
disconfirmed) in a future study. This can occur in many ways, including omission of important
predictor/independent/exogenous variables, the use or ordinal data with structural equation
analysis which assumes continuous data, etc.
8. An earlier version of this monograph commented on the well-known effects of this apparent
latitude in studies involving latent variables and survey data. However, informal reviewers
reacted negatively to statements such as "authors and reviewers... may apply idiosyncratic, rather
than generally accepted, standards in evaluating these studies," and, "this can produce an
unnecessarily prolonged and unpredictable review processes... in which errors of acceptance and
rejection can be higher than they ought to be." They stated that these problems were well known
and readers need not be reminded of them.
9. While survey model testing actually attempts disconfirmation of the proposed model, and thus
"invalidating the model" is perhaps the more appropriate phrase, I will use the phrase "validating
the model" in the monograph.
10. The terms concept and construct will be used to connote anything that can be measured. A
construct that is not directly observable but that has observable indicators is also termed an
unobserved or latent variable.
 2004 Robert A. Ping, Jr. 7
the ambient temperature, but we can measure only indirect evidence or indications of the concept or
construct "the attractiveness of the best alternative relationship." Thus, it is the practice in the Social
Sciences to observe several indications, or what is termed indicators, of each latent variable using
multiple-item measures. In the diagrammatic parlance associated with structural equation analysis
arrows point from the unobserved or latent variable to the observed indicators to suggest that the
latent variable makes manifest or 'causes' the observed variables or indicators (i.e., in Figure A as the
latent variable U changes the observed indicators ui change).
TYPES OF CONSTRUCTS
Unobserved constructs involved in survey model tests are of two general types: previously
investigated constructs and novel (i.e., new) constructs. (Interesting) model tests involving
previously investigated constructs usually involve linking these constructs in novel (i.e., new) ways.
For example, while relationship Satisfaction and Switching Cost (the "costs" of ending the current
relationship and establishing a replacement one) should obviously be (negatively) associated with
relationship Exiting, what is the association between Satisfaction and Switching Cost (i.e., does
Satisfaction perceptually increase Switching Cost? Does Switching Cost perceptually decrease
Satisfaction? Could both these latent variables affect each other?).
Because survey models seldom explain large amounts of variance in focal dependent or
endogenous variables, novel (i.e., new) constructs that explain additional, and hopefully large,
amounts of variance in focal dependent or endogenous variables are usually of considerable interest.
For example, Switching Costs might be re-conceptualized as two constructs, the cost to end the
incumbent relationship, and the cost to establish a replacement relationship. This might be
efficacious because there may be venues where a replacement relationship is not always desired.
Novel dependent or endogenous constructs are also occasionally proposed. These novel
constructs can be specified in the model as being "pointed to" by other model constructs and not
pointing to other constructs. For example the construct relationship Neglect (mental or emotional
relationship "exiting" without physical exiting--i.e., lack of emotional commitment to the
relationship) has been proposed in several contexts as an alternative to relationship Exit. A novel
endogenous construct can also be proposed as a mediator between two other constructs (i.e., the
novel construct is argued to belong on the path between two other previously investigated constructs,
and thus the novel construct "intervenes" between the two previously investigated constructs). For
example, Neglect may properly belong between Satisfaction and Exiting (i.e., Dissatisfaction
increases Exiting by first increasing Neglect, which in turn increases Exiting).
Several additional observations concerning the intrinsic nature of constructs may be of interest.
Unfortunately some constructs may be more "important" than others. Because we appear to be in an
"age of relevant research" these days, models, and thus constructs, that address "real-world"
problems, are usually deemed to be more important than those which do not. For example,
"atmoshperics" (e.g., lighting, music, aromas, etc.) is legitimate stream of research that could be,
perhaps unfairly, characterized as less-, to un-, important in many contexts (perhaps even in the
contexts in which atmospherics research is conducted). As a further example however, this could
 2004 Robert A. Ping, Jr. 8
change dramatically if atmospherics were found to be a "driver" of (i.e., strongly associated with)
some important outcome such as world peace or mental health.
Some constructs are also "politically correct" (i.e., fashionable), and thus of comparatively
greater interest. Unfortunately, there are constructs that are "politically incorrect" (i.e., out-offashion), and this "unfashionableness" may have little to do with explanation or prediction. For
example, I doubt that many would disagree that Matina Horner's Fear-of-Success (Horner 1968) is
just "out" these days. (Parenthetically, I will resist the urge to pontificate on the importance to
science of (circumstantially) unimportant and unfashionable science--my point is that not all
constructs are created equal.)
As previously mentioned, unobserved constructs are also called unobserved or latent variables
when they are accompanied (e.g., specified) by multiple items that measure (i.e., indicate) them. The
construction of the set of indicators of a latent or unobserved variable (i.e., the multiple items in a
measure of a latent or unobserved variable) should be guided by a written definition of the construct
or concept. These definitions were as a rule clearly stated in the articles that were reviewed. Because
these matters have received attention previously (e.g., Bollen , 1989:180 and Churchill, 1979), later
in this section I will simply summarize the two definitional requirements for the unobserved
variables typically involved in theoretical model tests: a conceptual definition and an operational
definition.
However, because several types of concepts or constructs appear to be underutilized in the Social
Sciences and they offer opportunities to add to the richness and descriptiveness of latent variablesurvey models, I will comment on second-order constructs, and interactions and quadratics. To
discuss second-order constructs I will begin with the notion of a first-order construct.
FIRST-ORDER CONSTRUCTS
A first-order concept or construct is a latent variable that has observed variables (i.e., the items in
its measure) as indicators of the construct or latent variable (e.g., U in Figure A). These constructs
were ubiquitous in the articles I reviewed. The relationship between indicators and their latent
variable in a first-order construct typically assumes the construct 'drives' or 'causes' the indicators
(i.e., the indicators are observable instances or manifestations of their unobservable construct, and a
diagram of the construct and its indicators would show the construct specified or connected to the
indicators with arrows from the construct to the indicators-- a reflexive relationship, see Bagozzi,
1980b, 1984 and Figure A). Less frequently, the indicators 'drive' the construct (i.e., the indicators
define the construct rather than being various instances of a construct, and a diagram of the construct
and its indicators would show the indicators connected to the construct with arrows from the
indicators to the construct-- a formative relationship, see Fornell and Bookstein, 1982). We will
restrict our attention to reflexive relationships between latent variable and their indicators.
In latent variable-survey models analyzed using regression, unidimensional items (i.e., items
having a single underlying concept/construct or unobserved/latent variable) are summed, then this
sum is analyzed in place of the construct (however see the cautions concerning analyzing latent
variables with regression in Step VI-- Validating The Model, Violations Of Assumptions,
Regression). In structural equation analysis the items' or indicators' relationship with their construct
is explicitly specified or modeled using structural equation analysis software such as AMOS, EQS
 2004 Robert A. Ping, Jr. 9
and LISREL, and in effect both the construct and the indicators are analyzed (however, see Step V-Single Indicator Structural Equation Analysis for a single summed indicator approach used in
structural equation analysis).
SECOND-ORDER CONSTRUCTS
A second-order construct is an unobserved or latent variable that has other unobserved variables
or constructs as its 'indicators.' For example in Figure J in Appendix J, the second-order construct F
had the 'indicator' constructs B, C and D that in turn had b's, c's and d's as observed indicators,
respectively. A second-order construct can be though of as a set of factors in an exploratory factor
analysis that is not particularly orthogonal. That is, when the items for each of these factors are
summed, an exploratory factor analysis of the resulting summed items is unidimensional. To use a
second-order construct in regression (e.g., for exploratory purposes-- see the cautions about
regression using variables measured with error in Step VI-- Violations of Assumptions), the items in
each first-order construct (factor) should be unidimensional. In addition the second-order construct
(factor) should be unidimensional using exploratory factor analysis with each first-order construct's
summed items as a single item per construct, and the second-order construct should be face or
content valid using the first-order constructs as 'items' (see Appendix J for an example).
While infrequently observed in the articles I reviewed, second-order constructs have been used in
survey models. For example, in Dwyer and Oh's (1987) study of Environmental Munificence and
Relationship Quality in inter-firm relationships, the second-order construct Relationship Quality had
the first-order constructs Overall Satisfaction, Trust, and Minimal Opportunism as indicators (see
Bagozzi, 1981a; Bagozzi and Heatherton, 1994; Gerbing and Anderson, 1984; Gerbing, Hamilton
and Freeman, 1994; Hunter and Gerbing, 1982; Jöreskog, 1970; and Rindskopf and Rose, 1988 for
accessible discussions of second-order constructs). Each of these first-order 'indicator' constructs
then had its respective observed indicators. Specifying second-order constructs, such as Relationship
Quality, with first-order constructs obviously simplified the structural paths of the model, and it
appeared to provide a richer description of the consequences of Environmental Munificence. A
similar approach was taken in Ping's (1997) study of the relationship between the cost of exiting a
buyer-seller relationship and Voice (e.g., complaints) in inter-firm relationships. Cost-of-exit was
itemized using several unobserved constructs (the attractiveness of alternative relationships,
Relationship Investment, and Switching Cost) which themselves had observed indicators.
As these examples may suggest, a second-order construct can be used to combine several related
constructs into a single higher-order construct to simplify the structural (i.e., hypothesized) paths in a
model. A second-order construct can also be used as an alternative to omitting items of a multidimensional measure to obtain model-to-data fit in structural equation analysis. This can be useful
with established measures, developed before the advent of structural equation analysis, that turn out
to be multi-dimensional using structural equation analysis (see Gerbing, Hamilton and Freeman,
1994). In addition, a second-order construct can be used to account for types of error other than
measurement error in structural equation analysis (see Gerbing and Anderson, 1984).
INTERACTIONS AND QUADRATICS
 2004 Robert A. Ping, Jr. 10
Unlike experiments analyzed with ANOVA where interactions (e.g., XZ in
Y = b0 + b1X + b2Z + b3XZ + b4XX + e
= b0 + b1X + (b2 + b3X)Z + b4XX + e )
(1
(1a
and quadratics (e.g., XX in Equation 1) are routinely estimated to help interpret significant main
effects (i.e., the X-Y and Z-Y associations), interactions and quadratics were rarely seen in the
articles reviewed. This may have been because authors have confused difficulties in detecting these
variables with their likelihood of occurrence (see Podsakoff, Tudor, Grover and Huber, 1984; also
see McClelland and Judd, 1993). In addition, until recently interactions and quadratics in latent
variable models have been difficult for researchers to specify and interpret (see Aiken and West,
1991; Ping, 1995, 1996a).11 Further, because they are mathematical, rather than mental, constructs,12
and have indicators that are products of observed variables (rather than indicators/items that can
actually be observed), interactions and quadratics may be judged by some substantive researchers as
inappropriate in latent variable models.13
Nevertheless, authors have called for more thorough investigations of interactions and quadratics
in survey research (e.g., Aiken and West, 1991; Blalock, 1965; Cohen, 1968; Cohen and Cohen,
1975, 1983; Darlington, 1990; Friedrich, 1982; Kenny, 1985; Howard, 1989; Jaccard, Turrisi and
Wan, 1990; Neter, Wasserman and Kunter, 1989; Pedhazur, 1982). Their argument is the same as
that used in ANOVA: failing to consider the possibility of interactions and quadratics in the
population model is likely to lead to erroneous interpretations of the study's results.
To explain, in Equation 1 the actual coefficient of Z is given by (b2 + b3X) (see Equation 1a). The
statistical significance of this moderated (factored) coefficient of Z could be very different from the
statistical significance of the coefficient of Z in Equation 1 without the XZ variable (i.e., Y = b0' +
b1'X + b2'Z ) (see Aiken and West, 1991). Specifically, if the interaction is significant (i.e., b3 is
significant) the coefficient of Z in Equation 1 without the XZ variable, b2', could be nonsignificant
while the moderated coefficient of Z due to the significant interaction, b2 + b3X, is significant over
11. There are a variety of techniques for detecting interactions and quadratics. However, because
many do not produce structural coefficients (e.g., b3 and b4 in Equation 1) that suggest the
direction and strength of a significant interaction or quadratic, and they do not permit detailed
interpretation such as that shown in Appendix C, this discussion will concentrate on techniques
that estimate structural coefficients for interactions and quadratics.
12. While not everyone would agree, mental constructs are variables such as affect, cognitions
and behavioral intentions. They are typically unobservable, and thus require observed indicators
(i.e., a measure) and are also latent variables. Mathematical constructs are variables that are
algebraic combinations of other constructs.
13. This research treats interactions as mathematical constructs. Because they are mathematical,
interactions and quadratics can, for example, be algebraically factored, as shown in Equation 1a.
However, because they are also latent variables (i.e., they have indicators), their psychometrics
(e.g., reliability and validity) are, or should be, important.
 2004 Robert A. Ping, Jr. 11
part(s) of the range of X in the survey (that b2 + b3X is significant over part(s) of the range of X in
the survey is guaranteed by the significance of b3--see Table C2 in Appendix C). Thus, failing to
include an interaction in a model when it is present in the population model could lead to a
misleading interpretation of the Z-Y association. While strictly speaking a nonsignificant b2' implies
the Z-Y association is disconfirmed, it is clearly not the case with a significant XZ interaction (i.e., b3
is significant) that Z is never associated with Y in the study. The Z-Y association simply depends on
the level of X in b2 + b3X. This has several implications, including that b2' could be observed to be
variously nonsignificant in one study then significant in another, and thus an interaction may explain
inconsistent findings (i.e., nonsignificant in one study but significant in another) in previous studies.
Alternatively b2' could be significant in the study while (b2 + b3X) could be nonsignificant over
part of the range of X in the study. In this event, failing to include the population interaction could
produce a false "confirmation"14 of the Z-Y association: the significant Z-Y association could be
nonsignificant over part(s) of the range of X in a study. This error is especially insidious in survey
model tests. Many of the studies I reviewed provided interventions (e.g., recommendations to
practitioners) based on significant associations in the study, many of which seemed to me could
easily have been contingent associations (i.e., associations moderated by an interaction).
The algebra and implications of failing to consider the possibility of a population quadratic are
similar.15 Thus, care should be taken to consider interactions and quadratics in survey model testing
studies. Obviously they should be considered when theory postulates their existence. However, the
above suggests that they should be considered in post-hoc probing (i.e., after the hypothesized model
is estimated), as is done in experiments analyzed with ANOVA, to aid either in interpreting and
providing the implications of significant associations, or as a possible explanation for hypothesized
but observed nonsignificant associations (see Appendix C for an example), or inconsistent results
across studies.
There has been considerable progress in estimating interactions in survey data using regression
(e.g., Aiken and West, 1991; Denters and Puijenbroek, 1989; Feucht, 1989; Heise, 1986; Jaccard,
Turissi and Wan, 1990; Ping, 1996b; Warren, White and Fuller, 1974) and structural equation
analysis (see Bollen, 1995; Hayduk, 1987; Jaccard and Wan, 1995; Jöreskog and Yang, 1996; Kenny
and Judd, 1984; Ping, 1995, 1996a; Wong and Long, 1987) (also see Appendix A). However,
estimating interactions using structural equation analysis is more difficult than using Ordinary Least
Squares regression (Aiken and West, 1991). Nevertheless latent variable interactions have been
estimated using structural equation analysis (e.g., Fullerton and Taylor, 2002; Hochwarter, Ferris and
Perrewe, 2001; Johnson and Sohi, 2003; Lee and Bae, 1999; Lee and Ganesh, 1999; Masterson 2001;
Osterhuis, 1997; Singh, 1998). In addition, interactions between first-order and second-order latent
variables have been estimated with structural equation analysis (see Ping, 1999). I will discuss the
14. Again, "confirmation" means failure to show that something is false, not necessarily that
something is true.
15. Equation 1 could be re-factored into Y = b0 + (b1 + b4X)X + b2Z + b3XZ + e, and thus the XY association is moderated or conditional on the existing level of X at which the association is
evaluated. Quadratics are discussed in more detail later.
 2004 Robert A. Ping, Jr. 12
estimation of interactions and quadratics later.
DEFINING CONCEPTS
Concepts are defined using words (e.g., "Overall Satisfaction" is the subject's overall attitude
toward an object). Then they are measured using one or more observed variables or items that "tap
into" or are instances of (i.e., fit) these definitions. Thus, conceptual definitions, definitions of the
concepts in the model, are required to provide meaning for the verbal label attached to each construct
in a model (e.g., verbal labels such as "Overall Satisfaction," "Equity," "Solidarity," "Intelligence,"
etc.). For example, a concept with the label "Irretrievable Investments" could have the conceptual
definition, "the subject's relationship investments that would be lost if the relationship were ended."
Conceptual definitions are obviously important for judging the adequacy or content (face) validity of
the observed items used to measure the concept--as we will see, these items will be judged to be
instances or indicators of the concept, and their adequacy or content validity will be judged based on
how well these items fit the conceptual definition.
Operational definitions of concepts can be used in addition to conceptual definitions. Conceptual
definitions can be general, while operational definitions can allow for contextual specificity or other
contingencies in a particular study. For example, Exiting (a label) could be conceptualized as, or
have the conceptual definition of, physically ending the relationship. However, this concept could be
operationalized or measured in many ways depending on the study context, the population being
sampled, the difficulty of tracking down subjects who have exited, etc. Thus the concept of exiting
might be operationalized as Exit-propensity, and have an operational definition of "the intention to
end the relationship."
Several observations pertaining to conceptual and operational definitions, and second-order and
interaction/quadratic concepts, may be of interest. In effect conceptual definitions describe a verbal
label such as "Equity" in terms of how the concept is to be measured (e.g., "Equity is the perceived
equality of relationship rewards and costs").
Clearly an operational definition should be consistent with its conceptual definition, and items
should be instances of or "tap into" (i.e., fit) their operational and/or conceptual definitions.
A second-order construct should be conceptually and operationally defined because the validity
of a second-order construct is, or should be, as important as the validity of a first-order construct.
Providing conceptual definitions for interactions or quadratics is difficult because, as previously
mentioned, they are mathematical concepts rather than mental constructs. However, operational
definitions of interactions/quadratics should be provided (e.g., "Z's moderation of the X-Y
association was operationalized as XZ, the product of X and Z) because there could be many
operationalizations of an interaction (e.g., X/Z, etc., see Jaccard, Turissi and Wan, 1990).
IDENTIFYING IMPORTANT ANTECEDENTS
Ideally, a model to be tested should include all the antecedents of each dependent or endogenous
variable. However, theoretical models in the social sciences typically account for only a small
portion of the variation in their dependent variables, and thus they seldom if ever include all the
antecedents of the dependent or endogenous variables in the model. The models I reviewed were no
 2004 Robert A. Ping, Jr. 13
exception. There usually are other antecedents of each dependent variable that are not included in the
model because they are simply not known. While accounting for all antecedents of every dependent
variable is frequently impossible, especially in early stages of theory development, not including
important antecedents (i.e., ones that are significantly related to the dependent variable, and that are
also correlated with the other independent variables included in the model) biases (i.e., inflates or
deflates) observed associations (see Duncan, 1975).
This places a great burden on theoretical model testing for the inclusion of important antecedent
variables. Knowing which unstudied antecedents are important obviously requires considerable
researcher knowledge and time. As a result, one objective of pretesting a model, which is discussed
later, should be to address the adequacy of the antecedents in the model (i.e., the amount of variance
they explain). This is important to theory testing because a model that does not explain much
variance in a dependent variable is vulnerable to subsequent research that includes variables which
together explain more variance in a dependent variable: disconfirmed (i.e., non significant) results in
the low explained variance model may later turn out to be significant because important antecedents
were not included in the original model (or vice versa).
Parenthetically, the requirement to adequately model the important antecedents of dependent
variables is not contrary to the notion of model parsimony because parsimony is intended to exclude
variables that are comparatively un important.
SUMMARY AND SUGGESTIONS FOR STEP I-- DEFINING MODEL CONCEPTS
Summarizing the above discussion, because of their importance in establishing the meaning of a
construct and their importance to evaluating the adequacy of the observed items that comprise the
measure of an unobserved construct (i.e., how well they "indicate" or are instances of the construct),
conceptual definitions should be carefully stated for each construct in latent variable model tests
involving survey data. Operational definitions could also be provided to allow for contextual
specificity of the sample or other contingencies in the study. These definitions should also be
provided for second order concepts, and interactions and quadratics.
To reduce bias in testing the associations in a model, and thus the likelihood of hypothesis false
rejection (Type I) or false acceptance (Type II) errors, the important antecedents of each dependent
variable should be specified in the model to be tested (i.e., the model should exhibit comparatively
high explained variance). While there is no hard and fast rule, significant associations that turn out to
explain less that 10% of the variance in a dependent or endogenous variable should probably be
ignored when interpreting the model test results because these associations (and the nonsignificant
associations with the dependent or endogenous variable) are likely to be materially different when
other important antecedents of this dependent or endogenous variable are included in the model.
For this reason (i.e., the potential for inference errors), interactions and quadratics should be
considered even if theory is mute on their possible existence (as they are in ANOVA studies). At a
minimum they should be estimated on a post hoc basis when an hypothesized association turns out to
be non significant, as a possible explanation for this lack of significance. I will discuss post-hoc
probing for interactions and quadratics later.
Survey model tests involve new constructs, previously tested constructs, or a combination of the
two. Based on the articles reviewed, models with previously tested associations should also include
 2004 Robert A. Ping, Jr. 14
untested associations (i.e., they should contain new constructs and/or untested associations between
previously tested associations).
Based on the articles reviewed, survey models containing "important" constructs may be of more
interest than models with constructs that are currently considered less "important." In addition, some
constructs, and thus the survey models that contain them, are more "politically correct" (i.e.,
fashionable) than others, and thus these models are of comparatively greater interest.
Because second-order constructs can combine dimensions of a multidimensional construct and
produce a more parsimonious structural model, among other reasons, plausible second-order
constructs should also be considered.
(end of section)
 2004 Robert A. Ping, Jr. 15