




Latent Variable Research 
Updated 9/24/14
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FOREWORDThis web site contains research concerning o Latent variables (LV's), and LV interactions (e.g., XZ) and quadratics o Testing these models. It is intended for Ph.D. students and researchers who are somewhat familiar with LV's NEWS: "Workarounds" for an unbiased estimate of an "endogenous" interaction or quadratic An annotated suggested format for substantive papers that may reduce the likelihood Specifications for an interaction form other than XZ are available by email. This may
not sound like much, but XZ is not the only form an interaction can take
X/Z and XZ^{2} are also interactions. So, a nonsignificant XZ may not mean an
hypothesized moderation (interaction) is disconfirmed.
Specifically, if the XZ>Y association is NS there still may be a significant
interactionit just doesn't have the form "X times Z". Experience suggests
that X/Z, XZ^{2} or other interaction forms may be significant when the XZ
is not.
Comments on the use of regression to test an hypothesized interaction are available.
(Please see "Why are reviewers complaining about the use of moderated multiple
regression in my paper? in Questions of the Moment.)
The suggestions for estimating an endogenous interaction have changed. An EXCEL A web site relocation has obsoleted many citations on this web site (and published papers) that reference http://home.att.net/~rpingjr/... Since that internet address no longer exists, these citations should be changed to http://www.wright.edu/~rping/... Coming Attractions: o Suggestions regarding reviewer comments about "Implications," and "(Management) o An EXCEL template that provides a simple and straightforward tabular interpretation Recent Additions and Changes in this Web Page (indicated by "New," "Revised" or "Updated" below): o Suggestions for estimating a mixed SEM model with the customary "Reflective" 

All the material on this web site is copyrighted, but you may save it and print it out. My only request is that you please cite any material that is helpful to you, either as a "book" (the APA citation for this website as a "book" is Ping, R.A. (2001). "Latent Variable Research." [online paper]. http://www.wright.edu/~robert.ping/research1.htm.), or using the individual citations for each of the papers, EXCEL templates, monographs, etc. shown below. Don't forget to Refresh: If you have visited this web site before, and the latest "Updated" date (at the top of the page) seems old, or, if you are actively estimating an interaction, etc., you may want to be sure you are viewing the Your questions are encouraged; just send an email to rping@wright.edu. Don't A Table of Contents or Index to this website is not available. With my apologies, 

INTRODUCTION 

Please Note: If you
are estimating an interaction involving one or
more endogenous variables please click herethings
are different in this case.
ALSO, please remember that interactions and quadratics should be hypothesized before data is ever collected. Please be aware that the only situations where one should search for significant interactions or quadratics after the data is collected are: 1) when one wishes
to explain a nonsignificant (NS) model association 2) where one
wishes to investigate an important significant
association(s) to see In the first case, any significant "suppressor" interaction XZ or quadratic ZZ could be offered as a possible explanation for the NS Z>Y association in the Discussion section of the paper. In the second case, any significant conditional (moderated) association could be the basis for noting that the significant Z>Y association was "supported" only for some levels of X in the study (see "Interpreting Latent Variable Interactions" in the SELECTED PAPERS...section below). Any of these moderated Z>Y association(s) discovered after the fact could be hypothesized in the Discussion section, for testing in a followup study, or replication (to investigate whether or not the "afterthefact" moderation simply was significant by chance in the present study) (see "Hypothesized Associations and Unmodeled Latent Variable Interactions/Quadratics: An FTest..." in the SELECTED PAPERS...section below for more). In absence of situations 1 or 2 above, hunting for significant interactions or quadratics, that were not hypothesized before the data was collected is considered "poor science." It can tempt one to add an interaction or quadratic hypothesis to the body of the paper as though it were hypothesized before the data was collected. This changes a confirmatory study into an exploratory study. Specifically, adding an interaction or quadratic that was discovered after the fact, changes one's "hypothesesbeforefirsttest" modeltest (confirmatory) study into an exploratory study where part of the model was unknown before data collection. Again, the proper approach is to put any interaction(s)/quadratic(s) discovered after the data was collected in the Discussion section, noting that they were discovered after the data was colected, and arguing for their disconfirmation in the next study. 

Questions of the Moment  
"What about the alternative specifications for a Latent Variable (LV) interaction?"  
An informal
review in 2005 of substantive Social Science journal
articles written since Kenny and Judd's (1984)
seminal proposal for specifying LV interactions
and quadratics found that the most frequently
encountered specifications for LV interactions in
substantive articles were: Jaccard and Wan (1995)
(which specifies a 4productindicator subset^{1}
of the Kenny and Judd interaction (product) indicators
to avoid the modeltodata fit problems that occur
when all of the Kenny and Judd indicators are
specified46 citations); Mathieu, Tannenbaum
and Salas (1992) (which has not been formally
evaluated for possible bias and inefficiency (see
Cortina, Chen and Dunlap 2001 for
other difficulties)51 citations); and Ping
(1995) (41 citations). (See FAQ's A, B and C
in the Frequently Asked Questions
section below for more.) ^{______________ 1} Unfortunately, in theoretical model tests, deleting ("weeding" out) all but 4 of the Kenny and Judd product indicators to attain modeltodata fit raises many questions, including, what is the reliability and validity of the resulting 4item interaction or quadratic? Reliability is necessary for the validity of all the LV's in a model, but the reliability of an interaction specified with nearly all of its indicators absent is unknown. (The formula for the reliability of the interaction of X and Z assumes XZ is operationally (unweeded) X times (unweeded) Z.) The face or contentvalidity of a 4item interaction or quadratic also is questionableif nearly all the indicators of X and Z are not represented in the itemization XZ, is XZ still the LV (unweeded) X times (unweeded) Z?. Further, it is easy to show that in realworld data a weeded XZ's structural coefficient varies with its indicators. Unfortunately, the "best" set of four indicators is unknown. Finally, an interaction with weeded Kenny and Judd product indicators cannot be "factored" to produce detailed interpretation because XZ is no longer (unweeded) X times (unweeded) Z. 

"Is there an example that shows all the steps
involved in estimating a latent variable
interaction/quadratic?"


(Re vis ed) 
"How does one estimate (truly) categorical variables in a model with LV's?"  
In the popular
structural equation analysis modeling (SEM) software
(e.g., LISREL, EQS, Amos, etc.), the term "categorical
variable" usually means an ordinal variable (e.g., an
attitude measured by Likert scales), rather than a
"truly" categorical (i.e., nominal) variable (e.g.,
Marital Status, with the categories Single, Married,
Divorced, etc.), and typically there is no provision
for "truly" categorical variables. In regression, a
(truly) categorical variable is estimated using
"dummy" variables with regression through the origin,
but a similar approach currently is not available
using the popular SEM software.
However, a "mixed SEM" approach for estimating
categorical variables and LV's with their measurement
errors is available here.
(There also is a working paper
available via email with a "workaround" alternative to the paper on
the "hot spot" just provided (categorical3.doc)
that might be useful for a theory test.) 

"Why are reviewers
complaining about the use of moderated multiple
regression in my paper?" (Please click here for comments on this matter.) 

"How should PRELIS or similar "preprocessor"
software be used with LISREL, EQS, AMOS, etc.
to create interactions/quadratics?" 

"Why should Applied
Researchers be interested
in interactions/quadratics?" 

Contrary to customary
practice, interactions and quadratics also may
be important in applied
modelbuilding/researchmodel building in
econometrics, epidemiology, market response models,
biostatistics, etc.not to explain additional
variance in a target variable, but to better
understand, explain and predict important
relationships in the model. 

"Is there any way to improve Reliability or Average Variance Extracted (AVE) in an interaction?"  
Please see the comments below in, "Is there any way to improve Average Variance Extracted (AVE) in a Latent Variable X?" (Interaction/quadratic reliability and AVE are improved by improving reliability and AVE in X and Z.)  
"When theory proposes that Z moderates the X>Y association, but theory is mute about a Z>Y association, why does one still include the Z>Y association, in addition to the X>Y and XZ>Y associations, in the model?"  
Excluding the
Z>Y (or the X>Y) association when
XZ>Y is hypothesized to be significant, biases
all structural coefficients and standard errors in
the proposed model EVEN WHEN THE ZY ASSOCIATION
SHOULD NOT EXIST IN THE POPULATION. 

"How is a cubic latent variable (LV)
estimated?" 

Please see the paper on estimating a LV cubic in the SELECTED PAPERS... section (below). The "EXCEL templates..." section (below) also includes a template to assist in calculating LV cubic loadings, error variances, etc.  
"Why is my hypothesized interaction or
quadratic nonsignificant?" 

"Is there any way to improve Average
Variance Extracted (AVE) in a Latent Variable
X?" 

"Is there any way to speed up "item weeding"
to find a set of items in a multiitem measure
that fits the data?" 

"How
might a 'mixed interaction' XZ, where X is a
manifest/observed/continuous/singleindicator,
etc. variable (not a latent variable), be
estimated?" (Please click here for suggestions.) 

"Why
is my hypothesized interaction significant
using a 'median split' of the data, or a '2group
analysis,' but not significant
when specified in my model?"
(Please click here for comments on this subject.) 

"Why
are most (or all) of my hypothesized interactions not
significant?" (Please click here for more on this matter.) 

"What
is the Average Variance Extracted (AVE) for
a Latent Variable Interaction (or
Quadratic)?" (Please click here for comments on this topic, then please email meI have more suggestions.) 

"What
is the 'validity' of a Latent Variable
Interaction (or Quadratic)?" (Please click here for more on this subject.) 

"How does one remedy a "not Positive
Definite" message?" (Please click here for suggestions.) 

"Why are reviewers
complaining about the use of PLS in my paper?" (Please click here for comments on this topic.) 

(N e w) 
"How
are "Formative" (indicatorspointingto) Latent
Variables estimated with LISREL, EQS, AMOS, etc.?" 

This may seem
uninterestingformative variables are rare in the
Social Sciences. But, a formative respecification
of a difficult measure (e.g., one with inadequate
AVE, excessive weeding was required to attain
modeltodata fit and it could now be judged to be
face invalid, it has unacceptable discriminant
validity, etc.) may be the only alternative
to abandoning the measure. 

Frequently Asked Questions  
(CLICK ON A RED DOT) 
Frequently Asked Questions (FAQ's) about Latent Variable Interactions and Quadratics in survey data E.g., The answer to FAQ D, "How does one test hypothesized interactions or quadratics?" contains stepbystep instructions for Ph.D. students, and 





EXCEL TEMPLATES 





Spreadsheets for expediting the specification of Latent Variable Interactions, Quadratics and cubics; for "weeding" measures to attain model fit; for Latent Variable Regression, etc. 




(CLICK ON A RED DOT) 
For specifying a Single Indicator LV Interaction or Quadratic using Direct (LISREL 8, CALIS, etc.) or "2Step" estimation (LISREL, EQS, AMOS, etc.) (see Ping 1995, JMRa revised version of which appears below). The template also calculates LV Interaction or Quadratic Reliability and Average Variance Extracted (AVE). More about the template. 


For specifying a Single Indicator LV Cubic using "2Step" estimation (LISREL, EQS, AMOS, etc.) (please also see "Notes on Estimating Cubics and other 'Powered' Latent Variables" below). 


For "weeding" a multiitem measure so it fits the data (i.e., finding a set of items that fits the data, so the measure is internally consistent). In realworld data, there frequently are many subsets of a multiitem measure that will fit the data, and this raises the question, which of these subsets is "best" from a validity standpoint? The template helps find at least one more subset of items, usually with a maximal number of items (typically different from the one found by maximizing reliability, for example), that will 

For Kenny and Judd (1984) multiple indicator specification with LISREL, EQS, AMOS, etc. (see Ping 1996, Psych. Bull.; a revised version appears below). This approach is useful with a consistent subset of product indicators (see Chapter VIII.SxA Unidimensionalization, in the monograph, LATENT VARIABLE INTERACTIONS... below). More about the template . 

For Latent Variable Regression, a measurementerroradjusted regression approach to Structural Equation Analysis, for situations where regression 



BIBLIOGRAPHY  

A Bibliography on Latent Variable Interactions and Quadratics. 

ONLINE MONOGRAPHS 

(CLICK ON A RED DOT) 
LATENT VARIABLE INTERACTIONS AND QUADRATICS IN SURVEY DATA: A SOURCE BOOK FOR THEORETICAL MODEL TESTING (2nd Edition) About Latent Variable Interactions and Quadratics, and their estimation, with examples. Potentially of interest to Ph.D. students and researchers who conduct or teach theoretical model (hypothesis) testing using survey data. It includes a "fast start" section on estimating a latent variable interaction, a section on estimating multiple interactions and quadratics, how to interpret a significant interaction or quadratic, and pedagogical examples (232 pp.). The APA citation for this online monograph is Ping, R.A. (2003). Latent variable interactions and quadratics in survey data: a source book for theoretical model testing, 2nd edition. [online monograph]. http://www.wright.edu/~robert.ping/intquad2/ toc2.htm. 





TESTING LATENT VARIABLE MODELS WITH SURVEY DATA (2nd Edition) About the results of a large study of
theoretical model (hypothesis) testing practices
using survey data, with critical analyses,
suggestions and examples. Potentially of interest to
Ph.D. students and researchers who conduct or teach
theoretical model testing using survey data. Its
contents include the six steps in theoretical model
(hypothesis) testing using survey data; scenario
analysis; alternatives to dropping items to attain
modeltodata fit; inadmissible solutions with
remedies; interactions and quadratics; and
pedagogical examples (177 pp.). The APA citation for this online monograph is Ping, R.A. (2004). Testing latent variable models with survey data, 2nd edition. [online monograph]. http://www.wright.edu/~robert.ping/lv1/toc1.htm . 





SELECTED PAPERS ON LATENT VARIABLES, AND THEIR INTERACTIONS AND QUADRATICS 

(CLICK ON A RED DOT) 
"Notes on 'Used Data'Reusing a Data Set to Create A Second 


"But what about Categorical (Nominal) Variables in Latent Variable Models?" 


"Interactions May Be the Rule Rather than the Exception, But...: A Note on Issues in Estimating Interactions in Theoretical Model Tests" (An earlier version of Ping 2008, Am. Mktng. Assoc. (Summer) Educators' Conf. Proc.). The paper critically addresses theorytesting questions on conceptualizing, estimating and interpreting interactions in survey data. For example, (before data is collected) what 

"On the Maximum of About Six Indicators per Latent Variable with RealWorld Data." (An earlier version of Ping 2008, Am. Mktng. Assoc. (Winter) Educators' Conf. Proc.). The paper suggests an explanation and remedies for the puzzling result that Latent Variables in theoretical model testing articles have a maximum of about 6 indicators. 

"SecondOrder Latent Variable Interactions, and SecondOrder Latent Variables." (An earlier version of Ping 2007, Am. Mktng. Assoc. (Winter) Educators' Conf. Proc.). The paper proposes several specifications for a SecondOrder Latent Variable interaction. 

"Notes on Estimating Cubics and other 'Powered' Latent Variables." (An earlier version of Ping 2007, Am. Mktng. Assoc. (Summer) Educators' Conf. Proc.). The paper discusses "satiation" and "diminishing returns," infrequently explored topics in theoretical model tests, and a Latent Variable (LV) that is related to a quadratic, a cubic. The paper suggests a specification for this difficulttospecify LV. 

"Estimating Latent Variable Interactions and Quadratics: Examples, Suggestions and Needed Research" (An earlier version of Ping 1998, in Interaction..., revised December 2006). The paper provides estimation examples, including LISREL and EQS code. The revision also corrects several errors. 

"Pseudo Latent Variable Regression: an Accessible Estimation Technique for Latent Variable Interactions," (An earlier version of Ping 2003, 2003 Acad. of Mktng. Sci. Conf. Proc., Miami: Acad. of Mktng Sci., revised October, 2003). The paper proposes a reliabilitybased regression estimator for Latent Variable Interactions and Quadratics. 

"Improving the Detection of Interactions in Selling and Sales Management Research" (An earlier version of Ping 1996, J. of Personal Selling and Sales Mgt., revised October 2003). Using Monte Carlo simulations, the paper evaluates nonstructural equation analysis approaches to detecting a Latent Variable Interaction such as median splits. 

"Interpreting Latent Variable Interactions" (An earlier version of Ping 2002, Am. Mktng. Assoc. (Winter) Educators' Conf. Proc., revised June 2002). The paper suggests an approach to developing detailed interpretation of a significant Latent Variable Interaction or Quadratic that reveals the regions of the domain of a moderated variable where it is significant and non significant. 

"A Parsimonious Estimating Technique for Interaction and Quadratic Latent Variables" (An earlier version of Ping 1995, JMR, revised July 2001). The paper proposes a single indicator specification for Latent Variable Interactions and Quadratics that addresses the modeltodata fit problem associated with specifying these variables in realworld data without omitting interaction items and thus impairing reliability and validity; the proposed specification can be used with LISREL, EQS, AMOS, CALIS, etc. 

"Latent Variable Interaction and Quadratic Effect Estimation: A Twostep Technique Using Structural Equation Analysis" (An earlier version of Ping 1996, Psych. Bull., revised July 2001). The paper proposes a "2step" Kenny and Judd (1984) estimation approach for Latent Variable Interactions and Quadratics with LISREL, EQS, AMOS, CALIS, etc. This approach is useful with a consistent subset of product indicators (see Chapter VIII.SxA Unidimensionalization, in the monograph, LATENT VARIABLE INTERACTIONS... above), and other subset itemizations, with software that does not permit direct estimation (e.g., EQS, AMOS, etc.). 

"Latent Variable Regression: A Technique for Estimating Interaction and Quadratic Coefficients" (An earlier version of Ping 1996, Multiv. Behav. Res., revised July 2001). The paper proposes a measurementerroradjusted regression technique for Latent Variables, including Interactions and Quadratics (the Standard Error is explained in the paper below, "A Suggested Standard Error..."). This approach is useful in situations where regression is valuable. These situations include (applied) model building (e.g., in market research, econometrics, epidemiology, biostatistics, etc.) where many candidate models are estimated using easily implemented "stepwise" and "backward elimination" procedures to determine the model that "best fits" the calibration data; and, theoretical model (hypothesis) testing of a model that combines nominal (categorical) variables with other latent variables. 

"A Suggested Standard Error for Interaction Coefficients in Latent Variable Regression" (An earlier version of Ping 2001, Acad. Mktng. Sci. Proc., revised September 2001). The paper suggests a Standard Error term for Latent Variable Regression (see above). 

WORKING PAPERS MENTIONED ELSEWHERE ON THE WEB SITE  
(CLICK ON THE RED DOT) 
"Hypothesized Associations and Unmodeled Latent Variable Interactions/ Quadratics: An FTest, Lubinski and Humphreys Sets, and Shortcuts Using Reliability Loadings." The paper proposes an approach to posthoc probing for Latent Variable Interactions and Quadratics in order to explain a non significant hypothesized association. The APA citation for this online paper is Ping, R.A. (2006). "Hypothesized Associations and Unmodeled Latent Variable Interactions/Quadratics: An FTest, Lubinski and Humphreys Sets, and Shortcuts Using Reliability Loadings." [online paper]. http://www.wright.edu/~robert.ping/Ftest10.doc . 

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