Latent Variable
Research

Updated 6/23/14 (Previous updates 2/10/14, 9/21/13, 6/4/13, 5/15/13, 2/12/13, 1/21/13, 12/26/12, 9/17/12, 5/10/12, 3/6/12, 2/6/12, 9/19/11, 8/22/11,3/21/11, 1/11/11, 11/18/10, 9/2/10, 6/24/10, 6/9/10, 1/27/10, 11/23/09, 11/10/09, 9/5/09, 8/26/09, 5/12/09, 4/3/09, 3/28/09, 2/9/09, 9/6/08, 7/21/08, 4/23/08, 1/30/08, 1/14/08, 10/25/07, 7/14/07, 6/14/07, 4/26/07, 2/3/07, 12/11/06, 11/29/06, 5/12/06, 2/22/06, 1/30/06, 12/11/05, 10/6/05, 7/12/05, 5/23/05, 10/5/04, 5/13/04, 2/26/04, 2/11/04, 10/20/03, 9/28/03, 5/23/03, 3/23/03, 2/14/03, 1/27/03, 9/18/02, 7/17/02, 5/6/02, 3/6/02, 10/03/01.)
                 Copyright (c) Robert Ping 2001-2014

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FOREWORD--This web site contains research concerning
              o Latent variables (LV's), and LV interactions (e.g., XZ) and quadratics  
(e.g., XX) in theoretical (hypothesis testing) models involving real world
survey data, and
              o Testing these models.
It is intended for Ph.D. students and researchers who are somewhat familiar with LV's 
and structural equation modeling software such as LISREL, EQS, AMOS, etc.,
and who may be just getting started with estimating substantive theory-testing
models with, for example, interactions or quadratics, (truly) categorical variables
(e.g., gender) with structural equation analysis, 3-way interactions (e.g., XZW),
one or more measures with low or unacceptable Average Variance Extracted (AVE), etc.
NEWS:
   "Workarounds" for an unbiased estimate of an "endogenous" interaction or quadratic 
with AMOS (and SIMPLIS) are available by e-mail. (AMOS, and possibly SIMPLIS, may
not allow the free (non zero) correlation(s) between an interaction, XZ, and its
endogenous constituent latent variable X (or Z) that are
required to adequately
estimate an interaction or quadratic in real-world data)
.

A paper about reusing one's data set to create a second theory-test paper is
available
. It turns out that an editor may not object to a theory-testing paper that
reuses data which has been used in a previously published theory-testing paper, if
the new paper's theory/model is "interesting" and materially different from
the
previously published paper
. The paper on reusing data discusses how submodels
from a previous paper might be found for a seco
nd paper that may not require
collecting new data
(to help reduce the "time between papers"). Please see "Notes on
Used Data
--Reusing a Data Set to Create A Second Theory-Test Paper
" in
SELECTED PAPERS... below.

A working-paper about peer-revewing a theory-test paper is available by e-mail. Its
suggestions could be used to help with peer-reviewing a (survey data) theory-testing
 paper. It might also be useful as a "check list" for a dissertation that tests a model
using survey data, or for a theory-test paper prior to its submission for review.
 
A warning about using "standardized loadings" (latent variable (LV) loadings that
are all free, so the resulting LV has a variance of unity) is available. (Standardized
loadings may produce an incorrect t-value for a structural coefficient, and
any
interpretation or implication of its significance or nonsignificance may be risky.)

Please see
"Why are reviewers complaining about my use of standardized loadings?"
in Questions of the Moment below.

Suggestions for estimating a "moderated quadratic" LV (i.e., XXZ) are available by
email. LV's such as XXZ may seem unlikely, but for an LV (X) that exhibits diminishing
returns (XX), its rate of return also may vary with the level of a moderator. (For
example, the Satisfaction-Exiting association can be quadratic--when Satisfaction is
high, further increases in Satisfaction are likely to produce fewer and fewer
reductions in the likelihood of Exiting. However, this quadratic (i.e., upside-down
horseshoe) Satisfaction-Exiting relationship
also may depend on the availability of
Alternatives: with lots of
Alternatives, increases in Satisfaction may produce
near-linear reductions to Exiting--i.e., the Satisfaction-Exiting association may no
longer be quadratic
--it may be linear.

The
procedure for estimating (truly) categorical variables in a mixed model with
(multiple indicator) LV's
has been expanded. Please see "How does one estimate
(truly) categorical variables in theoretical model tests using structural equation
analysis?" in Questions of the Moment below.)
   An annotated suggested format for substantive papers that may reduce the likelihood 
of paper rejection is available by email. (For example, a paper's customary
Abstract-Intro-Lit Review, etc. can be insufficient to "hook" the reviewers, and pique
their interest in the rest of the paper. The format suggests, in some cases,
subtle changes in wording and content for these and other paper sections.)

A suggested approach for improving a problem measure--one with a low Average
Variance Extracted, or one that has been excessively weeded or with low
model-to-data fit, etc.--also has been expanded. Please see "How are "Formative"
(indicators-pointing-to) Latent Variables estimated with LISREL, EQS, AMOS, etc.?"
in Questions of the Moment below.)
A suggested approach is available for estimating a 3-way interaction (e.g., XZW)
that has some surprising results.
A three-way interaction may seem unlikely, but a proper test of hypotheses
such as "H0: X-->Y is moderated by Z" and "H1: X-->Y also is moderated by W"
is to specify XZ, XW and XZW in the model. (Z and W may moderate X-Y jointly
via ZWX rather than separately (via ZX and WX). Stated differently, ZX and WX
could be NS while ZWX is significant (ZW moderates X).
However, there are at least 4 nonequivalent specifications of XZW. And, the
recommended specification used in regression is typically NS in real-world theory tests,
while one or more of the other three 3-way specifications can be significant, and
this difficulty appears to occur in latent variable specification as well. (Please see
"Why are the Hypothesized Associations not Significant? A Three-Way Interaction?"
in
SELECTED PAPERS... below.)

The suggestions for improving AVE have been expanded, also with some surprising
results (e.g., some of the popular "discriminant validity" tests are not trustworthy in
theory tests with real-world survey data). (Please see "I
s there any way to improve
Average Variance Extracted (AVE) in a Latent Variable X?" in
Questions of the Moment below.) A suggested approach is provided for remedying a "not Positive Definite" message
(an "Ill Conditioned" message in exploratory factor analysis) that typically occurs
when a full (unweeded) measure is factored, specified etc. for the first time.
(Please see "How
does one remedy a "not Positive Definite" message?" in
Questions of the Moment.)
Comments on the use of PLS in theory tests involving survey data are available. (Please see "Why are reviewers complaining about the use of PLS in my paper? in Questions of the Moment.)
   Specifications for an interaction form other than XZ are available by e-mail. This may 
   not sound like much, but XZ is not the only form an interaction can take--
   X/Z and XZ2 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 
   interaction--it just doesn't have the form "X times Z". Experience suggests 
   that X/Z, XZ2 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 
template for estimating the indirect effect of an endogenous interaction is available
by e-mail. (Please see "Please Note: If you are estimating an interaction involving 
an endogenous variable..." in the INTRODUCTION.) A paper on hypothesizing interactions is available (i.e., what evidence suggests
the presence of an interaction before data is collected, how an interaction might
be justified (argued for), etc.). (Please see "Interactions May Be the Rule Rather
than the Exception, But..." in
SELECTED PAPERS...below.)
   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) 
Recommendations" in a theory testing paper. (Please e-mail me for a progress
report.)
(Comments on handling reviewer comments in general are available on the
Higher Education web page--please click on "Home" above, then click on "Higher
Education.")
   o An EXCEL template that provides a simple and straightforward tabular interpretation 
of a significant Interaction (similar to crosstabs) that does not require
graphs (please e-mail me for a draft template).
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" 
(arrows-to-indicators) LV's, plus "Formative" (indicators-pointing-to) LV's are
provided.
This sounds like it would never be useful. Some may have been told this
can not, or should not, be done with SEM (e.g., with LISREL, AMOS, EQS, etc.).
However, a new measure (or an older, well established, measure developed
before the advent of SEM
) frequently requires substantial weeding in order to
attain model-to-data fit. And, the weeded measure may be missing so many
items that a reviewer might judge it to be no longer adequately face valid.
Or, no itemization can be found with adequate Average Variance Extracted
(AVE), or adequate discriminant validity, etc. Perhaps surprisingly, such
measures usually can be re-specified as Formative--typically using the full
(unweeded) measure--and estimated using LISREL, AMOS, EQS, etc. in
order to remedy these difficulties.
o The "Why is my hypothesized interaction or quadratic nonsignificant?" paper (below) is being revised to account for interaction forms besides XZ; o A paper on using regression to test an hypothesized interaction that links to a paper titled "What is Structural Equation Analysis?" is provided. o The cubics paper is revised, and an EXCEL template for specifying cubics is provided; and o Several EXCEL templates calculate reliability and Average Variance Extracted for XZ, XX and ZZ. (The quadratic XX, for example, could be viewed as the interaction of X with itself.).



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." [on-line 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 
current
version of this web page. For Internet Explorer, click on "Tools" (above),
 then click on "Delete Browsing History," "check" "Temporary Internet Files,"
uncheck everything else, and click on delete.
After that, close this browser
window, then re-launch it.
(The procedures for Firefox and Chrome are discussed
next in refreshing WORD documents.)
In adddition, many of the links on this web site are in Microsoft WORD. If you have viewed one or more of them before, the procedure to view the latest (refreshed) version of them is tedious ("Refresh" does not work for WORD documents on the web). With my apologies for the tediousness, to refresh any (and all) Word documents in Internet Explorer, please follow the above procedure.


To refresh all WORD documents in Chrome, please click on the 3 "bars" in the upper
right-hand corner of the screen, then click on "History." Next, click on "Clear
browsing data," check "Cashed images and files," and uncheck everything else.
Finally, click on
"Clear browsing data." After that, close this browser window,
then re-launch it so the latest versions of all the WORD documents are forced to
download.

To refresh WORD documents in Firefox, please click on "Tools" on the browser toolbar
(above), then click on "Options." Next, in the "Advanced" tab, find the "Network" tab,
and under "Cached Web Content" click on "Clear Now" Then, click "OK." After
that, close this browser window, then re-launch it.
If several browsers
are used, they all should be refreshed.
Your questions are encouraged; just send an e-mail to rping@wright.edu. Don't 
worry about being an expert in latent variables, structural equation modeling or structural equation analysis, or using "correct" terminology or perfect English.
A Table of Contents or Index to this website is not available. With my apologies, 
please consider using your browser's search capability to search for the topic in
which you are interested. For example, to find the EXCEL templates try
Ctrl+F. Or,
click on "Edit" on the browser toolbar (above) and click on "Find" (or click on the
three horizontal "bars" in the upper right-hand corner of Chrome, then click
"Find..."), and type the word "excel" in the find box.





INTRODUCTION



  Please Note: If you are estimating an interaction involving one or more endogenous variables please click here--things 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 non-significant (NS) model association
         (e.g., is there an interaction or quadratic that suppressed the hypothesized
         Z-->Y association?), or 

     2) where one wishes to investigate an important significant association(s) to see 
         if it is conditional (i.e., moderated by some other variable), and thereby provide
         a "finer grained" 
interpretation (as routinely done in experiments/ANOVA, for
         example). 

     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 follow-up study, or replication (to investigate whether or not the "after-the-fact" moderation simply was significant by chance in the present study) (see "Hypothesized Associations and Unmodeled Latent Variable Interactions/Quadratics: An F-Test..." 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 "hypotheses-before-first-test" model-test (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 4-product-indicator subset1 of the Kenny and Judd interaction (product) indicators to avoid the model-to-data fit problems that occur when all of the Kenny and Judd indicators are specified--46 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 model-to-data fit raises many questions, including, what is the reliability and validity of the resulting 4-item 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 content-validity of a 4-item interaction or quadratic also is questionable--if 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 real-world 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?"
   (Please click here for more.)

"Why are reviewers complaining about my use of standardized loadings?"

It turns out that standardized loadings (latent variable (LV) loadings specified as all free so the resulting LV has a variance of unity) may produce incorrect t-values for some parameter estimates, including structural coefficients. This presents a problem for theory testing: An incorrect (biased) t-value for a structural coefficient means that any interpretation of the structural coefficient's significance or nonsignificance versus its hypothesis may be risky.
   (Please click here for more.)    


 (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 e-mail 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?"
   (Please click here for suggestions.)



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


Contrary to customary practice, interactions and quadratics also may be important in applied model-building/research--model 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.
  
Anecdotally, it may not not widely known in applied research that important model predictors may not be "unconditional" (e.g., Y is unconditionally increased/decreased with X in the study). Instead, these effect may have been "conditional" (e.g., Y increased with X when Z was at a high level (strong), but Y was unrelated to X, or it decreased with X, when Z was at a lower level (weaker)).  
   (Please click here for more.)


 

"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 Z-Y ASSOCIATION SHOULD NOT EXIST IN THE POPULATION.
   (Please click here for more.)



 
 

"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?"
   (Please click here for a paper on this topic that is being revised to include other
    interaction forms (e.g., X/Z)--please e-mail me for more.)



 

"Is there any way to improve Average Variance Extracted (AVE) in a Latent Variable X?"
   (Please click here for a paper on this matter. Also, please consider reading "How are
   Formative Latent Variables estimated with LISREL, EQS, AMOS, etc.?", below, for
   more suggestions for "troubled" LV's--unacceptable AVE's, etc.)



 

"Is there any way to speed up "item weeding" to find a set of items in a multi-item measure that fits the data?"
   (Please see the EXCEL template "For 'weeding' a multi-item measure so it 'fits the
    data'..." below.)



 

"How might a 'mixed interaction' XZ, where X is a manifest/observed/continuous/single-indicator, 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 '2-group 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 e-mail me--I 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" (indicators-pointing-to) Latent Variables estimated with LISREL, EQS, AMOS, etc.?"


This may seem uninteresting--formative variables are rare in the Social Sciences. But, a formative re-specification of a difficult measure (e.g., one with inadequate AVE, excessive weeding was required to attain model-to-data 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. 
     T
his approach also may be useful when using older, well established, measures--developed before the advent of SEM--as they were intended (i.e., without weeding-out most of the items). 
   (Please click here for more.)




    
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 step-by-step instructions for Ph.D. students, and 
theoretical or applied researchers, interested in estimating their first Latent
Variable Interaction or Quadratic in a theoretical model test using survey data.



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 "2-Step" estimation (LISREL, EQS, AMOS, etc.)
      (see Ping 1995, JMR--a 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 "2-Step" estimation (LISREL,
      EQS, AMOS, etc.) (please also see "Notes on Estimating Cubics and other
      'Powered' Latent Variables" below). 


  For "weeding" a multi-item measure so it fits the data (i.e., finding a set of
       items that fits the data, so the measure is internally consistent).

      In real-world data, there frequently are many subsets of a multi-item
      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 
fit the data. The template then can be used to search for additional subsets of items that will also fit the data, and thus it may help to find the "best" face- or content-valid subset of items in a measure. More about the template.


  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 measurement-error-adjusted regression
      approach to Structural Equation Analysis, for situations where regression 
is useful (e.g., to estimate categorical variables with LV's) (see Ping 1996,
Multiv. Behav. Res., a revised version appears below).
More about the template.



BIBLIOGRAPHY


   A Bibliography on Latent Variable Interactions and Quadratics. 


ON-LINE MONOGRAPHS 

(CLICK
 ON A
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 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 on-line monograph is Ping, R.A. (2003). Latent variable interactions and quadratics in survey data: a source book for theoretical model testing, 2nd edition. [on-line    monograph]. http://www.wright.edu/~robert.ping/intquad2/ toc2.htm. 







Ping (2001) LATENT VARIABLE INTERACTIONS AND QUADRATICS IN SURVEY DATA: A SOURCE BOOK FOR THEORETICAL MODEL TESTING (Edition 1)



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 model-to-data fit; inadmissible solutions with remedies; interactions and quadratics; and pedagogical examples (177 pp.).

Of particular interest recently is how to efficiently and effectively "weed" items to attain a consistent measure (see STEP V, PROCEDURES FOR ATTAINING...).

The APA citation for this on-line monograph is Ping, R.A. (2004). Testing latent variable models with  survey data, 2nd edition. [on-line monograph].    http://www.wright.edu/~robert.ping/lv1/toc1.htm .







Ping (2002) TESTING LATENT VARIABLE MODELS WITH SURVEY DATA (Edition 1)





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 
Theory-Test Paper
"
(An earlier version of "Notes on 'Used Data'" (Ping 2013),
Am. Mktng. Assoc. (Summer) Educators' Conf. Proc.
).

The paper critically discusses an intriguing possibility: that one's data set
might be used in two
or more theory-test papers--which could reduce the time
and expense associated with gathering new data for an additional paper.
 
It turns out that an editor might not object to reviewing a paper that is based on
data which has been used in a previously published paper. I
t also turns out that a
published model is likely contain at least one submodel that might be a candidate
 for an additional paper.
The paper on reusing data discusses how submodels from
a previous paper might be found
, and issues that might arise.


"Why are the Hypothesized Associations not Significant? A Three-Way
Interaction?"
(An earlier version of Ping 2010, Am. Mktng. Assoc. (Winter) Educators'
Conf. Proc.
).


The paper suggests en approach for estimating a 3-way interaction (e.g., XZW) with
some surprising results. While a three-way interaction may seem unlikely, a proper
test of hypotheses such as "H0: X-->Y is moderated by Z" and "H1: X-->Y also is
moderated by W" is to specify XZ, XW and XZW in the model. (ZX and WX could be
NS while ZWxX is significant (ZW moderates X). The surprises are that there are at
least 4 nonequivalent specifications of XZW, the recommended regression
specification is typically NS in real-world theory tests (while one or more of the other
three 3-way specifications can be significant), and this difficulty appears to occur in
latent variables as well.



 
"But what about Categorical (Nominal) Variables in Latent Variable Models?"
(An earlier version of Ping 2009, Am. Mktng. Assoc. (Summer) Educators'
Conf. Proc.
).


In part because categorical variables almost always occur in the "Demographics" section
of a survey in the Social Sciences, the paper suggests a procedure for estimating
categorical variables in a structural equation model that also contains latent variables.



 
"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 theory-testing questions on conceptualizing, estimating and
   interpreting interactions in survey data. For example, (before data is collected) what 
evidence suggests that an interaction should be hypothesized? Is an interaction a
construct or a mathematical form, or both? Is specifying the interaction between X and
Z, for example, as XZ a sufficient disconfirmation test?


"On the Maximum of About Six Indicators per Latent Variable with Real-World
   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. 


"Second-Order Latent Variable Interactions, and Second-Order Latent
   Variables." (An earlier version of Ping 2007, Am. Mktng. Assoc. (Winter) 
   Educators' Conf. Proc.).

The paper proposes several specifications for a Second-Order 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 difficult-to-specify 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 reliability-based 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 non-structural 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 model-to-data fit problem associated with specifying
   these variables in real-world 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 Two-step
   Technique Using Structural Equation Analysis" (An earlier version 
   of Ping 1996, Psych. Bull., revised July 2001).
The paper proposes a "2-step" 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 measurement-error-adjusted 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 F-Test, Lubinski and Humphreys Sets, and Shortcuts 
   Using Reliability Loadings."
The paper proposes an approach to post-hoc probing for Latent Variable Interactions
   and Quadratics in order to explain a non significant hypothesized association.   
The APA citation for this on-line paper is Ping, R.A. (2006). "Hypothesized Associations
   and Unmodeled Latent Variable Interactions/Quadratics: An F-Test, Lubinski and 
   Humphreys Sets, and Shortcuts Using Reliability Loadings." [on-line paper]. 
   http://www.wright.edu/~robert.ping/Ftest10.doc .


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