Latent Variable
Research

Updated 5/15/13 (Previous updates 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 2001-2013

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FOREWORD--This web site contains research on

              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 in real-world 
   data).

   A paper about reusing a data set to create a second theory-test paper is available
   (to help reduce the "time between papers"). It turns out that an editor might 
   not object to a paper that reuses data which has been used in a previously 
   published 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 second paper that may 
   not require collecting new data. Please see "Notes on Used Data--Reusing a 
   Data Set to Create A Second Theory-Test Paper" in SELECTED PAPERS... below.


   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, the 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 "Is 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 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/~rpingjr/...

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 click on your browser's 
   "Refresh" or "Reload" button on the browser toolbar (above) to view the current 
   version of this web page.

   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 click on "Tools" on the browser toolbar 
   (above), then click on "Internet Options...." Next, in the "General" tab, find the 
   "Temporary Internet Files" section and click on "Delete Files...." Then, click in the 
   "Delete all offline content" box, and click "OK." 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 any (and all) Word documents in Firefox, please click on "Tools" on 
   the browser toolbar (above), then click on "Options." Next, in the "Advanced" tab, 
   find the "Offline Storage" section and click on "Clear Now" Then, click "OK." After 
   that, close this browser window, then re-launch it so the latest versions of all the 
   WORD documents are forced to download.
        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

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.

"Is there an example that shows all the steps involved in estimating a latent variable interaction/quadratic?"
   (Please click here for more.)

"How should PRELIS or similar "preprocessor" software be used with LISREL, EQS, AMOS, etc. to create interactions/quadratics?"
   (Please click here for suggestions.)

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.)

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?"

"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.)

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. 
     This 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.)

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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.

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

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  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.

   A Bibliography on Latent Variable Interactions and Quadratics. 

ON-LINE MONOGRAPHS 

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
 

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"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 data might be used in 
   two or more papers--which might reduce the time and expense associated with 
   gathering new data (and producing an additional paper for possible publication).
   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. It 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 several 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?

"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. 

"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).   

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"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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