Latent Variable
Interaction
Research

    
    
Updated 9/5/09 (Previous updates 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.)

    

  

    

    

    
      

(HOME)

  

    
FOREWORD--This web site concerns:
              o Latent variable interactions and quadratics in theoretical 
                 (hypothesis testing) models involving survey data,
 		and
              o Testing these models.
Its contents are intended for Ph.D. students, and theoretical and 
   applied researchers, who are familiar with latent variables and 
   estimation software such as LISREL, EQS, AMOS, etc., but are 
   just getting started with latent variable interactions and 
   quadratics.
Interactions are not only "fashionable" lately, experience suggests 
   that they tend to cloud survey-data model results (significances)
   when they are not specified (modeled) (please see "ALSO, just a
   reminder..." in the INTRODUCTION section below for more).
NEWS:
   A suggested approach for remedying a "not Positive Definite" 
   message ("Ill Conditioned" in exploratory factor analysis) in 
   theoretical model (hypothesis) tests with structural equation 
   analysis (LISREL, EQS. Amos, etc.) is available (please see 
   "How does one remedy a "not Positive Definite" message?" in the 
    Questions of the Moment section below).

   The paper on improving AVE has been completely revised, with 
   some surprising results (e.g., most of the "discriminant validity" 
   tests are not trustworthy in survey-data theory tests with 
   real-world data).
   A suggested approach for testing (truly) categorical variables in 
   theoretical model (hypothesis) tests with structural equation 
   analysis (LISREL, EQS. Amos, etc.) is available (please see "How 
   does one estimate categorical variables in theoretical model tests
   using structural equation analysis?" in the Questions of the 
    Moment section below).
 
   Specifications for interactions forms besides XZ are available by 
   e-mail. This may not sound like much, but XZ is not the only form 
   an hypothesized interaction can take. (E.g., X/Z and XZ2 are also 
   interactions--X/Z also can moderate the X-->Y association, so 
   can XZ2, etc.--and experience suggests they behave like XZ.) 
        So, testing an hypothesized interaction with just XZ
   is insufficient to disconfirm the typical interaction hypothesis 
   1) "Z moderates the X-Y association." An improved moderation 
   hypothesis for XZ might be "Z moderates the X-Y association as
   (in the form) XZ." The usual interaction hypothesis (e.g., 
   hypothesis 1 above) could be argued to be equivalent to the 
   hypothesis "The X-Y association is moderated by some form of 
   XZn" (e.g., X/Z, XZ2, etc., where n is a positive or negative 
   integer). 
        Thus, if XZ turns out to be nonsignificant, this does not 
   disconfirm hypothesis 1) above (i.e., there may still be an
   interaction, just not of the form "X times Z"). Experience 
   suggests that X/Z, XZ2 or another interaction form may be 
   significant when the XZ-Y association is nonsignificant and there 
   is strong theoretical support for Z moderating the X-Y association. 

   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 the Questions of the Moment section below.)
   The suggestions below for estimating an endogenous 
   interaction have materially changed (please see "Please 
    Note: If you are estimating an interaction involving an 
    endogenous variable..." in the INTRODUCTION section
   below).

   A paper on hypothesizing interactions is available (how 
   interactions might be "theorized"--what types of evidence 
   suggest there might be an interaction in a model, how an 
   hypothesized interaction might be theoretically justified 
   (argued for), etc.). (Please see "Interactions May Be the Rule 
   Rather than the Exception, But..." in the SELECTED 
   PAPERS ON LATENT VARIABLE INTERACTIONS 
   AND QUADRATICS section below.)

   A suggested approach for testing (truly) categorical (i.e., 
   nominal) variables in theoretical model (hypothesis) tests with 
   structural equation analysis (LISREL, EQS. Amos, etc.) is 
   available (please see "How does one estimate categorical 
   variables in theoretical model tests" in the Questions of 
   the Moment section below).

Coming Attractions:
   An EXCEL template to help provide a detailed interpretation of a 
   significant Interaction or Quadratic (pls. e-mail me for a draft 
   template).
Recent Additions and Changes in this Web Page (indicated by 
      "New," "Revised" or "Updated" below):
   o A paper on hypothesizing interactions; how they are 
      "theorized." (E.g., what forms of evidence suggests there is 
      an interaction in one's model? Also, how is a suspected 
      interaction theoretically justified (argued for)?) 
   o The "Why is my hypothesized interaction or quadratic 
      nonsignificant?" paper (below) is being revised to account for the 
      interaction forms besides XZ,
   o A paper on the using of regression to test an hypothesized 
      interaction that links to a paper titled "What is Structural 
      Equation Analysis?"
   o The cubics paper is revised, and an EXCEL template for specifying 
      cubics is provided, 
   o Suggested remedies for low Average Variance Extracted are 
      provided, 
   o FAQ's A, B and C address all the interaction/quadratic estimation 
      techniques, 
   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.)
   o The Bibliography has been updated, and
   o Several new working papers in various stages of review have been
      added.
Please note: If you have visited this web site before, and the latest
   "Updated" date (above) seems old, or, if you are actively estimating
   an interaction or quadratic, you may want to click on your browser's
   "Refresh" or "Reload" button on your toolbar to view the current version 
   of this web page.

  

    

  

    
    
    
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 "book" is Ping, R.A. (2001). "Latent Variable 
   Research." [on-line paper]. http://home.att.net/~rpingjr/
   research1.htm.), or using the individual citations for each of the 
   papers, spreadsheets, monographs, etc. shown below.
Don't forget to Refresh: This web site is purposefully "low tech." 
   Many of the links below are in Microsoft WORD, and if you 
   have viewed 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 jointly refresh all the Word documents, please click
   on "Tools" on your toolbar, 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 latest WORD documents 
   are forced to download.
Your questions are encouraged; just send an e-mail to
   rping@wright.edu. Don't worry about being an expert in latent 
   variables or structural equation analysis, or using "correct" 
   terminology or perfect English.
A Table of Contents or Index to this website is not yet available.
   In the meantime, please consider using your browser's search 
   capability to go to the relevant material. For example, to find
   the EXCEL templates click on "Edit" on your toolbar, then click on 
   "Find..." and type the word "EXCEL" in the "Find what:" box.

  

    
    
  

  

    

  

    
    
    
INTRODUCTION

    

  

    
    

  

    (N

       e

       w) 		
    Please Note: If you are estimating an interaction 
      involving an

         endogenous variable please click here--things 
      are different

         in this case. 
      
ALSO, just a reminder: Please remember that in 
      theoretical model (hypothesis) testing using survey data (survey data models), interactions or quadratics should be hypothesized 
      before  the model is ever tested with data. Please be 
      aware that the only situations in survey data models where one can
      search for significant interactions or quadratics is 1) the 
      case where one wishes to probe for an explanation for a non-significant (NS) 
      first-order model association (e.g., is there an interaction, XZ, 
      or quadratic, ZZ, "causing" the hypothesized 
      Z-->Y association to be non-significant?), or 2) where one wishes to check
      on significant model associations that actually may be conditional in 
      the study (i.e., moderated) (as is routinely done in ANOVA studies), and
      thus improve interpretation of 
      the model estimation results.  
      
     In the first case, any significant 
      "suppressor" interaction XZ or quadratic ZZ found by trial and error 
      could then 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 
      discovered by trial and error could then be the basis for a note in the 
      Discussion section of the paper that the hypothesized Z-->Y 
      association actually depended on the levels of X in the study for 
      its strength and significance.  
      
     Because any suppressed or conditional 
      Z-->Y association in the study also may be that way in the population,
      a conditional Z-->Y 
      association could be hypothesized, then tested in a follow-up 
      study or replication, to investigate whether or not it was 
      significant by chance in the previous study (see "Hypothesized 
      Associations and Unmodeled Latent Variable 
      Interactions/Quadratics: An F-Test..." (below) for more). 
      
     In absence of situation (motivation) 1 or
      2 above, hunting for significant 
      interactions or quadratics that were not hypothesized before the 
      model was tested with the data at hand (i.e., to "improve" a
      paper's contribution) is considered "poor 
      science" in theoretical model testing. It can tempt one to state 
      a significant interaction or quadratic's hypothesis as though 
      it was hypothesized before the data was collected. And, it changes one's
      "hypotheses-before-first-test" model-testing ("confirmatory") study into a
      "test-before-hypothesis"  model-building (exploratory) study. This 
      in turn increases the likelihood that significant 
      associations in the model exist only by chance (they are 
      spuriously significant, and they exist in this sample 
      only). 

  

    

  

    Questions of the Moment

  

    
    "What about the 
      alternative specifications for a Latent Variable (LV) 
      interaction?" 
  

  

    
    
    A  recent review of 
      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 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 Frequently 
      Asked Questions A, B and C below for 
      more.) 
  

  

    
    
      

  

    (N

      e

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

      

         (Please click here for a paper on this
      matter.)
 		
  

  

    
    
     

  

    (N

      e

      w) 		
    "How
      does one estimate categorical variables in
      theoretical model tests using structural equation analysis?"
 
  

  

     
    The quick
      answer is, with considerable care and effort. In covariance structure
      analysis (e.g., using LISREL, EQS, Amos, etc.), the term "categorical
      variable" is usually used to mean an ordinal variable (e.g., an
      attitude measured by Likert scales), rather than a nominal or "truly
      categorical" variable (e.g., Marital Status, with the categories
      Single, Married, Divorced, etc.), and typically there is no provision for
      "truly" categorical variables. In regression, (truly)
      categorical variables are estimated using "dummy" variables, and
      a similar approach might be used in covariance structure analysis.
      However, there are several issues in theoretical model testing using
      covariance structure analysis.

      

      (Please click here
      for a paper on this matter, then please e-mail me--I have more
      suggestions.)
    		
  

  

    
    
    
  

  

    (N

      e

      w) 		
    "Why
      are reviewers complaining about the use of moderated multiple regression
      in my paper?"

      

        
      (Please click here for a paper on 
      this subject.) 		
  

  

    
    
    
  

  

    (N

      e

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

      

         (Please click here for a paper on 
      this subject.)
 		
  

  

    
    
     

  

    (N

      e

      w) 
    		
    "Why would applied 
      researchers be interested in 
interactions/quadratics?" 		
  

  

    
    
    

      Contrary to 
      conventional wisdom, interactions and    quadratics 
      also may be important in applied research    (model 
      building in econometrics, epidemiology, marketing models,
      biostatistics, etc.)--not to explain additional variance in a target variable, but to 
      better understand, explain and predict 
      important relationships in a model. Apparently it is 
      not well understood in applied research that 
      important model effects may not be "Imperative" 
      (e.g., A increases/decreases with B). 
      These effects instead may be "Conditional" (e.g., A 
      increases with B when C is at a high level, but 
      A is unrelated to B, or it decreases with B, when C 
      is at a lower level.  
 
      

      (Please click here for a paper on 
      this subject.)		
  

  

    
     

  

    (N

      e

      w) 		
    "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.)  
  

  

    
    
     

  

    (N

      e

      w) 		
    "When theory proposes an X-Y association and it also proposes that Z
      moderates this association, but theory is mute about or doesn't propose a
      Z-Y association, why does one still include Z in addition to X and XZ in
      the model to be tested?"  
    
  

  

    
     
    
    In short, excluding the Z-Y (and/or the X-Y) association when XZ-Y
      is hypothesized to be significant, can bias all structural coefficients
      and standard errors in the proposed model. This in turn casts a shadow on
      the trustworthiness of the test of the proposed model. In addition, and
      perhaps surprisingly, if XZ is significant, excluding Z from the model
      biases the (significant) factored ("true" contingent)
      association of Z with Y, EVEN WHEN THE Z-Y ASSOCIATION IS HYPOTHESIZED TO
      NOT EXIST IN THE POPULATION.

      

      (Please click here for
      more on 
      this subject.) 		
  

  

    
    
     
  

  

    (N

      e

      w) 		
    
      
"How is a cubic LV estimated?" 

  

  

    
    
    
      Please see the paper on 
      estimating a LV cubic in the "Selected Papers on 
      Latent Variable Interactions and Quadratics" 
      section (below). The "EXCEL templates..." section (below) includes a template to
      assist 
      in calculating loadings, error variances, 
      etc. for LV cubics.
  

  

    
    
     

  

    
    "Why is my hypothesized 
      interaction or 
      quadratic 
 nonsignificant?"

      

         (Please click here for a paper on this 
      topic that is being

          revised--please e-mail me with 
      any questions you may have

          on this topic.) 
    		
  

  

    
    
      

  

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

      

         (Please click here for a paper on 
      this matter.)

  

  

    
    
     

  

    
    
      
"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 a paper on this
      topic.) 		
  

  

    
    
      

  

    (Re-
vis-

      ed) 		
    "Why is my hypothesized 
      interaction significant using a 'median split' of the data, 
      but not significant when  specified  in my model?" 


        
      (Please click here for a paper on 
      this subject.) 		
  

  

    
    
      

  

    (Re-
vis-

      ed) 		
    "Why are most (or all) 
      of my hypothesized interactions not
significant?" 


         (Please click here for a paper on this
      matter.) 		
  

  

    
    
      

  

    
    "What is the Average 
      Variance Extracted (AVE) for a 
Latent Variable Interaction 
      (or Quadratic)?" 


         (Please click here for a paper 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 a paper on 
      this subject.)

         		
  

  

    (N

      e

      w) 		
    "How
      does one remedy a "not Positive Definite" message?"
      

         (Please click here for a paper on 
      this matter.)

        

  

  

    
    ______________
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 several issues, including the reliability 
      and validity of the resulting 4-item interaction or quadratic.
      Reliability is necessary for validity, and the 
      reliability of an interaction specified with nearly all of its indicators
      deleted is unknown. (The formula for 
      the reliability of XZ assumes XZ is operationally (unweeded) X times (unweeded) Z.) The face- or
      content-validity of a 4-item interaction or quadratic also is 
      questionable (e.g., if nearly all the indicators of X and Z are
      unrepresented in the itemization XZ, for example, is XZ still the
      latent variables X and Z"?). Further, it is easy to show 
      that in real-world data a weeded XZ's structural coefficient 
      varies with the set of four indicators that are selected as its 
      indicators. Unfortunately, the "best" four weeded indicators are 
      unknown. Finally, an interaction with weeded Kenny and Judd 
      product indicators cannot be "factored," which produces detailed interpretation problems because XZ is 
      no longer (unweeded) X times (unweeded) Z.  

  

      
         

  

    Frequently Asked Questions

  

    
    
(CLICK
 ON A
 RED
 DOT)
(Re-
vised)

      

      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?" may be a 
   useful "cookbook" for Ph.D. students, and theoretical 
   researchers interested in estimating their first Latent 
   Variable Interaction or Quadratic in a theoretical model 
   test using survey data.
FAQ
      D also may be of interest to applied researchers interested in specifying
      their first Latent Variable    Interaction or Quadratic in
      a model.   

  

    

  

    

    

    EXCEL 
      TEMPLATES
    

  

    

    

    

    
EXCEL Templates for expediting the specification of Latent 
Variable (LV) Interactions, Quadratics and cubics; for "weeding" 
measures to attain model fit; for Latent Variable Regression, etc.

  

    

    

    

    
(CLICK
 ON A
 RED 
 DOT)

(Re-
vised)

    
  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 that paper 
      appears below).

      The template also calculates LV Interaction or 
      Quadratic Reliability and Average Variance 
        Extracted (AVE). More about the template.

  

    
    
(New)

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

  

    
    
(New)

    
  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 several 
      subsets of a multi-item measure that "fit the data,"
      and this raises the question of which of these subsets 
      is "best" from a validity standpoint. The template helps
      find at least one more subsetof items, usually with a
      maximal number of items (typically different from the 
      one found by maximizing reliability), 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 helps 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 OLS regression
      approach to Structural Equation Analysis for those
      situations where regression is useful (see Ping 
      1996, Multiv. Behav. Res., a revised version appears 
      below). More about the template .

  

    
    
    

  

    BIBLIOGRAPHY

  

    
    
(Up-
dated)

    
   A Bibliography on Latent Variable Interactions and 
     Quadratics. 

  

    
    

  

    
ON-LINE 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.
      Includesa "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://home.att.net/~rpingjr/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. 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://home.att.net/~rpingjr/lv1/toc1.htm
      .

  

    
    
    
    
    

       		
    
      


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

      

      

  

  

    
    
    
    

  

    
SELECTED PAPERS ON LATENT VARIABLE 
   INTERACTIONS AND QUADRATICS

  

    
    
    
    
(CLICK
 ON A
 RED 
 DOT)

     
    
"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, 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?
   (Pls. be patient, the download is a bit long). 

  

    
    
     
    
      
"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. 
   (Pls. be patient, the download is a bit long).

  

    
    
     
    
      
"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. (Pls. be patient, the 
   download is a bit long).

  

    
    
     
    
      
"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. (Pls. be patient, the download is a bit long).

  

    
    
    
    
"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.
   (Pls. be patient, the download is a bit long).

  

    
    
    
    
      
"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 OLS Regression 
   estimator for Latent Variable Interactions and Quadratics. 
   (Pls. be patient, the download is a bit long).

  

    
    
    
    
"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. 
   (Pls. be patient, the download is a bit long).

  

    
    
    
    
"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 subset(s) of the domain of a 
   moderated variable where it is significant and non 
 significant. (Pls. be patient, the download is a little long).

  

    
    
    
    
"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. (Pls. be patient, the download is a little 
   long).

  

    
    
    
    
    
    
"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.).
   (Pls. be patient, the download is a bit long).

  

    
    
    
    
    
    
"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" 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 ordinal or continuous latent 
   variables.
   (Pls. be patient, the download is a little long).

  

    
    
    
    
    
    
"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). (Pls. be patient, the download 
   is a bit long).

  

    

  

    

  

    
    
    WORKING PAPERS MENTIONED ELSEWHERE ON THE WEB 
      SITE

  

    
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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. (Pls. be 
   patient, the download is a little long.)
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 
   HumphreysSets, and Shortcuts Using Reliability Loadings." 
   [on-line paper]. http://home.att.net/~rpingjr/Ftest10.doc .

  

    
    
    
      
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