Latent Variable
Interaction
Research

    
    
Updated
      1/27/10
      (Previous updates 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.)

    

  

    

    

    
      


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The web site move will create broken links. Please e-mail me 
if you find one--I will provide the desired material some other way 
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FOREWORD--This web site concerns:
              o Latent variable (LV) interactions and quadratics (e.g., XZ and XX), and 
                 (LV) 3 way interactions (e.g., XZW) and cubics (e.g., XXX), 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 structural equation analysis software such
   as LISREL, EQS, AMOS, etc., but are just getting started with latent variable
   interactions, quadratics and their relatives (e.g., XZW, XXX, etc.).
NEWS:
   ATT is no longer web hosting, so, with my apologies for any inconvenience this
   may create, the web site has been moved. Unfortunately, the new server 
   occasionally may not be available, but most outages are brief (infrequently, over 
   a weekend, an outage can be longer). Links are now broken, and some unbroken 
   links will be broken after mid-March. Please e-mail me with any difficulties this 
   creates and I will try to provide a remedy.
   A suggested approach for estimating a 3-way interaction (e.g., the three-way 
   interaction XZW) for the LV's X, Z and W is available by e-mail. XZW may seem 
   unlikely, but a proper test of the hypothesis "X-Y is moderated by Z and W" is to
   specify XZ, XW and XZW in the model. (Z and W may moderate X-Y jointly (via XZW) 
   rather than separately (via XZ and XW). Or, a combination of XZ, XW and XZW may be 
   significant, possibly including all three.)

   A suggested approach for remedying "troubled" LV's--ones with poor psychometric
   properties (i.e., low reliability or AVE, or discriminant invalidity), low external 
   consistency, underspecification, etc.--is available. (Please see "How are 
   Formative Latent Variables estimated with LISREL, EQS, AMOS, etc.?" in 
    Questions of the Moment below.)

   The paper on improving AVE has been revised, with some surprising results (e.g.,
    some of the "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 for testing (truly) categorical variables in theoretical model 
   (hypothesis) tests with structural equation analysis is available. (Please see "How
   does one estimate categorical variables in theoretical model tests using structural
   equation analysis?" in Questions of the Moment.)
 
   A suggested approach for remedying a "not Positive Definite" message ("Ill 
   Conditioned" in exploratory factor analysis) in theoretical model (hypothesis) tests 
   with survey data and structural equation analysis is available. (Please see "How does
   one remedy a "not Positive Definite" message?" in Questions of the Moment.)

   Comments on using 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 interactions forms besides XZ are available by e-mail. This may 
   not sound like much, but XZ is not the only form an interaction can take. (E.g.,
   X/Z and XZ2 are also interactions.) So, a nonsignificant XZ may not mean the 
   hypothesized moderation 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 using 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 below for estimating an endogenous interaction have changed.
   An EXCEL template for estimating the indirect effect of an endogenous interaction
   is available. (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., how interactions might be 
   "theorized"--what evidence might suggest there is an interaction in the model 
   (before data is collected), how an hypothesized interaction might be theoretically 
   justified (argued for), etc.). (Please see "Interactions May Be the Rule Rather than 
   the Exception, But..." in SELECTED  PAPERS ON LATENT VARIABLE INTERACTIONS
    AND QUADRATICS.)

Coming Attractions:
   o An EXCEL template to help provide a simple straightforward tabular interpretation of a 
      significant Interaction or Quadratic (similar to crosstabs) that does not require graphs
      (please e-mail me for a draft template).
   o More on estimating a mixed structural equation analysis model containing a truly 
      categorical variable.
Recent Additions and Changes in this Web Page (indicated by "New," "Revised" or
   "Updated" below):

   o Suggestions for estimating mixed SEM models with Formative and Reflective LV's
      (these models may be an option when an LV, or a 2nd order LV, has poor 
      psychometric properties--low reliability or AVE, discriminant invalidity, low 
      external consistency, underspecification, etc.) (please see "How are Formative 
      Latent Variables estimated...?" below); 

   o Suggested remedies for remedying low Average Variance Extracted (AVE) are 
      provided; 

   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?"

   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 "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,
   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, 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.
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. 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 the browser toolbar (above), and type the word "EXCEL" in
   the "Find what:", then click on "Find...".

  

    
    
  

  

    

  

    
    
    
INTRODUCTION

    

  

    
    

  

    (N

       e

       w) 		
    Please Note: If you are estimating an interaction 
      involving one or more endogenous variables please click here--things 
      are different in this case. 
      
ALSO, and at the risk of overdoing it, please remember that interactions and quadratics should be
       hypothesized before     the model is ever tested with data. Please be 
      aware that the only situations where one should search for significant interactions or quadratics is  
      
    
      1) when one wishes to
      explain a non-significant (NS) 
      first-order model association

               (e.g., is there an interaction, XZ, 
      or quadratic, ZZ, masking the hypothesized

               Z-->Y association?), or  
      
    
      2) where one wishes to investigate a
      significant association(s) to see if it is

               conditional (i.e.,
      moderated by some other variable), and
      thus improve

               interpretation of 
      the model estimation results (as is routinely done in ANOVA).  
      
     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 hypothesized Z-->Y 
      association actually depended on the levels of X in the study for 
      its strength and significance (see "Interpreting Latent Variable
      Interactions" in the SELECTED
      PAPERS...section below).  
      
     Any of these moderated Z-->Y 
      association(s) discovered after the fact also 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 motivation 1 or
      2 above, hunting for significant 
      interactions or quadratics, not hypothesized before the 
      model was tested (i.e., to "improve" a
      paper's contribution), is considered "poor science." It can tempt one to
      add 
      an interaction or quadratic hypothesis 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 is discovered after the fact, changes one's
      "hypotheses-before-first-test" model-test (confirmatory) study into a
      "test-before-all-hypotheses"  model-building (exploratory) study. This 
      in turn increases the likelihood that the significant 
      associations observed in the model test exist only by chance (i.e.,
      they are 
      spuriously significant, and they exist in the present sample 
      only). 

  

    

  

    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 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.) 
  

  

    
    
      

  

    (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
      short
      answer is, with considerable effort. In the popular structural equation analysis
      software (e.g., LISREL, EQS, Amos, etc.), the term "categorical
      variable" usually is 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 structural equation
      analysis models with latent variables and a truly
      categorical variable.
      However, there are several issues in this mixed model.

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

      

       		
  

  

    
    
    
  

  

    (N

       e

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

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

      

      
 		
  

  

    
    
     

  

    (N

       e

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

  

    
    
    
      Contrary to 
      conventional practice, interactions and quadratics 
      also may be important in applied 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 a model.
      Anecdotally, it is 
      not widely known in applied research that 
      important model effects may not be "Imperative" 
      (e.g., the latent variable Y increases/decreases with X). Instead, these
      effect may be "Conditional" (e.g., Y 
      increases with X when Z is at a high level (strong), but Y is unrelated to
      X, or it decreases with X, when Z is at a lower level (weaker)).  
 
         (Please click here for a paper on 
      this matter.)		
  

  

    
     

  

    (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 latent variable (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. Also, consider reading "How are

         Formative Latent Variables estimated with LISREL, EQS, AMOS,
      etc.?", below, to

         remedy really "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 a paper on this
      topic.)

       		
  

  

    
    
      

  

    (Re-
vis-

      ed) 		
    "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 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.)

        		
  

  

     
     
  

  

    (N

       e

       w) 		
    "Why
      are reviewers complaining about the use of PLS
      in my paper?"

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

       		
  

  

     
     
  

  

    (N

       e

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

       		
  

  

     
     
    This
      may not seem like much--formative variables are comparatively rare in the
      Social Sciences. But, a formative respecification of a reflective LV may be a remedy for "problem"
      LV's (e.g., low reliability or AVE, unacceptable discriminant
      validity, low external consistency, etc.). This material also may be a "must read" for using
      older, well established (often multidimensional), 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 on 
      this subject.)

       		
  

  

     
     
     
  

  

    
    ______________
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
      a 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.


  

    
    

    
  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). 


  

    
    

    
  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 subsets of 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

  

    
    

    
   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. 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://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 time may be 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 time may be 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 time may be 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 time may be 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 time may be 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 time may be 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 time may be 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 time may be 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 time may be 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 time may be 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 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 ordinal or continuous latent 
   variables.
   (Pls. be patient, the download time may be 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 time may be a bit long).

  

    

  

    

  

    
    
    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. 
  (Pls. be patient, the download time may be 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 
   Humphreys Sets, and Shortcuts Using Reliability Loadings." [on-line paper]. 
   http://home.att.net/~rpingjr/Ftest10.doc .

  

    
    
    
      
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      (c) Robert Ping 2001-2010