Latent Variable
Interaction
Research

Updated 5/12/09 (Previous updates 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.)

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:

   The paper on improving AVE has been completely revised, with 
   some surprising results (e.g., most of the "discriminant validity" 
   tests do not work in real-world survey-data theory tests).

   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--1/Z also can moderate the X-->Y association, so 
   can Z2, 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 (hypothesis 1 
   above) could be argued to be equivalent to the hypothesis "The 
   X-Y association is moderated by some form of Z (e.g., 1/Z, Z2, 
   etc."). 
        Thus, if XZ turns out to be nonsignificant, this does not 
   disconfirm hypothesis 1) above (i.e., there may still be an
   XZ 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 X moderating the Z-Y association. 

   Comments on the use of regression to test an hypothesized 
   interaction. (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 forms of evidence 
   suggests 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 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 makes one suspect there is an 
      interaction in one's model before any data is collected? 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,
   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 (above) 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 use 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" (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 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" (above), then click on "Find..."
   and type the word "EXCEL" in the "Find what:" box.

INTRODUCTION

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).

"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.)

"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.)

"How is a cubic LV estimated?"

"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.)

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(Re-
vised)

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

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(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 .

(Up-
dated)

   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. 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

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"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 types of evidence suggest 
   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). 

"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).

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

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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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