Mediating Variable Specification

[tem3]

Hello,

In a July 2003 Journal of Marketing article (White, Varadarajan, and Dacin), the authors used the approach described below to provide support for speficiation of a mediating variable in their model. (Their model had 3 predictor LVs with each loading on one "mediating" LV, which loaded on one dependent LV.)

The authors stated, "First, we reviewed the theta matrix to check for potentially significant nonspecified paths. Second, we added those paths to the model and the model reran to assess signficance" (p. 73)

Essentially, the authors are trying to determine if they should have specified paths from the predictor variables directly to the dependent variable without going through the mediating variable. I would like to try this with my model, but I don't know where to find the theta matrix. In LISREL, the theta matrix is the covariance of measurement errors. I didn't think this was produced in PLS. Can you tell me if/where I can find a theta matrix? Also, once I have the theta matrix, how would I determine what is a "potentially significant" covariance?

I would be very greatful for any comments!

Best regards,
Teresa

[Diogenes]

Hi Teresa,

the Theta matrix is produced in PLSGraph software and your meaning is the same of the LISREL (Theta-delta or Theta-Epslon).

For instance, if we have a measurement model like this

Delta1 ==> X1 <== LV1 ==> X2 <== Delta2 (see a correction in my next box, below)

In the standardized way we will have:

Theta-delta12 = r12 - load1 * load2
Theta-delta11 = 1 - load1 * load1

Theta-delta (TD) is our ignorance about a likely cause of the correlation between the indicators (not explained by the LV).

The 2nd question: how would I determine what is a "potentially significant" covariance?

I didn´t remember me any reference, but I think that:
If the Theta-delta (TD) between two indicators is high (>0,1), you could:
- to try link these variables (one to another) or
- to include a new LV conected between them;
- to remove one of these indicators.

In any case you should see if the modification (improvement X parsimony) in the model is significant and if it has practical and theorical meaning.

Best regards

Bido

[jjsailors]

Theta-delta (TD) is our ignorance about a likely cause of the correlation between the indicators (not explained by the LV).

[

But, LVs in PLS are always formative, and there so there is never a requirement, assumption, or necessity that indicators loading on the same construct be correlated. So in the PLS context, correlation between indicators is neither explained nor particularly interesting, it would seem to me. A theta matrix (is it really even appropriate to produce in a PLS model?) isn't informative in this environment.

Or am I wrong? ;-)

John

[Diogenes]

Hi prof. John ,

If we agree that "LVs are always formative", we shoudn´t compute AVE or Cronbach´s Alpha too (it´s supposed that the indicators of the same LV are correlated for these measures).

Some quotations:

LVPLS Program Manual (p.1-03 Lohmöller, 1984)
"...leads to the estimation of path equation (01) by minimizing the residual variances."

Lauro & Vinzi < http://w3.uniroma1.it/sis/pdf/atti/RSMi ... e%3Apdf%22 >
page 203-4: "...PLS aims at minimising the trace (variances) of Ψ and, in case of reflective indicators, also the trace of TE (theta-epsilon) while the trace of TD (theta-delta) is minimised in case of formative indicators."

Finally, if PLS minimize the residual variances, but yet remains some big covariances between the errors, probably the model is with error in its specification.

I need to make a correction in my example, I´ve used the notation from LISREL, but in PLS is quite different: [/b]Delta is the residual from regression when we use formative way and Epsilon when outer-directed (reflective). PLSGRAPH always shows as the Theta matrix.


Best regards.

Bido

[jjsailors]

If we agree that "LVs are always formative", we shoudn´t compute AVE or Cronbach´s Alpha too (it´s supposed that the indicators of the same LV are correlated for these measures).

I think that's correct. Cronbach's alpha--that is to say, reliability based on a factor or true score model--is not important when dealing with formative indicators. Take the often given example of a formative LV, socio-economic status. Here we have no reason to assume or even desire that income, education, and occupation prestige correlate so a low Alpha, for example is not important.

A low AVE is perhaps another matter, focussed as it is on discriminant validity and not reliability, but even here this may not be the best way of assessing discriminant validity within a PLS framework. (?)

LVPLS Program Manual (p.1-03 Lohmöller, 1984)[/b]
"...leads to the estimation of path equation (01) by minimizing the residual variances."

Yes, variances, not covariances.

page 203-4: "...PLS aims at minimising the trace (variances) of Ψ and, in case of reflective indicators, also the trace of TE (theta-epsilon) while the trace of TD (theta-delta) is minimised in case of formative indicators."

Same as above.

Finally, if PLS minimize the residual variances, but yet remains some big covariances between the errors, probably the model is with error in its specification.

Again, not sure this is the case. Want to pursue this idea and see if there's something interesting here?

Best regards,

John

[Diogenes]

John,

thank you for your aswers and questions.

I´ll think about a didactical example for us.


Best regards.

Bido

[Diogenes]

Hi Teresa,

I´ve read the article that you quoted and:
1) In p.71 we have: " ...an examination of the theta matrix confirmed that no item loaded higher on another construct than it did on its associated construct." ==> In fact we could do this assessing the matrix of cross-loads.

2) In p.73 we have: "First, we reviewed the theta matrix to check for potentially significant nonspecified paths. Second, we added those paths to the model and the model rerun to assess significance." ==> The matrix of correlations between LVs could be used for this purpose.

Then the term "theta matrix" in the article seems quite stranger.!?

===================

Hi John,

I really try to find a way to use the theta matrix from PLS-Graph, but I didn´t find !! (You were right.)
1) Even the TD (theta-delta = covariation between errors) were big we don´t have anything to do, in LISREL we could model this, but in PLS.
2) The things that were quoted in the article could be done with matrix of cross-loadings and the matrix of correlations between LVs.


Thank you.

Bido

[tem3]

Hello Bido,

Thank you very much for your reply and for taking the time to read the article I referenced. Your explanation clarified the issue for me. Just one more question: Can I assume there are no "potentially significant" cross loadings if the cross loadings range from .013 to .559, and none are above the specified loadings in the model? Based on these numbers, do I need to specify any paths to assess significance or am I justified in assuming the cross-loadings are not significant?

Thanks very much,
Teresa

[Diogenes]

Hi Teresa,

this question seems "How much weight should we put on data or theory?"

1) If you are working in a confirmatory context (theory is available), you shoud put less weight on your data. I mean, is it expected that a indicator (MV) has loads from two LVs? If not, you could use cross loadings just to justify the discriminant validity (the loadings from the LVs to their indicators is bigger than the loadings from another LVs).

2) If you are working in a exploratory context (where the theory is been building) you could try modifications in the model, but it is a good idea to justify why a indicator has loadings from some LVs. It´s useless a model with good results (AVE, Communality, R2...), but without a theoretical support.

Best regards.

Bido

[Andrea Altmann]

Hello Bido,

I have just read your comment to model modification:

"In any case you should see if the modification (improvement X parsimony) in the model is significant and if it has practical and theorical meaning.

Best regards

Bido[/quote]"


Could you please explain how I can measure the significance of a modification? I do not understand what you mean with improvement X parsimony.

Thanks a lot.

Andrea