Moderators in PLS: Which modeling approach is appropriate?

[Honest]

Hi all,

I have a question concerning the analysis of moderator effects with PLS that differs from previous questions on the topic discussed in this forum:

Let´s say I have a structural model with four exogenous variables and two endogenous variables. Each exogenous variable is connected via paths to both endogenous variables, other paths (for example between endogenous variables) are not included.

Now I assume that ONE of the exogenous variables is moderated by a moderator variable which was not included in the original model. Let´s assume further that the moderator increases the strength of the relationship between the moderated exogenous variable and both endogenous variables in the original model.

My question now is: What is the appropriate model to test the hypothesized moderation effect? Do I use

a) a model with ONLY the moderator variable Z, the moderated exogenous variable X and their interaction construct X x Z as exogenous variables or

b) the WHOLE original model (including also the exogenous variables not affected by the moderator), extended by the moderator variable Z and the interaction term X x Z as exogenous variables?

I have already read the basic article of Baron and Kenny (1986) on moderators and especially the article of Eggert, Fassott, Helm (2005) dealing with the issue of modeling moderator effects in PLS without finding a satisfying answer...

Help from one of the PLS-Experts in this forum would be highly appreciated!

Regards,

Oliver Ehrlich

[ghozali]

Hi,
Actually there are two moderator variable, pure moderator or quasi moderator. If you put Z on the equation as predictor and also as moderator (Z*X) the problem we do not know whether the moderator variable is X or Z. If you want to assumed that ther moderator variable is Z then do not put Z as predictor.
See Sharma, S et al .1981. "Indentification and Analysis of Moderator variable". Journal of Marketing Research. 18. 291-300

[cringle]

Dear Oliver,

besides the proper use of 2nd-oder-models (viewtopic.php?t=145), the issue you addressed on moderating effects bothers me as well and I came to the same questions you asked. For myself, I came to the following solution: Do not account for moderating effects in PLS, unless a statistical sound foundation is presented.

The suggestions, so far, come from a practical side. However, you should not use moderating effects in an overall model (indirect reuse of indicators, increase of R² of the latent endogenous variable). Just single it out and analyze it in a separate PLS model.

I also have my doubts regarding the use of a single indicator (as manifest variable for the interaction term, created by multiplying the latent variable scores of exogenous and moderator variable) when formative measurements are involved. In that case one could argue just to take the latent variable scores of the endogenous variable and the moderator variable in the traditional regression analytic way (get the latent variable scores of exogenous, endogenous and moderator variable into SPSS and analyze the moderator effect the “classical” multiple regression analytical way).

A last point must be addressed with respect to sign problem in the inner model. Different PLS software applications lead to different results due to their initialization (see Temme/Kreis in the publications area). Therefore, you cannot be sure if the PLS operationalzed computation of the interaction effect has indeed a positive or negative sign, which is important for further interpretation. Since SmartPLS is the only application that uses only +1 value as initial weights, you can get reasonable results, if you make sure that the manifest variables in the outer models of the exogenous and moderator variable have the same direction plus positive weights.

Oliver, beside this collection of thoughts, your question on moderating effects remains and I would be glad, if somebody could give some answers to these important issues.

Best
Christian

[stefanbehrens]

Oliver,

I'm struggling with similar issues in my path model. Not that I'm any closer to a definitive answer to your questions, but here's my current thinking on the available options:

(A) Singling out the effect into a simple 4 LV-PLS-model: Y=b1*X+b2*Z+b3*X*Z
This is only valid if the indicator weights for X and Y remain similar to the ones in your full model. If they don't, you are in effect testing a moderator-relationship between different variables because the X and the Y in your reduced model no longer coincide with the X and Y in your full model. This is particularly critical if X or Z are measured formatively and X*Z is measured by an LV-product-indicator.
One way to fix this issue is to export the LV-scores from your full model and then run the moderator-analysis in SPSS or SmartPLS to determine the significance of the product term. Either way you have to be careful about interpreting the resulting coefficients as they may be very different from those in your full model.
However, if you are working with LV-scores anyway, you may as well proceed to (B)...

(B) Including Z and X*Z in your full PLS-model:
Using LV-scores from your full model in testing an expanded full model (adding Z and X*Z) will ensure that the inclusion of 2 more constructs will not alter any weights and thus distort your variables. Also, this allows you to directly interpret all the path coefficients at the same time. Nonetheless, there is a drawback. If you incorporate the two additional constructs into your model (one of which is by nature collinear with two other constructs) your statistical power goes way down. This loss in power is even worse, if X and Z are correlated. So if your sample is not exactly very large, you may observe that some coefficients will become insignificant.
Using Chin et al's product-indicator approach instead of the LV-scores is another option if X and Z are both reflective. However, it suffers from the same statistical power drawbacks. In addition, I've experienced that using this approach may result in individual negative weights/loadings of the product-indicators on the interaction-LV. I am not sure as to how to interpret this and whether it is acceptable from a theoretical point of view (my guess is: no). Unfortunately, Chin et al fail to discuss this issue in their article.

To sum it all up, I would probably recommend B) using the LV-scores approach.

As always, I'm happy to hear the experts' opinion on the above. Jörg may be able to shed some light on this.

Kind regards,
Stefan

[Honest]

Ghozali, Christian, Stefan, thank you for your helpful advice on the moderator issue.

Some comments on your answers:

Stefan wrote:
"(A) Singling out the effect into a simple 4 LV-PLS-model: Y=b1*X+b2*Z+b3*X*Z
This is only valid if the indicator weights for X and Y remain similar to the ones in your full model. If they don't, you are in effect testing a moderator-relationship between different variables because the X and the Y in your reduced model no longer coincide with the X and Y in your full model."

Your argument on the changing weights of X and Y, Stefan, seems fundamental to me: the problem does not only occur in the singled-out moderator model, but also in the extended full model (B), if Z and X*Z are added. Furthermore, if I use the isolated moderator model, the weights of the X*Z-indicators (my X and Z are reflective, so I have used the product-indicator-approach of Chin et al. to obtain the indicators) are drastically different from those I get if I use the extended full model, which cannot be beneficial for the validity of my results either.
In fact: if the weights of the X*Z-indicators do not reflect the weights of the underlying X- and Z-indicators, and furthermore the weighting scheme changes significantly depending on my modeling approach, the disturbing question arises when the product-indicator-approach of Chin et al. to obtain the X*Z-indicators is appropriate at all???

Stefan wrote:
"One way to fix this issue is to export the LV-scores from your full model and then run the moderator-analysis in SPSS or SmartPLS to determine the significance of the product term."

Using LV-scores appears to me like an elegant way to keep the weighting scheme fixed, as intended. However, if you suggest that the X*Z-indicator should be determined by multiplying the LV-scores of X and Z, a new problem occurs:
In the scenario outlined in my first post, the moderator Z is not included in the original model - therefore I´m not able to obtain a LV-score for Z. I could include Z in my full model, of course, but then the weighting scheme of X and Y would be affected again... Or would you multiply the LV-score of X with each indicator of Z instead to get indicators for XZ?
Apart from that, thank you for new ideas on moderation modeling in PLS!

Finally,

Ghozali wrote:
"If you put Z on the equation as predictor and also as moderator (Z*X) the problem we do not know whether the moderator variable is X or Z. If you want to assumed that ther moderator variable is Z then do not put Z as predictor."

Whether to include Z in the moderated regression is another important issue. Although you´re right in saying that in an equation with X, Z and X*Z, X could be as well the moderator as Z, I think an exclusion of Z may be problematic, too, because of two reasons:

1. In a moderated regression equation Y = b1*X + b2*Z + b3*XZ, b3 shows how much b1 changes given a one-unit change of Z (Jaccard, Wan, Turrisi 1990, p. 469), thereby indicating the direction and strength of the interaction effect. I´m not quite sure if the same interpretation of b3 holds if Z is eliminated from the equation, but my guess would be no.

2. X*Z is usually highly correlated with X and Z (although standardization of X and Z before calculating X*Z is decreasing the multicollinearity problem). The correlation has the effect that X*Z contains variance accounted for by X and Z, which distorts the view on the "pure" interaction effect (Cohen 1968, 1978). One way of dealing with this problem is hierarchical regression, the comparison of the R-square of Y = b1*X + b2*Z with the R-square when X*Z is included, Y = b1*X + b2*Z + b3*XZ. The results can be used to calculate the effect size of the interaction. Again, without Z in the equations, hierarchical regression and the interpretation of the R-square-increase caused by X*Z would not be meaningful.



I´d be happy to read further comments on this topic, especially as I´m obviously not the only one who faces problems with the moderator analysis in PLS!

Regards,

Oliver

[stefanbehrens]

Oliver,

now we're really delving into the issues... ;-) Some comments on your comments:

(A) Weight changes in independent LVs
Let me clarify my earlier post a little bit here. I was not saying that any change (even the slightest) in the indicator weights is necessarily problematic. I was simply saying that the weights should be "similar enough" (whatever exactly that means) to make sure you are not in fact comparing two different concepts and thus arrive at faulty conclusions. The following article touches on this issue (Error 9) and may be an interesting read:
Carte, T.A., and Russell, C.J. "In Pursuit of Moderation: Nine Common Errors and Their Solutions," MIS Quarterly (27:3), September 2003, pp. 479-501.

(B) Weight changes in the interaction LV
I would not worry very much about weight changes of the product-interaction-indicators between the singled-out and the full model. As long as all weights are positive, I wouldn't care if there are any differences. If only the measurement models of X and Y are relatively stable. I believe Chin's approach is appropriate, if the weights of all product-indicators are positive for the interaction LV.

(C) Where to get LV-scores for Z
You don't necessarily need LV-scores for Z: Run your full model. Save the LV-scores. Include them in your data file and also calculate X*z1, X*z2, etc. for all indicators of Z. Then run your model using LV scores for X, Y, and all other LVs besides Z and X*Z. Use z1, z2, ... for Z and X*z1, X*z2, ... for X*Z. Voila.

(D) Leaving out Z in the regression equation
I would strongly argue against removing Z from the regression from a purely algebraic point of view. If we start with a typical MMR equation:
(1) Y = a + b1*X + b2*Z + b3*X*Z
Factor X:
(2) Y = a + (b1 + b3*Z)*X + b2*Z
From (2) we can easily derive the statement from Jaccard et al. that you cited: If Z is one, X's coefficient becomes b1+b3 and thus goes up by b3 as compared to when Z is zero.
Now if you leave the b2*Z term out of the equation, you are effectively setting b2=0. The interpretation will still hold, but estimation bias may be an issue: Your coefficient estimates for b1 and b3 will only be unbiased, if Z and Y are completely uncorrelated.


Hope this helps,
Stefan

[sergeja]

I would like to estimate interaction between a formative and a reflective variable. I haven't come across any paper dealing with the interaction between two types of LV. The formative variable is a function of only one indicator. The reflective one has three indicators.

If I treat the interaction as a product of indicators, I do not know whether I should look at weights or loadings on the interaction term in the PLS output. In my case both are strange. Loadings are over 1, whereas weights are even as small as 0.19. I do not know now, whether I can safely go on with the interpretation of the results. Interaction term is not significant anyway. I included it into a full model.

Thank you for your answers.

Sergeja

[stefanbehrens]

Dear Sergeja,

if your formative LV only has one indicator, things are not that difficult. Simply standardize your 4 indicators (f1, r1, r2, r3) and create 3 product-interaction-indicators (f1*r1, f1*r2, f1*r3) prior to loading the data into SmartPLS. You can then use the reflective mode to create an interaction-LV with these 3 indicators.

This way, the loadings for the 3 indicators of your interaction-LV should all be <1 (if they're not, something else is seriously wrong, since the loadings are simply correlation coefficients that - in theory - cannot exceed 1). The weights should all have the same sign, but otherwise don't matter.

Good luck,
Stefan

[Paolo Pinkel]

Hi all,

I digged out this old discussion as I have a question regarding the correct interpretation of an interaction effect in my PLS model.

I have one exogenous formative LV (X), a formative moderator (Z), and one reflective endogenous LV (Y). Now, in my model I hypothesize the relationship X -> Y and that Z moderates this relationship. The model does not contain any relationship of Z -> Y.

The question is now, what constitutes my main effect model?
Alternative 1) X -> Y without Z, or is it
Alternative 2) X -> Y and Z -> Y

If I go for alternative 2 the following problem arises:

In the main model both X and Z show positive and significant path coefficients to Y, although the relationship Z -> Y is not part of the theory. After extending the main effect model with Z*X -> Y, the original paths X -> Y and Z -> Y become insignificant and only the path X*Z -> is significant. Furthermore, the resulting f² shows only a weak interaction effect. With this extension I have no idea how to interpret these results and they don't seem logical to me either, as the path Z -> Y was never hypothesized.

Yet, if I go with alternative 1, everything is fine. But, the papers I've seen so far and the discussion above all hint to alternative 2.

I would really appreciate it if some of you could come up with some helpful ideas or suggestions on this matter, as the above discussion is already a couple of years old.

Greetings
Sasa