They diagram is rather strange, so I would closely examine the actual coefficients that the original authors reported. In particular, a mediating viable would typically be located directly between the independent variable and the dependent variable. In that case, "mediated moderation" means that the nature of the mediation differs between the categories of the moderating variable. Since the moderating variable is always the source of the interaction, your null model is that the same specification fits equally across each category in your moderating variable. You can test that by "freeing up" the coefficients associated with the mediation, and assessing the improvement in fit for that second model.

The easiest way to do this is to model the data elements (predictors/outcomes; exogenous/endogenous) in AMOS. This will give you the right information and visual presentation. Please see the 2015 paper I wrote extracted from my original 2011 dissertation using Baron and Kenny to determine mediation/moderation in my SEM model. While there are many papers about B&K and many statements in papers that a variable is mediating or moderating, there is often no evidence to support this claim as the term is used very loosely to demonstrate location of a variable instead of the conduct of statistical analysis to determine if the variable does, in fact, have this relationship. Further, you may have to reevaluate the position of your presumed moderators variables W and Z. In fact, the direction of your arrows may also change or be completely eliminated between the moderators and the relationship between your primary items X-.Y. For example, the AMOS layout would show arrows to X-> W and X->Z. Also, Z->Y and W->Y. Further, you should never assume or identify the intervening models as moderating/mediating before you have established the statistical relationship between and among the variables. Eliminate the second down arrow on the relationship of Z and W pointing at X and Y. Maintain the first down arrow between Z and W. Rerun the model to establish overall relationships. Then, you will have to test the basic relationship between X and Y; then test XZY, XWY first. You will have to follow the same logic to establish the relationship, if any, between Z and W. (You may also want to test the model with a double line between Z and W but this would require another complete set of tests.) There are other tests to determine mediation and moderation but I am confident the application of B&K to SEM models in AMOS is valid.

This can be done in AMOS. However, the PROCESS Macro (Hayes, 2018) would also work quite well. The Macro is free as an add on to the SPSS Regression function. It appears that Model 14 is a good fit for you. See Hayes' recent book entitle Introduction to Mediation, Moderation, and Conditional Process Analysis: A Regression-based Approach (2018), The Guilford Press. I have the Kindle edition but I'd suggest getting the book (or both versions). Thanks...Nick

Dear Professor David L Morgan , Professor Beth Ann Fiedler , and Professor Nicholas J. Beutell ,

I highly appreciate your answer to my confusion and most of it is clear now, however, the question of concern for me is that the moderating effect a variable is mediated by an other variable. Let me illustrate it.

IV= A

MOD V=B

Med= C

DV= D

AxB ==> C ==> D

As it is explained in the model given above. However, this is the simplest form of my model. It includes two mediation between the interaction term and DV. I totally understand this is very complex situation but the literature is leading towards these relationships. I hope you understand my concerns.

My suggestion is to stop the thought process on mediation and moderation and keep the process as simple as possible. You must understand that modeling the data by placing it between major X1....and Y1....is not going to determine if it is moderating or mediating. You cannot predetermine mediation and moderation. So, model the data in in AMOS as you believe the literature guides you. If X1 influences Y1 and then Y2, then draw it that way. Y1 would be an example of an intermediate variable that may or may not be moderating or mediating. When you successfully run the data and review outputs that are significant, then you can run the model with just the major X1... and Y1.... (eliminating the intermediating variables). If there are losses to significance of the primary X1>Y1, then you can determine if there is mediation/moderation.

PS Those mediating variables, as I have mentioned before, must have an arrow from the X leading to them and from them. An intermediate variable is both an outcome and a predictor. I hope this helps. Now I must get back to my own work.

Hi

You can run the PROCESS macro in SPSS and run the analysis. Hayes work is very helpful.

Tou may visit his site (http://www.processmacro.org/index.html) for more on this.

Regards,

Sushanta

Ubedullah. Please do not call out my specific name on your questions. I am unable to give any further assistance due to my own project commitments and deadlines. But I will only ask the following. Do you have literature to support the relationships you are suggesting? Does one item apparently moderate another? OR are they thought to be a predictor of that predictor? That is a big difference STATISTICALLY. You see, you are trying to form your diagram with the assumption that they ARE moderating/mediating but NO! You cannot even begin to use those words mediator/moderator until you have established a sound model. Your predictor, outcome and intermediate variables (which you like to erroneously call mediating/moderating) must be modelled ACCORDING TO EVIDENCE IN LITERATURE. I sent the article I did on Moderating/Mediating but it does not look like you read it. Read the paper I sent. You are drawing conclusions before there is any sound model or statistics. ONCE THE MODEL IS DEVELOPED ACCORDING TO LITERATURE PROVING PREDICTOR RELATIONSHIPS TO OUTCOMES AND ANY INTERMEDIATING RELATIONSHIPS, RUN THE MODEL FOR STATISTICS OF MODEL FIT, ETC. IF YOU HAVE A RELATIONSHIP BETWEEN PREDICTOR AND OUTCOME, THEN REMOVE THE ITEMS AND TEST THE INTERMEDIATE MODELS ONE AT A TIME. YOU MAY HAVE TO CONSIDER THE ITEMS THAT PREDICT A PREDICT AS AN X=Y, TOO. If you can't do this then you have bitten off more than you can chew with too many inputs on inputs which may only lead to a bad model. Good luck.

Hi folks,

I am not very familiar with the PROCESS macro but let us think about the testable issues from your model. The implications of your model would be that Z (the distant variable) moderates the relationship between X and Y but as its moderator effect is mediated by W, controlling for the moderator effect of W should run the first moderation by Z "spurious" (since controlling for W blocks the effect of Z and this works for direct effects in the same way as for moderator effects. In a statistical model (regression or SEM) this should work via two steps

1) run the model with an X*Z product term. This should be significant (the "distant moderation)

2) then add the X*W product to the model. This should lead to a significant product and run the X*Z product non-significant.

I created a small simulation which proved that this process works. I can share that on demand.

The Achilles heel, however, is that this is only partial evidence for your model as you still assume that Z affects W and and X affects Y and your model is only weak support for this. Actually, if Z and W were correlated for any other reason (other than Z-->W) than the above procedure would also "work".

Hence, you plagued by the same task to increase the causal interpretation by thinking about control variables that may act as confounders of the X-Y effect as well as the Z-W affect.

Having said that, I am rather puzzled to still see disciples of the Baron-Kenny (BK) approach. BK made many claims that are false and unnecessary (for instance that you must have a significant relationship between X and Y before addressing mediation). I can only speculate that the reason is the historically caused tendency (especially among psychologists) to avoid causal concepts and rules in favor of a statistical fetish. The problem however is that mediation is a causal concept (and no statistical) -- that is, you assume that the causal effect of X on Y is transmitted via a mechanism which M represents. There is no statistical magic bullet that transforms correlation into causation and this is still true in more complex forms of correlation. As a sign of causal avoidance, inspect the works by BK and count the number of "relationship"-terms they use. The most apparent implication of this basic dilemma is that you cannot differentiate mediation from confounding (M acting as a common cause of X and Y). The BK approach still "works" if there is no effect at all.

Of course, an indirect effect in a SEM shows some support for the mediation process (in the same way that a significant correlation provides some support for the effect of X on Y) but the whole interpretation hinges on your theoretical /causal assumptions that are inherent in an innocent indirect effect (for instance, no confounding of the X-M and M-Y links). Therefore, the basis of a mediation model is theoretical, not empirical (from past research) and not obtained by playing around with sub forms of a model. This is especially dramatic in the BK approach that most often still results in a partial mediation model. Such a model has around 40 alternative structures that imply the same data (and you believe in one of them :). That is, you can switch around arrows and will not notice that you messed up. Incorporating just one restriction (i.e., fixing the direct effect) reduces the number of equivalent models to 7! (still bad). That is if you want to increase the causal interpretation, a) think hard about potential confounders and control for them, and b) add restrictions (this may be resulting from incorporating other variables, such as instrumental variables).

HTH

Holger

Kline, R. B. (2015). The mediation myth. Basic and Applied Social Psychology, 37(4), 202-213. doi:10.1080/01973533.2015.1049349

Thoemmes, F. (2015). Reversing arrows in mediation models does not distinguish plausible models. Basic and Applied Social Psychology, 37(4), 226-234. doi:10.1080/01973533.2015.1049351

Hayes, A. F. (2009). Beyond baron and kenny: Statistical mediation analysis in the new millennium. Communication Monographs, 76(4), 408-420.

Pearl, J., & MacKenzie, D. (2018). The book of why. New York: Basic books.

MacKinnon, D. P., Krull, J. L., & Lockwood, C. M. (2000). Equivalence of the mediation, confounding and suppression effect. Prevention Science, 1(4), 173-181.

Zhao, X., Lynch Jr, John G., Chen, Q., Lynch, J. G., Jr., & Chen, Q. (2010). Reconsidering baron and kenny: Myths and truths about mediation analysis. Journal of Consumer Research, 37(2), 197-206. doi:10.1086/651257

Hi Ubedullah Khoso

It is unfortunate that this question did not show up in my feed, so I apologize for the extremely late reation :).

Most of the answers above seem to miss the point of your research question completely, and show why there is a need for the indirect moderation model.

In Mediation analysis, a conceptual representation of usually corresponds to the SEM/path model based on the regression equations. This, however, does not generally hold for moderation analysis (e.g., product terms are not shown in conceptual models, while clearly needed in the regression equations).

Due to popular demand I am currently thoroughly revising the original report. In the revision I am making sure that the graphical path models are shown, next to the conceptual ones where neccesary. I've attached some placeholders to this answer.

Hope this helps

Please feel free to let me know if you require a detailed path model.

Hi Geert,

interesting model but I have some unclarities that you can possibly solve. From your lower conceptual diagram I would expect that in a model without W, Z should interact with X as this is the indirect interaction of interest. I understand, why in the model including W, Z has a direct effect on W but why should a product term between Z and W exist?

Still, my hunch would be that (without W), the ZX interaction is significant (as in the upper model) but in the moment in which W and WX is entered, the latter becomes significant and the XZ product becomes non-significant...Where's my error?

All the best,

--Holger

Dear Ubedullah Khoso

First, what is the nature of your model (exploratory or confirmatory)? If you adapted a previous model, probably you’re in the exploratory phase. Second, are your constructs and their relationships grounded in firmly theoretical assumptions ? Third, why necessarily using Amos? Finally, given that we don’t know what are the constructs, there is a chance that the model‘s relationships could be rearranged in a more conventional model.

Kind regards

Holger Steinmetz Thanks for pointing out the error in my too-hastily made representation :).

Z*W should of course be W*X as in the regression equations

This would be work. I am not sure

Abdallah Alsaad the interaction between X and Z should be in the prediction of Y, not that of W.

See updated picture attached here.

For the question poster, the arrow with the cross may also be significantly different from zero. (I'm updating the preprint now, btw).

Thanks Greet. As I understood the model, Z moderates the effect of X but indirectly through W. So why should we romove that interaction between X and Z in predicting W?

If X influences W (which you specificied) and W interacts with X, your model states that X indirectly interacts with itself ;)

@Geert van Kollengburg

I found that Hays has considerd in his software the case in which a predictor interacts with a mediator

See model 74.

See also this discussion on research gate

https://www.researchgate.net/post/Model_74_SPSS_Process_Macro