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I'm referring to a method called PLS PM:

http://cran.r-project.org/web/packages/semPLS/vignettes/semPLS-intro.pdf
http://gastonsanchez.com/PLS_Path_Modeling_with_R.pdf

Not going into detail, I just want to know, if it is possible (and statistically correct) to multiply outer loadings of the model with the path coefficients of the inner model in the PLS-PM approach.

E.g. if you look into the first link (page 5), I not only want to know how the latent construct "Image" is related to it's manifest variables (e.g. "IMAG1") or how the latent constructs are connected to each other (e.g. the influence of Image on Expectation), but I also want to assess the influence of the manifest variable IMAG1 on Expectation.

The "problem" is that the PLS-PM approach calculates outer loadings (if I'm not mistaken, these are regression coefficients between the latent construct and their manifest variables) and inner path coefficients (OLS regressions between latent constructs).

Since both seem to be regression coefficients, I thought it would be ok, to multiply them, but honestly I'm not sure about the statistical appropriateness of this (e.g. there must be a reason why loadings are called loadings and coefficients are called coefficients).

Again the above mentioned example:

  • The outer loading between IMAG1 and Image is 0.74 (bottom of page 19)
  • The path coefficient of the latent construct Image on Expectation is 0.505 (page 18).

Would it be ok to say that the influence of IMAG1 on Expectation is 0.74*0.505=0.3737 ?

Unfortunately I didn't find any explanation in the original papers, so I'm wondering.

I also know that I didn't provide my own example and I'm neglecting the differences between reflective and formative models, but I hope my question will become clear with the reference to the links at the beginning.

Thanks for your help!

Update: I'm aware that in any case I could only figure out the influence of IMAG1 on Expectation that goes through the "image (LV)" path. Since the IMAG1 variable isn't fully reflected by the LV "Image", there is a IMAG1-part that will not be covered by the model.

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First, note that the entire model is specified in reflective Mode A. Now take a look at the equations (page 6):

Structural (inner) model:

Image = Image + 0 (Note: exogenous variable)

Expectation = $\beta_{12}Image + z_{2}$

Measurement (outer) model:

IMAG1 = $\lambda_{11}Image + \epsilon_{11}$

There is no way, to figure out the influence of IMAG1 on Expectation, as can be seen by the equations.

Or even more simple: look at the visual model (page 5). Notice the arrows and their directions!

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  • $\begingroup$ Thx jannic. Do I understand you correctly that you can't assess the influence of IMAG1 on expectation because the outer model is measured in a refelctive way? If so, would it be possible, if the outer model is measured in a formative way (Mode B)? The reason Im asking is because the above mentioned is just an example- In may real dataset, I indeed have a formative model and since the formative outer model and the inner model are both basically multiple regressions,multiplying the coefficients might be possible? $\endgroup$
    – deschen
    Jul 29, 2015 at 15:30
  • $\begingroup$ Please note: I also updated my question (and sorry for my late response after over a month only!) $\endgroup$
    – deschen
    Jul 29, 2015 at 15:30
  • $\begingroup$ If your outer model is formative, the latent variable is a multiple regression with the indicators as predictors, and the loadings are actually regression weights not correlations. To be honest I really don't know if you can multiply regression weights with each other or with correlations and what the outcome would tell us. But why don't you take a look at the cross loadings instead? $\endgroup$
    – jannic
    Jul 29, 2015 at 15:42
  • $\begingroup$ Not sure, if we mean the same with corss loadings. In terms of PLS: "Besides checking the loadings of the indicators with their own latent variables, we must also check the so-called cross-loadings. That is, the loadings of an indicator with the rest of latent variables." That's not exactly, what I'm interested in, because it will only tell me, if IMAG1 is best "connected" to Image and not to another LV. $\endgroup$
    – deschen
    Jul 29, 2015 at 15:51
  • $\begingroup$ It would help a lot if you provide an example, maybe an artificial model which matches your real one. To my understanding the cross loadings are nothing but the correlations of the manifest variables to other latent variables and thats what you wanted or not? Edit: I'm not sure of the formative case here but thats what they are in the reflective case $\endgroup$
    – jannic
    Jul 29, 2015 at 15:58

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