Higher-order model: explained variance

[Rebeka]

Hello,

I am working with a reflective-formative third-order model and aiming to clarify the variance explained by each second-order construct on the third-order construct. Could you confirm if the following calculation accurately explains the proportion of variance from a lower-order construct to a higher-order construct?

The Calculation:
I conducted the consistent PLS algorithm with a factor focus. I then computed the squared path coefficient for each second-order construct concerning the third-order construct and subsequently summed these coefficients. Can I correctly infer that the X second-order construct accounts for approximately Z% of the variance in the Q third-order construct?

Thank you!

[jmbecker]

First, for a formative construct in PLS, regardless of it being higher-order or not, you will always have an r-squared of 1 (or at least very close to one). This is, because the indicators define (compose) the construct and thus they should and will explain all the variance of the construct because the variance comes on from themselves.

Second, the sum of squared path coefficients does not give you the R² and thus a squared path coefficient is also not the contribution to R² (except in the case of uncorrelated / orthogonal predictors).
Tenenhaus et al. (2005) show on page 179 that the case of correlated standardized variables, the R² is the sum of path coefficient multiplied with the correlation between predictor and outcome. Accordingly, they define the R² contribution of a predictor as its path coefficient multiplied with the correlation divided by the R².
However, they also note that this only works, if the coefficients and the related correlations have the same sign.

Tenenhaus, M., Vinzi, V. E., Chatelin, Y. M., & Lauro, C. (2005). PLS path modeling. Computational statistics & data analysis, 48(1), 159-205.


However, there are also other ways of defining R² contribution. For example, for the calculation of the f² effect size, the R² conrtibution is calculated as R² included - R² excluded, where included means with the focal predictor and excluded without the focal predictor.
You could calculate something like this by saving the latent variable scores and running separate regressions with the focal predictors included or excluded. This will likely give your some slightly different results.

[Rebeka]

Thank you, your response is greatly appreciated! Your assistance proves to be of immense value to my PhD thesis.

Based on the information you've provided, I have two more questions:

1) In your statement (and also in Tenenhaus et al., 2005), "However, they also note that this only works if the coefficients and the related correlations have the same sign," does this imply that the sign should be consistent for individual paths, or do all predictors need to share the same sign to enable this calculation? For instance, should both the path coefficient and correlation coefficient of my second-order constructs with third-order constructs share the same sign? Or is it sufficient for each to have the same sign separately (so I can have a predictor with "-" for both, path and correlation coefficient and the other predictor with "+" sign and still employing the calculation)?

2) Can I figure out how much all the predictors together contribute to forming my third-order construct? is there a formula available that allows me to calculate the extent to which, for instance, two second-order constructs contribute to the creation of my third-order construct? I attempted summing up each predictor's contributions (I calculated each predictor contribution as you suggested: path coefficient multiplied with the correlation divided by the R²), but this resulted in an R2 value exceeding 1.

Once again, I extend my gratitude for your invaluable time and insights!

Kind regards,
Rebeka

[jmbecker]

1) You need to have the same sign for coefficient and correlation, so that after multiplication everything is always positive.

2)

Can I figure out how much all the predictors together contribute to forming my third-order construct?

All predictors together, i.e., the sum of products (path coefficients multiplied with the correlations) gives the R².
This should be 1 for a formative higher-order construct.

I attempted summing up each predictor's contributions (I calculated each predictor contribution as you suggested: path coefficient multiplied with the correlation divided by the R²), but this resulted in an R2 value exceeding 1.

Path coefficient multiplied with the correlation divided by the R² gives you the relative contribution (a percentage). The relative contributions of all predictors should sum up to 100%.