Assessing reciprocal relationships in PLS-SEM: An illustration based on a job crafting study

The article below explains how one can test reciprocal relationships in the context of structural equation modeling via partial least squares (PLS-SEM).

Morrow, D. L., & Conger, S. (2021). Assessing reciprocal relationships in PLS-SEM: An illustration based on a job crafting study. Data Analysis Perspectives Journal, 2(1), 1-5.

A link to a PDF file is available ().

Abstract:

Over the last 25 years two types of job crafting have emerged with similar quantitative measurement scales. This paper describes the process used in determining the presence of reciprocal relationships between the two job crafting constructs using WarpPLS.

There is no need for two-way arrows in WarpPLS

In covariance-based structural equation modeling (SEM) software tools, often one has to explicitly model correlations between predictor latent variables (LVs) to obtain related estimates. In WarpPLS, correlations between predictor LVs are automatically taken into consideration in the calculation of path coefficients.

It should be noted that when we say "two-way arrows" we are not referring to reciprocal relationships. Reciprocal relationships are usually represented via two straight arrows, each arrow pointing in a different direction. What we call "two-way arrows" here are representations of correlations.

The path coefficients calculated by WarpPLS are true standardized partial regression coefficients, of the same type as those calculated through multiple regression analysis. The difference is, of course, that in WarpPLS the model variables are LVs, which are usually measured through more than one indicator. With multiple regression, only one measure (or indicator) is used for each variable in the model.

Nonlinearity and type I and II errors in SEM analysis

Many relationships between variables studied in the natural and behavioral sciences seem to be nonlinear, often following a J-curve pattern (a.k.a. U-curve pattern). Other common relationships include the logarithmic, hyperbolic decay, exponential decay, and exponential. These and other relationships are modeled by WarpPLS.

Yet, the vast majority of statistical analysis methods used in the natural and behavioral sciences, from simple correlation analysis to structural equation modeling, assume relationships to be linear in the estimation of coefficients of association (e.g., Pearson correlations, standardized partial regression coefficients).

This may significantly distort results, especially in multivariate analyses, increasing the likelihood that researchers will commit type I and II errors in the same study. A type I error occurs in SEM analysis when an insignificant (the technical term is "non-significant") association is estimated as being significant (i.e., a “false positive”); a type II error occurs when a significant association is estimated as being insignificant (i.e., an existing association is “missed”).

The figure below shows a distribution of points typical of a J-curve pattern involving two variables, disrupted by uncorrelated error. The pattern, however, is modeled as a linear relationship. The line passing through the points is the best linear approximation of the distribution of points. It yields a correlation coefficient of .582. In this situation, the variable on the horizontal axis explains 33.9 percent of the variance of the variable on the vertical axis.

The figure below shows the same J-curve scatter plot pattern, but this time modeled as a nonlinear relationship. The curve passing through the points is the best nonlinear approximation of the distribution of the underlying J-curve, and excludes the uncorrelated error. That is, the curve does not attempt to model the uncorrelated error, only the underlying nonlinear relationship. It yields a correlation coefficient of .983. Here the variable on the horizontal axis explains 96.7 percent of the variance of the variable on the vertical axis.

WarpPLS transforms (or “warps”) J-curve relationship patterns like the one above BEFORE the corresponding path coefficients between each pair of variables are calculated. It does the same for many other nonlinear relationship patterns. In multivariate analyses, this may significantly change the values of the path coefficients, reducing the risk that researchers will commit type I and II errors.

The risk of committing type I and II errors is particularly high when: (a) a block of latent variables includes multiple predictor variables pointing and the same criterion variable; (b) one or more relationships between latent variables are significantly nonlinear; and (c) the predictor latent variables are correlated, even if they clearly measure different constructs (suggested by low variance inflation factors).

Why are pattern cross-loadings so low in WarpPLS?

I have recently received a few related questions from WarpPLS users. Essentially, they noted that the pattern loadings generated by WarpPLS were very similar to those generated by other PLS-based SEM software. However, they wanted to know why the pattern cross-loadings were so much lower in WarpPLS, compared to other PLS-based SEM software.

Low cross-loadings suggest good discriminant validity; a type of validity that is usually tested via WarpPLS using a separate procedure, involving tabulation of latent variable correlations and average variances extracted.

Nevertheless, low cross-loadings, combined with high loadings, are a "good thing" (generally speaking) in the context of a PLS-based SEM analysis.

The pattern loadings and cross-loadings provided by WarpPLS are from a pattern matrix, which is obtained after the transformation of a structure matrix through an oblique rotation (similar to Promax).

The structure matrix contains the Pearson correlations between indicators and latent variables, which are not particularly meaningful prior to rotation in the context of measurement instrument validation (e.g., validity and reliability assessment).

In an oblique rotation the loadings shown on the pattern matrix are very similar to those on the structure matrix. The latter are the ones that other PLS-based SEM software usually report, which is why the loadings obtained through WarpPLS and other PLS-based SEM software are very similar. The cross-loadings though, can be very different in the pattern (rotated) matrix, as these WarpPLS users noted.

In short, the reason for the comparatively low cross-loadings is the oblique rotation employed by WarpPLS.

Here is a bit more information regarding rotation methods:

Because an oblique rotation is employed by WarpPLS, in some (relatively rare) cases pattern loadings may be higher than 1, which should have no effect on their interpretation. The expectation is that pattern loadings, which are shown within parentheses (on the "View indicator loadings and cross-loadings" option), will be high; and cross-loadings will be low.

The combined loadings and cross-loadings table always shows loadings lower than 1, because that table combines structure loadings with pattern cross-loadings. This obviates the need for a normalization step, which can distort loadings and cross-loadings somewhat.

Also, let me add that the main difference between oblique and orthogonal rotation methods (e.g., Varimax) is that the former assume that there are correlations, some of which may be strong, among latent variables.

Arguably oblique rotation methods are the most appropriate in PLS-based SEM analysis, because by definition latent variables are expected to be correlated. Otherwise, no path coefficient would be significant.

Technically speaking, it is possible that a research study will hypothesize only neutral relationships between latent variables, which could call for an orthogonal rotation. However, this is rarely, if ever, the case.

See the most recent version of the WarpPLS User Manual (linked below) for more details.

http://www.scriptwarp.com/warppls/#User_Manual