Regression Analysis with Latent Variables by Partial Least Squares and Four Other Composite Scores: Consistency, Bias and Correction

Abstract

Compared to the conventional covariance-based SEM (CB-SEM), partial-least-squares SEM (PLS-SEM) has an advantage in computation, which obtains parameter estimates by repeated least squares regression with a single dependent variable each time. Such an advantage becomes increasingly important with big data. However, the estimates of regression coefficients by PLS-SEM are biased in general. This article analytically compares the size of the bias in the regression coefficient estimators of the following methods: PLS-SEM; regression analysis using the Bartlett-factor-scores; regression analysis using the separate and joint regression-factor-scores, respectively; and regression analysis using the unweighted composite scores. A correction to parameter estimates following mode A of PLS-SEM is also proposed. Monte Carlo results indicate that regression analysis using other composite scores can be as good as PLS-SEM with respect to bias and efficiency/accuracy. Results also indicate that corrected estimates following PLS-SEM can be as good as the normal-distribution-based maximum likelihood estimates under CB-SEM.

Keywords:

• partial least squares
• bias-correction
• unweighted composite scores
• factor scores

Notes

¹ In PLS-SEM, the vectors of weights are estimated first, and estimates of other model parameters depend on the estimated weights. In CB-SEM, factor scores are not needed in estimating the model parameters, and can be computed after model parameters are estimated.

² PLS-SEM also has a mode C, which is a combination of modes A and B.

³ Bartlett-factor-scores further divide each $w_j$ by $\lambda {}^{\prime }{\mathbf{\Psi }}^{-1}\lambda$ and regression-factor-scores further divide each $w_j$ by $(1+\lambda {}^{\prime }{\mathbf{\Psi }}^{-1}\lambda )$.

⁴ In practice, we do not need to rescale the unweighted composites unless there is a need for the estimated $\beta$ to be in alignment with a known unit.

⁵ Normal-distribution-based maximum likelihood also faces the issue of non-convergence at small $N$, we thus use the population value of $\lambda$ to rescale the unweighted composite. This does not create a problem since the purpose is to compare with the results of PLS-SEM, and no such rescaling is needed when unweighted composites are used in practice.

Additional information

Funding

This work was supported by Grant 31971029 from the Natural Science Foundatoin of China, and in part by the Institute for Scholarship in the Liberal Arts, College of Arts and Letters, University of Notre Dame