Retrieving relevant factors with exploratory SEM and principal-covariate regression: A comparison
Behavioral researchers often linearly regress a criterion on multiple predictors, aiming to gain insight into the relations between the criterion and predictors. Obtaining this insight from the ordinary least squares (OLS) regression solution may be troublesome, because OLS regression weights show only the effect of a predictor on top of the effects of other predictors. Moreover, when the number of predictors grows larger, it becomes likely that the predictors will be highly collinear, which makes the regression weights’ estimates unstable (i.e., the “bouncing beta” problem). Among other procedures, dimension-reduction-based methods have been proposed for dealing with these problems. These methods yield insight into the data by reducing the predictors to a smaller number of summarizing variables and regressing the criterion on these summarizing variables. Two promising methods are principal-covariate regression (PCovR) and exploratory structural equation modeling (ESEM). Both simultaneously optimize reduction and prediction, but they are based on different frameworks. The resulting solutions have not yet been compared; it is thus unclear what the strengths and weaknesses are of both methods. In this article, we focus on the extents to which PCovR and ESEM are able to extract the factors that truly underlie the predictor scores and can predict a single criterion. The results of two simulation studies showed that for a typical behavioral dataset, ESEM (using the BIC for model selection) in this regard is successful more often than PCovR. Yet, in 93% of the datasets PCovR performed equally well, and in the case of 48 predictors, 100 observations, and large differences in the strengths of the factors, PCovR even outperformed ESEM.
KeywordsPrincipal-covariate regression Exploratory structural equation modeling Multicollinearity Dimension reduction
The research leading to the results reported in this article was supported in part by the Research Fund of KU Leuven (GOA/15/003); by the Interuniversity Attraction Poles program, financed by the Belgian government (IAP/P7/06); and by a postdoctoral fellowship awarded to M.V. by the Research Council of KU Leuven (PDM/17/071). For the simulations, we used the infrastructure of the VSC–Flemish Supercomputer Center, funded by the Hercules foundation and the Flemish government, Department EWI.
- Ceulemans, E., & Kiers, H. A. L. (2006). Selecting among three-mode principal component models of different types and complexities: A numerical convex hull based method. British Journal of Mathematical and Statistical Psychology, 59, 133–150. doi: https://doi.org/10.1348/000711005X64817 CrossRefPubMedGoogle Scholar
- Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Erlbaum.Google Scholar
- Gurden, S. (n.d.). Multiway covariates regression: Discussion facilitator. Unpublished manuscript.Google Scholar
- Kline, R. B. (2015). Principles and practice of structural equation modeling (4th ed.). New York, NY: Guilford.Google Scholar
- Marsh, H. W., Liem, G. A., Martin, A. J., Nagengast, B., & Morin, A. J. (2011). Methodological-measurement fruitfulness of Exploratory Structural Equation Modeling (ESEM): New approaches to key substantive issues in motivation and engagement. Journal of Psychoeducational Assessment, 29, 322–346.CrossRefGoogle Scholar
- Pearson, K. (1901). On lines and planes of closest fit to systems of points in space. Philosophical Magazine, 2, 559–572.Google Scholar
- Tibshirani, R. (1996). Regression, shrinking and selection via the Lasso. Journal of the Royal Statistical Society: Series B, 58, 267–288.Google Scholar
- Velicer, W. F., & Fava, J. L. (1998). Effects of variable and subject sampling on factor pattern recovery. Psychological Methods, 3, 231–251. https://doi.org/10.1037/1082-989X.3.2.231, see https://www.researchgate.net/publication/232509045_Effects_of_Variable_and_Subject_Sampling_on_Factor_Pattern_Recovery
- Yates, A. (1987). Multivariate exploratory data analysis: A perspective on exploratory factor analysis. New York, NY: SUNY Press.Google Scholar