Blocked-Error-R ²: A Conceptually Improved Definition of the Proportion of Explained Variance in Models Containing Loops or Correlated Residuals

Abstract

Bentler and Raykov (2000, Journal of Applied Psychology 85: 125–131), and Jöreskog (1999a, http://www.ssicentral.com/lisrel/column3.htm, 1999b http://www.ssicentral. com/lisrel/column5.htm) proposed procedures for calculating R ² for dependent variables involved in loops or possessing correlated errors. This article demonstrates that Bentler and Raykov’s procedure can not be routinely interpreted as a “proportion” of explained variance, while Jöreskog’s reduced-form calculation is unnecessarily restrictive. The new blocked-error-R ² (beR ²) uses a minimal hypothetical causal intervention to resolve the variance-partitioning ambiguities created by loops and correlated errors. Hayduk (1996) discussed how stabilising feedback models – models capable of counteracting external perturbations – can result in an acceptable error variance which exceeds the variance of the dependent variable to which that error is attached. For variables included within loops, whether stabilising or not, beR ² provides the same value as Hayduk’s (1996) loop-adjusted-R ². For variables not involved in loops and not displaying correlated residuals, beR ² reports the same value as the traditional regression R ². Thus, beR ² provides a conceptualisation of the proportion of explained variance that spans both recursive and nonrecursive structural equation models. A procedure for calculating beR ² in any SEM program is provided.