Seemingly unrelated regression models as a generalized least squares solution to path Analysis

Background: The popularity of path analysis (PA) and its generalization to latent variables, Structural Equation Models, has steadily increased over the last few decades. Both approaches have been crucial in theoretical developments in public health (e.g., validation of health belief models; quality of life assessments).

Methods: Seemingly Unrelated Regression (SUR) models are formulated as p correlated regression equations. The regression equations are “seemingly unrelated” because taken separately the error terms would follow standard linear OLS linear model form. Calculating p separate standard OLS solutions ignores any correlation among the errors across equations; however, because the dependent variables are correlated and the design matrices may contain some of the same variables there may be “contemporaneous” correlation among the errors across the equations.

Results: SUR models allow each of the p dependent variables to have a different design matrix with some of the predictor variables being the same. SUR models also allow for a variable to be both in the Y and X matrices, which has particular relevance to path analysis. We explicate how SUR models can be used to solve PA.

Conclusions: PA can be performed using SUR models; however, there are many other situations in public health research in which multiple dependent variables are of interest. However, it is commonplace to conduct separate analyses for multiple dependent variables even though they are likely to be correlated and have non-identical design matrices. We contend that SUR models are underutilized and should be give more consideration as an analytic technique.