Analyzing Individual and Average Causal Effects via Structural Equation Models

Although both individual and average causal effects are defined in Rubin’s approach to causality, in this tradition almost all papers center around learning about the average causal effects. Almost no efforts deal with developing designs and models to learn about individual effects. This paper takes a first step in this direction. In the first and general part, Rubin’s concepts of individual and average causal effects are extended replacing Rubin’s deterministic potential-outcome variables by the stochastic expected-outcome variables. Based on this extension, in the second and main part specific designs, assumptions and models are introduced which allow identification of (1) the variance of the individual causal effects, (2) the regression of the individual causal effects on the true scores of the pretests, (3) the regression of the individual causal effects on other explanatory variables, and (4) the individual causal effects themselves. Although random assignment of the observational unit to one of the treatment conditions is useful and yields stronger results, much can be achieved with a nonequivalent control group. The simplest design requires two pretests measuring a pretest latent trait that can be interpreted as the expected outcome under control, and two posttests measuring a posttest latent trait: The expected outcome under treatment. The difference between these two latent trait variables is the individual-causal-effect variable, provided some assumptions can be made. These assumptions – which rule out alternative explanations in the Campbellian tradition – imply a single-trait model (a one-factor model) for the untreated control condition in which no treatment takes place, except for change due to measurement error. These assumptions define a testable model. More complex designs and models require four occasions of measurement, two pretest occasions and two posttest occasions. The no-change model for the untreated control condition is then a single-trait–multistate model allowing for measurement error and occasion-specific effects.