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Common effect models: Positive or negative manifold?

Observing exclusively positive associations among a set of variables (i.e., a positive manifold) is a robust finding in many areas in psychology. These positive associations can be explained by positing an underlying common cause or, alternatively, through positive direct effects among the variables. Recently, the Kruis-Maris model has been proposed as a third alternative mechanism to explain a positive manifold. In the Kruis-Maris model, observed variables are causes of a latent variable, such that conditioning on the latent variable induces associations between the observed variables. In this paper, we analyze the difference between the Kruis-Maris model and the more familiar causal indicator model. Both are examples of common effect models, but they imply different relations between the response variables and the effect variable. Whereas conditioning on the common effect in the Kruis-Maris model implies a positive manifold among the response variables, we show that conditioning on the common effect in the causal indicator model implies a negative manifold. We demonstrate this result for two specific cases: (1) in which both the response variables and the effect variable are normally distributed variables, and (2) in which the effect variable is normally distributed but the response variables are dichotomous variables.