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I'm wondering how to test the significance of factor(s) and/or covariate(s) along with modeling the causal relationship among responses. Let me explain this with a concrete example.

Example:

Suppose a researcher observed four responses Y1, Y2, Y3, and Y4 along with three covariates X1, X2, and X3 from an experiment involving ab treatment combinations from a fixed factor A with a levels and a random factor B with b levels. Based on past experience, it is assumed four responses are correlated and Y1 is also influenced by the other three (Y2, Y3, and Y4).

Question:

  • How can I test the significance of Factors (A and B) and covariates (X1, X2, and X3) and causality among responses (Y1, Y2, Y3, and Y4) using a single model?
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  • $\begingroup$ As far as the 'causality' part goes, this sounds like testing for mediation, but as the lead of that Wikipedia article says, the possibility of / methods for testing for mediation using empirical data have been challenged. I haven't made a thorough study of this, but i'd number myself among the skeptics. $\endgroup$ – onestop Apr 12 '11 at 12:18
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I don't believe you can definitively do so. The most difficult aspect will be the relationships among the 4 Y variables. That there are "assumed" causal relationships among them does not help us to test causality. In my limited understanding, a structural equation approach would assess the plausibility of an hypothesized model but would not test the correctness of such a model in any strict sense.

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