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I'm wondering if this is possible to fit structural equation model for experimental design data.

Problem

Suppose a researcher observed four responses $Y_1$, $Y_2$, $Y_3$, and $Y_4$ along with three covariates $X_1$, $X_2$, and $X_3$ 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 $Y_1$ is also influenced by the other three ($Y_2$, $Y_3$, and $Y_4$).

Is it possible to model causality among responses $Y_1$, $Y_2$, $Y_3$, and $Y_4$ as well as to asses the effects of factors A , B and their interaction AB on responses $Y_1$, $Y_2$, $Y_3$, and $Y_4$?

Thanks

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There is no simple yes or no answer. People constantly attempt to make inferences about causal relationships. The question is what assumptions you have to make, and how sensitive your inferences are to changing those assumptions.

The causal effects you can identify with the fewest assumptions are the effects of the things you randomly assign: A, B, and the interaction A*B, on Y1, Y2, Y3, and Y4.

I'm likely to be skeptical of a claim to have identified the causal effect of any of the non-randomized variables on anything else. The scientific context (which you have not provided) will shape what is considered a reasonable inference.

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