Comparing regression coefficients between models with dependent predictors and dependent criterions

Question

I would like to compare regression coefficients between two models as part of a within-subject study setup. The problem is that both the predictors and criterions are likely to be correlated and that my understanding of repeated measurement analysis is thus far limited.

Basically, I know (from theoretical considerations) that there exists a linear relationship: $$ DV_{ij}=\beta_0 + \beta_{1j}IV_{ij}+\epsilon_{ij} $$ where $i \in 1,..,400$ (subjects) and $j \in 1,2$ (conditions).

I measure both variables two times (2 conditions): $DV_{i1}, DV_{i2}$ and $IV_{i1}, IV_{i2}$. I expect the variables' values to change.

I want to test whether $\beta_{11} \ge \beta_{12}$.

I was thinking about using dummy coding to describe it as an interaction effect as described in this question and I thought about using SUR as described in this question, yet I believe that both approaches would not acknowledge the correlation between the criterions. Likely, my biggest problem right now is that I have too little knowledge about regression and I do not know what to call this "statistical problem" and which "tools" to look for.

I appreciate any remarks and comments.

Answer 1

You can use a repeated measures regression model, with an unstructured covariance (2x2) covariance matrix to account for the dependence between (and possible heteroscedasticity across) the two conditions. Then use a contrast test provided by the software. Any good software would allow this contrast test, and would automatically incorporate the unstructured covariance matrix in the result.

As always, read "the manual" to be sure that the test is correctly calculated. If they show a formula for the standard error that involves this estimated covariance matrix, then they are probably doing it right.