I'm currently performing the following analysis :

  1. Computing $r_j(Y_{ij}, X_{ij})$ for each design cell (factor with level 1 or 0 for each unit)

  2. Estimating effect of factor on $r(Y,X)$ with a linear model :
    $r_j(Y,X) \sim \mathcal{N}(\mu, \sigma) $
    $\mu = \alpha + \mathbf{\beta} \mathbf{X}$ with $\mathbf{X} = \matrix{1, 0}$
    Additionally I include random effects but for the sake of simplicity I don't write it down.

My question is whether one can build an integrated model simultaneously computing the correlation (preserving its uncertainty) and test for the effect of the factor. I have guessed the following but I can't find any relevant literature to back up my thought :

$\mathbf{Y} \sim \mathcal{MVN}(\mathbf{M}, \Sigma)$ with $\mathbf{Y} = \matrix{X, Y} $
$\mathbf{M} = \mathbf{A} + \mathbf{B} \mathbf{X}$ with $\mathbf{A}$ and $\mathbf{B}$ being vectors of corresponding parameters for $X$ and $Y$
and $\Sigma = \pmatrix{\sigma_X & \sigma_X \sigma_Y \rho \\ \sigma_X \sigma_Y \rho & \sigma_Y}$ where (at least) $\rho \sim \alpha + \mathbf{\beta} \mathbf{X}$

I would like to make inferences for $\mathbf{A}$, $\mathbf{B}$ and $\rho$ in the same model. Naively the logic seems OK but I guess that I might miss some assumptions or other potential source of problems by building such a model.


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