Instead of reliability factor (Error factor), let: log(sigma_ji) = nu_j + f_j(eta_i). Not individual-specific anymore, but still identified.
Will require J GPs though (good luck). Direction should be identified due to eta.
The benefit is that the residual variance pattern can vary across eta differently for each item. Currently, if you are high in etaSigma, and etaSigma is f(eta) + e, then areas where f(eta) is high will have greater residual variance for all items. If you plot y_j by eta, then all y_j's will have a similar residual variance pattern along the line, across eta. This would in a sense, 'cut out the middle man' at the expense of much more complexity; but is also similar to IRT in the sense that 1) It's item-analysis rather than person-analysis 2) loosening a rating scale model to a GRM [error thresholds are varying, rather than fixed along the line].
Omega is computed the same way; the only change is how shat's are computed. They would be computed from f(eta)_j rather than nuSigma_j + lambdaSigma_jetaSigma, which incidentally is nuSigma_j + lambdaSigma_j(f(eta) + e)
Instead of reliability factor (Error factor), let: log(sigma_ji) = nu_j + f_j(eta_i). Not individual-specific anymore, but still identified. Will require J GPs though (good luck). Direction should be identified due to eta.
The benefit is that the residual variance pattern can vary across eta differently for each item. Currently, if you are high in etaSigma, and etaSigma is f(eta) + e, then areas where f(eta) is high will have greater residual variance for all items. If you plot y_j by eta, then all y_j's will have a similar residual variance pattern along the line, across eta. This would in a sense, 'cut out the middle man' at the expense of much more complexity; but is also similar to IRT in the sense that 1) It's item-analysis rather than person-analysis 2) loosening a rating scale model to a GRM [error thresholds are varying, rather than fixed along the line].
Omega is computed the same way; the only change is how shat's are computed. They would be computed from f(eta)_j rather than nuSigma_j + lambdaSigma_jetaSigma, which incidentally is nuSigma_j + lambdaSigma_j(f(eta) + e)