Dear all,
I have an identifiability problem with my latent variable hierarchical model and am hoping that someone will be able to point me in the direction of some references which have experienced a similar problem...
Usually in the case of a hierarchical latent variable model, the latent variable acts as a covariate in the sense that;
Y(t) = beta0 + beta1*latent(t) + beta2*x1(t) + beta3*x2(t) + epsilon(t)
latent(t) = alpha0 + alpha1*x3(t) + alpha2*x4(t) +epsilon.latent(t)
In which case, when substituting the latent variable into the equation for Y(t) you get;
Y(t) = beta0 + (alpha0*beta1) + (alpha1*beta1)*x3(t) + (alpha2*beta1)*x4(t) + beta1*epsilon.latent(t) + beta2*x1(t) + beta3*x2(t) + epsilon(t)
However, my problem is that my latent variable is not multiplied by the parameter, beta1. Instead it is multiplied by a covariate;
Y(t) = beta0 + latent(t)*x5(t) + beta2*x1(t) + beta3*x2(t) + epsilon(t)
Hence, when substituting out for Y(t), we get;
Y(t) = beta0 + alpha0*x5(t) + alpha1*(x5(t)*x3(t)) + alpha2*(x5(t)*x4(t)) + x5(t)*epsilon.latent(t) + beta2*x1(t) + beta3*x2(t) + epsilon(t)
Which includes covariates being multiplied by covariates (alpha1*(x5(t)*x3(t)) ) and a covariate multiplying a random error term, (x5(t)*epsilon.latent(t)).
What I would like to know is, is this a special case of latent variable model or are there still standard ways of checking identifiability of such a model? So far I have been using the method of obtaining the Jacobian of the Full Model and the Reduced Model (Skrondal and Rabe-Hesketh, 2004) but this doesn't seem to work in this case.
Any advice would be much appreciated, thanks!
Jenny
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