I would like to fit a bivariate probit model, i.e. I observe Y=(Y1,Y2),
where Y1 and Y2 take values 0 or 1,
and I assume that there is a latent bivariate Gaussian vector Z=(Z1,Z2), and
* if (Z1<s1,Z2<s2) then Y=(0,0)
* if (Z1>s1,Z2<s2) then Y=(1,0)
* if (Z1<s1,Z2>s2) then Y=(0,1)
* if (Z1>s1,Z2>s2) then Y=(1,1)
The point here is that I assume that the noise E=(E1,E2) can be correlated
between the two components...
I have some covariates X1,...Xk, and I assume that there is a hierarchical
structure (and random effects), i.e. Y_i is actually a Y_{i,j} where j is a
subgroup index.
So this yield two correlations levels,
- within subgroup correlation between Y1_i's (if they belong to the same
subgroup)
- between equations, i.e. Y1_i and Y2_i
I have found packages for logistic hierarchical regressions (eg *lmer*), and
others for bivaraite logistic (eg *zelig*)
without the hierarchical structure, I can compute the log likelihood and
optimize it (and it works)
but with the hierarchical structure, I can still get the likelihood of the
latent vector (written in dimension 2n), but I have trouble when I want to
go from the latent model to the observed one since Poincarré equation
involves 2^n summations (e.g. if I observe ((0,1),(1,0),(1,1)) I have to get
the probability that ((Y1<s.Y2>s),(Y1>s.Y2<s),(Y1>s.Y2>s)) which is related
to cdf's with a lot a summations) which cannot be computed here
So I was wondering if someone has already computed such a thing ?
thanks

	[[alternative HTML version deleted]]


