Apologies -- you are being more subtle than I thought. Nevertheless, I think
that the censoring language isn't quite right.
You are thinking of a hierarchical model:
z ~ N(Xb, sigma), where Xb is the linear predictor, whatever covariates you
think belong in the model. Whether the distribution should be Gaussian or
somthing else depends not on the overall distribution of z, but on distribution
of (z | Xb). We could have a skewed predictor leading to skewed z, even if the
distribution about any given expectation is symmetric.
y = F(z) is what you observe. The classic tobin model is y= max(0,z), which
does lead to censored data.
In your case y_i = Binomial(n_i, p_i = H(z)). Note a binomial is k heads
out of n tries with a coin of probability p, a "Bernouli" is a binomial
restricted to a single coin flip. From the way you wrote the problem I assumed
that there is some number of n "looks" at the subject and then you count them
up. Note that var(y) = n p (1-p)
H describes how the probability changes with z. In biology we very rarely
use H(z)= max(min(z,1),0) because it gives a hard threshold, and the probability
of nearly anything doesn't go all the way to zero or one.
If H were as above and
var(y) = constant and
n is sufficiently large so that Binomial dist is approx Gaussian and
var(y |p) << var(z| Xb)
then your y will fit a censored Gaussian. Since at least the second is false,
it doesn't.
A censored model may still be an ok first cut at fitting the data, but I
would be suspicious of variance estimates and particularly of any p-values. The
bootstrap could help that.
Terry T.