Specifying cluster or robust does not seem to change the predicted
probabilities from oprobit. Does it? Shouldn't it?
Intuitively/naively, I am thinking that for an observation for which
the variance of the random part of the latent index is high, there is
a greater chance of ending up further away from what the
deterministic part of the latent index alone might suggest.
One other clarification: by "variance of the random part of the
latent index" I assume you mean the residual. In Probit the
residual is assumed to have a Normal(0, 1) distribution. The linear
prediction is an estimate and is subject to sampling variability. I
suppose if one were so inclined, you could compute the confidence
interval for that estimate, and then based on that estimate come up
with a range for the predicted probabilities. I don't recall ever
having seen that done. Nor am I exactly sure how you would do it,
since the estimates of the cutpoints are themselves subject to
sampling variability.