In the simplest logistic regression we model the (logit) probability of a 0/1 valued outcome as a (linear) function of some given 0/1 valued predictor (dummy) variables encoding membership in certain discrete categories (gender, blood type, etc).

The parameters of the model (coefficients of the linear function) for the variables representing membership in just one of the categories are mutually orthogonal, in the sense that the relevant terms in the Fisher information matrix are zero (the entry at the row for blood type A and column for blood type AB, will be zero). They are not orthogonal to the variables representing other categories.

Generally the parameter estimates will be asymptotically normal. THE QUESTION: In this multivariate normal distribution, ** are the parameters within a single category distributed independently **? In other words, is the estimation of the "blood type A" independent of the estimate of the "blood type O" or the "blood type AB" coefficients?

If the answer is No, are there other useful consequences of the partial orthogonality?