Ah yep, I forgot it's the canonical link. That's more of a small computational convenience though, right, at least when fitting a straightforward GLM -- it should be very cheap to fit regardless.
I suppose the logistic having heavier tails than the normal is probably the main consideration in motivating one or the other as the better model for a given situation.
Logistic being is heavier-tailed, is potentially more robust to outliers. Which in terms of binary data, means that it might be a better choice in cases where an unexpected outcome is possible even in the most clear-cut cases. Probit regression with its heavier normal tails, might be a better fit in cases where the response is expected to be pretty much deterministic in clear-cut cases, and where quite severe inferences can be drawn from unexpected outcomes in those cases. Sound fair?
I suppose the logistic having heavier tails than the normal is probably the main consideration in motivating one or the other as the better model for a given situation.
Logistic being is heavier-tailed, is potentially more robust to outliers. Which in terms of binary data, means that it might be a better choice in cases where an unexpected outcome is possible even in the most clear-cut cases. Probit regression with its heavier normal tails, might be a better fit in cases where the response is expected to be pretty much deterministic in clear-cut cases, and where quite severe inferences can be drawn from unexpected outcomes in those cases. Sound fair?