@ttnphns can you please elaborate how you got from assumption that exp(logit) = 1 for reference variable to the fact that probability from multinomial reduces to one from binomial regression? Do you mean to say sum of exp(logit) for all categories from j to r (excluding i) is equal to 1?

@ArtemMavrin - Thanks for being patient and trying. I am trying to figure out probability of time spent in stage 2 conditioned on fact that stage 1 is completed. May be my notation is off the mark.

@ArtemMavrin: I have simplied the notation. Is it better? let me know what other detail you need?

Will add it would be great if someone can help derive it rigorously instead of using intuition / constraint

@MikeP - you may be correct here. I am having difficulty in internalizing this. If that's true, isn't basic assumption that pdf of t1 and t2 are independent, incorrect? In any case, will it be possible for you to expand on what you have written or help me to derive it from the last step I have left things at?

Understood. If I were to use that partial area, will it give me a measure of average performance if the classifier was to operate with FPR in the range of 0% and 8% instead of at 8%?

Got it - and then should I look up partial AUC between 0% and 8% FPR of various classifiers and pick the one with the largest area in that range in this specific example?

Thanks. I am following up on an old answer from you: So if I know that FPR is 10% with a margin of error +- 2%, then should I look at the partial area under curve between 8% and 12% and pick the classifier that has most AUC under that limited range of FPR?