I'm not sure what topic to ask this question in. How would you treat an outcome that was operationalized as the percent of two behaviors in a 24 hour period (breast vs formula feeding), with lots of zero percents (30% of the distribution, lots of 100% (50% of the distribution) and the rest evenly distributed between the two extremes. Looks sort of like this:
XXXXXXXXXXXXXXX X X XX X XX XX X XXXXXXXXXXXXXXXXXXXXXXXXXXXXX
It's not zero-inflated, unless it's also 100% inflated, and the continuum between the two extremes is pretty flat.
Seems like a complicated outcome to consider - maybe it should be simplified. One possibility is to consider two outcomes at a time: the extent of breast feeding and the extent of formula feeding. Using the example of the extent of breast feeding you could then apply Mplus two-part modeling. The binary part would predict breast feeding (any amount) or not, while the continuous part would predict the extent of breast feeding for those who do any breast feeding.
This seems like a very good idea to me. Much better than dichotomizing or trichotomizing the distribution -- which was the student's plan, and according to her, the practice in that literature. I'll suggest this to the student and her advisor when we meet next week.