Background/how this came up:
A student has numeric ratings on a variety of items in questionnaires and sports performance data and would like to model the performance on questionnaire scores. He wanted to do an analysis (don't ask me why) where some of the data are split into quantiles so he can compare high, medium, and low performance groups. I suggested he do a linear regression instead so that he wouldn't lose power by throwing away the continuous data - but I realized that the only papers I've seen on this being bad were about doing median splits.
Intuitively, it seems that quantizing your data will sacrifice power (based on arguments against median splits)- but how is this affected by the number of quantiles you choose to use? Are there circumstances where using, say, quartiles isn't much worse than doing a regression? For a given sample size, can a minimum number of quantiles be calculated to stay above a given power level?