# regression with rounded covariates

A basic assumption of something as simple as OLS regression is that the covariates are continuous. Often this is only partly the case, e.g. heart rate is often reported in integer number of bpm, yet the underlying value is clearly continuous, not discrete.

When doing kernel density estimation, there are suggestions for how to deal with this, e.g. the kernelheaping package in R. (which seems to rely on an explicit model for how rounding is done in things like questionnaires)

I'm wondering if there are ways of formally handling the effects of rounding in regression scenarios, and also in general for arbitrary (supervised) learning algorithms? In particular, for the heart rate example, I intuitively think that rounding doesn't matter too much, if I have enough data...can this intuition be formalised? e.g. "if the average rounding amount is so and so small compared to <some measure of spread of the data set>, then don't worry"? At what point does the rounding become so severe that I should treat the data as categorical?

• Rounding is a type of interval censoring. A nice discussion of the problem and possible solutions is given in this Sandia National Lab report: prod.sandia.gov/techlib/access-control.cgi/2007/070939.pdf Feb 23, 2018 at 12:02
• I don't think continuity is is an assumption at all. For example, nothing in regression rules out one, some or all of the predictors being binary (dichotomous, indicator, dummy). Continuity is not even an ideal condition. Whether discrete predictors clash with what is most familiar and/or how regression is first taught or how it is experienced are perhaps interesting educational, psychological and sociological questions, but nothing in the mathematics rules out discreteness. Feb 23, 2018 at 12:02
• continuity is not an assumption, but there is an assumption that the perdictors are measured without error, the errors are in the response only Jul 5, 2018 at 9:00