regression with rounded covariates

Question

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?

Answer 1

There are a couple of sides to this question.

Based on looking at the data and working backwards:

If the rounding effect induces larger coefficient error in the equation than unrounded data then it is introducing significant distortion into the model. Obviously if the data comes rounded (which it sounds like is your situation) you can't check this but you can estimate it from the errors for the rounded data coefficients. If they are of similar magnitude it is difficult to rule out rounding confounding the issue. If the errors are much bigger, then rounding is unlikely to contribute much.

Based on building simulated variables: Also, based on unpublished (sorry) experiments based on stochastic simulation problems involving correlated multinomials I found that if there are 10 or more levels in the multinomial it behaved very similarly to continuous variables. By that I mean that the end result was with the confidence interval that would be achieved by a comparable continuous variable. If there were 8 or fewer variables it behaved significantly differently from a continuous variable. This was specifically in terms of achievable correlation. The impact may also depend on the statistical test being carried out (I had no reason to generalise the work I was doing at the time). This may have a bearing on whether the assumptions of continuity can be considered violated.

In any case the rounding error WILL compound with the inherent error and there will be overall more noise. Whether the noise increase is crippling or not depends on your circumstances.