# Regression: zeros in heavy-tailed independent variable from quantization

This question is about handling zeros in an independent variable for a regression.

In particular, the zeros are not missing data or true zeros, but occur because of quantization. As a concrete example, lets say the observations are cities, and the variable is the number (or fraction) of people in some category, based on a sample. If the sample for a particular city is small, it might have zero people in a category, even if the true number in the city's population is nonzero.

In this case, what are possible ways to work with zeros if the variable is heavy tailed? Normally I would log-transform, but I can't do that when zeros are present, and because many of the observations are zero, excluding them would introduce a large bias.

Some things I'm considering: other transformations, replacing the variable with a bayesian estimate of the fraction, switching from regression to ANOVA with people as observations and city as a categorical variable. Are these valid approaches? Am I missing any? What are the pros and cons?

• Square root should work ok – probabilityislogic Dec 24 '17 at 1:32

• By 'conditioned on' I mean that the values are known and that we do not entertain any distributional variation in them. For example, conditioning on yearly rainfall of 30in would mean that we are evaluating properties of $Y$ when rainfall=30in. We are interested in the distribution of $Y$ given $X$, and the distribution of $X$ is not at issue. Where the distribution of $X$ does have a major effect is that for certain poorly represented values of $X$ we don't have a sample size large enough to study the distribution of $Y$. – Frank Harrell Dec 25 '17 at 13:56