# Using linear regression for count data - will this introduce bias?

Say I am fitting a model to Poisson count data, but I am only interested in estimating the mean of the count variable. I understand a ordinary linear regression is a good approximation when the Poisson mean is large (see Why is Poisson regression used for count data?), but I am interested in zero inflated Poisson models with small means.

To be specific, for each sample we have factors $X$, the Poisson mean $Y \sim X$, and the counts $\mu \sim Y$.

I constructed a toy example, and fitted two OLS models: one to $Y \sim X$ and one to $\mu \sim X$. The two models seem to have similar coefficients, and I didn't find a statistically significant difference when repeating many times. I also tried a zero inflated Poisson version.

Mathematically speaking, am I introducing any bias by doing this? (I understand OLS will not give the same result as Poisson regression due to the variable transformation. Eventually I want to use an arbitrary positive valued function, e.g. a random forest.)

• Linear regression is usually used for intensive variables. A count is an extensive variable. See the discussion in my answer here: stats.stackexchange.com/questions/142338/… – kjetil b halvorsen Sep 12 '15 at 17:22
• There are lots of models for fitting zero heavy, count data. Poisson is one, negative binomials are another, hurdle models, Tobit models, etc. If you were to transform the counts using a natural log, OLS regression would deliver more "reasonable" results but there would still be bias. – Mike Hunter Nov 17 '17 at 20:14

## 1 Answer

A couple of points:

• Since the mean of a Poisson is necessarily positive, a linear regression - which with nonzero slope is certain to cross into negative territory somewhere - cannot be the mean of the Poisson there, so the regression cannot be unbiased there: • However, since you're interested in significance, it's not just bias in the mean that matters. The Poisson has variance proportional to mean, while inference for the least squares regression line assumes constant variance. If the variance estimate at a given value of the predictor(s) is biased, CIs will be biased. Similarly, correct p-values (and so correct rejection rates) in hypothesis tests rely on a correct variance specification.
• Thank you, you bring up interesting points! I understand the OLS model produces problematic negative predictions, and that any CI's on means won't be of much use when using OLS for this data. Actually I think my question is unclear, because I'm more interested in whether regression on the "counts" can be safely used as a proxy for regression on the "mean" (even through the "mean" is not known) when using methods other than Poisson regression. In fact OLS is a bad example because of the negativity problem; the question applies equally to e.g. non-negative functions. Does this make sense? – user157969 Jul 2 '15 at 14:05
• It's not quite clear to me what you're getting at now. If the question you have doesn't change much you could edit to clarify. Alternatively, if it changes relatively more, perhaps you can ponder how to explain it, and ask it as a new question? – Glen_b Jul 2 '15 at 14:52
• OK thanks. I've tried to clean up the question, it ended up being somewhat different so I have posted it here: stats.stackexchange.com/questions/159665/… – user157969 Jul 2 '15 at 16:13