# 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
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