# Non-normal variable: multiple vs poisson regression [duplicate]

If my outcome variable is count data, people often recommend using some type of poisson regression. I'm struggling to understand why this is the preferred method.

Lets say for this scenario that my mean count is low, so the distribution is skewed and does not look normal.

Given that regression makes no assumptions about the normality of the outcome variable, why does it matter that it is skewed? Why could I not run OLS regression anyway?

If my outcome mean was large the poisson distribution of the variable would look more normal, and I've heard people say in that case OLS with no transformation is acceptable. So there seems to be something special about a low mean count and the skew of the resulting distribution

Is it because the likelihood of a non-normal residual distribution increases with highly positively skewed data, and the reason for using poisson regression is to circumvent that normality violation?

Assuming the residuals are distributed normally, what is the actual problem with modelling (untransformed) count data using OLS regression (barring the decimal predictions)?

## 1 Answer

This is ultimately a somewhat common thing to wonder about generalized linear models with non-normal data (i.e., why can't I just use linear regression), which is more general than the specific case with count data. It most commonly comes up when discussing logistic regression. @Kjetil b halvorsen's answer to Goodness of fit and which model to choose linear regression or Poisson contains a lot of good information, but it may be helpful to put the basics in a simpler and more concise form.

1. If the response data are counts, the residuals from an OLS regression cannot be distributed normally, no matter what. A linear (OLS) regression would assume they are, which could cause some problems.
2. The conditional variance of count data almost always changes as a function of the conditional mean. This means that you would have heteroscedasticity. A linear (OLS) regression assumes homoscedasticity, which could cause some problems.
3. The decimal predictions is not really a problem, because the predicted values ($\hat y$s) are supposed to be conditional means, not necessarily values you will observe. Instead, a big problem is that the model will imply negative predicted values for allowable predictor ($X$) values, whether or not they actually exist in your dataset. Note that a negative mean count is nonsensical.

The above is the standard justification for using a generalized linear model (e.g., a Poisson regression or a logistic regression) instead of OLS regression with a non-normal variable. But I've always thought the biggest problem is just that a linear model is the wrong way to think about your data, and that in the end, we are supposed to be thinking about our data and using the model to help us do so.