# Why is linear regression results so much different from Poisson regression?

I ran both a linear regression and Poisson regression on count data (data ranges from 0-54) with two continuous predictors and the p values were very different between them.

m <- glm(Count ~ Age + Days, data = dat, family = 'poisson')

m <- lm(Count ~ Age + Days, data = dat)


Am I doing something wrong here with one of those models? The p values was significant for age and days (around .02) in the Poisson regression and non-significant for both age and days (around .5) in the linear regression. The standard errors were also lower in the Poisson regression.

• There's nothing wrong with your seeing a "significant' effect in one model but not the other. As kjetil b halvorsen and @Glen_b has pointed, out these are two very different models that made different assumptions. As a result, they should not necessarily have agreements on all statistical tests. – StatsStudent Jun 12 at 1:09

The biggest difference there will be caused by the fact that the Poisson GLM by default will be using the log link while the regression model uses an identity link. That is, it will fit a model

$$\log(E[Y|x_1,x_2]) = \beta_0 + \beta_1 x_1 + \beta_2 x_2$$

for the Poisson GLM (by default). This model is linear in the log of the conditional mean. Meanwhile the regression model fits

$$E[Y|x_1,x_2] = \beta_0 + \beta_1 x_1 + \beta_2 x_2\,.$$

which is linear in the mean.

These are very different models. The coefficients mean different things. The shape of the fit is different.

The two models also make different variance assumptions.

Why do you expect the results from those two models to be even similar? Those are very different models, linear regression is an additive model while Poisson regression is multiplicative. See Goodness of fit and which model to choose linear regression or Poisson for a comparison.

For count data, mostly the Poisson regression model is indicated.

• Thanks so much for the answer. So everything looks ok with my code? Also, im new to stats, can you tell me what you mean by multiplicative and how its different from the additive model? – Ryan Jun 11 at 21:59
• There is no problem with your code. Else, see the link above. And tell us what you know about glm's (generalized linear models.) Do you know about link functions? For additive/mult see stats.stackexchange.com/…*+model+answers%3A1 – kjetil b halvorsen Jun 11 at 22:56