Count data: Linear model vs generalized linear model

Question

To play a little bit with GLMs, I tried this simulation:

I generated a variable $x$ with two levels, $A$ and $B$.
I generated a variable $y$ by drawing from a Poisson distribution with parameter $0.1$ if $x=A$ and $10$ if $x=B$.

Then I applied a linear model and a GLM with Poisson family. The results I get are very similar: the linear model says that when $X=A$ the mean of $Y$ is 0.06, while when $X=B$ the mean is $10.02$. The GLM gives me the means $0.06$ and $10.02$ respectively.

I also tried different combinations of parameters for the Poisson distribution (e.g. 0.1 and 0.5 to avoid that the Poisson is approximated by a Gaussian), but the results of the linear model always agreed with the ones of the GLM.

So my question is: am I doing something wrong? If the linear model gives similar results as the GLM, why using a GLM (Talking from a practical point of view... of course from a theoretical point of view the GLM is better in this case)?

In the GLM, is the log link being used by default? (I assume your linear model is implicitly using an identity link.) Can you share your code? You might also see more of a difference if you sampled from a continuous X over a wider range. — gung - Reinstate Monica, Feb 03 '20 at 19:35
See https://stats.stackexchange.com/questions/142338/goodness-of-fit-and-which-model-to-choose-linear-regression-or-poisson/142353#142353 — kjetil b halvorsen, Feb 03 '20 at 19:36
Does this answer your question? Goodness of fit and which model to choose linear regression or Poisson — kjetil b halvorsen, Feb 03 '20 at 19:36

Count data: Linear model vs generalized linear model

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