Poisson Regression and Linear Regression give the same error

Question

I'll use the data azdrg112 from COUNT package. The los will be the response variable while the remaining three variables gender, type1 and age75 are the explanatory variables. Here is the plot of the response

install.packages('COUNT')
library(COUNT)
data("azdrg112")
barplot(table(azdrg112$los))

The responses seem to be drawn from the Poisson distribution. I modelled it and calculated the absolute mean square error as

pois_model = glm(los ~ ., data = azdrg112, family = poisson)
pois_pred = predict(pois_model, azdrg112, type = 'response')
mean(abs(pois_pred - azdrg112$los)) # 2.330834

I then tried using Linear Regression

gaus_model = glm(los ~ ., data = azdrg112, family = 'gaussian')
gaus_pred = predict(gaus_model, azdrg112, type = 'response')
mean(abs(gaus_pred - azdrg112$los)) # 2.343426

I expected that the Poisson Regression would be a better fit then yield a smaller error but why both models output the same error?

Answer 1

Your expectation is wrong. It's all about evaluation metrics and loss functions:

• Poisson regression minimizes the average unit Poisson deviance.
• lm() minimizes the mean-squared error (MSE).

Thus: If you evaluate the models with MSE, you can expect lm() to "win". Similarly, if you evaluate with mean unit Poisson deviance, glm() will win.

Now enters Juan the game and chooses mean absolute error as evaluation metric. Neither of the two models tried to minimize this, so either of them might win.

Try: Run a linear median regression via quantreg::rq(...). I'd expect it to do better because it was fitted to minimize mean absolute error.