Consider the following:
foo = 1:10
bar = 2 * foo
glm(bar ~ foo, family=poisson)
I get results
Coefficients:
(Intercept) foo
1.1878 0.1929
Degrees of Freedom: 9 Total (i.e. Null); 8 Residual
Null Deviance: 33.29
Residual Deviance: 2.399 AIC: 47.06
From the explanation on this page, it seems like the coefficient of foo should be log(2), but it's not.
More generally, I thought the output of this is supposed to mean that lambda = 1.187 + .1929 * foo where lambda is the parameter for the Poisson distribution, but that doesn't seem to fit with the data.
How should I interpret the output of this regression?
Poisson models are multiplicative. What this is saying is that as a result of some sort of averaging process that an increase of 1 in the order (increments in the foo predictor), will be associated with ratio of adjacent even integers in the range seq( 2, 20, by 2) that is exp(0.1929). I don't think the prediction is very good but when you look at the possible values, not bad.
> exp(0.1929)
[1] 1.212762
> seq(4,20,by=2)/seq(2,18,by=2)
[1] 2.000000 1.500000 1.333333 1.250000 1.200000 1.166667 1.142857 1.125000 1.111111
> mean( (2:11)/(1:10) )
[1] 1.292897