How is it possible that Poisson GLM accepts non-integer numbers?

Question

I am really stunned by the fact that the Poisson GLM accepts non-integer numbers! Look:

Data (contents of data.txt):

1   2001    0.25  1
1   2002    0.5   1
1   2003    1     1
2   2001    0.25  1
2   2002    0.5   1
2   2003    1     1

R script:

t        <- read.table("data.txt")
names(t) <- c('site', 'year', 'count', 'weight')
tm       <- glm(count ~ 0 + as.factor(site) + as.factor(year), data = t, 
                family = "quasipoisson")  # also works with family="poisson"
years    <- 2001:2003
plot(years, exp(c(0, tail(coef(tm), length(years)-1))), type = "l")

The resultant year index is as "expected", i.e., 1-2-4 in years 2001-2003.

But how is it possible that Poisson GLM takes non-integer numbers? The Poisson distribution has always been integer-only!

Answer 1

Of course you are correct that the Poisson distribution technically is defined only for integers. However, statistical modeling is the art of good approximations ("all models are wrong"), and there are times when it makes sense to treat non-integer data as though it were [approximately] Poisson.

For example, if you send out two observers to record the same count data, it may happen that the two observers do not always agree on the count -- one might say that something happened 3 times while the other said it happened 4 times. It is nice then to have the option to use 3.5 when fitting your Poisson coefficients, instead of having to choose between 3 and 4.

Computationally, the factorial in the Poisson could make it seem difficult to work with non-integers, but a continuous generalization of the factorial exists. Moreover, performing maximum likelihood estimation for the Poisson does not even involve the factorial function, once you simplify the expression.

Answer 2

For a response $y$, if you assume the logarithm of its expectation is a linear combination of predictors $\renewcommand{\vec}[1]{\boldsymbol{#1}}\vec{x}$ $$\operatorname{E}Y_i=\exp{\vec\beta^{\mathrm{T}}\vec{x}_i}$$ & its variance is equal to its expectation $$\operatorname{Var}Y_i=\operatorname{E}Y_i$$ then consistent estimates for the regression coefficients $\vec\beta$ can be obtained by solving the score equations for the Poisson model: $$\sum_i^n{\vec{x}_i\left(y_i-\exp{\vec\beta^{\mathrm{T}}\vec{x}_i}\right)}=0$$ Of course consistency doesn't imply validity of any tests or confidence intervals; the likelihood has not been specified.

This follows on from the method-of-moments approach we learnt at school, & leads on to that of generalized estimating equations.

@Aaron's pointed out you're actually using a quasi-Poisson fit in your code. That means the variance is proportional to the mean

$$\operatorname{Var}Y_i=\phi\operatorname{E}Y_i$$

with a dispersion parameter $\phi$ that can be estimated from the data. The coefficient estimates will be the same, but their standard errors wider; this is a more flexible & therefore more generally useful approach. (Note also that sandwich estimators for the parameters' variance–covariance matrix are often used in these sort of situations to give robust standard errors.)