# Can I model incidence per 1000 people per month using poisson regression without an offset or using weights?

I am trying to model incidence rates (number of malaria cases per 1000 people per month) over time. I only have the data in the form of rates per 1000 people per month, i.e. I do not know the total population size each month. i.e. in January, there were 3.3 cases per 1000 individuals, in February, there were 5.6 cases per 1000 individuals etc. etc.

What I am unsure about is whether I can use a Poisson regression model to model the rates i.e 5.6 and 3.3, without using an offset, or weighted Poisson regression? I haven't done either of these things as the population would always be 1000 so I don't think using an offset or a weighted Poisson regression approach would achieve anything.

I am just not sure whether my approach is valid when the "count data" aren't the actual number of events, and indeed aren't count data as they are not integers.

Short: You can use a quasi-Poisson regression.

You have data for $$n$$ regions/populations/months, each of size $$N_i$$ (which could vary from month to month.) The number of malaria cases is $$Y_i$$, but that is not given to you, neither is the $$N_i$$'s known to you. Apart from trying to obtain that missing information, you have to work with what is given, the rates $$\DeclareMathOperator{\E}{\mathbb{E}} R_i =1000\frac{Y_i}{N_i}.$$ We start with a Poisson regression model in terms of $$Y_i$$ $$\E [Y_i \mid x_i] = \lambda_i, \quad \log\lambda_i= \log N_i + x_i^T\beta$$ which we cannot use directly, since $$Y_i$$ is not known.

Rewriting in terms of $$R_i$$ we get $$\E [R_i \mid x_i] = 1000\frac{\lambda_i}{N_i}$$ with logarithm $$x_i^T\beta +\log 1000$$. This is a quasi-poisson model for $$R_i$$ with offset $$\log 1000$$, but since the offset is the same for all the observations it can just be absorbed in the intercept. Note that even with $$N_i$$ unknown we have no problem with expectation structure, since the expectation do not depend on $$N_i$$. But there will be one problem: The quasi-poisson model do use the assumption that the variance is proportional to the expectation. But that will no longer be true, because if the unknown $$N_i$$ are nonconstant, that will not hold here. See: Assume that $$Y_i \mid X_i=x_i$$ is Poisson, then we find that $$\DeclareMathOperator{\V}{\mathbb{V}} \V [R_i \mid x_i]=\frac{1000^2 \lambda_i}{N_i^2}$$. So while the expectation structure is right, the variance structure is not. So maybe use robust standard errors.

This is a version of Poisson rate regression, see for instance How is a Poisson rate regression equal to a Poisson regression with corresponding offset term?

• (+1) This is probably a dumb question but why is the offset $+\log(1000)$ instead of $-\log(1000)$ in the logarithm, because the $1000$ is not in the denominator? – COOLSerdash Apr 21 at 19:15
• Because the data is rates per thousand, so the denominator is $N_i/1000$. – kjetil b halvorsen Apr 21 at 19:17