Binomial(n,p) GLM - Can n be passed as an explanatory variable?

Question

For example, lets say I'd like to fit a binomial GLM to predict D, the number of deaths in a hospital on a given day, where n is the number of patients, and p is the probability a patient dies.

What I'm really interested in is the value of p. It seems this is what R tries to model when fitting a binomial GLM anyway..?

What I'd like to know is whether it makes any sense to pass n as explanatory variable into our binomial GLM for D? (eg it may be that as n increases, the hospital's facilities are more stretched and p, the probability of dying, increases). My hunch is that in this case maybe a binomial model shouldn't be used at all?

Any help appreciated, thanks.

Answer 1

@mkt points out that you might be mixing up two different questions. I'll tackle one of them (which you might not even be asking):

Usually when fitting a binomial GLM, $n$ is part of the specification of the dependent variable. From the help for glm:

For binomial and quasibinomial families the response can also be specified as a factor (when the first level denotes failure and all others success) or as a two-column matrix with the columns giving the numbers of successes and failures.

It's the two-column matrix form of the response that you want to use. The first column will be the number of deaths, say $d$, and the second will be the number of non-deaths, $n-d$.

If you already knew that, apologies, but you seemed unsure how it all works:

It seems this is what R tries to model when fitting a binomial GLM anyway..?

The other question--does it still make sense to use $n$ as an explanatory variable, for the reason you state--is a fair one, to which I don't know the answer, but I think you're right to worry about it.

Answer 2

Using n as an explanatory variable to predict a binary outcome (life or death) is reasonable. There are also ways to utilize a binomial distribution for multinomial regressions (e.g. having three possible outcomes).

I would say you can't use a glm with a binomial to predict D (the number of deaths in a hospital on a given day) because that's not a binary response and instead its in a form of count data. Therefore a glm with a poisson or a negative binomial would be better suited when using D as your response variable.