Transforming the expected value of $Y_i$ in binomial regression

Currently, I'm learning generalized linear regression (GLM). There is something troubling me concerning binomial regression.

In this text, in the part about the structure of a GLM, the random component is defined as a response variable $$Y_i$$ conditioned on the explanatory variable $$X_i$$. And the link function is a function that transforms the expectation of the response variable $$\mu_i = E(Y_i)$$.

However, according the same document and this post, in binomial regression, it seems that the linked function transforms $$p_i (Y_i \sim Bi(m_i, p_i))$$, instead of the mean of $$Y_i$$, $$m_ip_i$$.

And in this post, it's also only mentioned that "we are interested in fixing the relation between the conditional expectation of the probability π of a single Bernoulli trial". In this paper, this is similar argument without further explanation:

...$$Y\sim Binomial(n, \pi)$$. Let $$P_i = \frac{Y_i}{n_i}$$ for i,...,m is a proportioni of success from m independent binomial observations, with $$E(Y_i) = n_i\pi_i$$ and $$E(P_i) = \pi_i$$....Binomial regressoin models assume that $$\pi_i = F(x_i^T\beta), i=1,...,m$$.

Here $$F^{-1}$$ is defined as the link function.

To summarize, my question is: in binomial regression, why does the link function transform the probability/parameter ($$p_i$$), instead of the mean ($$m_i p_i$$), to the linear predictor ($$x_i^T\beta$$), as by the definition of link function?

• Oct 28 '21 at 13:47

Usually, as for instance in chapter 7 of MASS4, the response in a binomial generalized linear model is taken as $$Y_i = S_i/n_i$$, that is, we use a scaled binomial distribution ($$S_i$$ is number successes, $$n_i$$ the fixed sample size). Then the expectation is $$\DeclareMathOperator{\E}{\mathbb{E}} \mu_i = p_i$$. It is just more natural to use $$p_i$$ as parameter than to use $$n_i p_i$$.