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?
Best Answer
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 $n_i p_i$.
See also Logistic Regression: Bernoulli vs. Binomial Response Variables