What is the dispersion parameter of binomial distribution? Binomial distribution is a member of exponential dispersion models, but I can not find the dispersion parameter of it.
Could anyone help me find it out?
IMO the Binomial distribution only has an ordinary parameter p which describes both the mean and variance of this distribution, and I think there can't be the second parameter, neither does the dispersion parameter!
I find an answer in page 213 of Generalized Linear Models With Examples in R. But it changes the form of binomial distribution.

Using the definition of an exponential dispersion family at Definition of exponential family with dispersion parameter, we find that for the binomial family it is $a(\phi)=1$. Copying that definition here:
$$
   f(y|θ,ϕ)=exp\left(yθ−b(θ)a(ϕ)+c(y,ϕ)\right)
$$
For the binomial distribution with mean $np=\mu$, say, the variance is $\sigma^2=np(1-p)=\mu(1-\mu/n)$, so is a function of the mean (assuming $n$ a fixed constant). Still, sometimes it is estimated as if a free parameter (if the binomial is seen as a model just for the mean, not for the dispersion, called under/over-dispersion). That must be done in the framework of quasi-likelihood, see Mean-variance relationship in the quasi-likelihood, What is the difference between logistic regression and Fractional response regression?
Giving more details, showing how the binomial probability mass function can be written in the above form:
\begin{align}
f(y \mid p) &= \binom{n}{y} p^y (1-p)^{n-y} \\
 &= \exp\left\{ \log\binom{n}{y} + y\log(p/(1-p)) - n\log(1-p)  \right\} \quad\text{Now writing $\theta=\log(p/(1-p))$} \\
 &= \exp\left\{ y\theta - n\log(1-p) + \log\binom{n}{y}  \right\} \\
&= \exp\left\{ \frac{y\theta + n\log(1+e^\theta)}{1} + \log\binom{n}{y}  \right\}
\end{align} making it clear that we have to choose $a(\phi)=1$ to match that particular definition.  In this case we could clearly simplify the definition, by replacing $a(\phi)$ by $\phi$, which is what is done in forinstance the R implementation glm for logistic regression.
