# Comparing performance of Quasi-binomial model and Beta-binomial model

I read some books in biostatistics about fitting binary date with Beta-Binomial regression model and Quasi-Binomial regression model. It proposes a setting:

Setting:
Assuming we have a sequence of binomial trials of size $$N_i$$. $$Y_i$$ is the number of events of interest, $$x_i$$ is a predictor associated with trial $$i$$, and $$\pi_i$$ is the proportion of the events of interest ($$i = 1,2,...,n$$). Books mentioned that it is very likely that data in reality doesn't have binomial variance so we have to use models that adjusted Overdispersion. And therefore we can consider two models:

1. Beta-Binomial regression model with an overdispersion parameter: $$P_i|\pi, \tau_i^2 \sim Beta(a_i, b_i), \quad \text{where} \quad \pi_i = \frac{a_i}{a_i + b_i} \quad \text{and} \quad \tau_i^{2} = \frac{1}{a_i + b_i + 1}$$ $$Y_i|P_i \sim Binomial(N_i, P_i)$$ $$\pi_i = \frac{e^{\beta_0 + \beta_1 x_{i}}}{1 + e^{\beta_0 + \beta_1 x_{i}}}, \quad \text{for} \quad i = 1,2,...,n$$ In here, $$\pi_i$$ and $$\tau_i^{2}$$ are fixed constants for each observation rather than random variables.

2. Quasi-Binomial regression model:
We need to assume a mean-covariance structure for quasi-likelihood function. $$E(Y_i) = N_i \pi_i \quad \text{and} \quad Var(Y_i) = N_i\pi_i(1-\pi_i)\alpha$$ $$log(\frac{\pi_i}{1-\pi_1}) = \beta_0 + \beta_1x_i, \quad \text{for} \quad i = 1,2,...,n$$

Question:
I noticed that I can use the Pearson residual $$R_i = \frac{y_i - N_i\hat{\pi}_i}{\sqrt{N_i\hat{\pi}_i(1-\hat{\pi}_i)}}$$ to compare performance of two models. But I don't know how I can build up the connection between the Pearson residual with the models' variances. I am wondering does anyone know how the Pearson residual related with models?

Attempt:
For Beta-Binomial regression model, I find that the variance of the $$Y_i$$ can be obtained by the law of total variance $$Var(Y_i) = Var[E(Y_i|P_i)] + E[Var(Y_i|P_i)] = [\tau_i^2(N_i - 1)+1]N_i\pi_i(1-\pi_i).$$ But how can I obtain the $$\hat{\pi}_i$$ under this model.

For the Quasi-Binomial regression model, how can I see the connection among $$\hat{\pi}_i$$, the variance of model, and $$R_i$$?

• Noticed that when $\tau_i^2 \to 0$, $Var(Y_i) \to N_i\pi_i(1-\pi_i)$ which is the binomial variance, the denominator in $R_i$ is just the squared root of a binomial variance. Nov 10 at 20:42