# Beta-binomial logistic regression model for binomial data with small samples

I have fitted a nonlinear beta-binomial logistic regression model on data y_i:

y_i ~ beta-binom(n_i,mu_i,\Phi)

where mu_i = exp(\eta_i)/(1+exp(\eta_i)) , and \eta_i=\beta_0+\beta_1/(1+exp(\beta_2x_i +\bata_3)).

Some graphical representations tell me that the model works very well.

The observed data (grouped data) includes 45052 rows in sum. 21972 rows have n_i=1, 11222 rows have n_i=2, 4794 rows have n_i=3, 2669 rows have n_i=4, 2669 rows have n_i=5 and the rest have 5 < n_i <69. From 21972 rows with n_i=1, there are 21366 rows with p_i=0 and 606 rows with p_i=1. Also, from 11222 rows with n_i=2 there are 10851 rows with p_i=0, 144 rows with p_i=0.5 and 227 rows with p_i=1 and so on.

Collett 2003, in Modelling Binary Data book p. 196 says:

"When the data are sparse, i.e., there are proportions based on small numbers of individual experiments, the chi square approximation of the deviance breaks down."

and

"It is vital that all of these possible explanations of apparent overdispersion be eliminated before concluding that the data are overdispersed."

As you see, in some rows there are proportions based on small numbers of individual experiments (I suppose this means e.g. n_i=1 and n_i=2, isn't it?) . In addition, many observed proportions are equal to zero and one.

I have some questions:

1- This model fits very well to my data (by looking at some plots). This is an observational study and not an experimental one to let me design it in a way to have no small n_i. Shouldn't I use a beta-binomial based on Collett? Am I misunderstood about this?

2- How should I calculate the saturated likelihood in this example where there exist proportions equal to zero and one? I need this because I want to calculate the deviance. I have already calculated the maximum likelihood estimations for the beta-binomial model.

3- Does the deviance here have an approximate chi-square distribution?

4- If I can calculate the above deviance, would it be possible to compare it with the deviance of the binomial model?