Estimating beta-binomial distribution

Question

Suppose that I culture cancer cells in n different dishes g₁, g₂, … , g_n and observe the number of cells n_i in each dish that look different than normal. The total number of cells in dish g_i is t_i. There is individual differences between individual cells, but also differences between the populations in different dishes because each dish has a slightly different temperature, amount of liquid, and so on.

I model this as a beta-binomial distribution: n_i ~ Binomial(p_i, t_i) where p_i ~ Beta(α, β). Given a number of observations of n_i and t_i, how can I estimate α and β?

Answer 1

You have a hierarchical bayesian model. Brief details below:

Likelihood Function:

$$f(n_i | p_i, t_i) = (t_i n_i) p_i^{n_i} (1-p_i)^{(t_i - n_i)}$$

Priors on $p_i, \alpha, \beta$:

$$pi \sim Beta(\alpha, \beta)$$

$$\alpha ~ N(\alpha_{mean}, \alpha_{var}) I(\alpha > 0)$$

$$\beta ~ N(\beta_{mean}, \beta_{var}) I(\beta > 0)$$

Posteriors are:

$$p_i \sim Beta(\alpha + n_i, \beta + t_i-n_i)$$

$$\alpha \propto I(\alpha > 0) \prod p_i^{(\alpha-1)} exp(-(\alpha-\alpha_{mean})^2) / (2 \alpha_{var})$$

$$\beta \propto I(\beta > 0) \prod (1-p_i)^{(\beta-1)} exp(-(\beta-\beta_{mean})^2) / (2 \beta_{var})$$

You can then use a combination of Gibbs and Metropolis-Hastings to draw from the posterior distributions.

Answer 2

To fit the model you can use JAGS or Winbugs. In fact if you look at the week 3 of the lecture notes at Paul Hewson's webpage, the rats JAGS example is a beta binomial model. He puts gamma priors on alpha and beta.

Answer 3

You don't necessarily have to go Bayesian on your model, plain maximum likelihood estimation works just fine (though has no explicit solution). Multiple R packages (eg. aod or VGAM) will fit the distribution for you.

Alternatively, you can use the quasi-likelihood based overdispersed binomial model that does not assume a beta-binomial distribution, just adjusts for the overdispersion. The glm function with the quasibinomial family will fit this model in R.