Remark(Correlated Bernoulli trials)

For \(n_i>1\) trials, let \(Y_i=(X_{i1}+\cdots+X_{i,n_i})/n_i\) where \[\begin{eqnarray*} P(X_{ij}=1)&=&\pi_i=1-P(X_{ij}=0),\\ \text{Corr}(X_{ij},X_{ih})&=&\rho, \mbox{ }j\ne h,\\ \text{Var}\left(\sum_jX_{ij}\right)&=&n_i\pi_i(1-\pi_i)+n_i(n_i-1)\rho\pi_i(1-\pi_i),\\ \implies\text{Var}(Y_i)&=&\frac{n_i\pi_i(1-\pi_i)+n_i(n_i-1)\rho\pi_i(1-\pi_i)}{n_i^2}\\ &=&\frac{\pi_i(1-\pi_i)+(n_i-1)\rho\pi_i(1-\pi_i)}{n_i}=\frac{\pi_i(1-\pi_i)}{n_i}\left(1+ (n_i-1)\rho\right).\\ \end{eqnarray*}\]

Since \(n_i>1\), there is overdispersion in this correlated bernoulli data.

  

Remark(Beta-Binomial Model)

Suppose given \(\pi\), \[ Y|\pi\sim \text{Bin}(n,\pi)\mbox{ }\mbox{ and }\mbox{ } \pi\sim \text{Beta}(\alpha,\beta), \] where \[ f(\pi;\alpha,\beta)=\frac{1}{B(\alpha,\beta)}\pi^{\alpha-1}(1-\pi)^{\beta-1}, \mbox{ } 0<\pi<1, \mbox{ }\alpha,\beta>0. \] Let \(\mu=\frac{\alpha}{\alpha+\beta}\), \(\theta=\frac{1}{\alpha+\beta}\). Then, \[ E(\pi)=\mu,\mbox{ } \text{Var}(\pi)=\mu(1-\mu)\frac{\theta}{1+\theta}. \] Unconditionally, \[\begin{eqnarray*} P(Y=y)&=&E[P(Y=y|\pi)]\\ &=&\int_0^1{n \choose y}\pi^y(1-\pi)^{n-y}\frac{1}{B(\alpha,\beta)}\pi^{\alpha-1}(1-\pi)^{\beta-1} d\pi\\ &=&\cdots={n \choose y}\frac{B(a+y, n+\beta-y)}{B(\alpha,\beta)},\mbox{ }y=0,1,\ldots,n. \end{eqnarray*}\] This is Beta-Binomial distribution and \[ E(Y)=n\mu, \mbox{ } \text{Var}(Y)=n\mu(1-\mu)\{ 1+(n-1)\theta/(1+\theta)\}. \]

  

back