Estimation of intra-cluster correlation coefficient via the failure of Bartlett's second identity

A new means of estimating the correlation coefficient for cluster binary data in the regression settings is introduced. The creation of this method is founded upon the violation of Bartlett's second identity when adopting the binomial distributions to model binary data that are correlated. The new methodology applies to any sensible link functions that connect the success probability and covariates. One can easily implement the procedure by using any statistical software providing the naïve and the sandwich covariance matrices for regression parameter estimates. Simulations and real data analyses are used to demonstrate the efficacy of our new procedure.