Interval Estimation For The Beta-Binomial Dispersion Parameter

ABSTRACT

The dispersion parameter in proportions occurring in toxicology, biology, clinical medicine and epidemiology is important in making inference regarding the regression parameters on the mean. Most data in form of proportions often exhibit excess variation (extra-dispersion). This can arise when the data is from different sub-populations (clusters) or when the assumption of independence is violated. The Beta-Binomial distribution has been applied to model over-dispersion in binary responses in clustered samples. This parametric procedure involves numerical methods of finding MLEs. Many authors have also proposed among other non-parametric procedures, the Quasi-likelihood and Method of Moments for estimation of the over-dispersion parameter. However, much literature focuses discussion on point estimation only. Interval estimation for the over-dispersion parameter in proportions is yet to be done. In this thesis, estimates for the construction of asymptotic confidence interval for the over-dispersion parameter based on the Beta-Binomial distribution, Method of Moments and Quasi-Likelihood procedures were first derived using: the likelihood function in the case of MLE and the quadratic estimating equations for Quasi-likelihood procedure and the Method of Moments. We then apply Monte‟ Carlo simulation technique to perform bootstrapping procedures for the case of equal and un-equal cluster sizes. The asymptotic coverage probabilities with the lengths of confidence intervals were computed for small and large cluster samples. It is apparent from simulation results that confidence interval lengths reduce with the increase in the mean response probability or increase in the cluster size. The asymptotic CIs based on these three estimators have coverage below the nominal coverage probability (0.95). This shows that these confidence intervals are completely inadequate. Moreover, when the over-dispersion parameter is small, the resulting coverage probabilities are high. These coverage probabilities decrease as the over-dispersion parameter exceeds 0.3. It was observed that when the cluster size is greater than or equal to , the over-dispersion parameter estimate performs well in the coverage probabilities except when this parameter is greater than 0.2. It is concluded that bootstrapping technique reduces the width of confidence intervals and improves coverage probabilities significantly for the case of unequal cluster sizes in over-dispersed data. An example of real biological proportions data was presented to demonstrate the above results.