Interval estimation of the over-dispersion parameter in the analysis of one-way layout of count data

The over-dispersion parameter is an important and versatile measure in the analysis of one-way layout of count data in biological studies. For example, it is commonly used as an inverse measure of aggregation in biological count data. Its estimation from finite data sets is a recognized challenge. Many simulation studies have examined the bias and efficiency of different estimators of the over-dispersion parameter for finite data sets (see, for example, Clark and Perry, Biometrics 1989; 45:309-316 and Piegorsch, Biometrics 1990; 46:863-867), but little attention has been paid to the accuracy of the confidence intervals (CIs) of it. In this paper, we first derive asymptotic procedures for the construction of confidence limits for the over-dispersion parameter using four estimators that are specified by only the first two moments of the counts. We also obtain closed-form asymptotic variance formulae for these four estimators. In addition, we consider the asymptotic CI based on the maximum likelihood (ML) estimator using the negative binomial model. It appears from the simulation results that the asymptotic CIs based on these five estimators have coverage below the nominal coverage probability. To remedy this, we also study the properties of the asymptotic CIs based on the restricted estimates of ML, extended quasi-likelihood, and double extended quasi-likelihood by eliminating the nuisance parameter effect using their adjusted profile likelihood and quasi-likelihoods. It is shown that these CIs outperform the competitors by providing coverage levels close to nominal over a wide range of parameter combinations. Two examples to biological count data are presented.