Profile likelihood-based confidence interval for the dispersion parameter in count data

Abstract

The importance of the dispersion parameter in counts occurring in toxicology, biology, clinical medicine, epidemiology, and other similar studies is well known. A couple of procedures for the construction of confidence intervals (CIs) of the dispersion parameter have been investigated, but little attention has been paid to the accuracy of its CIs. In this paper, we introduce the profile likelihood (PL) approach and the hybrid profile variance (HPV) approach for constructing the CIs of the dispersion parameter for counts based on the negative binomial model. The non-parametric bootstrap (NPB) approach based on the maximum likelihood (ML) estimates of the dispersion parameter is also considered. We then compare our proposed approaches with an asymptotic approach based on the ML and the restricted ML (REML) estimates of the dispersion parameter as well as the parametric bootstrap (PB) approach based on the ML estimates of the dispersion parameter. As assessed by Monte Carlo simulations, the PL approach has the best small-sample performance, followed by the REML, HPV, NPB, and PB approaches. Three examples to biological count data are presented.

Keywords:

• asymptotic confidence interval
• dispersion parameter
• hybrid profile variance approach
• profile likelihood approach
• non-parametric bootstrap approach

Acknowledgements

The first author of this research was partially supported by the CSU-AAUP University Research Grant. This paper was presented at the 2011 Eastern North Atlantic Region (ENAR) meeting, held in Miami, Florida. The authors thank the two referees for their helpful suggestions that led to the improvement of this paper. The authors also thank Roger Bilisoly and David LaPierre for reading the manuscript.