Practical Aspects of Corrected Likelihood Ratio Confidence Intervals: A Discussion of Jeng–Meeker and Wong–Wu

Abstract

Modern statistical analysis of censored life data has been greatly facilitated by the recent availability of specialized routines in mainstream software packages. For point estimation, maximum likelihood (ML) seems to be the preferred method due to its versatility and good statistical properties. The articles by Jeng and Meeker (JM) and Wong and Wu (WW) use ML estimation. For confidence intervals, nearly all major statistical software packages used asymptotic normal (AN) theory until as recently as five or six years ago. In general, for good power, the relevant quantity is the number of failures rather than the total number of observations. Unless the number of failures in the sample reaches 50 or more (often but not always realistic in practical investigations), one-sided and even two-sided AN intervals can be grossly misleading. Numerous studies, including the present ones by JM and WW, have shown that AN intervals suffer from deficiencies in terms of both symmetry of coverage probabilities between upper and lower confidence bounds and actual confidence levels being close to stated coverage probabilities. The chi-squared approximation to the large-sample distribution of the log-likelihood ratio (LLR) statistic has been shown to be highly accurate and thus provides LLR confidence intervals that are superior to AN intervals. For samples containing as few as 15-20 failures, two-sided LLR intervals will perform fairly well. In recent years, several of general-purpose software packages [e.g., Proc Reliability in SAS/QC (SAS Institute Inc. 1997), JMP (1996)] as well as a few specialized ones [e.g., WinSMITH (Abernethy 1996)] have included LLR intervals as standard features. Other packages, notably S-PLUS (MathSoft 1999), have programming capabilities that allow special functions to incorporate LLR confidence intervals with little effort but much specialized knowledge. The availability of LLR intervals in software packages makes it much easier for us to convince clients to use them instead of the traditional AN intervals. In our experience, the LLR intervals are gaining acceptance in applications, and we hope that the trend will continue. There are also other factors contributing to acceptance of LLR intervals that we will discuss.