The Likelihood Ratio Test for the Presence of Immunes in a Censored Sample

The mixed trunsored model with applications to SARS

The trunsored model, which is a new incomplete data model regarded as a unified model of the censored and truncated models in lifetime analysis, can not only estimate the ratio of the fragile population to the mixed fragile and durable populations or the cured and fatal mixed populations, but also test a hypothesis that the ratio is equal to a prescribed value with ease. Since SARS showed a severe case fatality ratio, our concern is to know such a case fatality ratio as soon as possible after a similar outbreak begins. The epidemiological determinants of spread of SARS can be dealt with as the probabilistic growth curve models, and the parameter estimation procedure for the probabilistic growth curve models may similarly be treated as the lifetime analysis. Thus, we try to do the parameter estimation to the SARS cases for the infected cases, fatal cases, and cured cases here, as we usually do it in the lifetime analysis. Using the truncated data models to the infected and fatal cases with some censoring time, we may estimate the total (or final) numbers of the patients and deaths, and the case fatality ratio may be estimated by these two numbers. We may also estimate the case fatality ratio using the numbers of the patients and recoveries, but this estimate differs from that using the numbers of the patients and deaths, especially when the censoring time is located at early stages. To circumvent this inconsistency, we propose a mixed trunsored model, an extension of the trunsored model, which can use the data of the patients, deaths, and recoveries simultaneously. The estimate of the case fatality ratio and its confidence interval are easily obtained in a numerical sense. This paper mainly treats the case in Hong Kong. The estimated epidemiological determinants of spread of SARS, fitted to the infected, fatal, and cured cases in Hong Kong, could be the logistic distribution function among the logistic, log-normal, gamma, and Weibull models. Using the proposed method, it would be appropriate that the SARS case fatality ratio is roughly estimated to be 17% in Hong Kong. Worldwide, it is roughly estimated to be about 12-18%, if we consider the safety side without the Chinese case. Unlike the questionably small confidence intervals for the case fatality ratio using the truncated models, the case fatality ratio in the proposed model provides a reasonable confidence interval.

Cure model analysis in cancer: an application to data from the Children's Cancer Group

The most commonly used statistical methods for evaluating treatment or prognostic effects on cancer outcome--the logrank test and Cox regression analysis--rely on the proportional hazards (PH) assumption in that they have maximal power in this circumstance. Implicitly, these methods emphasize covariate effects on failure times rather than their effects on the proportion of long-term survivors ('cures'), which may be of equal or primary interest. In paediatric cancer, treatment has progressed dramatically in recent decades, and in many diagnoses cures are obtained in a large fraction of patients. A primary focus of clinical research is therefore the achievement of cure. Parametric cure model (PCM) analysis, introduced 50 years ago, is arguably better suited to the analytic requirements of clinical research in paediatric and other cancers where cure is achieved. In this paper two classes of PCMs are described and used to analyse examples from the Children's Cancer Group. These are compared to analyses using Cox regression analysis. Results from PCM analyses are similar or identical to Cox regression analysis when the PH assumption is appropriate. When it is not, PCMs can provide a coherent way to investigate and report covariate effects on the proportion cured separately from their effect on time to failure. Despite their reliance on explicit parametric forms, PCMs often provide a good description of cancer outcome, and are insensitive to lack of fit provided that follow-up is sufficient.

Testing for the presence of cured patients: a simulation study

An important but difficult problem in clinical trials is to determine the presence of cured patients when long-term survivors are observed. The likelihood ratio test has been studied for this purpose in the gamma mixture model. However, its asymptotic null distribution is not readily available due to a violation of boundary conditions in the standard asymptotic theory. In this paper, a simulation study is employed to examine a proposed asymptotic null distribution of the likelihood ratio test. We find that the distribution can also be used to approximate the asymptotic null distribution of the likelihood ratio test in the Weibull and log-normal mixture models when the censoring rate is not too light. However, the simulation study also shows that null distribution of the likelihood ratio test deviates significantly from the suggested distribution under moderate sample sizes when the censoring rate is small or the hazard rate is large. Consequently caution is needed in this case to determine the presence of cured patients. Finally, the results are used to confirm the presence of cured patients in a leukaemia study.