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.