The statistical analysis of truncated data: application to the Sverdlovsk anthrax outbreak

Abstract

An outbreak of anthrax occurred in the city of Sverdlovsk in Russia in the spring of 1979. The outbreak was due to the inhalation of spores that were accidentally released from a military microbiology facility. In response to the outbreak a public health intervention was mounted that included distribution of antibiotics and vaccine. The objective of this paper is to develop and apply statistical methodology to analyse the Sverdlovsk outbreak, and in particular to estimate the incubation period of inhalational anthrax and the number of deaths that may have been prevented by the public health intervention. The data available for analysis from this common source epidemic are the incubation periods of reported deaths. The statistical problem is that incubation periods are truncated because some individuals may have had their deaths prevented by the public health interventions and thus are not included in the data. However, it is not known how many persons received the intervention or how efficacious was the intervention. A likelihood function is formulated that accounts for the effects of truncation. The likelihood is decomposed into a binomial likelihood with unknown sample size and a conditional likelihood for the incubation periods. The methods are extended to allow for a phase-in of the intervention over time. Assuming a lognormal model for the incubation period distribution, the median and mean incubation periods were estimated to be 11.0 and 14.2 days respectively. These estimates are longer than have been previously reported in the literature. The death toll from the Sverdlovsk anthrax outbreak could have been about 14% larger had there not been a public health intervention; however, the confidence intervals are wide (95% CI 0-61%). The sensitivity of the results to model assumptions and the parametric model for the incubation period distribution are investigated. The results are useful for determining how long antibiotic therapy should be continued in suspected anthrax cases and also for estimating the ultimate number of deaths in a new outbreak in the absence of any public health interventions.