Regression analysis of censored and truncated data: estimating reporting-delay distributions and AIDS incidence from surveillance data.

AIDS surveillance provides a vital source of information for health departments to assess the AIDS epidemic and to plan for future health-care needs. However, the use of surveillance data requires proper adjustments for the underreporting of AIDS cases caused by the delay in reporting diagnosed AIDS cases to the surveillance system. The statistical problem of adjusting for this underreporting concerns making inferences about an unobservable random sample of which only a portion is observed in a chronologic time interval defined by the analysis. Most regression methods for making inferences using right-truncated data employ a reverse-time hazard function, which requires that the observed data be transformed so that methods for left-truncated data can be applied. In this paper, we discuss fitting regression models to data that can be truncated and even censored in arbitrary intervals. The proposed methodology was applied to the national AIDS surveillance data provided by the Centers for Disease Control to analyze the trend of delays over chronologic time and variation among different geographic regions as well as across risk groups.