The AIDS Epidemic: Estimating Survival After AIDS Diagnosis From Surveillance Data

Abstract

Survival analysis based on reported acquired immune deficiency syndrome (AIDS) cases from the surveillance data maintained by the Centers for Disease Control (CDC) leads to severe bias because of a sizable fraction of unreported deaths. One approach to this problem is to condition the analysis on reported deaths, but this approach presents several difficulties. First, only the individuals who die within a chronologic time interval defined by the analysis may be observed. Second, this right-truncated sampling process is complicated by the delay in reporting death to the surveillance system, and as a result, only a proportion of those who die within the defined time interval are reported. Third, the time each death is reported to the surveillance system, which is essential for the construction of a joint likelihood of survival and reporting delay, may not be available. Fourth, deaths that occurred before 1984 all had unknown times of death. As a consequence, the direct implementation of the joint likelihood approach appears to be complicated and difficult. On the other hand, this data set is worth analyzing; it is the only data base that comes close to covering the entire AIDS population and thus provides invaluable information to assess the current and predict the future AIDS epidemic. In this article we apply several missing–data techniques to deal with these difficulties. In particular, we discuss how to estimate the delay and survival distributions separately, and then combine the two sources of information to make valid inferences for the survival distributions using multiple imputation. The EM algorithm is used to facilitate computations. The methodology developed here can also be applied to other studies that share similar data structures, such as data from local and other national surveillance systems. Analysis of the CDC surveillance data as of March 1991, for some high-risk groups considered, indicates that there has been a steady and notable increase in survival after an AIDS diagnosis over chronologic time, especially for those who had Pneumocystis carinii pneumonia (PCP) as one of the AIDS–defining diagnoses.

KEY WORDS:

• Acquired immune deficiency syndrome
• Censoring
• Expectation-maximization algorithm
• Incomplete data
• Missing data
• Multiple imputation
• Proportional hazards model
• Supplemented-expectation-maximization algorithm
• Truncation