Estimating the cumulative risk of a false-positive test in a repeated screening program

The goal of screening tests for a chronic disease such as cancer is early detection and treatment with a consequent reduction in mortality from the disease. Screening tests, however, might produce false positive and false-negative results. With an increasing number of screening tests, it is clear that the risk of a false-positive screen, a finding with potentially significant emotional, financial, and health costs, also increases. Elmore et al. (1998, New England Journal of Medicine 338, 1089-1096), Christiansen et al. (2000, Journal of the National Cancer Institute 92, 1657-1666), and Gelfand and Wang (2000, Statistics in Medicine 19, 1865-1879) investigated this problem under the somewhat unrealistic assumption that the choice of making the decision to drop out at the kth screen does not depend upon the results of the earlier k - 1 screens. In this article we obtain sufficient and necessary conditions for their assumption to hold and use one of them to provide a method for testing the validity of the assumption. A new model which does not depend on their assumption is introduced. The maximum likelihood estimator of the cumulative risk of receiving a false-positive screen under the new model is derived and its asymptotic normality is proved. The extension of the new model by incorporating covariate information is also considered. We apply our testing method and the new model to data from the breast cancer screening trial of the Health Insurance Plan of Greater New York.