Sidak-type simultaneous prediction intervals for the mortality measures RSRR i about the corresponding S ePMR i for several competing risks of death in an epidemiologic study

Abstract

Simultaneous prediction intervals are obtained involving the externally standardized proportional mortality measures S ePMR i which include the corresponding RSRR i with probability 0.95, or better, for several competing risks of death, under a reasonable assumption, namely, that the total number of deaths d.. due to all risks in the mortality study be large. The results are obtained under a very general situation when several risks compete for an individual's life in the study, irrespective of any covariance structure among the risks of death, known or unknown, positive or negative, or zero. As a particular case, if the risks of death may be assumed as independent, then the simultaneous prediction intervals reduce to those of the present authors (1985a). It is demonstrated that asymptotically (as d.. → ∞) the joint c.d.f. of the statistics Z 1, Z 2, ·, Z M , where Z i = RSRR i /S ePMR i , may be approximated by means of the joint c.d.f. of the multivariate ( M-variate) normal distribution. This is the basic result from which the prediction intervals are derived, using Sidak's (1967, 1968) multivariate normal probability inequalities. Our results are empirically evaluated using real data from a recent retrospective cohort mortality study of Enterline and Marsh (1983), concerning respiratory cancer among man-made mineral fiber workers. These prediction intervals do indeed quantify the observation that for each cause i of death, the measure S ePMR i is close in value to the corresponding RSRR i .