A Probabilistic Approach to the Construction of Competing‐Risk Life Tables

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Abstract
This paper aims to indentify net and partial‐crude probabilities in the competing‐risk life table context, by using probabilistic approaches. Five types of lifelength random variables are defined to formulate these nonidentifiable probabilities. General expressions for net and partial‐crude probabilities are first derived under independent risks assumptions. Two sets of explicit formulas for estimating the net and partial‐crude probabilities are then derived in terms of the identifiable overall and crude probabilities by making the additional assumption of piecewise uniform distribution of the lifelength random variables. A study of the degree to which nonidentifiability can affect the net and partial‐crude probabilities in a variety of situations is developed. An example from cross‐sectional studies is employed to illustrate the methodology developed.