The Multistate Life Table with Duration-Dependence

Wolf, D.A. (1987). The Multistate Life Table with Duration-Dependence. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-87-046

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Abstract

The classical linear multi-state model is represented by an equation due to Kolmogorov, and applied to demography by Andrei Rogers. For many purposes it gives a realistic representation of phenomena, especially in problems in which the population is nearly homogeneous. In that respect it resembles the ordinary life table, of which it is a generalization. But like the life table it acts as though all of the individuals of a given category have the identical probability, so the statistically observed average represents each and every individual in its category.

No demographer has ever regarded this as quite satisfactory; all recognize that individuals within a given cell are different from one another and the average of the cell does not apply to individuals. In a given group every couple may have one chance in three of divorcing; or else 1/9 of couples may divorce 3 times each. The overall probability that a couple will divorce is the same in the two cases, but the inference about what will happen to a random couple in the future is very different for the two. Yet to take into account this distinction involves difficulties, both of data and of the model for dealing with the data.

James Vaupel and Anatoli Yashin of this program have made great progress in dealing with this question, and their work will be brought together in a volume now being prepared.

The present paper sets out the theory of a procedure for taking account of a particular kind of heterogeneity -- that associated with the length of time in a state. Insofar as people are less likely to divorce the longer they have been married, and if divorce rates by duration are known, separate transition matrices can be set up for different durations. Douglas Wolf ingeniously shows how these separate transition matrices can be combined in a single matrix, and the analysis carried out simply and without further reference to duration.

Thus what follows has a special significance for IIASA's population program, in that it combines lines of thought that go back to the multi-state model introduced by Rogers, and on which many IIASA papers were based in the period 1975-83, and the work on heterogeneity of Vaupel and Yashin, that has been central to IIASA's program in more recent years.

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