Mortality and causes of death: matrix formulation and sensitivity analysis

Hal Caswell, University of Amsterdam
Nadine Ouellette, Institut National d'Études Démographiques (INED)

The analysis of mortality by cause of death (COD) is a classical problem of competing risks. The canonical questions addressed by COD analysis include the properties of deaths when specified subsets of the set of all causes are operating, or are removed from operation, and interaction between the probabilities of death due to different competing causes. Properties of interest include life expectancy, the distribution of deaths by cause and by age, and the life lost to each of the different causes operating. Whenever it is possible to formulate a demographic calculation in matrix terms, certain advantages often accrue, including notational simplicity, computational efficiency, the generalization from age-classified to stage-classified populations, and the availability of sensitivity analysis. Here, we present a new matrix formulation of COD analysis. The analysis proceeds from specification of a matrix of age- and cause-specific hazards of mortality. The life course is described by an absorbing Markov chain with absorbing states given by causes and ages of death. The transient matrix and the absorbing matrix of this chain are computed from the hazard matrix. The result gives all the moments (not just the mean) of longevity when any subset of causes is operating, the probabilities of eventual death due to each cause, the joint distribution of ages and causes of death, and all the moments (not just the mean) of life lost due to each cause. Sensitivity analysis provides the effect, on any of these outputs, to specified patterns of perturbation, including additive or proportional perturbations of mortality at specified ages, from specified causes, or from any interesting combination of ages and causes.