PAPP104 - S03: Multiple decrement and current status life tables

Competing risks

Multiple decrement probabilities of dying from various causes are often referred to as competing risks because they are dependent on each other: a decline in the death rate of cause -i will induce an increase the probability of dying from cause i. Mathematically, this can be derived by substituting _nm_x = _nm_xⁱ + _nm_x^-i in the denominator of the conversion formula:

{}_{n}q_{x}^{i} = \frac{n \cdot {}_{n}m_{x}^{i}}{1 + (n - {}_{n}a_{x}) . n . ({}_{n}m_{x}^{i} + {}_{n}m_{x}^{- i})}

Let i be cancer and –i all other causes, then a decline in the death rate of heart disease will lead to an increase in the probability of dying from cancer (the denominator in the above formula will shrink and _nq_xⁱ will increase). Note that this does not necessarily imply that the rates from different causes are dependent on each other.

\frac{l_{50}^{} - l_{50}^{e}}{l_{50}} = \frac{87,156 - 3,319}{87,156} = 0.9619