Construction of multi-decrement tables can be difficult.
Alternative – construct several single decrement tables and use these to build a multi-decrement table. Idea behind the construction of theses tables: From a population, take the subset of individuals who eventually fail due to cause (j) and construct a decrement table from this subset.
To begin, we define the single decrement functions
begin{eqnarray*}
~ _t q_x^{prime(j)} = 1- exp left{ – int_0^t mu_{x+u}^{0j} ~ du right}
end{eqnarray*}
with the corresponding survival probabilities, ( ~ _t p_x^{prime(j)}=1- ~ _t q_x^{prime(j)}).
Some authors use the alternative notation ( ~ _t q_x^j =~ _t q_x^{ast(j)}=~ _t q_x^{prime(j)}) and ( ~ _t p_x^j =~ _t p_x^{ast(j)}=1 – ~ _t q_x^{prime(j)}).
Note that this decrement uses only the transition force ({mu_{x}^{0j}}). In this sense, it is “independent” (or unrelated) to the other transition intensities – it is not independent in the usual stochastic sense.
Also known as the “net probability of decrement” and the “net rate of decrement” – the later definition emphasizes that the quantities (~ _t q_x^{prime(j)}) are not probabilities in the usual sense.
Because
begin{eqnarray*}
~ _t p_x^{(tau)} &=& exp left{ – int_0^t mu_{x+s}^{(tau)} ~ ds right} \
&=& prod_{j=1}^n exp left{ – int_0^t mu_{x+s}^{(j)} ~ ds right} = prod_{j=1}^n ~
~ _t p_x^{prime(j)},
end{eqnarray*}
we think of these decrements as “competing” with one another. In biostatistics, multi-decrement models fall under the rubric “the Theory of Competing Risks.”