Construction of multi-decrement tables can be difficult.<\/p>\n
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.<\/p>\n
To begin, we define the single decrement functions
\n\\begin{eqnarray*}
\n~ _t q_x^{\\prime(j)} = 1- \\exp \\left\\{ – \\int_0^t \\mu_{x+u}^{0j} ~ du \\right\\}
\n\\end{eqnarray*}
\nwith the corresponding survival probabilities, \\( ~ _t p_x^{\\prime(j)}=1- ~ _t q_x^{\\prime(j)}\\).<\/p>\n
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)}\\).<\/p>\n
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.<\/p>\n
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.<\/p>\n
Because
\n\\begin{eqnarray*}
\n~ _t p_x^{(\\tau)} &=& \\exp \\left\\{ – \\int_0^t \\mu_{x+s}^{(\\tau)} ~ ds \\right\\} \\\\
\n&=& \\prod_{j=1}^n \\exp \\left\\{ – \\int_0^t \\mu_{x+s}^{(j)} ~ ds \\right\\} = \\prod_{j=1}^n ~
\n~ _t p_x^{\\prime(j)},
\n\\end{eqnarray*}
\nwe think of these decrements as “competing” with one another. In biostatistics, multi-decrement models fall under the rubric “the Theory of Competing Risks.”<\/p>\n