Re-cap: Life Table Processes. Single Decrement Assumptions: -Only two states: Alive (1) => Death (2) -All individuals transit from state 1 to state 2 ›Destination state is “absorbing”: no backflow Alive Death

Multiple Decrement A generalisation of the single-decrement life table. However, a multiple decrement life table has competing destination states. Alive (1) => Cause of death (2) => Cause of death (3)

Life Table Processes Cont’d. Multiple Decrement Married Single Death

Multiple Decrement Continued Assumptions: There exist at least two state space. Event is proper: all individuals transit from state 1 to either state 2 or state 3. Destination states are “absorbing”: no backflow

Multiple Decrement Cont’d It provides two ways of obtaining information paying particular attention to the importance of the mode of exit, e.g. cause of death:

Multiple Decrement Cont’d Multiple-decrement life table is also used when one is interested in finding out: Probability of leaving the initial state due to decrement i between age x and x+n (nqix); and Proportion that will eventually leave the initial state due to decrement i, etc.

Multiple Decrement Cont’d Cause of Death 1 Alive Cause of Death 2

Multiple Decrement Cont’d Construction of multiple decrement life table is simple adjustment of construction single decrement life table ›Add columns equivalent to other columns in a single decrement life table which pertain exclusively to particular causes of decrement.

Multiple Decrement Cont’d ndxi = number of decrements from cause i in the exact age interval x to x+n. nqxi = probability of leaving the table from cause i in the age interval x to x+n for some one who reached age x. nqxi = ndxi lx

Multiple Decrement Cont’d nmxi =rate of decrement from cause i in the age interval x to x +n = number of decrements from cause i in the age interval x to x +n nmxi = ndxi nLx l xi= number of people reaching age x who will eventually succumb to cause i lxi = ∞ Σndx a=x

Multiple Decrement Cont’d Interpretation multiple-decrement lifetable › lxi / lx = proportion of persons aged x who will eventually leave the table from cause i › li0 / l0 = probability that a newborn will eventually die from cause i nqix; npix => like in normal lifetable

Multiple Decrement Cont’d eix => no straightforward interpretation. In one sense it would be ‘’life expectancy at age x for persons who will succumb to cause i. However, persons who eventually exit from decrement i (lix) can not yet be identified at age x. They will only be identifiable at later stages as the competing risks work themselves out.

Multiple Decrement cont’d Thus, it is point less to compute an expectation for unidentifiable people. Therefore also nLix, Tix, naix are ignored. lix still very useful column. lix / lx = proportion of persons aged x who will eventually leave the table from decrement i .

Multiple Decrement and Cancer Mortality. Computational Procedures 1. Calculate the probabilities of dying and surviving. npx = lx+n/lx OR 1- nqx. nqx = lx-lx+n/lx OR 1- npx

Computational Procedures Cont’d 2. Proportion Dying of Cancer nDxi / nDx. 3. Probability of dying of Cancer. nqxi = nqx* nDxi / nDx.

Computation;la Procedures Cont’d 4. Cancer Deaths ndxi = nqxi * lx 5. lxi = ∞ Σndix a=x

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Exercise 1. Using life lines in Preston (2002:73), design a multiple decrement life table similar to the one on Preston (2002:74). 2. Calculate the other sections of the multiple decrement life table using data on Excel sheet 1.

REMINDER Make sure you are able to do the following: Complete any given single decrement life table; Calculate the probabilities of dying/surviving from any single decrement life table data; and Calculate the CBR, CDR, Infant mortality, child mortality and under 5 mortality using life table data.