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**Competing Risks & Multiple Decrement Tables**

(Session 17)

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**Learning Objectives – this session**

At the end of this session, you will be able to understand ideas of multiple decrements from the LT explain and utilise the concept of competing risks appreciate the distinction between dependent and independent rates and their roles in calculation approach further topics in actuarial studies

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Introduction The issue addressed here concerns LT uses where death is not the only exit from the pop.n. One well-known example is a “net nuptiality table”: say original pop.n are all single females. Marriage or death are 2 ways of leaving the pop.n in question. Working population examples above were restricted by assumption that workforce never left one specific employment: reality involves more complex pathways.

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**Multiple decrement tables**

The LT has a single cause of pop.n losses – death. The loss of individuals is described as “decrement” ~ a negative increment. Where individuals can be lost to two or more causes the table is described as a double or multiple decrement table respectively. In some cases, losses from one category e.g. interviewer, re-appear as gains to another e.g. supervisor. This may be described as an increment-decrement table.

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Competing risks: 1 The medical concept of “competing risks” has important similarities. A person who has a long life free of any fatal infectious disease remains “available” for a long time to the slowly-developing non-transmissible diseases e.g. degenerative heart/circulatory problems, e.g. cancers. So there are higher rates of these causes of death in longer-lived pop.ns where people have not already died of something else.

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Competing risks: 2 The phrase “competing risks” is based on the rather bizarre idea that the causes of death knowingly “compete” with each other to see which can kill the individual first, rather than just existing. The phrase is very commonly used despite this oddly fanciful attribution of intelligence to viruses, bacilli, rogue genes etc!

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**Dependent & independent rates**

This notion of competing risks explains the technical idea below that the death or other rates are in reality “dependent” e.g. the cancer death rate is dependent on the prevailing “force of mortality” due to other conditions. The “independent” cancer death rate would be higher in an unreal world where the other causes were removed.

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**Numerical example Deaths of men aged 35 to 85 of**

HIV/AIDS; 2. Cancer; 3. All other causes; were measured in a population in 2001 (upper-case Q’s are dependent rates):- Age-group Q1x Q2x Q3x 35-44 0.0496 0.0055 0.0248 45-54 0.0413 0.0190 0.0579 55-64 0.0387 0.0596 0.1123 65-74 0.0092 0.1248 0.3002 75-84 0.0014 0.2550 0.5643

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**Multiple decrement table**

A part-of-life multiple decrement life table for this limited age-range is computed as for a normal LT. Each death rate applies to the overall starting population of the age-group:- Age lx d1x d2x d3x 35 10000 496 55 248 45 9201 9201 X =380 9201 x = 175 533 8113 314 484 911 65 6404 59 799 1922 75 3624 5 924 2038 85 657

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**Towards independent rates**

An approximation to a rate if all but one of the causes of mortality were eliminated is e.g.:- Q1x = q1x(1 - ½q2x - ½q3x)* where the lower-case qix are the independent death rates. The thinking is that “on average” the independent probability of dying, q1x, applied on average for half the period to those who died in the period from the other causes.

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**Approximate independent rates**

An approximate solution, if the death rates are not too large is:- Q1x 1 - ½Q2x - ½Q3x Of course the formulae for the other two independent rates are of the same form, with 1s, 2,and 3s moved appropriately. * The fact that the independent rates are higher is evident from either form of these formulae. q1x* =

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**Independent death rates**

We can calculate the corresponding “non-competitive” rates from the above data and formulae (below lower-case q’s are independent rates (dependent rates in red)):- Age-group q1x q2x q3x 35-44 .0504 (0.0496) .0057 (0.0055) .0255 (0.0248) 45-54 .0430 (0.0413) .0200 (0.0190) .0597 (0.0579) 55-64 .0423 (0.0387) .0645 (0.0596) .1181 (0.1123) 65-74 .0117 (0.0092) .1476 (0.1248) .3218 (0.3002) 75-84 .0024 (0.0014) .3556 (0.2550) .6473 (0.5643)

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What-if calculations Looking at the effect of a change in mortality due to change in treatment of a condition uses the independent rates. For example HIV/AIDS death rates by age can be expected to change for all age-groups as time goes by. This could be reflected by assuming (or deriving from data) new figures to put into the q1x column. To figure out the what-if-world effects then requires working back from revised {qix} to corresponding dependent rates {Qix}.

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Example: 1 To illustrate a rather unlikely suggestion, suppose cancer death rates were reduced by 90%. The independent rates would then be:- Age-group q1x q2x q3x 35-44 .0504 .0006 .0255 45-54 .0430 .0020 .0597 55-64 .0423 .0065 .1181 65-74 .0117 .0148 .3218 75-84 .0024 .0356 .6473

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Example: 2 Using Q1x = q1x(1 - ½q2x - ½q3x) on these numbers we can return to the “what if” dependent rates, not “real-life” ones. Below compare with previous prob.s:- Age-group Q1x Q2x Q3x 35-44 .0497 (.0504) .0005 (.0006) .0249 (.0255) 45-54 .0416 (.0430) .0019 (.0020) .0584 (.0597) 55-64 .0397 (.0423) .0059 (.0065) .1152 (.1181) 65-74 .0497 (.0117) .0123 (.0148) .3175 (.3218) 75-84 .0016 (.0024) .0240 (.0356) .6350 (.6473)

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**Original Revised Age-group Q1x Q2x Q3x 35-44 0.0496 0.0055 0.0248**

45-54 0.0413 0.0190 0.0579 55-64 0.0387 0.0596 0.1123 65-74 0.0092 0.1248 0.3002 75-84 0.0014 0.2550 0.5643 Revised Age-group Q1x Q2x Q3x 35-44 .0497 .0005 .0249 45-54 .0416 .0019 .0584 55-64 .0397 .0059 .1152 65-74 .0123 .3175 75-84 .0016 .0240 .6350

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**Revised LT The LT corresponding to the revised rates is:- Age lx d1x**

35 10000 497 5 249 45 9249 385 18 540 55 8307 330 49 857 65 6970 68 86 2937 75 3880 6 93 2464 85 1317

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**Original LT “What if” LT**

Age lx d1x d2x d3x 35 10000 496 55 248 45 9201 380 175 533 8113 314 484 911 65 6404 59 799 1922 75 3624 5 924 2038 85 657 Original LT Age lx d1x d2x d3x 35 10000 497 5 249 45 9249 385 18 540 55 8307 330 49 857 65 6970 68 86 2937 75 3880 6 93 2464 85 1317 “What if” LT

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Summary The big reduction in cancer death rates of course allows rather more LT pop.n members to survive longer ~ 256 more to age 75, 660 more (twice as many) to 85. It also leaves more “available” to die of other causes, especially in this case the general diseases (category 3) that affect the elderly with deaths increased from 3960 to 5401 (54% of the LT population of 10,000 at age 35) between 65 and 85.

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In conclusion In this session, attention has been focused on a medical “competing risk” example to show the means of manipulating multiple decrement data. Much insurance & actuarial calculation develops from more detailed application of ideas covered briefly here. This tends to involve voluminous arithmetic, difficult to assimilate in lecture conditions.

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**Some practical work follows …**

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