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SADC Course in Statistics Competing Risks & Multiple Decrement Tables (Session 17)

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Presentation on theme: "SADC Course in Statistics Competing Risks & Multiple Decrement Tables (Session 17)"— Presentation transcript:

1 SADC Course in Statistics Competing Risks & Multiple Decrement Tables (Session 17)

2 To put your footer here go to View > Header and Footer 2 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

3 To put your footer here go to View > Header and Footer 3 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.

4 To put your footer here go to View > Header and Footer 4 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.

5 To put your footer here go to View > Header and Footer 5 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.

6 To put your footer here go to View > Header and Footer 6 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!

7 To put your footer here go to View > Header and Footer 7 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.

8 To put your footer here go to View > Header and Footer 8 Numerical example Deaths of men aged 35 to 85 of 1.HIV/AIDS; 2. Cancer; 3. All other causes; were measured in a population in 2001 (upper-case Qs are dependent rates):- Age-groupQ1 x Q2 x Q3 x

9 To put your footer here go to View > Header and Footer 9 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:- Agelxlx d1 x d2 x d3 x X.0413 = x.0190 =

10 To put your footer here go to View > Header and Footer 10 Towards independent rates An approximation to a rate if all but one of the causes of mortality were eliminated is e.g.:- Q1 x = q1 x (1 - ½q2 x - ½q3 x )* where the lower-case qi x are the independent death rates. The thinking is that on average the independent probability of dying, q1 x, applied on average for half the period to those who died in the period from the other causes.

11 To put your footer here go to View > Header and Footer 11 Approximate independent rates An approximate solution, if the death rates are not too large is:- Q1 x 1 - ½Q2 x - ½Q3 x 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. q1 x * =

12 To put your footer here go to View > Header and Footer 12 Independent death rates We can calculate the corresponding non- competitive rates from the above data and formulae (below lower-case qs are independent rates (dependent rates in red) ):- Age-groupq1 x q2 x q3 x (0.0496).0057 (0.0055).0255 (0.0248) (0.0413).0200 (0.0190).0597 (0.0579) (0.0387).0645 (0.0596).1181 (0.1123) (0.0092).1476 (0.1248).3218 (0.3002) (0.0014).3556 (0.2550).6473 (0.5643)

13 To put your footer here go to View > Header and Footer 13 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 q1 x column. To figure out the what-if-world effects then requires working back from revised {qi x } to corresponding dependent rates {Qi x }.

14 To put your footer here go to View > Header and Footer 14 Example: 1 To illustrate a rather unlikely suggestion, suppose cancer death rates were reduced by 90%. The independent rates would then be:- Age-groupq1 x q2 x q3 x

15 To put your footer here go to View > Header and Footer 15 Example: 2 Using Q1 x = q1 x (1 - ½q2 x - ½q3 x ) on these numbers we can return to the what if dependent rates, not real-life ones. Below compare with previous prob.s:- Age-groupQ1 x Q2 x Q3 x (.0504).0005 (.0006).0249 (.0255) (.0430).0019 (.0020).0584 (.0597) (.0423).0059 (.0065).1152 (.1181) (.0117).0123 (.0148).3175 (.3218) (.0024).0240 (.0356).6350 (.6473)

16 To put your footer here go to View > Header and Footer 16 Age-groupQ1 x Q2 x Q3 x Age-groupQ1 x Q2 x Q3 x OriginalOriginal RevisedRevised

17 To put your footer here go to View > Header and Footer 17 Revised LT The LT corresponding to the revised rates is:- Agelxlx d1 x d2 x d3 x

18 To put your footer here go to View > Header and Footer 18 Agelxlx d1 x d2 x d3 x Agelxlx d1 x d2 x d3 x Original LT What if LT

19 To put your footer here go to View > Header and Footer 19 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.

20 To put your footer here go to View > Header and Footer 20 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.

21 To put your footer here go to View > Header and Footer 21 Some practical work follows …


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