Re: Naive question

shiling99@xxxxxxxxx wrote:
I think the general formula is,

P(A + B + C) = P(A) + P(B) + P(C) - P(A and B) - P(A and C) - P(B and
C) + P(A and B and C)

It's not totally clear to me what A, B and C represent here. If they are "death from ..." or "death induced by risk factor ..." , then, clearly, P(A and B) must be zero unless you allow death from multiple causes.

If A, B and C are mutually independent, then it is,

P(A + B + C) = P(A) + P(B) + P(C) - P(A)P(B) - P(A )P(C) - P(B)P(C) +
P(A)P(B)P(C)

If A,B,C are "death from cause ...", then 1-P(A) is the probability of not dying from cause A, and the above says that the probability of not dying from any of these causes

(1) 1-P(A+B+C) = (1-P(A))*(1-P(B))*(1-P(C)).

However, reducing the risk of dying from one cause, will neccessarily increase the risk of dying from something else: at a later time, true, but still. E.g. if by some means we're able to eliminate accidents and heart attack, it will probably lead to a longer life, and longer life means that the chance of getting cancer increases since you have more time to develope cancer, and in addition the risk of developing it increases with time. Hence, (1) only makes sense if you are only looking at a very limited time span.

The simplest model I can think of, is one where at any time, the mortality rate, M_X, from cause X is given. This is a more meaningful measure of mortality than the probability of death: it is possible to reduce the total mortality rate of a population, but the probabilty of dying will still be 100%.

Of course, the mortality rate will change with age. However, to keep things simple, let's assume that mortality rates M_A, M_B, ... always have the same proportion: i.e. ratios M_A/M_B are constant over time. If that's the case, the probability of dying from cause X is

(2) P(X) = M_X/M where M=M_A+M_B+...

and the list A, B, ... list all possible causes of death. If you assume that some action or intervention changes M_A only, another M_B only, etc., the joint effect on P(A+B+C) will not be additive (and certainly not multiplicative). Though a formula can be calculated for this particular model, it seems to me it will be slightly more complicated ... and I don't find that formula interesting enough to go through with that calculation.

Now, this model is not a truely realistic one: e.g. someone who contracts one disease will often experience an increased risk of other diseases or complications. However, the main problem is to get some idea of what the expected mortality is for a given person: i.e. the chance of dying of a given cause at a given time.

The mortality risks (and probabilities) you find in the literature are population averages. People, however, are very different. Heredity (or what you happened to be born with) as well as various life-style factors (many of which are unknown or unmeasurable and analysed through their assumed correlation with socioeconomic factors) influence life expectancy. Also, the effects of risk factors are neither linear nor additive: a combination of risk factors typically have greater impact than each one separately.

Thus, a simple health question, does not reduce to simple maths.

Einar
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