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Fisher's fife-table mistakes

When reading about mathematical models of evolution you will probably inevitably bump into The Genetical Theory Of Natural Selection[1]. This book praised by many authors as the fundamental work on reconciling genetics and evolutionary theory[2]. Also on wikipedia Fisher is praised as a blessed mathematician and the inventor of the probability F-distribution. After finding the book whose copyright fortunately has expired on the Internet I started reading it and became disappointed. Here is my report summarizing the second chapter (the first is dispensable anyway).

The second chapter bears the title “THE FUNDAMENTAL THEOREM OF NATURAL SELECTION”.

Fishers life-table arithmetics

In order to explain how natural selection effects mortality rates Fisher gives a short introduction of life-tables, or at least what he believes to be the life-table arithmetic.

First he gives his definition of death rate:

"

Consequently if lx is the number living to age x, the death rate at age x is given by:

"

\mu_x = - \frac{1}{l_x} \frac{dl_x}{dx}

I did not find this definition elsewhere. What comes close to the term death rate is mortality rate which is the number of deaths per 1000 individuals per year. Anyway everybody is free to give his own definitions based on which a theory is developed provided the theory is correct. Therefore let us follow his thoughts first before our final judgment.

"

Just as a person alive at the beginning of any infinitesimal age interval dx has a chance of dying within that interval measured by

"

\mu_x dx

Given the first equation and doing the algebra this expression yields:
\mu_x dx= - \frac{dl_x}{l_x}

This exactly corresponds to the wikipedia definition of

"

the probability that someone aged exactly x will die before reaching age x+1.

"

q_x = \frac{\Delta l_x}{l_x} = \frac{l_x - l_{x+\Delta x}}{l_x}

if dx is one year. Or any other unit of time.

Next he proceeds:

"

Again, just as the chance of a person chosen at birth dying within a specified interval of age dx is

"

l_x \mu_x dx

Again taking the first equation and doing the algebra this expression yields:
l_x \mu_x dx= - dl_x

This is the absolute number of people dying at age x and has nothing to to with chance or probability or such things.

Interestingly Fisher transposes this formula to the rate of reproduction. It cannot be but wrong.

The problem is that Fisher with his further deductions relies exactly on this error, so we have to conclude at least this chapter has to be completely rewritten. In order to accomplish that, we have to find out what he was probably up to and then try to put mathematics it straight.

Force of mortality

The closest term of what resembles Fisher's rate of mortality that I can think of is force of mortality defined by wikipedia as the conditional probability of dying at a given age interval \Delta x of a person that has attained age x.

Let assign
A = \{X > x\}
B = \{X \leq x + \Delta x\}
A^C = \{ X \leq x\}

then
P_{\Delta x}(x) = P(x < X \leq x + \Delta x | X > x) = P((A \cap B) | A)

Applying the product rule of probability we get.

P((A \cap B) | A) = \frac{P((A \cap B) \cap A)}{P(A)}

Because of the associative law and the fact that P(A \cap A) = P(A) we may simplify.

P((A \cap B) | A) = \frac{P(A \cap B)}{P(A)}

Now applying the theorems of probability

P(A) = 1 - P(A^C)

and

P(A \cap B) = P(B) - P(A^C \cap B)

we get

P((A \cap B) | A) = \frac{P(B) - P(A^C \cap B))}{1 - P(A^C)}

Applying the product rule of probability again this can be simplified

P((A \cap B) | A) = \frac{P(B) - (P(A^C) P(B | A^C)))}{1 - P(A^C)}

Because P(B | A^C) = 1, this term can be omitted.

Finally we get

P((A \cap B) | A) = \frac{P(B) - P(A^C)}{1 - P(A^C)}

Now if we apply this to a continuous function we get

P_{\Delta x}(x) = \frac{F_x(x + \Delta x) - F_x(x)}{1 - F_x(x)}

where F_x(x) is the cumulative probability functions of the random variable X age-at-death.

If we use an infinitesimal small age interval (\Delta x \rightarrow dx) we get the derivation of F'_x(x) which is the probability density function f_x(x).

\mu (x) = \frac{F'_x(x)}{1 - F_x(x)} = \frac{f_x(x)}{1 - F_x(x)}

As 1 - F_x(x) can be written as survival function S_x(x) The force of mortality function sometimes is written as

\mu (x) = \frac{F'_x(x)}{1 - F_x(x)} = \frac{f_x(x)}{S_x(x)}

This function bears some resemblance to Fisher's rate of death function listed above, but it is by far not the same. The term l_x is replaced by S_x(x). That is the probability to survive to age x is replaced in Fishers formula by the mere number of survivors at age x. Of course there is a relatioship between these figures. They are related by the number of births x years before N_x.

S_x(x) = \frac{N_x}{l_x}


Tags: Genetics Society Statistics Theory


Categories: Mathematics

 
   

(c) Mato Nagel, Weißwasser 2004-2013, Disclaimer