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June 01, 2003


I’ve just been reading Steve Jones’s enjoyable book Y: The Descent of Men. More on that some other time, but first I want to comment on one passage: “In China, with its one-child policy, the ratio of boys to girls has risen from about even to a 20 per cent excess of males. Much of this comes from abortion and infanticide, but many farming peoples follow the same rule in a less murderous manner, as they continue to have children until a son is born. All over the developing world a last child is more likely to be a boy than a girl, which again biases life in a masculine way” (p. 33).

Now, I hesitate to accuse the great Professor Jones of a fallacy, but it certainly looks as if he thinks that the practice he describes - continuing to have children until a son is born - leads to an excess of males in the population. I have also seen similar assumptions elsewhere, so it is worth pointing out the error.

Suppose the probability that a birth is male is K. I assume that the probability is the same for all births in all families (I’ll come back to this point). Now consider the total number, A, of first-born children in the population, the total number, B, of second-born children, and so on. The probability K applies independently to all births, so the expected number of first-born males is KA, the expected number of second-born males is KB, and so on. The total expected number of all males is therefore K(A + B + C....). But the expected total number of all children is simply A + B + C...., so the ratio of all males to all children reduces to K. There is no ‘excess’ of males. This reasoning applies quite regardless of whether parents ‘stop at a boy’ or follow any other such rule.

If not convinced by this, consider a population of N families, all of whom apply the simple rule: ‘have children until a son is born, then stop’. For simplicity I will assume that the probability that a birth is male is exactly 1/2. We therefore expect half of the families (N/2) to have a son at the first attempt, half of the remainder (N/4) to have a son at the second attempt, and so on. The total expected number of male births is therefore N(1/2 + 1/4 + 1/8...), and the expected number of all births is N(1/2 + 2/4 + 3/8 + 4/16 + 5/32....). It can be proved that the ratio of male births to all births converges on 1/2 as these series are extended (proof available if required.)

The probability of male births is of course, like any probability, a long run expectation, and there will be stochastic fluctuations both above and below the expected levels. And in practice, in a finite population all families will either terminate with a boy or reach their reproductive limit with no boys at all. But these factors produce no bias one way or the other. More seriously, in reality the probability of a male birth will not be exactly the same in every family. It is likely that some families have a tendency to produce more girls, and some to produce more boys (though the evidence suggests that such variations are slight). But note, the effect of any such tendency is in the opposite direction to ‘male bias’. If families keep ‘trying for a boy’, and stop when they have got one, then those families with a tendency to produce girls will, on average, have to keep going for longer, and will have more offspring (mainly female) than those with a tendency to produce boys. The ‘stopping rule’ will therefore tend to produce a female-biased population.

While on the subject of sex ratio fallacies, it may also be worth mentioning infanticide. Obviously, selective infanticide of one sex produces an immediate distortion of the sex ratio, but it is sometimes argued that it it also produces a selective pressure on the sex ratio at birth. For example, Darwin’s Descent of Man included a discussion of the sex ratio. Following the publication of the first edition, Darwin received a letter from an anthropologist, Colonel Marshall, in which Marshall tried to account for an excess of male births by the practice of female infanticide. Darwin appears to have been impressed by his argument, and included an extract from Marshall’s letter in the second edition of Descent. Unfortunately, Marshall’s argument is not so much fallacious as unintelligible.

Viewing the matter from the perspective of R. A . Fisher’s theory of the sex ratio, one’s immediate thought is that female infanticide will produce a female-biased sex ratio. Since infanticide occurs before parental investment in the offspring is complete, selective mortality of females will reduce the total investment in females below that in males, and by Fisher’s well-known argument, the proportion of females at birth will tend to rise to restore equilibrium.

However, on second thoughts, there are complications. Fisher’s argument (which concerns natural mortality, rather than deliberate infanticide) implicitly assumes that the risk of increased mortality falls equally on all members of the disadvantaged sex. But it is possible to imagine patterns of infanticide where this is not the case. To take an extreme scenario, suppose the practice is to kill all female offspring in the family until at least two sons have been born. The selective mortality would then fall disproportionately on families with a tendency to produce females. These families would produce fewer surviving offspring in total, and possibly even fewer surviving females, than those with a tendency to produce males. So maybe Colonel Marshall had a point after all.

None of the above takes account of W. D. Hamilton’s theory of ‘extraordinary sex ratios’. Notably, where there is close inbreeding, selection may favour a female-biased ratio. In the extreme case where females mate exclusively with their brothers, and males exclusively with their sisters, selection would favour production of the minimum number of males capable of fertilising the females. However, this assumes zero effects of inbreeding on fitness, and zero parental care, so it is largely irrelevant to humans.


Posted by David B at 04:22 AM

funny-because the fallacy you site is one of the first problems in hartl's principles of population genetics (which i'm reading right now).

Posted by: razib at June 1, 2003 01:13 PM

One real consequence of continuing to have children until the boy shows up is that there are a considerable number of Chinese men and boys with many older systers, but not too many Chinese women and girls with many older brothers.

A sadder consequence of Chinese custom is that there are a fair number of smart uneducated Chinese women with dumb educated younger brothers.

Posted by: zizka at June 1, 2003 02:44 PM

Another way to see that the "stop if it's a son" method won't produce more boys is to imagine the sex of each baby is determined by a god who flips a (fair) coin at the moment of conception. Thus if a million babies are born, you have a million independent coin flips, which will certainly be close to 50% heads and 50% tails.

Posted by: BobJohnson at June 1, 2003 10:02 PM