We thank Dr James for his always stimulating comments on variations in the sex ratio at birth. Dr James makes some specific comments and queries about our interpretation of the Lexis variations that we found in data from both Europe and Africa. These Lexis variations (different probability of bearing a boy among couples) were shown primarily by a statistical analysis of the distribution of boys and girls according to the number of males and female already born in the family (Garenne, Reference Garenne2008b, Reference Garenne2009a). In another paper, we showed Poisson variations by age of mother and birth order using the same African data (Garenne, Reference Garenne2008b). We fully agree with Dr James that any systematic Poisson variation will have only a minor effect on the variance of the distribution of boys and girls in families. We have also addressed the issue of behavioural factors in a comment to a study based on American data (Garenne, Reference Garenne2009b).
Statistical evidence for asymmetry
Dr James argues that what we interpreted as Lexis variations could also be interpreted as random or chaotic (non-systematic) Poisson variations (the sex ratio varies among couples for a variety of factors over time and differently for different couples), or by an effect of birth order. His main argument is based on a re-analysis of our regression model linking the sex ratio with the number of boys and girls already born to the couple. This model was:
Logit (sex ratio) = A + B×Boys + C×Girls,
which can also be written as:
Logit (sex ratio) = A + B′×(Boys+Girls)+C′×(Boys−Girls),
with B′= (B+C)/2 and C′= (B−C)/2.
The two models are equivalent. If there is symmetry, then B∼−C, and (B+C)∼0. If only parity matters and not the family composition, then (B−C)∼0 and B+C<>0. Otherwise, this means asymmetry and effects other than parity.
This was the case in our analysis of African data. Since our original publication, we added some more data to the database from recent Demographic and Health Surveys (DHS) and found:
B=+0.001756 (SD=0.000968)
C=−0.009676 (SD=0.000996).
Therefore:
B′=−0.003960 (SD=0.000550)
C′=+0.005716 (SD=0.000814).
This shows clearly the effect of family composition and the asymmetry of the relationship, with high statistical significance (p<2.2×10−16). The same effect was found in France, when re-analysing the data gathered by E. Malinvaud (Garenne, Reference Garenne2008a).
Rationale for asymmetry
There are many reasons to expect heterogeneity and asymmetry in the sex ratio, because of the complexity of the conception and birth processes, and because the two sexes are involved at various times in the development of the embryo. At time of meiosis and spermatogenesis, the sex ratio (M/F) is assumed to be 1 to 1, although this has been questioned. Just after fertilization and implantation in utero, the sex ratio is about 1.6, and this increase is assumed to be primarily coming from the father's side. At birth, the sex ratio is about 1.05, and what happens in between can only be under the control of the mother. Many things could happen between the development of sperm cells and delivery, and a great variety of factors could influence the sex ratio, some coming from the father's side, other from the mother's side. These are likely to be influenced by genetic factors (as exemplified by differences between races or population groups) as well as by a large array of environmental factors including diseases, so one can only expect a large heterogeneity and some asymmetry. We do agree with Dr James that this could be either genetic (pure Lexis variations) or due to variations in the environment (chaotic Poisson variations). However, the strong effect of number of boys (or girls) in unisexual sibships (families with only boys or only girls) indicates that most of the effect seems to be of the Lexis type (couples have either a low or high sex ratio throughout their reproductive period).
Behaviours
We have addressed the issue of the stopping rule in another document (Garenne, Reference Garenne2009b). Of course, the stopping rule (preventing pregnancy after the number of desired males or females is achieved) does not change the sex ratio at birth. By behaviour, we rather meant ‘sex selective abortion’ or other radical means of controlling the sex ratio before delivery. In the case of the two data sets analysed (France in the 1950s and Africa since 1950), such behaviour simply did not exist, and could not be evoked as a potential explanation. However, in theory, assuming that couples have another attitude, and have differential preferences for boys and girls, so that a large number of couples choose to have mostly boys and another large number choose to have mostly girls, one could find a variance in the distribution of boys and girls similar to that found under the hypothesis of heterogeneity due to biological factors. This hypothesis remains purely speculative, and no population in the real world has been shown with this behaviour.
Conclusion
With our log-gamma fitting, we hope to have provided statisticians with an appropriate tool to include heterogeneity among couples in their calculations and models, whatever the causation. The similarity between France and Africa suggests that these are likely to be global patterns.
