The Sex Composition of Two-Children Families: Heterogeneity and Selection for the Third Child: Comment on Stansfield and Carlton (2009)

The study presented by Stansfield and Carlton (2009) (in this issue of Human Biology, pp. 3–11) raises several issues about the sex ratio at birth and family composition and focuses on the sex composition of two-children families in the context of family limitation.

Under the assumption of homogeneity (each couple has the same probability of bearing a male or a female each time), the number of male and female children ever born is provided by a binomial distribution:

• For one birth: p for one male and (1 − p) for one female.

• For two births: p² for two males, 2p(1 − p) for one male and one female, and (1 − p)² for two females.

• For three births: p³ for three males, 3p₂(1 − p) for two males and one female, 3p(1 − p)² for two females and one male, and (1 − p)³ for three females.

Using French vital registration data, Malinvaud (1955) showed that there was heterogeneity and asymmetry in the probability of bearing a boy or a girl, most likely for biological reasons. We have reanalyzed the same data set and have estimated the extent of the heterogeneity (Garenne 2009a).With an average of p = 0.51440, the coefficient of variation is σ/p = 0.09007, and the skewness equals −0.58605. The same coefficient of variation and the same skewness could be applied to U.S. data, with p = 0.51187 (U.S. vital registration data for 1987–2002). Under the assumption of heterogeneity, the formulas listed earlier still apply but p is replaced with E(p), p² with E(p²), and p³ with E(p³), which are all values that can be derived from the coefficient of variation and the skewness.

The distribution of two-children families in the population depends on these sex distributions of births and on parity progression ratios (a_n), that is, on the probability of having (n + 1) births given n births already. In populations practicing contraception, parity progression ratios might depend on family composition, that is, on the number of previous males and females already born, for behavioral reasons. For the United States, published values of parity progression ratios, on average for all family compositions, are a₀ = 0.846, a₁ = 0.659, and a₂ = 0.309 (Frejka and Sardon 2007).

Altogether, if we assume the sex ratio at birth observed in the U.S. vital registration, the heterogeneity observed in France, and a constant parity progression ratio, then we are not able to fit the 1987–1993 National Health Interview Survey (NHIS) data on the distribution of sexes in two-children families. There are too many balanced families of one boy and one girl (51.4% observed against 49.5% expected, P <10^–13) and not enough two-girl families (22.2% observed against 24.0% expected, P <10^–99), whereas the proportion of two-boy families was as expected (P = 0.095) (Table 1). This has already been noted by Will Lassek and Stansfield and Carlton (2009).

Table 1.

Observed and Expected Distribution of Two-Children Families Under Various Assumptions, United States

We solved the equations and found that the data could be fitted by assuming differential parity progression ratios: a₂ = 0.282 when one boy and one girl are already born, a₂ = 0.308 when two boys are already born, and a₂ = 0.361 when two girls are already born. This shows that U.S. families seem to have a net preference for stopping after two children when they have a balanced family (one boy and one girl) and are more likely to have a third child if they already have two girls than if they already have two boys.

This pattern of differential parity progression ratios produces a deficit of females in two-children families, simply by selection bias. The sex ratio becomes 1.088 instead of 1.049 for all births combined. This could explain the...