Analysing data on the sex ratio of human births by cycle day of conception

Abstract

Walters (2000) is correct: pooling data from a set of 2×2 frequency tables is (except under some conditions) invalid. Whether these conditions are fulfilled in the present instance will not be discussed though (as will be seen) they may be suspected of being approximated. He describes a method for simultaneously analysing a number of such tables where there is heterogeneity of unknown origin. In the present case, I suggest that the origin of the heterogeneity may be identified, and that a different form of analysis is appropriate and valid. It may be noted that the heterogeneity of sex ratios (proportions male) exists only within the two categories of births (those reportedly conceived on `most fertile' and on `other' days). In contrast, the five total sex ratios (one for each study cited by Walters) shows remarkably little heterogeneity (χ2 = 2.2, with 4 degrees of freedom, P = 0.7). Moreover the heterogeneity within the two categories would be expected if the hypothesis under discussion were true (i.e. that cycle day of conception were associated with offspring sex). This may be illustrated as follows. Suppose the regression of sex ratio on cycle day of conception were as in Table I. The overall sex ratio is 0.5. But the sex ratios of births associated with `most fertile' and `other' days would vary according to the chosen definitions of `most fertile'. If day 0 alone is chosen as `most fertile' the sex ratios would be respectively 0.45 and 0.53: in contrast, if days –1 to +1 are chosen as `most fertile', the corresponding sex ratios would be 0.475 and 0.6. And it is clear that different definitions have been used in the five data sets because the proportion of births in the `most fertile' category vary from 0.23 in sample 1 to 0.5 in sample 3. I suggest that that is the explanation of the heterogeneity of the sex ratios within the `most fertile' and `other' days. Dr Walters writes: `the question to be answered is: if these five trials represent a random sample from the entire population, is there evidence of a genuine relationship between sex ratio and time of conception?' But as indicated above, these five samples are not to be taken as from the same population. Instead, I suggest that we merely want to know whether there is a difference in sex ratio that is consistently in one direction (even though it may not show up clearly in the individual tables). For this purpose, one may use a Mantel–Haenzel method (e.g. Snedecor and Cochran, 1967). For the five data sets, the Mantel–Haenzel test statistic z (a normalized deviate) was evaluated at 3.05, P < 0.005. Since Gray (1991) had also pooled data, I disaggregated the data in his presentation and analysed those six data sets (i.e. those of Guerrero 1974; Harlap 1979; France et al., 1984; WHO, 1984; Perez et al., 1985; Gray 1991) together with the data sets sets subsequently located (von Koller and Degenhardt 1983; France et al., 1992; Wilcox et al., 1995; Gray et al., 1998;). Table II reproduces the sets pooled in Gray (1991). For the 10 data sets, the Mantel–Haenzel test statistic z = 2.87, P < 0.005. Table III gives the details of the calculation, and may provide a useful basis for additions from further data sets. Thus, regardless of whether or how the data are pooled, there seems strong evidence that human offspring sex ratio is reportedly associated with the cycle day of conception. It would be useful if Dr Walters were to indicate the source of the disparity between his P value of 0.12 and that of.005 assessed here. In my previous letter (James 1999), I offered a number of qualifications which should be held in mind when attempting to interpret the above result. Sex ratio by cycle day of conception (illustrative data) Sex ratio by cycle day of conception (illustrative data) The six data sets pooled within the table of Gray (1991) The six data sets pooled within the table of Gray (1991) The Mantel–Haenzel calculation for the 10 data sets The Mantel–Haenzel calculation for the 10 data sets I fear that Dr James has misinterpreted my letter, as regards the word heterogeneity. I used the word to denote heterogeneity of the `effect' being investigated; that is the day of conception. In the five papers quoted in Table I this effect is in one direction for four of the papers and in the reverse direction for one, quite large, study. The remainder of Professor James's response does not therefore address my objections. If one uses the Log-Odds ratio statistic to quantify the effect, the figures for the five papers are +0.224, –0.048, +2.485, +0.129, +0.132, where positive figures indicate a preponderance of males for `Other days'. Dr James then illustrates his approach with the Mantel-Haenzel test. This test is in fact a means of pooling several 2×2 tables, when it is assumed that there is a CONSTANT population statistic. Many analysts test for heterogeneity of the effect before applying this pooling technique. In answer to Dr James's question regarding P values (0.005 versus 0.12), the difference arises because he, implicitly, uses `within trial' errors to make the test, whereas I have effectively used variation between trials of the effect in question. In the philosophy of hypothesis testing, it is a matter of whether we are considering the finite model of the infinite model; that is, do we wish to make inferences about the general population, based on the evidence available. I suspect that analysts are almost invariably interest in the wider, global, inferences.

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