The variation of the probability of a son within and across couples

William H. James

The Galton Laboratory, University College London, Wolfson House, 4 Stephenson Way, London NW1 2HE, UK


    Abstract
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 Abstract
 Introduction
 Theoretical background
 Previous evidence on Markov,...
 History of the estimation...
 Data which impugn the...
 The inference
 Estimating chaotic Poisson and...
 References
 
It is suggested that there is a flaw in the currently accepted account of the variation of P, the probability of a boy, within and across couples. It was previously suggested that P has a mean (for Caucasian couples) of ~0.514 with an SD of ~0.05 across couples: and that the variation within couples is rather less. Grounds are offered here for suspecting that this formulation underestimates both SDs by a factor of as much as 4. It is suggested that in estimating these sources of variation, earlier workers did not consider the possibility that within-couple variation might be random and substantial. In view of the established epidemiology of human sex ratios, it now seems likely that such variation exists, and that there is a substantial measure of counterbalancing across-couple variation.

Key words: human sex ratio/Lexis/Markov/Poisson/variation


    Introduction
 Top
 Abstract
 Introduction
 Theoretical background
 Previous evidence on Markov,...
 History of the estimation...
 Data which impugn the...
 The inference
 Estimating chaotic Poisson and...
 References
 
The present topic has been extensively studied by mathematical statisticians, but there are two reasons for reconsidering it: (i) previous authors have not discussed the possibility that P (the probability of a boy) varies randomly and substantially within couples (here called `chaotic Poisson variation'); and (ii) previous workers have not considered the implications of the markedly sub-binomial variance of collections or series of sex ratios when taken in conjunction with the small magnitude of the correlations between the sexes of sibs within sibships.

It will be argued that, contrary to current opinion, the available data imply that P varies substantially within and across human couples. Notes will be given on the theoretical background; the previous evidence on Lexis, Markov and Poisson variation; the history of estimating this variation; the data which impugn these estimates; the inference underlying this claim; and future prospects for estimating this variation.


    Theoretical background
 Top
 Abstract
 Introduction
 Theoretical background
 Previous evidence on Markov,...
 History of the estimation...
 Data which impugn the...
 The inference
 Estimating chaotic Poisson and...
 References
 
The probability P that a birth will be male is potentially subject to three different forms of variation, namely Poisson, Lexis and Markov variation. These will be described later. Moreover analysis is made complicated by the fact that couples operate `stopping rules' by which decisions to reproduce further are based on the sex(es) of existing offspring. This falsifies the assumption of randomness that would otherwise simplify analysis.

These three forms of variation have been most comprehensibly described (Edwards, 1960Go).

Lexis variation
P is constant within a given couple, but varies across couples.

Markov variation
P varies within couples according to the sex (or sexes) of previous births. Where P increases with previous male births, Markov variation may be called `positive': where it decreases with previous male births, it may be called `negative'.

Poisson variation
P varies from one pregnancy to the next within couples regardless of the sexes of the existing sibs, and has the same mean for all couples. Poisson variation may be called `chaotic' where P varies randomly within individual couples. Poisson variation may be called `systematic' where P varies from one pregnancy to the next in parallel across all mothers (as in a birth order effect). It may be remarked that in his account, Edwards (Edwards, 1960Go) confined himself to systematic Poisson variation. However, chaotic Poisson variation (which I will suggest almost certainly must exist) is not readily detectable and, if present, may invalidate estimates of the other forms of variation.

There are two important forms of stopping rule, viz. type I where couples wish for one or more children of one sex, and cease reproducing when they have arrived. The type II rule is where couples wish for representatives of both sexes among their progeny, and cease reproducing when they have arrived.

All these forms of variation and stopping rule may co-exist and interact, and no statistical test has been devised for the independent presence of each. Crouchley and co-workers (Crouchley et al., 1984Go) showed how to estimate these forms of variation on the assumption that any Poisson variation present was systematic and not chaotic.

In comparison with binomial expectation, the variances of the distributions of the combinations of the sexes within sibships (and the correlations between the sexes of sibs within sibships) are independently increased or decreased by the above forms of variation in the absence of stopping rules as follows (e.g. Gini, 1951; Edwards, 1962) (see Table IGo).


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Table I.
 
It should be emphasized that chaotic Poisson has the same effect as systematic Poisson variation in this context.

Yamaguchi described the effects of reproductive stopping rules on sex ratio and its variance (Yamaguchi, 1989Go). The two types of stopping rule are both associated with decreased variances (and correlations) in small sibships, and increased variances (and correlations) in large sibships. However, these stopping rules can have no effect on the sort of data considered below (variances of series of sex ratios). They will therefore be ignored in the following discussion.


    Previous evidence on Markov, Lexis and Poisson variation
 Top
 Abstract
 Introduction
 Theoretical background
 Previous evidence on Markov,...
 History of the estimation...
 Data which impugn the...
 The inference
 Estimating chaotic Poisson and...
 References
 
Before proceeding to formal argument, it may be useful to indicate some of the pre-existing (mainly non-theoretical) evidence for the existence of these forms of variation in the human being.

The question of Markov variation is a vexed one. After a series of studies, it was concluded (Edwards, 1970Go) that there are positive Markov associations between successive sibs within human sibships, but no Lexis variation. However, research on another monotocous mammalian species might be expected to throw light on the human situation. Slight Lexis variation has been reported in the sex ratios of the offspring of bulls (McWhirter, 1956Go; Bar-Anan and Robertson, 1975Go; Powell et al., 1975Go; Totey et al., 1996Go) and cows ( Astolfi et al., 1995) and the latter authors also failed to find significant evidence for Markov variation in the offspring of a sample of 266 000 dams.

In regard to human beings, some (admittedly unquantifiable) measure of Lexis variation must exist in view of the overwhelming evidence for skewed offspring sex ratios following parental exposure to many chemical compounds, e.g. the pesticide dibromochloropropane (Potashnik and Yanai-Inbar, 1987Go), boron (James, 1999aGo), dioxin (Mocarelli et al., 1996Go) and a variety of unspecified pesticides (see James, 1998 for references), and to non-ionizing radiation (James, 1997Go) and diseases, e.g. testicular cancer (Moller, 1998Go), multiple sclerosis (James, 1994Go), prostatic cancer (James, 1990Go), non-Hodgkin's lymphoma (Olsson and Brandt, 1982Go) and several HLA-related rheumatic conditions (James, 1991Go). Psychological factors operating around the time of conception also reportedly alter the offspring sex ratio, e.g. psychological distress (Fukuda et al., 1998Go; Hansen et al., 1999Go) and the nature of domicile (harem or separate dwelling) of polygynous women (James, 1995Go); moreover, maternal dominance reportedly affects offspring sex ratio (Grant, 1994Go) as it undoubtedly does in a number of other mammalian species (James, 1985Go). Further variables reportedly associated with offspring sex ratio are parental occupation, occupational hazard, and induced ovulation (for references, see James, 1996).

In short, there must be some Lexis variation in the human being but the evidence for Markov variation is, at best, equivocal. Moreover, I know of no biological mechanism that might underlie Markov variation so it will be provisionally discounted here. This decision is made merely in the interest of simplicity and is not crucial to the ensuing argument.

Lastly, systematic Poisson variation, if it exists, must be of small magnitude (Crouchley et al., 1984Go). This leaves chaotic Poisson variation. Table IIGo illustrates how the co-existence of Lexis and chaotic Poisson variation could lead to the appearance of binomial sampling. It will later be suggested that that table may, if Markov variation really does not exist, very roughly exemplify the situation in human couples.


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Table II. Hypothetical data to show the effect of the simultaneous existence of Lexis and chaotic Poisson variation. All the matings show chaotic Poisson variation. The matings in rows 1 and 2 have high average sex ratios. Those in rows 3 and 4 have low average sex ratios. Taken together, the data exemplify Lexis variation. The overall column totals would suggest binomial sampling with P = 0.5, yet the data seethe with underlying variation that could not be detected in the absence of external indicators
 

    History of the estimation of the variation of P
 Top
 Abstract
 Introduction
 Theoretical background
 Previous evidence on Markov,...
 History of the estimation...
 Data which impugn the...
 The inference
 Estimating chaotic Poisson and...
 References
 
Earlier efforts to estimate the Lexis variation between couples, using various methods, led to remarkable agreement.

Edwards (Edwards, 1958Go)
An SD was estimated across couples of 0.05 in P (the probability, assumed constant through a couple's reproductive life, of producing a son). He made this estimate by fitting the distributions of the combinations of the sexes in a large number of sibships on the assumption that P varied as a ß variate.

Malinvaud (Malinvaud, 1955Go)
Data were given on the sexes of 4x106 current infants by their numbers of prior brothers and sisters. On the basis of these data, it was estimated that P had an SD across couples of 0.045 (James, 1975Go).

Pickles et al. (Pickles et al., 1982Go)
An estimate was offered of 0.051 on the basis of the sequences of the sexes in two large series of sibships, but making the assumption that the `stopping rules' do not exist.

As noted above, these three estimates (of 0.05, 0.045 and 0.051) show remarkable agreement. So it was tentatively suggested that for practical purposes, P might be conceptualized as being distributed across (Caucasian) couples with a mean of ~0.514 and an SD of 0.05 (James, 1987aGo). The data impugning this estimate will now be considered.


    Data which impugn the above estimate of the variability of P
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 Abstract
 Introduction
 Theoretical background
 Previous evidence on Markov,...
 History of the estimation...
 Data which impugn the...
 The inference
 Estimating chaotic Poisson and...
 References
 
These data are of two sorts, viz.: (i) the variances of series or collections of sex ratios; and (ii) the correlations between the sexes within sibships.

When considered jointly, they suggest that something is wrong. Each of these two sorts of data will now be considered.

Variances of series of sex ratios
Arbuthnott (Arbuthnott, 1710Go)
The variance of the yearly sex ratios of births in London 1629–1710 was reported to be sub-binomial. In other words, these sex ratios lay too close to their mean for the sampling to have been binomial. Arbuthnott attributed this to Divine Providence, but a more mundane explanation is available. It is that P varied from birth to birth within each year's births.

von Mises (von Mises, 1957Go)
A similar phenomenon was reported. He gave the 24 monthly sex ratios of births in Vienna 1908–1909. For such data, he suggested that if there were binomial sampling, the expected variance of such a series of sex ratios is E(V) where

where n is the number of sex ratios, z is the mean number of births each month, and P is the overall sex ratio.

In von Mises' data, n = 24, z = 3903 and P = 0.514. The expected variance is therefore 0.000061336. The observed variance was 0.000053. So {sigma}2P (the estimated variance of P) may be taken as

(Weatherburn, 1949Go)

It will later be suggested that this variance may provisionally be thought of as the sum of the (roughly equal) variances within and between couples. So it is anomalously large in contrast with the above estimated variance across couples of 0.0025 (viz. the estimated SD of 0.05).

Dickinson and Parker (Dickinson and Parker, 1997Go)
The sex ratios of offspring of men in 367 different occupational categories were examined. In only 11 of these (3%) was the sex ratio significantly different from the overall mean at the 5% level. These authors noted that McDowall in a similar earlier study (McDowall, 1985Go) had found such significance in only 10 (2.9%) occupational categories out of 350. The Poisson probability of observing (10 + 11) = 21 or fewer events when 717x0.05 = 35.8 were expected is close to 0.005. So taken together, these deficits are highly significant. In short there is also significantly sub-binomial variance in samples of sex ratios chosen by paternal occupational category.

Further data
For this purpose, I chose the legitimate live births for England and Wales 1950–1969 inclusive (as given in the OPCS Birth Statistics for 1974, Table 1.1). These dates are not arbitrary. The period was deliberately chosen: (i) to start after the wartime perturbations of the sex ratio had ceased and (ii) to end before the decline in sex ratios starting in 1973.

It was known that sex ratios remained roughly stable across these two decades: but there was no prior ground for supposing them to have sub-binomial variance. The overall sex ratio was 0.5144, with a variance of 0.00000023. The average number of births per year was 718 317. Using the method above, the expected binomial variance was 0.00000033036 and the estimated Lexis variance {sigma}2P was estimated at 0.0721 for these data.

In short, five sets of data on series (or collections) of sex ratios show sub-binomial variance. Estimates of {sigma}2P have been made in respect of two of them and they seem anomalously large in contrast with the usually accepted estimate of Lexis variance (0.0025) across couples. This point will be elaborated on later.

Correlations between the sexes within sibships
Two studies (Maconochie and Roman, 1997Go; Jacobsen et al., 1999Go) concluded from their analyses (of >0.5x106 births and 800 000 births respectively) that there is no appreciable correlation between the sexes within sibships.


    The inference
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 Abstract
 Introduction
 Theoretical background
 Previous evidence on Markov,...
 History of the estimation...
 Data which impugn the...
 The inference
 Estimating chaotic Poisson and...
 References
 

(i)The variances of series of sex ratios of randomly chosen births are substantially sub-binomial. Hence P, the probability of a male, must vary within such samples.
(ii)Hence there is either Lexis variation of P (between couples) or Poisson variation of P (within couples) or both. (The possibility of Markov variation is discounted for reasons given above.)
(iii)But there cannot be only Lexis variation (because if there were, there would be a positive correlation between the sexes within sibships).
(iv)Moreover, there cannot be only Poisson variation (because if there were, there would be a negative correlation between the sexes of sibs within sibships).
(v)Hence there must be both Poisson variation within sibships and Lexis variation across sibships of P.

I have adduced very substantial quantities of data to support the hypothesis that the sexes of mammalian (including human) offspring are partially dependent on the hormone concentrations of both parents around the time of conception (James, 1996Go, 1999bGo). So it is worth pointing out that both these forms of variation (Lexis and chaotic Poisson) would be expected if this hypothesis were true.

This is so because these hormone levels vary substantially between couples (thus ex hypothesi causing the Lexis variation) and within couples (e.g. Kemper, 1990) (thus ex hypothesi causing the chaotic Poisson variation).


    Estimating chaotic Poisson and Lexis variation
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 Abstract
 Introduction
 Theoretical background
 Previous evidence on Markov,...
 History of the estimation...
 Data which impugn the...
 The inference
 Estimating chaotic Poisson and...
 References
 
In view of the lack of persuasive evidence for the existence of Markov variation, a method of estimation will be offered on the assumption that it does not exist. If a biological (e.g. immunological) mechanism were to be discovered, that might imply it: or if it were to be plausibly described in other mammalian species, then the following treatment would need revision.

As a first approach to the estimation process, note that the variance of the binomial is nPQ, and that in the Lexis binomial, this value is augmented by n(n – 1){sigma}2b, while in the Poisson binomial, this value is diminished by n{sigma}2w, where n is the sample (sibship) size, P is the probability of a boy, Q = 1 – P, {sigma}2b is the variance of P between couples, and {sigma}2w is the variance of P within couples (Weatherburn, 1949Go).

When P is distributed binomially, the variances are nPQ and the correlations between the sexes at different birth ranks will be expected to be zero. If both Poisson and Lexis variation are present and roughly counterbalance such that the variances are roughly nPQ, the resulting sequences of the sexes would be indistinguishable from those generated by binomial sampling (as in Table IIGo) and the correlations between the sexes will be roughly zero. This, it is suggested, underlies the above-mentioned failures to find significant correlations between the sexes in very large samples of sibships.

If that were so, the two values above would be roughly equal, viz.

Lastly, let W be the value by which the variance of a set of sex ratios falls short of the binomial. Then , whence both may be evaluated.

There is a difficulty. It is not clear how to choose the series of random samples of sex ratios from which W is to be estimated. Annual series of population sex ratios at birth show slow, significant, unexplained, secular movements (Gini, 1955Go). Sex ratio has also sometimes been reported to show seasonal variation too (James, 1987b, p. 723). Such phenomena would cause W to be underestimated. So one might wonder whether the annual (roughly) 600 000 births in England and Wales could be arbitrarily formed into, say, 600 samples of 1000, whence W could be estimated. It is not clear whether such estimates of W would be stable. The data presented above (in variances of series of sex ratios) suggest that W is positive and large compared with the value of 0.0025; but it would be interesting to get other estimates of W from other sources.

Lastly, it is worth emphasizing that this note is provisional and raises more questions than it answers. It suggests that something is seriously wrong with the accepted formulation of the variation of P within and across couples. It is not clear whether that is because W has been overestimated here. So there is an urgent need for fresh estimates of W. Meanwhile if the present formulation were substantially correct, it might suggest a genetic explanation for the mysterious but significant secular meanderings of population sex ratios described previously (Gini, 1955Go). Such an explanation would require a substantial genetic component of the Lexis variation in P (Bodmer and Edwards, 1960Go).


    References
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 Abstract
 Introduction
 Theoretical background
 Previous evidence on Markov,...
 History of the estimation...
 Data which impugn the...
 The inference
 Estimating chaotic Poisson and...
 References
 
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Submitted on October 28, 1999; accepted on January 14, 2000.