Is sex at birth a biological coin toss? Insights from a longitudinal and GWAS analysis

Abstract

Some families consistently have offspring of only one sex, raising questions about whether sex at birth is truly random. This study investigated whether offspring sex follows a simple binomial distribution within families and identified maternal factors associated with unisexual sibships. We analyzed 58,007 US women with two or more singleton live births (146,064 pregnancies, 1956–2015). Offspring sex followed a beta-binomial rather than a simple binomial distribution, indicating that each family may have a unique probability of male or female births, akin to a weighted coin toss. Deviations from simple binomial distribution were more pronounced when we excluded each woman’s last birth to reduce the influence of sex-based stopping behavior. After excluding the last birth, older maternal age at first birth was associated with higher odds of having offspring of only one sex. A genome-wide association study identified maternal SNPs linked to having female-only (NSUN6) and male-only (TSHZ1) offspring. Our findings suggest maternal factors influence offspring sex distributions.

Offspring sex may not be random—maternal age and genetics influence sex clustering within families.

INTRODUCTION

Sex at conception, determined by the sex chromosomes in the sperm, has long been considered a textbook example of a simple binomial distribution, implying that each fertilization event is random and independent, like a coin toss (1). This understanding is rooted in meiosis, during which sperm are produced in equal numbers containing either an X or a Y chromosome (2–6). Several coauthors, however, observed cases of friends, colleagues, first-degree relatives, or themselves that produce offspring of only one sex (offspring N range = 2 to 9), raising questions about chance. There is a well-established evolutionary biology theory suggesting that heritable traits, both genetic and environmental, may affect the sex determination process to pass on characteristics that improve offspring reproductive fitness (7). For example, some have reported that good-looking parents are more likely to have daughters; whereas wealthy, aggressive, big, and tall parents are more likely to have sons (8–12). Others have suggested that maternal factors such as vaginal pH, temperature, length of the follicular phase, or the menstrual cycle phase during conception may affect viability of sperm with X or Y chromosome differently (13–16). However, none of these hypotheses have been confirmed in large epidemiological studies, and genetic factors related to offspring sex remain underexplored (17, 18).

The choice of analytical unit may be critical in addressing research questions regarding offspring sex (19). Previous research has identified paternal alleles associated with the sex of their offspring (20), but a meta-analysis of genome-wide association studies (GWAS) failed to find autosomal variants associated with the sex of individuals (i.e., the offspring themselves) (21). Therefore, we hypothesized that genetic or individual determinants of offspring sex can only be identified when investigated within families. In other words, in the general population, there are factors that predispose each couple to producing offspring of a certain sex. However, the distribution of these factors is essentially random, which may cause researchers to miss patterns of sex clustering within families if the study design does not clearly define the analytical units (22).

Using data from two large ongoing cohort studies in the United States, the Nurses’ Health Study II (NHSII) and Nurses’ Health Study 3 (NHS3), we aimed to answer two questions at the maternal level: (i) whether sex at birth follows a simple binomial distribution or beta-binomial distribution, and (ii) whether there are highly heritable traits (e.g., natural hair color, ABO blood type, height, and chronotype) and/or reproductive factors (e.g., age at menarche and age at first birth) associated with having offspring of a single sex. We also conducted a GWAS to determine if any genetic loci were associated with having offspring of a single sex. Of note, this study includes data solely from the mother, and the term “family” is used only in a contextual sense to describe the grouping of offspring based on maternal data.

RESULTS

Participant characteristics

The 58,007 women included in the analysis were 95.4% (n = 55,333) white (fig. S1), with a mean (SD) age of 25.5 (4.4) at first birth, and a total number of 2.5 (0.8) births. The range of sibship size was from 2 to 12, although less than 0.5% women had more than 5 children and less than 0.02% had 10 or more children. The overall male-to-female sex ratio by each sibship size ranged from 1.05 to 1.09, and the combined sex ratio was 1.08 (Table 1).

Table 1. Distribution of offspring sex at birth by sibship size, NSHII and NSH3 (offspring year of birth: 1956–2015).

Sibship size	N (%) of mother	N of male offspring	N of female offspring	Male/female ratio
2	35,221 (60.7)	36,430	34,012	1.07
3	17,148 (29.6)	26,839	24,605	1.09
4	4,469 (7.7)	9,274	8,602	1.08
5	891 (1.5)	2,284	2,171	1.05
>5	278 (0.5)	959	888	1.08
Total	58,007 (100.0)	75,786	70,278	1.08

Among participants in our study population, sibship size steadily declined across maternal birth cohorts (table S1, with comparable final number of sibship size regardless of menopausal status), reflecting the broader national trend toward smaller family sizes. The male-to-female sex ratio at birth showed slight and gradual declines over the study period, with values fluctuating between 1.03 and 1.13 (fig. S2). This trend is consistent with historical data from the US general population, supporting the representativeness of our cohort for assessing sex ratio patterns over time.

Sex clustering compared to binomial and beta-binomial distribution at the maternal level

We observed that a balanced offspring sex [i.e., FM (female-male) or MF] was the most frequent family composition in sibship size of 2, but a clustering of single sex (e.g., MMM and FFFFF) was generally more frequent in sibship size 3 or larger (Fig. 1). When comparing the observed distribution to an emulated simple binomial distribution with P_boy = 0.519, the combined sex ratio at birth in the study population (Fig. 2 and fig. S3), we found that single-sex families were underrepresented in sibship size of 2 but overrepresented in sibship sizes of 3, 4, and 5. Moreover, the magnitude of relative deviation from expectation increased with sibship size. Chi-square goodness-of-fit test suggested a statistically significant deviation from a simple binomial distribution for all sibship sizes (all P < 0.001). The beta-binomial distribution provided better fitness to the data compared to a binomial distribution, both in fitness statistics and in magnitudes of relative deviations (table S2 and figs. S4 and S5, all chi-square P > 0.05).

Fig. 1. Distribution of offspring sex sequencing by sibship size (range: 2 to 5), in descending order by number and percentage of mothers within each sibship size.

Fig. 2. Relative deviation of offspring sex distribution from binomial expectation by sibship size (range: 2 to 5).

Note that expected distribution was calculated as a simple binomial distribution with a probability of having a male offspring = 0.519 at any single birth, according to the male/female ratio in the study population. Relative deviation = (observed counts − expected counts)/expected counts. All chi-square tests P < 0.001.

Sensitivity analysis

We used several methods to reduce the influence of parental decision-making for having both sexes among their offspring (termed as “coupon collection” henceforth) (1), either because of the desire for a certain sex or a balanced-sex family or both (23, 24). In such cases, the last-born child’s sex is dependent on the previous births but not vice versa. Therefore, in two separate analyses, we (i) excluded obvious “coupon collectors” (i.e., women who only stopped producing offspring after both offspring sexes are reached) and (ii) more conservatively excluded the last birth of every woman. We also conducted several additional sensitivity analyses. First, we tried to account for other factors that run in the family by excluding women who had a history of divorce or miscarriage. Second, we also conducted the binomial distribution comparison according to the sex ratio at birth in the US (P_boy = 0.512), instead of the ratio in the study population (P_boy = 0.519) (25). Third, we restricted analysis to postmenopausal women at the time of reproductive history assessment, who would have completed their reproductive years. We observed the same pattern of deviation from simple binomial distribution as in the primary analysis in sensitivity analyses accounting for coupon collection behaviors, excluding women with a history of divorce, excluding women with a history of miscarriage, using sex ratio at birth in the US, and restricting analyses to postmenopausal women (figs. S6 to S11).

Within-mother sex clustering disregards sibship size

To evaluate the distribution of offspring sex in the entire study population (irrespective of sibship size), we combined all women into the analysis and assumed they all stopped having children after the second child. The observed counts were almost identical to the expected counts for a simple binomial distribution (chi-square P = 0.79, relative deviation range = 0.00 to 0.00).

Risk factors for within-mother sex clustering

To reduce influence of parental preference of offspring sex, we focused on analyses excluding the last birth from all women (Table 2 and Fig. 3). Older age at first birth was associated with higher odds of producing only males or only females [>28 versus <23 years: risk difference = 0.07; odds ratio (OR) = 1.13, 95% confidence interval (CI): 1.04, 1.24; P trend = 0.002]. The associations were similar when we included the last birth from all women (table S3). None of the other factors examined [race, natural hair color, ABO blood type, height, and body mass index (BMI) at age 18 years] were associated with having offspring of a single sex (Table 2 and table S3).

Table 2. Adjusted ORs and 95% CIs of having offspring of only one sex (versus having offspring of both sexes) according to maternal characteristics, excluding the last birth of all women.

Note that numbers may not add up to 100% due to missingness. BMI at age 18 years, chronotype, and blood type were only assessed among NHSII participants. P value < 0.05/8 = 0.00625 was considered as statistically significant after adjusting for multiple comparisons. P trend analyses were conducted treating categorical variables as continuous. Multivariable models adjusted for sibship (continuous), ethnicity (white, non-white), and cohort (NHSII, NHS3). Ref, reference.

Characteristics		Multivariable model
		Mixed sex	Single sex*	Male only†	Female only†
		(n = 11,845)	(n = 10,941)	(n = 5,971)	(n = 4,970)
	N total	OR (95% CI)
Race
White	21,776	Ref	Ref	Ref	Ref
Non-white	1,010	Ref	0.96 (0.84, 1.09)	0.95 (0.81, 1.12)	0.97 (0.82, 1.14)
Age at first birth (years)
<23	6,728	Ref	Ref	Ref	Ref
23–25	7,037	Ref	1.07 (1.00, 1.15)	1.11 (1.02, 1.21)	1.02 (0.93, 1.12)
26–28	5,541	Ref	1.11 (1.03, 1.19)	1.13 (1.04, 1.24)	1.07 (0.98, 1.18)
>28	3,342	Ref	1.13 (1.04, 1.24)	1.11 (1.00, 1.23)	1.16 (1.05, 1.30)
P trend			0.002	0.02	0.004
Age at menarche (years)	21,377	Ref	1.01 (0.99, 1.03)	1.01 (0.98, 1.03)	1.01 (0.98, 1.03)
P trend			0.55	0.66	0.58
Height, per inch (2.54 cm)	22,665	Ref	0.87 (0.79, 0.96)	0.86 (0.77, 0.96)	0.88 (0.79, 1.00)
P trend			0.004	0.007	0.04
BMI at 18 years old
<18.5	540	Ref	1.07 (0.99, 1.16)	1.05 (0.95, 1.15)	1.11 (1.00, 1.22)
18.5–24.9	12,852	Ref	Ref	Ref	Ref
25–29.9	3,443	Ref	1.07 (0.96, 1.19)	1.06 (0.94, 1.21)	1.07 (0.94, 1.23)
≥30	1,744	Ref	1.02 (0.83, 1.26)	1.18 (0.94, 1.50)	0.83 (0.63, 1.10)
P trend			0.58	0.60	0.12
Natural hair color at 20 years old
Red	825	Ref	1.03 (0.89, 1.20)	1.03 (0.86, 1.22)	1.04 (0.87, 1.26)
Blonde	3,451	Ref	1.03 (0.95, 1.12)	1.01 (0.92, 1.11)	1.06 (0.96, 1.18)
Light brown	8,144	Ref	Ref	Ref	Ref
Dark brown	8,273	Ref	0.99 (0.93, 1.06)	0.95 (0.88, 1.03)	1.04 (0.96, 1.13)
Black	619	Ref	1.03 (0.85, 1.24)	1.00 (0.80, 1.25)	1.06 (0.84, 1.34)
Chronotype
Definitely a morning person	5,302	Ref	Ref	Ref	Ref
More of a morning person	4,332	Ref	1.04 (0.96, 1.13)	1.07 (0.97, 1.17)	1.01 (0.91, 1.12)
More of an evening person	3,920	Ref	0.91 (0.83, 0.99)	0.93 (0.84, 1.03)	0.88 (0.79, 0.98)
Definitely an evening type	1,769	Ref	1.09 (0.97, 1.21)	1.12 (0.98, 1.27)	1.05 (0.91, 1.21)
Neither	820	Ref	1.05 (0.91, 1.23)	1.13 (0.95, 1.35)	0.96 (0.79, 1.17)
ABO blood type
A	6,171	Ref	Ref	Ref	Ref
B	1,994	Ref	0.97 (0.87, 1.07)	0.94 (0.83, 1.07)	1.00 (0.87, 1.14)
AB	801	Ref	1.04 (0.89, 1.21)	1.05 (0.88, 1.25)	1.02 (0.85, 1.24)
O	6,978	Ref	0.98 (0.91, 1.05)	0.97 (0.90, 1.06)	0.98 (0.90, 1.07)
Unknown	2,635	Ref	1.00 (0.91, 1.10)	1.01 (0.90, 1.13)	0.99 (0.88, 1.12)

*Results from a binary logistic regression comparing families with offspring of only one sex to those with both sexes.

†Results from a multinomial logistic regression comparing male-only and female-only offspring separately to those with both sexes.

Fig. 3. Manhattan plot showing results of GWAS of having offspring of only one sex (versus having offspring of both sexes), excluding the last birth of all women.

Note that GWAS analysis was only conducted among NHSII participants with genome information. (A) Having offspring of only one sex (versus having offspring of both sexes), N = 2933. (B) Having offspring of only females (versus having offspring of both sexes), N = 2015. (C) Having offspring of only males (versus having offspring of both sexes), N = 2135. Multivariable models adjusted for sibship, ethnicity, and top four principal components.

After excluding the last birth, GWAS identified seven single-nucleotide polymorphisms (SNPs) associated with having children of only one sex at P < 1 × 10⁻⁵ (λ = 1.000; Fig. 3 and table S4), mainly in the gene CYP2U1 (chr4: 107947570 to 107949568). One SNP in chromosome 10 (rs58090855, in NSUN6) was significantly associated with having only female offspring (P = 2.7 × 10⁻⁸; table S5). This analysis also identified one SNP (rs1506275, chr18: 75553217, near TSHZ1; table S6) associated with having only male offspring (P = 4.6 × 10⁻⁸). When all births were included, no loci reached genome-wide significance (λ = 1.007; fig. S12 and tables S7 to S9).

So, will it be a boy or a girl?

We calculated the conditional probabilities of the next birth being a boy or a girl, given that previous births were of the same sex. Using the fitted beta-binomial distribution, the predicted conditional probability for the sex of the next child being the same as the previous ones increased with sibship size (Fig. 4). Notably, in families with three boys (MMM), the probability of having another boy was 61%; in families with three girls (FFF), the probability of having another girl was 58%. Results were comparable when we used the observed counts in our dataset (fig. S13), although the statistical power was limited for families with more than three children.

Fig. 4. Conditional probability of the sex of the next birth in families with offspring of only one sex.

Note that conditional probability was calculated on the basis of the fitted beta-binomial distribution after excluding the last birth from all women. For example, the conditional probability of the next birth being a boy for a beta-binomial distribution is given by $(\text{the} n+1 \text{birth is} \mathrm{a} \text{boy})=\frac{\mathrm{\alpha }+k}{\mathrm{\alpha }+\mathrm{\beta }+n}$ , where n is the current sibship size, and k is the current number of boys in the family.

DISCUSSION

We identified a deviation from a simple binomial distribution of sex at birth, with the effect more prominent in larger families. The deviations remained significant in several sensitivity analyses to account for coupon collection behaviors. The sex ratio data were compatible with a beta-binomial distribution in our study sample, suggesting that P_boy was not constant in the entire population but instead varies from mother to mother (or couple to couple). Older maternal age may be associated with higher odds of having single-sex offspring, but other heritable, demographic, and/or reproductive factors were unrelated to offspring sex. Further, GWAS suggested that there may also be maternal biological factors influencing the distribution of sex at birth within families.

The human sex ratio has long been of interest of biologists, statisticians, demographers, sociologists, and economists (23, 26). Here, we showed that within each sibship size, sex at birth did not conform with a simple binomial distribution and identified a significant intramother correlation in offspring sex. Several groups analyzing European datasets (N family range = 70,000 to 5 million) have also found a deviation from binomial distribution and correlation between successive births, which increases with sibship size (27–30). On the other hand, our study suggested that the notion of offspring sex determination is a random process that may also be true in the context of the whole population, if the possibility of maternal/familial clustering is ignored. The similar overall sex ratio across all sibship sizes and the exceptionally good fit of a simple binomial model to the first two children in all families validate the seemingly independence of each birth on a population scale. Prior research that pooled data at the offspring level but ignored potential maternal clustering found no evidence of a genetic predisposition for sex determination, and studies investigating predictors of population sex ratios have not generated consistent findings (26, 30, 31).

There has been an ongoing debate about whether variations of sex ratio at birth are genetic or facultative (e.g., coupon collection behaviors) (29). While it is challenging to disentangle these factors, as they may negate each other’s effects, our study suggests that both may play a role. For families of sibship size of 2, the overrepresentation of MF/FM families suggest that couples are more inclined to stop reproducing when a balanced sex was reached, as has been previously reported (27, 30, 32, 33). Furthermore, the deviations were stronger when we excluded coupon collectors, suggesting that some humans may behave to counteract their genetic propensity to having offspring of only one sex. Historical datasets also provide insights from periods before the development of family planning methods and medical advances which have made sex selection possible. These datasets might also represent periods before the evolving preference for children of both sexes in Western societies, influenced by factors such as gender equity, changing social roles, and improved economic circumstances (23, 24, 29, 34). An analysis of 332,788 UK Biobank families (children born between 1940 and 1970) showed a coupon collection behavior which explained an underrepresentation of single-sex offspring families (35). In contrast, an overrepresentation of single-sex families was observed among more than 200,000 Dutch families with children born between 1600 and 1939 (35). Therefore, in our study, the fitness of beta-binomial rather than binomial distribution and the within-mother correlation conditioned on sibship size were probably due to two main reasons: (i) the existence of women at high risk of producing offspring of a certain sex (e.g., due to factors affecting sperm survival in a Y chromosome–specific manner) and (ii) procreation behavior strongly influenced by the sex of existing children (34, 36, 37), and this behavior may be particularly strong among families with high biological risk for producing offspring of only one sex.

We found that older maternal age at first birth may be a risk factor for repeatedly giving birth to children of only one sex, even after adjustment for sibship size. Maternal aging during the reproductive years is associated with several physiological changes, such as a shorter follicular phase and lower vaginal pH (38–41). A shorter follicular phase tends to favor Y chromosome survival, whereas a more acidic vaginal environment favors X chromosome survival (14, 16). Each woman may have a different predisposition to each of these factors as they age, which could lead to a higher probability of consistently producing same-sex offspring. However, these mechanisms remain speculative, and more detailed data are required to confirm these hypotheses.

Recent GWAS have explored the genetic underpinnings of reproductive behaviors and success, including age at first birth (42–45). However, none of these studies have investigated SNPs associated with sex clustering within families. Notably, the maternal SNPs identified in our study, which are associated with having offspring of only one sex, did not replicate those previously identified as linked to age at first birth, the strongest reproductive risk factor for having single-sex offspring in our study (42–45). In addition, these SNPs had not been previously reported as being implicated in reproductive traits, such as age at menarche, menopause, or fertility. Therefore, the genetic associations we observed may not be mediated by age at first birth but by other mechanisms, such as hormone regulation, mate selection behaviors, or family composition preferences (42–47). Future research is essential to replicate our findings, explore new risk factors (e.g., lifestyle, nutritional status, and exposure to environmental chemicals), and investigate the possibility of gene-gene/gene-environment interactions.

Strengths and limitations

The strengths of our study include the large sample size, detailed within-family information about birth order and offspring sex, data on infertility treatments, and extensive covariates. This comprehensive dataset allowed us to investigate a wide range of factors associated with human reproduction. We were able to conduct a GWAS analysis that provides insights about the maternal genetic determinants of sex clustering within families.

There are several limitations of the study. First, we did not collect information about biological fathers. Although results were similar when we excluded women with a history of divorce, it is likely that there are paternal factors that we failed to include (e.g., paternal age and identity). However, it is even more fascinating that we found maternal risk factors for offspring sex beyond the paternal (i.e., sperm) contribution, which underscores the complexity of factors influencing offspring sex. Second, the study population is predominantly white women (95%) residing in the United States. Since sex preferences and reproductive behaviors vary across cultures, religions, or countries (34), the sex ratio distribution pattern observed in our study may not apply to other societies. We also do not know whether our study population, composed exclusively of nurses, is predisposed to having a unisex brood due to their occupational exposures. Third, our data indicated that sibship size is informative, thus the definition of complete reproductive history is important. Despite most of our study participants being past their reproductive age, the inference of our findings is limited because expanding sibship size is constrained by the short reproductive span of human beings and the decline in birth rates over the past decades. Fourth, as with other large databases, we were not able to identify half-siblings. However, these events are relatively rare and thus unlikely to meaningfully bias the result. Last, although GWAS analyses were pooled across four different genotyping platforms using an inverse-variance weighted random-effects meta-analysis approach, we cannot rule out the possibility of residual confounding or platform-specific heterogeneity that may not have been fully accounted for.

Conclusions

In summary, we found that sex at birth did not follow a simple binomial distribution when women, rather than pregnancies, was the unit of analysis. Both biological factors and reproductive decisions to delay childbearing may contribute to the observed phenotype of sex clustering within families. Future research is warranted to study the extent to which each of these factors explains the sex clustering within families. Until then, families desiring offspring of more than one sex who have already had two or three children of the same sex should be aware that when trying for their next one, they are probably doing a coin toss with a two-headed coin.

MATERIALS AND METHODS

Study design

The NHSII was established in 1989, when 116,429 nurses aged 25 to 42 years living in 1 of 14 US states were enrolled. Participants were asked to report their demographic, reproductive, and medical information on biennial follow-up questionnaires. The NHS3 is a web-based open cohort launched in 2010 that includes nurses born after 1 January 1965 from throughout the US and Canada. The follow-up schedule is every 6 months, and more than 50,000 participants have been enrolled. We used data from NHS3 participants enrolled 2010–2021. The study was approved by the institutional review boards of Brigham and Women’s Hospital and Harvard T.H. Chan School of Public Health. Return of complete questionnaire implied consent.

Assessment of pregnancy history

In NHSII, lifetime pregnancy history was reported on the 2001 and 2009 questionnaires. History of infertility and infertility treatment was first asked in 1989 and updated every 2 to 4 years till 2009. In NHS3, lifetime pregnancy history and infertility treatment associated with each pregnancy were reported at cohort enrollment. NHSII participants were likely to have completed their reproductive years at the time of pregnancy history assessment (97% over age 45 years) but not for most of the NHS3 participants (7% over age 45 years). Birth order was derived from the reported year of delivery. The validity of self-reported reproductive events has been found to be high in our cohorts and others (48–52).

All gravid women were eligible for the study. We excluded participants who had any missing information about offspring birth sex and birth year because the birth order could not be determined. We also excluded pregnancies that were not singleton live births (e.g., miscarriage, stillbirths, and twins) and women who had a history of infertility treatment because of their known relation to sex ratio variation (fig. S1) (53–55). Last, we excluded women who had only one live birth because they cannot contribute offspring clusters to the analysis. The final analytical sample included 58,007 women and their 146,064 offspring born between 1956 and 2015.

Covariates

At enrollment in NHSII, participants self-reported birthday, race, weight at age 18 years, height, and age at menarche. BMI was calculated as weight/height × height (kilograms per square meter). Natural hair color at age 18 years (response options: red, blonde, light brown, dark brown, and black) was reported on the 1991 questionnaire. Chronotype (response options: “definitely a morning type,” “more of a morning type,” “more of an evening type,” “definitely an evening type,” or neither), blood type (A, B, AB, O, or unknown), and menopausal status were asked in 2009. In NHS3, birthday, race, weight at age 18 years, height, marital status, natural hair color at age 20, and menopausal status were self-reported at cohort enrollment. Age at menarche was collected at the second follow-up questionnaire.

Statistical analysis

We assessed secular trends in sibship size and the male-to-female sex ratio at birth across the study period. Sibship size trends were evaluated on the basis of the total number of live births per mother, grouped by maternal birth cohort in 5-year intervals. We included maternal birth cohorts with more than 100 women (mothers born between 1946 and 1990). The male-to-female sex ratio was calculated as the number of male live births divided by the number of female live births for each calendar year. To minimize random fluctuations, we restricted the sex ratio analysis to calendar years with more than 100 recorded births (offspring born between 1965 and 2013) and applied a 3-year moving average (current year and two preceding years). Observed trends were compared with national patterns reported in the US National Vital Statistics Reports (56).

We compared the distribution of offspring to a binomial distribution (P_boy = 0.519, the combined sex ratio at birth in the study population). We stratified the analyses by sibship size (2, 3, 4, and 5), with an additional analysis combining all mothers together assuming that they all stopped reproducing after having the second child. The relative deviation from a binomial distribution was calculated as (observed count − expected count)/expected count. We also compared the fitness of a beta-binomial distribution model where higher-than-expected sex clustering was observed. The beta-binomial distribution estimated a different P_boy for each mother and was supposed to account for potential overdispersion of sex ratio data (57). Model fitness was evaluated by Akaike information criterion, Bayesian information criterion, and chi-square goodness-of-fit test (58).

To identify potential risk factors for producing offspring of a single sex, we fit logistic or multinomial logistic regression models with several highly heritable and reproductive factors each as the independent variable to estimate the OR and 95% CIs of having single-sex offspring compared to having mixed-sex offspring, adjusting for race, cohort, and sibship size. In addition, among 7530 NHSII participants with genome data (59), we performed a GWAS using logistic regression modeling SNPs as ordinal variables and having same-sex offspring as the outcome, adjusting for race, sibship size, and the top four principal components to account for the population structure. Samples for NHSII participants were genotyped on four different platforms, including HumanCoreExome, Illumina HumanHap, OncoArray, and Global Screening Array (59). These genotype data were imputed using 1000GP phase 3 version 5 as a reference panel. Results for either rare variants with minor allele frequencies <20/(2 × number of cases) or variants with poor imputation quality (coefficient of determination < 0.3) were removed from each association result. A random-effects meta-analysis was carried out by running METAL to sequentially process filtered results from different platforms (60).

Last, we calculated the conditional probabilities of the next birth being a boy or a girl, given that previous births were of the same sex. For example, the conditional probability of the next birth being a boy for a beta-binomial distribution is given by $P (\text{the} n+1 \text{birth is} \mathrm{a} \text{boy})=\frac{\mathrm{\alpha }+k}{\mathrm{\alpha }+\mathrm{\beta }+n}$ , where n is the current sibship size, and k is the current number of boys in the family. We also calculated the conditional probabilities using the observed counts in the dataset by including all families with a sibship size of >n.

Analyses were performed using SAS (9.4 for UNIX) and R (version 4.2.2). All statistical tests were two-sided, and a P value of <0.05 was considered statistically significant. For the exploratory analyses of eight risk factors for within-mother sex clustering, we adjusted for multiple comparisons using Bonferroni correction (P < 0.05/8 = 0.00625). Genome-wide significance was considered as P < 5 × 10⁻⁸ (61).

Supplementary Materials

The PDF file includes:

Figs. S1 to S13

Tables S1 to S3

Legends for tables S4 to S9

Other Supplementary Material for this manuscript includes the following:

Tables S4 to S9