Responding to a 100-Year-Old Challenge from Fisher: A Biometrical Analysis of Adult Height in the NLSY Data Using Only Cousin Pairs

Abstract

In 1918, Fisher suggested that his research team had consistently found inflated cousin correlations. He also commented that because a cousin sample with minimal selection bias was not available the cause of the inflation could not be addressed, leaving this inflation as a challenge still to be solved. In the National Longitudinal Survey of Youth (the NLSY79, the NLSY97, and the NLSY-Children/Young Adult datasets), there are thousands of available cousin pairs. Those in the NLSYC/YA are obtained approximately without selection. In this paper, we address Fisher’s challenge using these data. Further, we also evaluate the possibility of fitting ACE models using only cousin pairs, including full cousins, half-cousins, and quarter-cousins. To have any chance at success in such a restricted kinship domain requires an available and highly-reliable phenotype; we use adult height in our analysis. Results provide a possible answer to Fisher’s challenge, and demonstrate the potential for using cousin pairs in a stand-alone analysis (as well as in combination with other biometrical designs).

This paper has three goals, which converge. The first goal is to resurrect, and discuss, a 100-year-old challenge suggested by Fisher (1918), to explain why cousin correlations may be broadly inflated. The second goal is to document the existence of a remarkably large cousin database from the National Longitudinal Survey of Youth, which may have the potential to support a number of interesting studies from a behavior genetic perspective (using cousins exclusively, or especially when they are combined with twin and sibling data from the same data source). The third goal is to study adult height in a biometrical analysis that uses only cousin data. These three different goals will be combined into an interesting modern puzzle in the last section of the paper, one that resurrects and expands Fisher’s original challenge.

Responding to Fisher’s Challenge

Fisher (1918, p. 433) noted: “… the hypothesis of cumulative Mendelian factors seems to fit the facts very accurately. The only marked discrepancy from existing published work lies in the correlation for first cousins. … but until we have a record of complete cousinships measured accurately and without selection, it will not be possible to obtain satisfactory numerical evidence on this question.” Fisher referenced Snow, and Miss Elderton, as separately finding surprisingly high cousin correlations.

His 1918 monograph focuses on three different phenotypes, “stature,” “span,” and “cubit,” with particular attention to the first. Of course “stature” references height, a particular focus of the current paper. In his reference to Snow’s findings, he states that Snow’s cousin correlations -- presumably on those three traits, or perhaps a subset, and with a genetic coefficient of R=. 125-- were as high as an uncle-niece/nephew correlation, with a genetic coefficient of R=.25. (Note that the genetic coefficient – also called the “coefficients of relationship” – is an estimate of degree of consanguinity, or genetic relatedness, among two individuals. Originally defined by Wright, 1922, R estimates the average number of segregating genes that are shared by individuals of a given relatedness, assuming random mating.) Fisher stated, furthermore and without elaboration, that Snow’s correlations were possibly in error. In the case of Miss Elderton’s findings, he noted that the correlation was “certainly extremely high” (p. 433), and again the phenotypes he was referencing were likely all or some of the stature, span, and cubit measures on which he focused throughout his paper. Earlier in the paper he provided more detail about Miss Elderton’s findings. The goal of his paper was to derive estimates for variance components for many different levels of genetic relatedness under a Mendelian model. Concerning his own derived estimates for cousin correlations under a purely Mendelian model, Fisher noted that “Certainly they do not approach some of the values found by Miss Elderton in her memoir on the resemblance of first cousins … . Series are there found to give correlations over .5, and the mean correlation for the measured features is .336. From special considerations this is reduced to .270, but if the similarity of first cousins is due to inheritance, it must certainly be less than that between uncle and nephew. No theory of inheritance could make the correlation for cousins larger than or even so large as that for the nearer relationship” (p. 427). Although many of Fisher’s references are vague, and we would wish for more details about specific phenotypes and the “special considerations” that are referenced, it is clear that Miss Elderton’s cousin correlations were consistently large. If we assume these kinship correlations apply to heights (the most heritable of the three phenotypes on which Fisher focused), then these results virtually match (or may slightly exceed) half-sibling correlations and, as Fisher noted, even exceed “avuncular correlations.”

Fisher’s goals had nothing to do with environmental variance. The title of his paper noted “the supposition of Mendelian inheritance.” Obviously, if he had expected meaningful and significant environmental influence on height – that is, if he thought that environment influences actually acted on height -- he should not have been so surprised by first cousin phenotypic correlations as high as r=.27. But knowing that the genetic coefficient for first cousins was R=.125, and failing to recognize the potential for other (e.g., environmental) influences, caused Fisher to view r=.27 as a surprising and substantially inflated value as an observed phenotypic cousin correlation. His only explicit effort to account for the apparent inflation considered whether the large correlations for first cousins could be due to the effect of Mendelian dominance. That particular interpretation was rejected, as he derived theoretically that first cousins cannot have a dominance component (though “double first cousins” – those related by two parental sibling relationships, rather than just one, would be expected to have inflated cousin correlations due to dominance). We are not aware of additional treatment of Fisher’s interesting comments on the inflation of first cousin correlations. Obviously, Miss Elderton – with whom Fisher published a number of papers – had done work on cousins prior to Fisher’s (1918) monograph, but there did not seem to be any satisfactory resolution for why these correlations were higher than his estimates for genetic correlations could accommodate. Those century-old findings set the stage for more modern work, and for the current paper.

In 2005, after doing biometrical work with the National Longitudinal Survey of Youth (NLSY) for many years, the first author organized a small conference in Philadelphia at which he noted (without knowledge of the above quote from Fisher) that several past NLSY findings included surprisingly large cousin correlations. The last author, who was attending the conference, brought the Fisher quote forward for discussion, and it has informed our NLSY biometrical work ever since. For example, Rodgers et al (2016, p. 540) stated: “… our research team has observed several times that the cousin correlations obtained from the NLSY kinship links appear unusually high (or sometimes we have felt that the half-sibling correlations were surprisingly low, thus leading to the appearance of high cousin correlations).” Following the statement above is the following (p. 540): “The NLSY data provide two different generations of cousin data, and the Gen2 cousins may be as close as can ever be obtained to a large cousin sample, emerging from probability sampling mechanisms and providing, in Fisher’s words, ‘a record of complete cousinships measured accurately and without selection.” Up to now, Fisher’s challenge appears to be largely untreated.

However, there exist cousin studies that have been published in the psychology (and related) literature (though without attention to explaining inflated cousin correlations). The use of cousins in research settings began many years ago, in designs that took advantage of comparing cousin pairs to both other cousin pairs and to non-cousin kinship categories. Rollins (1929), a decade after Fisher’s comments, used select New England college graduates matched to non-college-graduate cousins in a cousin-comparison design to study fertility, and concluded “The college graduates have consistently had less children than their cousins who did not go to an institution of higher learning” (p. 535). In more recent designs, mothers who are twins or siblings have been compared in terms of the differential outcomes associated with their offspring; these offspring, obviously, are cousins (see, among many other studies: D’Onofrio et al, 2007; D’Onofrio et al, 2009; Garrison & Rodgers, 2016; Geronimus, Korenman, & Hillemeier, 1994; Hadd & Rodgers, 2017; Jaffee, van Hulle, & Rodgers, 2011; Lahey & D’Onofrio, 2010; Turley, 2003). Coyne (2014) used the NLSY in a cousin-comparison design to study the risk factors that underlie teenage pregnancy and childbearing. Goodnight et al (2012) used a cousin comparison design in the NLSY to study the relation between neighborhood risk and adolescent conduct problems. Most of these kinship-comparison study designs have used the NLSY79 maternal data combined with the NLSY-Children/Young Adults to estimate models. In a slightly different type of design, van den Oord and Rowe (1999) used an earlier version of the cousin data in the NLSY to study the cross-cousin correlations between family functioning and childhood IQ. It is important to motivate our current study that most of these cousin-comparison designs did not use data that are directly genetically-informed, that is, most of the studies cited above were not based on biometrical analysis.

There do exist a few biometrical studies in the literature that have used cousins in the NLSY as a genetic category, and cousin correlations were used alongside other kinship correlations to estimate biometrical models (e.g., D’Onofrio et al, 2008; Miller et al, 2010; Rodgers, Bard, & Miller, 2007; Rodgers et al, 2015, Garrison & Rodgers, 2019). Rodgers et al (2006) created a design called the Mother-Daughter-Aunt-Niece (MDAN) design that directly compared mother-daughter correlations to aunt-niece correlations; in addition, however, the within-generational-within-nuclear-family correlations among the NLSY79 mothers (who are sisters, and a very few who are cousins), and among their NLSYChildren/Young Adult children (who are siblings and cousins) are available and usable in biometric estimation as well.

The NLSY Cousins

Starting in the early 1990s, our research team has developed a set of kinship links using the National Longitudinal Survey of Youth data, for the purposes of supporting our own and others’ biometrical analysis within the NLSY. We documented that effort and cited some of the 75+ papers that have used the NLSY kinship links in past research, in Rodgers et al (2016). There are in fact three different NLSY datasets for which we have generated kinship links: the NLSY79 data, the NLSYChildren/Young Adult (NLSYC/YA) data, and the NLSY97 data. The kinship links from each of these datasets, as well as an open-source master file that combines the kinship links can be freely downloaded from http://liveoak.github.io/NlsyLinks/, and are also available in the R-package NLSYLinks (Beasley et al, 2015). A bibliography of kinship linking articles, theses, and dissertations, and both R and SAS code useful to biometrical researchers, can also be found in these web sources.

Each dataset contains cousin pairs. The NLSY79 cousins lived together in the same house and were of similar ages (14 to 21) on December 31, 1978, when the household probability sample was drawn. In the NLSY79 there were 96 full cousins and 47 half cousins in this sample, for a total of 143 different cousin pairs (a few of which contained the same individual; i.e., some individuals in the NLSY79 were members of more than one cousin pair).

The NLSYC/YA respondents were, by definition, the biological children of the 6,283 females in the NLSY79 survey. Because sisters are naturally represented among the NLSY79 females, and their children are related as cousins, there are 4,995 cousin pairs identified in the NLSYC/YA data. There are a number of different levels of cousin relatedness, an important feature to support the analyses in the current paper.

The NLSY97 dataset is an 18-year approximate replication of the NLSY79 data. Respondents in the NLSY97 were between ages 12 and 16 on December 31, 1996, when the household probability sample was drawn. There were 90 cousin pairs who were living together in the same house on December 31, 1996 (of still yet unidentified levels of cousin relatedness).

Thus, in the overall NLSY datasets, there are a total of 5,228 cousin pairs from the three datasets. The cousin pairs in the NLSY79 and NLSY97 datasets are unusual in that they lived together in the same household for some period of time. The cousins in the NLSYC/YA dataset are an approximately-representative sample of the cousin pairs produced by the NLSY79 females (for whom childbearing was completed in 2011). The most valuable cousin sample comes from the NLSYC/YA, and that sample will be used in the study reported in this paper.

The NLSYC/YA cousins are related across their mothers, who themselves have several different levels of relatedness. NLSY79 mothers who are full sibling sisters have children who are related as full cousins (with a genetic coefficient of R=.125). NLSY79 mothers who are half-sibling sisters have children who are related as half-cousins (R=.0625). NLSY79 mothers who are related as cousins have children who are related as quarter cousins (R=.03125). In Table 1, we reproduce a table of the cousin information from the NLSYC/YA that is published in Rodgers et al (2016), documenting those and several other different levels of relatedness.

Table 1:

Kinship links for various types of cousins, categories and sample sizes (number of pairs) in the NLSY-C/YA linking files

Cousin’s R Coefficient	Description	Sample size	Mothers’ relatedness	Mother’s R Coefficient
R = 0	Genetically unrelated cousins	314	Adoptive siblings	0
R = .015625	Eighth cousins	61	Half cousins	.0625
R = .03125	Quarter cousins	204	Full cousins	.125
R = .0625	Half cousins	309	Half siblings	.25
R = .09375	Unknown half/full cousins	12	Ambiguous siblings	.375
R = .125	Full cousins	3941	Full siblings	.50
R = .1875	Full cousins, mothers ambiguous twins	5	Ambiguous twins	.75
R = .25	Full cousins, mothers mz twins	18	MZ twins	1.0
R = ?	Cousins, mothers relatedness unknown	131	Unknown	??
	Total	4995

Note: cousin pairs identified across links to their mothers; cousins in NLSYC/YA do not live together in the same house

R Coef refers to the coefficient of genetic relatedness; R = 1.0 for MZ twins, R = .50 for full siblings, R = .25 for half siblings, R = .125 for full cousins, etc

Note: This table is reproduced from Table 4 in Rodgers et al (2016)

An ACE Analysis of Adult Height Using NLSY Cousin Pairs

Because large sample sizes exist in the NLSY of cousin pairs that differ in genetic relatedness, there is the basis for a biometrical study using only cousin pairs. But such a study has several challenges that we note at the outset. First, and most importantly, the differences between the genetic relatedness underlying different types of cousin pairs is very small. Unlike a comparison of MZ versus DZ twins (R=1.0 versus R=.5), or even a comparison of full- and half-siblings (R=.50 versus R=.25), the cousin R coefficients in Table 1 are R = .125, R=.0625, R=.03125, etc. Differences between these genetic coefficients provide the analytic leverage to estimate biometrical ACE models (e.g., Neale, Boker, Xie, & Maes, 2003). These cousin categories are similar enough in their genetic coefficients (e.g., R=.125 and R=.0625 only differ bv .0625, whereas R=1.0 and R=.5 differ by .50) to require relatively large sample sizes and/or a highly reliable phenotype before the parameters in such models can be estimated.

A second problem is that these cousins do not live together in the same household, and so the usual ACE model approach used for siblings in the same household of assigning a latent shared environmental correlation of 1.0 to sibling pairs must be adjusted. We use (as a starting point) a fixed parameter value of .80 for this purpose, because these cousins live in households with mothers who lived together in households growing up, and so it is reasonable to assume a relatively high shared environmental correlation (though not 1.0). We will present sensitivity analyses to do an empirical evaluation of this question in the Results section.

A third challenge in defining a cousin analysis in the NLSY goes back to Fisher’s (1918) statement quoted above. He only referenced full cousins in that statement as being anomalous in the kinship correlation. However, to run a biometrical analysis requires multiple levels of kinship relatedness. We selected three categories from Table 1 that have substantial sample sizes – we will use full cousins (N=3,941 pairs), half cousins (N=309 pairs), and quarter cousins (N=204 pairs). We ruled out using cousins with moderately large sample sizes from mothers who were adoptive siblings and who had genetic coefficients of R=0, and also from mothers of unknown relatedness, because of potential imprecision in estimating those categories. Other cousin categories (R=.09375 with N=12, R=.015625 with N=61, etc, were dropped because we could not estimate stable cousin correlations from categories with such small sample sizes.

Method

Using the three cousin categories defined above (full cousins, half cousins, and quarter cousins) from the NLSYC/YA data, we had an available sample size of 4,086 cousin pairs. As noted, to achieve appropriate power requires both a large sample size, and a reliable phenotype. The outcome “adult height” is available for nearly all of the NLSY79 and NLSYC/YA respondents, and has been extensively studied in past biometrical research. These height measures were gender standardized, using CDC height norms (Flegal & Cole, 2013).

NLSY-79 subjects self-reported their height on multiple occasions, both early in the study (annually from 1981 through 1985) and more recently (biannually since 2006). Within-person height correlations were high (median .93; IQR = .03), and at least one reported height score was available for nearly all subjects (98.5%; 12541/12686).

The primary NLSY sample that we used for analysis was the NLSYC/YA. Unlike the NLSY79, height was recorded at virtually every time point in the NLSYC/YA. However, the method used for height varied across subject ages, including interviewer tape measure, mother-report, and self-report. On select occasions (typically when subjects were in adolescence), multiple methods were used in the same survey. Inter-method reliability was high: r= .91 for self-report and interviewer-measure (n=1494); r=.85 for mother-report and interviewer-measure (n = 2207); and r=.78 for self-report and mother-report (n = 2337). On average, mothers underestimated the child’s self-reported height by 1.1 inches (median = 0.5, n = 2337), and interviewer-measured height by 1.77 inches (median = 1, n= 2207). The mothers had larger underestimates for older children, suggesting that much of the maternal unreliability was likely caused by mothers who simply didn’t know how tall their adolescent/adult children were. In contrast, subjects tended to slightly over-estimate their own height compared to interviewer measurements, by 0.02 inches (median = 0.0, n = 1494). Accordingly, whenever available we used self-reported or interview-measured height, and avoided the maternal measures. For adult height, we took the median value of all self- and interview-measured heights after the respondents turned 18. There existed some missing data; 67% of the NLSYC/YA respondents had adult height measures (7,735/11,521).

Most past research on adult height has been based on ACE models and twin studies. Johnson et al (2010), in summarizing such studies, noted a broad range in height h² of between .70 and .95, with h² generally around .90. Visscher et al (2006) studied data from the Australian Twin Registry, and found adult height to have h² = .80. Silventoinen et al (2003) evaluated estimated ACE models for adult height using 31,000 twin pairs from eight different countries, and found male height h² ranged from = .87 to .93, and female h² ranged from .68 to .84. Studies that have stratified by race have found somewhat lower heritabilities than these for African-American samples, and especially for Hispanic samples (e.g., Lai, 2006; Roberts et al., 1978). In these studies, estimated shared environmental proportion of variance has been consistently near zero. (We note that twin studies and others that are not based on probability sampling likely have less diverse subjects than in the NLSY and other studies that begin with probability sampling).

We computed kinship correlations within cousin categories to evaluate the NLSYC/YA data for the same patterns found by Fisher 100 years ago. We estimated those kinship correlations using R (version 3.5.1, R Core Team, 2018). We estimated several different ACE models of adult height, using Mplus (version 7.4, Muthen & Muthen, 2014) and the Mplus automation R package (Hallquist & Wiley 2018). All models were estimated using robust full information maximum likelihood (RFIML) to account for missing data. Sample Mplus syntax, adapted from Prescott (2004), is available from the authors. We adjusted standard errors for clustering at the household sample level. For correlations, we cluster-adjusted our standard errors using the CR0 estimator (Laing & Zeger, 1986) from the estimatr package, and calculated confidence intervals using the Fisher (1915) transformation. A reviewer correctly noted that there exist alternatives to the ACE model that could be effectively used with such data (the reviewer suggested an “A and correlated E* model,” for example). We agree and recognize the value of fitting different modeling approaches. In the current study, our goal is to evaluate whether even cousin categories with close genetic coefficients can be used to estimate height heritabilities; because most of the past literature against which we leverage our estimates is based on ACE modeling, we will restrict our analyses in the current studies to similar modeling approaches. We do, however, support the reviewers interest in investigating different models in future research.

A challenging component of this analysis was setting the latent environmental correlation for the cousin pairs. We don’t expect that correlation to be 0, even though they were raised in separate family environments, because their mothers by definition shared a family environment when they were children; we expect those maternal shared environments to substantially inflate the underlying C parameter toward 1.0. We defined C=.80 for our basic analyses, which was a starting point for understanding the appropriate parameter estimate. Because this was an important assumption, we conducted a careful sensitivity analysis of this decision, by evaluating the shared-environmental correlation in values ranging from C=0 to C=1 in intervals of 1/1000 (.001, .002, .003, …, .998, .999. 1.0). Thus, we estimating the full model 1000 times, each an identical analysis except for adjusting the shared-environmental parameter for each iteration. We summarize this sensitivity analyses using a ternary plot (see Fowler, Baker, & Dawes, 2008), which was constructed using ggtern (Hamilton & Ferry, 2018).

Results

We present our basic kinship correlations in Table 2, which also includes standard errors, a 90 % confidence interval, and the sample size. We note that our hypotheses are one-tailed hypotheses throughout; we are specifically evaluating kinship correlations for inflation. As a result, we report 90% confidence intervals throughout. Of note from that table is that the full-cousin kinship correlation in the overall sample was r=. 183, 46% larger than the genetic coefficient for full cousins of R=.125. The half cousin kinship correlation was r=.089, 42% larger than the R coefficient for half cousins of R=.0625. We estimated a negative kinship correlation for quarter cousins. These first two correlations dropped when we estimated kinship correlations separately for non-minority and minority samples, as reported in Table 2. It appears that the inflation in cousin correlations emerges primarily from the non-minority sample; for full- and half-cousins, height kinship correlations in the minority samples were consistently below the coefficient of genetic relatedness. This finding likely matches Fisher’s results fairly closely, because he was surprised at the “inflated” cousin correlations, suggesting that he expected them to be not larger than, and quite possibly below, the coefficients of genetic relatedness.

Table 2 :

Adult height kinship correlations, overall NLSY-C/YA dataset, with 95 % CI and sample size of (double entered) cousin pairs

Kinship (cousin) correlations	r (se)	90% CI	N
Overall sample
Full Cousins (R=.125)	.183 (.035)	(.127,.239)	4902
Half Cousins (R=.0625)	.089 (.096)	(−.068,.242)	444
Quarter Cousins (R=.03125)	−.130 (.090)	(−.272,.018)	270
Non-minority sample
Full Cousins (R=.125)	.159 (.052)	(.075,.240)	2816
Half Cousins (R=.0625)	.068 (.105)	(−.105,.236)	350
Quarter Cousins (R=.03125)	−.162 (.105)	(−.324,.009)	230
Minority (Hispanic or Black) sample
Full Cousins (R=.125)	.097 (.043)	(.026,.167)	2086
Half Cousins (R=.0625)	−.063 (.166)	(−.325,.207)	94
Quarter Cousins (R=.03125)	−.635 (.172)	(−.775,−.436)	40

We will develop and interpret these findings in more detail in the Discussion section. However, we note at this point that for both the full- and half-cousin categories in the overall sample, the height cousin correlations are substantially larger than quantitative genetic theory suggests should be possible under a purely Mendelian model (i.e., an ACE model that is an AE model without a shared environmental component).

The ACE analysis of adult height among cousins in the NLSYC/YA data is presented in Table 3. Using only two cousin groups (full- and half-cousins) and the full available sample, we estimated h² = .91, c² = .08, and e² = .00, with an RMSEA below .000 and a chi-square that could not reject the model. Results were almost identical using all three cousin categories, though the RMSEA showed a slightly worse (though still very good) fit. The estimated h² values were consistently higher for the non-minority sample than for the minority sample, c² values were consistently low throughout, and fits were generally quite good.

Table 3:

ACE Analysis of adult height scores among the NLSYC/YA sample, including h², c², e^2, standard errors, RMSEA, and 90% CI

Total sample
	h2	c2	e2	RMSEA (90%CI)	χ2 Test (DF);	p
Two Cousin Groups	.914 (.823,1.01)	.082 (.016,.198)	.0003 (−.061, .128)	.000 (0,.029)	4.323 (5)	.504
Three Cousin Groups	.933 (.848,1.02)	.067 (.008,. 183)	0	.021 (0,.040)	14.22 (9)	.115
Non-minority sample
	h2	c2	e2	RMSEA (90%CI)	χ2 Test (DF);	p
Two Cousin Groups	.956 (.848,1.07)	.043 (−.004,.228)	0	.005 (0,.042)	5.17 (5)	.395
Three Cousin Groups	.978 (.922,1.04)	.023 (.002,.116)	0	.080 (.061,.101)	56.0 (9)	<.001
Minority (Hispanic or Black) sample
	h2	c2	e2	RMSEA (90%CI)	χ2 Test (DF)	p
Two Cousin Groups	.762 (.264,1.518)	0	.238 (.024,1.277)	.000 (0,.033)	2.56 (5)	.768
Three Cousin Groups	.741 (.249,1.50)	0	.258 (.011, 1.257)	.040 (.001,.069)	16.9 (9)	.0498

As an additional analysis, we evaluated the biometrical properties of the mothers of the NLSYC/YA cousin pairs. Mothers of the full cousins were themselves full siblings (by definition), mothers of the half cousins were themselves half siblings, and mothers of the quarter cousins were themselves cousins. Our reasoning for this analysis was that we wanted to evaluate whether the inflated cousin correlations were a feature of the families in which the cousins were defined, or whether they were particular to the cousin relationship themselves. These maternal kinship correlations are reported in Table 4. It should be noted that these sister (mother) correlations are based on substantially smaller sample sizes. The reduction in sample sizes is because we used all possible cousin pairs, which resulted in clustering within families. For example, two NLSY79 mothers who are sisters, who each have 2 children, contribute four cousin pairs but only one mother pair for the kinship correlations. The maternal kinship correlations are slightly inflated compared to the kinship coefficients for the mothers of full cousins (r=.522, 4% higher than the R=.50), and the other two categories are not. As before, the correlations for the non-minority sample is higher than for the minority sample.

Table 4:

Kinship correlations of the mothers of the cousins in Table 1, overall sample

Kinship correlations	r (se)	(90% CI)	N
Overall sample
Mothers of Full Cousins (R=.50)	.522 (.035)	(.478,.563)	1292
Mothers of Half Cousins (R=.25)	.211 (.118)	(.021,.387)	90
Mothers of Quarter Cousins (R=.125)	−.134 (.236)	(−.480,.248)	48
Non-minority sample
Mothers of Full Cousins (R=.50)	.547 (.046)	(.492,.599)	636
Mothers of Half Cousins (R=.25)	.078 (.127)	(−.130,.280)	70
Mothers of Quarter Cousins (R=.125)	−.203 (.259)	(−.559,.216)	38
Minority (Hispanic or Black) sample
Mothers of Full Cousins (R=.50)	.456 (.052)	(.385,.521)	652
Mothers of Half Cousins (R=.25)	.271 (.186)	(−.029,.525)	20
Mothers of Quarter Cousins (R=.125)	245 (.329)	(−.283,.659)	10

A threat to the validity of this previous analysis was the different sample sizes for the mother sample and the cousin sample, caused by clustering within NLSYC/YA family due to multiple cousin pairs within families and more than two NLSY79 sister pairs in a few families. To equate these sample sizes, we ran an additional analysis that defined the cousin correlations for the cousins who were the first child of the first two mothers in each NLSY79 family. Then, following, we computed sister (mother) correlations for the mothers of exactly those cousin pairs. This method eliminated the family clustering noted above (which was accounted for through the analysis above by computing standard errors and confidence intervals using the approach described in the Methods section), and provided a very direct comparison between the cousin correlations and the mother correlations. These results are reported in Table 5. For full cousins, we found r=.137 (against an R=.125), and for the mothers of these cousins, we found r = .487 (against an R=.50). For half cousins, we found r= .044 (against an R=.0625), and for the mothers of these cousins we found r = .309 (against an R=.25). There were very large 90% CIs around the second set of estimates (for half cousins and their mothers), because of the relatively small sample size. Thus, we do not strongly interpret the apparently inflated mother correlation in this case (the equivalent mother sample in the original much larger dataset was r=.21, against an R=.25, and based on a sample of size n=90 rather than n=56).

Table 5:

Kinship (Cousin) and Mother (Sibling) correlations from Matching Sample, with mothers matched exactly to their children

Kinship correlations	r (se)	(90% CI)	N
Overall sample
Mothers of Full Cousins (R=.50)	.487 (.032)	(.446,.526)	714
Full Cousins, children of mothers (R=.125)	.137 (.036)	(.078,.195)	714
Mothers of Half Cousins (R=.25)	.309 (.109)	(.139,.461)	56
Half Cousins, children of mothers (R=.0625)	.044 (.117)	(−.147, .232)	56
Non-minority sample
Mothers of Full Cousins (R=.50)	.495 (.042)	(.441,.545)	382
Full Cousins, children of mothers (R=.125)	.126 (.050)	(.044, .206)	382
Mothers of Half Cousins (R=.25)	.119 (.131)	(−.096,.323)	38
Half Cousins, children of mothers (R=.0625)	.047 (.147)	(−.192, .281)	38
Minority (Hispanic or Black) sample
Mothers of Full Cousins (R=.50)	.422 (.050)	(.352,.487)	332
Full Cousins, children of mothers (R=.125)	.115 (.054)	(.027, .202)	332
Mothers of Half Cousins (R=.25)	.271 (.191)	(−.036,.531)	18
Half Cousins, children of mothers (R=.0625)	−.533 (.136)	(−.674, −.355)	18

The sensitivity analysis that we ran involved estimated ACE models under different specifications of the environmental correlation C. For this analysis, we used the overall sample, and fit an ACE model. C was set to .80 in the analysis reported in Table 3 (and 1.0 for the sisters/mothers of the cousins, who grew up in the same household). In Figure 1, we present a ternary plot that shows the estimates of h², c², and e², for values of C that ranged from 0.0 to 1.0. We expect ACE estimates for height to be around h² = .80, c²=.05, e² = .15. Points in the ternary plot show the combination of these three estimated ACE parameters, and the colors reflect the value of C. The models that were approximately in range – with h² around .80, c² less than .20, and e² = 1 – h² – c², had C values between .80 and 1.0, with most very close to .80. This finding across a whole simulation of model results supports our original decision to fix C=.80.

Figure 1:

Ternary plot, as a sensitivity analysis of the choice of using a C=.80 environmental correlation in the ACE analysis reported in Table 3

Discussion

Findings

One of the two meaningful results from the current study is that we can successfully run an ACE analysis of adult height, using only cousin kinship pairs. This is, as far as we know, the only study in the literature to rely exclusively on cousin correlations at different levels of cousin relatedness to support a biometrical decomposition. In one analysis we used full cousins (R=.125) and half cousins (R=.0625), in the second analysis we added quarter cousins (R=.03125). Both ACE modeling exercises estimated height parameters that were consistent with a rather large previous biometrical literature studying height. The second finding – discussed below – is to document and explain inflated cousin correlations. It is perhaps even more notable that our ACE analyses estimated h² and c² values that were consistent with previous literature using twins and full family data; the ACE results suggest that if there is inflation in the cousin correlations, it was consistent in such a way that it did not degrade the overall ACE analysis (though another interpretation exists that is more likely, which we will review shortly). Indeed, the full and half cousin kinship correlations showed almost exactly the same level of apparent inflation (and these two cousin categories carried most of the load in the estimation process, even when we added quarter cousins, because of the differential sample sizes).

Fisher’s Challenge

We have replicated Fisher’s empirical findings of surprisingly large cousin correlations, under an assumption of Mendelian inheritance, with a 100-year gap between the original statement of the result (Fisher, 1918) and the presentation of the cousin correlations from the NLSYC/YA data (which were first presented at the Behavior Genetic Association meetings in Boston in summer, 2018). Fisher stated clearly that he didn’t know why the cousin correlations were higher than expected under a standard quantitative genetic model; he speculated that the results might even have been in error, though he also noted that the inflated cousin correlations had been identified in multiple samples. As discussed in our introduction, during the 35-years in which the first author of this article has been using the NLSY datasets (the NLSY79, the NLSYC/YA, and the NLSY97), and 25 years of doing biometrical analyses with those datasets, our research team has noted a number of times that cousin correlations were surprisingly high. We’ve found correlations suggestive of that pattern not only for height and weight, but also for age at first intercourse, age at menarche, completed fertility, and delinquency. Those reports were always in the context of cousin correlations being compared to kinship correlations from other non-cousin kinship categories, such as twin, full-sibling, or half-sibling correlation.

Despite noticing this anomaly in past research, this is the first concerted attention that we have given to this interesting and surprising finding. Using height as an excellent phenotype to address this question – a phenotype with high heritability, high reliability and validity of measurement, and many previous studies to which to compare our results – we find kinship correlations for a large sample of full- and half-cousins that are over 40% larger than they could possibly be under a purely Mendelian model. This finding was statistically stable in the overall analysis with a large sample size; the inflated kinship correlations were no longer statistically stable in subgroup analyses with smaller sample sizes, but the general patterns were repeated, and were apparent primarily for the non-minority samples. When we estimated kinship correlations for the mothers of the cousins, they appeared to be only slightly inflated, if at all. We view the mother analyses as still a bit ambiguous, because it is not clear whether we found slightly inflated correlations by chance, or whether they are slightly inflated because of underlying (and still unidentified) processes.

Before we began this study, we brainstormed some possible explanations for why cousin correlations might be inflated. We review several of those speculative suggestions, and then turn to a much simpler and more straightforward solution suggested by our analyses. The first suggestion is a “data explanation” (which we ultimately reject as implausible). If the full- and half-cousins in the NLYSC/YA dataset are contaminated by having some misdiagnosed full- or half-sibling pairs included among the cousins, these pairs with higher R coefficients would help explain the inflation. The inflation of the full cousin correlations resulted in cousin correlations that were approaching the magnitude of the expected half-sibling correlations, suggesting that there would have to be a large number of half- and full-siblings (or perhaps even some MZ twins), mixed in with correctly diagnosed full-cousins, to achieve this level of kinship correlation. We can reject this as a likely explanation, on logical and procedural grounds. The cousins are identified, and assigned cousin status, using the NLSY79 sister-sister pairs – which are themselves identified with relatively high reliability and validity. These sister-sister pairs are the mothers of the NLSYC/YA cousins, and we know their kinship relatedness with high confidence. We invite careful scrutiny of our kinship linking approaches – developed over a 25-year effort to build effective methods to define those pairs – as they are described in Rodgers et al (2016). Of particular note is that the cousin correlations among the NLSY79 mother have been themselves occasionally slightly inflated in past research, although the half-sibling, full-sibling, and twin correlations are typically consistent on both internal (to the NLSY) and external (in relation to other studies) grounds.

A second potential explanation is assortative mating. If the NLSY79 sisters – mothers of the NLSYC/YA cousins – mated with males in producing the NLSYC/YA respondents, with relatively high correlations between the mother and father heights, this correlation could inflate the cousin correlations. We have no information in the NLSY files about the heights of the mates of the NLSY79 females. Past research on height assortative mating suggests that there is a positive coefficient, though not one high enough to cause full-cousins to appear to be half-siblings, and half-cousins to appear to be full-cousins.

A third potential explanation reflects potential extra-pair copulation/childbearing between an NSLY79 female and her sister’s mate. In this case, those categorized as full-cousins are more highly related than it appears, because they share mothers who are sisters (as typical cousins), but also share the same biological father (i.e., R=.375, halfway between half-siblings and full-siblings). Not only would this phenomenon produce artificially higher-related cousin pairs, it also would open the possibility of a dominance effect (as Fisher, 1918, noted in his original specification of the cousin-inflation-effect). We believe that this mating phenomenon is highly unlikely to occur at high enough levels to explain our kinship correlations. We make this suggestion on logical grounds, because if this phenomenon were widespread enough to cause the cousin correlation inflations, it would likely be noted and recognized. Early research on extra-marital childbearing suggested rates as high as 10% (i.e., 10% of men were raising children who were biologically fathered by another man, which is a much more general case than a woman mating with her sister’s spouse), although reliable surveys of this phenomenon have only been available during the past couple of decades. More recent research based on such surveys (e.g., Anderson, 2006; Larmuseau, Matthijs, & Wenseleers, 2016) suggested that this rate is actually closer to 1-2%, which supports our skepticism that this process is likely to be the single – or even a dominant – explanation to resolve Fisher’s challenge. This suggestion does belong in the list of potential and logical explanations, even if the empirical data render it unlikely.

A fourth potential explanation is a genetic one, associated with mitochondrial DNA (mtDNA), which is transmitted separately from nuclear DNA and only through the maternal line (see Burt, Verhulst, & Neiderhiser, 2018, who discussed mtDNA from a biometric perspective). If mtDNA did influence human height, it could inflate cousin correlations – but only for maternal cousins. But all NLSYC/YA cousins are defined through the maternal line, by design, which supports the potential for mtDNA as a possible contributing factor.

Finally, we draw on the results of our analyses for what we feel to be the best resolution of Fisher’s challenge. This explanation, which is both plausible and supported by our new empirical analyses, is that Fisher’s study that focused solely on “Mendelian inheritance” was missing an important component of variance. He recognized (correctly) that height is a phenotype that is highly heritable. His surprise at large cousin correlations reflected his expectation that virtually all of the important variance underlying height is genetic and not environmental. A century of further research has supported this expectation; h² is consistently large (especially for Caucasian samples such as most of our non-minority sample), around h² = .70 to .90, and c² is consistently small, usually in the range of c² = .05 to .20.

The key to resolving at least part of Fisher’s challenge is to recognize the critical role that the C parameter played in our ACE model. Our sensitivity analysis supported that C should be around C= .80 (and not C=0, as some definitions would suggest, because the NLSYC/YA cousins were not reared in the same family environment). When we set C = .80, it has a substantial impact on the analysis, even with a relatively low estimated c² parameter. We were able to very closely reproduce the most inflated of the kinship correlations – for the overall sample using the two cousin categories (with a cousin correlation of r = .183), and the model reproduced the other cousin correlations quite accurately as well. The model estimates (see the first row of Table 2) show that 0.125 * 0.914 + 0.80 * 0.082 + .0003 = .180, which is very close to the actual correlation of r = .183. If, as Fisher assumed, we only looked at the genetic portion of this model, we would predict a correlation of r = 0.125 * 0.914 = 0.113, which is smaller than the genetic coefficient of R=.125, as Fisher would have expected. But it is both plausible, and empirically supported, that the mothers who are sisters and grew up in a common home environment likely create environmental circumstances that are relatively highly correlated. Even in the presence of a low c², a high environmental multiplier creates an important role for the correlated environments across the mothers. In this case, the “shared environment” is the environment extended across related family members, and not simply the nuclear family environment. To summarize the substantive finding suggested by our results, the inflation of cousin height correlations is plausibly caused by the mothers of cousins – who are themselves sisters who grew up in the same family environment – creating substantially correlated environments that contribute a small but meaningful component to the magnitude of cousin correlations. Fisher simply didn’t account for this potential source of environmental variance in his analysis assuming only Mendelian inheritance.

We conclude by noting the value – beyond addressing Fisher’s challenge – of the NLSY cousin data. These data may be used in innovative ways to address Fisher’s challenge, or may be combined with other kinship categories in the NLSY datasets to do standard biometrical analysis (see Rodgers, Rowe, & May, 1994 for an early biometrical analysis of achievement patterns, Boutwell et al, 2017, for a recent study of the link between intelligence and criminal behavior, and Garrison & Rodgers, 2019 for a recent bivariate ACE analysis of the SES-health gradient). Or the NLYS kinship links even may be used in research based on only cousin pairs, as in the current empirical study. We invite the behavior genetic community to join us in using the NLSY cousin data to address interesting biometrical and family-based problems.