Overview of method

We consider three approaches to estimating heritability for a phenotype with a narrow-sense heritability of 80%. First, the classic approach to estimating heritability is to divide the phenotypic covariance of related individuals by the fraction of the genome they share IBD¹³. In this instance, the phenotypic covariance of pairs of related individuals will be 0.80 times the fraction of genome shared IBD (Figure 1a). The second approach, developed by Yang et al.¹⁰, is to estimate the genetic relationship of unrelated individuals over genotyped SNPs and applied a linear mixed model with the genetic relationship matrix to estimate phenotype. To illustrate this approach we simulated 2 million independent pairs of individuals, regressing their normalized genetic similarity over the product of their normalized phenotypes giving a regression coefficient of 0.79±0.014 (Figure 1b). This Haseman-Elston regression¹⁶ shows how genetic similarity of unrelated individuals can be used to estimate heritability of genotyped SNPs (h_g²). In general, the heritability explained by genotyped SNPs is less than the total narrow-sense heritability (h²) since phenotypic variation determined by poorly tagged SNPs such as rare variants will not be captured¹⁰.

The approach used in this work is similar to that of Yang et al.¹⁰, but instead of using genotypes to estimate genetic similarity we use the number of copies of local ancestry in an admixed population. A crucial element of our approach is that the phenotypic variation described by variation in local ancestry ( $h_{\gamma }^2$) is a function of all causal variation, not just that tagged by SNPs on the genotyping platform. This is because local ancestry tags both common and rare variation. To illustrate this approach we simulated 4 million unrelated admixed individuals from ancestral populations with genetic distance F_STC= 0.08 and an equal proportion of ancestry from each ancestral population θ = 0.5 (see Online Methods). Applying Haseman-Elston regression to regress the product of normalized phenotypes against genetic similarity of local ancestry, we observe a regression coefficient 0.033±0.007 ≈ 2F_STCθ(1−θ)h²= 0.032, corresponding to h² = 0.83 (s.e. = 0.18) (Figure 1c). The Haseman-Elston regression used in generating these figures is for illustrative purposes (as in Figure 3 of [¹⁰]). In practice, we use a mixed model approach due to its lower standard errors¹⁰.

We first construct a local ancestry based kinship matrix K_γ, which is constructed similarly to the genotype-based kinship matrix K in previous methods¹⁰, but with local ancestry substituted for genotypes at each SNP. We use a variance components approach to estimate the phenotypic variance explained by variation in local ancestry ( ${\sigma }_{\gamma }^2$) and the residual phenotypic variance ( ${\sigma }_{\epsilon }^2$)^10,17. We included genome-wide ancestry proportion θ and the top five principal components as fixed effects when fitting the mixed model (see Online Methods). The heritability explained by local ancestry is given by $h_{\gamma }^2=\frac{{\sigma }_{\gamma }^2}{{\sigma }_{\gamma }^2+{\sigma }_{\epsilon }^2}$. Finally, to estimate h², we use the formula h_γ² = 2F_STCθ(1−θ)h², where F_STC is a specific measure of weighted allele frequency differences between ancestral populations at causal loci (see Online Methods). For dichotomous phenotypes we applied the same approach, but converted the observed scale estimates to a liability scale estimate of heritability using [¹⁸], and the published disease prevalence in African Americans. In our previous work¹¹, this conversion was not possible because non-randomly ascertained individuals in multiple relatedness classes (e.g. siblings, first cousins, avuncular) were studied, and there is currently no method for accounting for ascertainment in such complex pedigrees. A complete description of the approach, along with an analytical derivation, is given in Online Methods.

Simulations with Simulated Genotypes

We first verified the analytical derivations and examined the properties of the approach under a simple simulation framework. We simulated the genotypes and local ancestry of 4,000 unrelated diploid individuals at 1,000 SNPs from a two-way admixed population with causal variant genetic distance F_STC, and either normally or uniformly distributed ancestry proportion θ. Each local ancestry segment contained exactly one SNP and all segments were generated independently. Phenotypes were simulated under an additive model with heritability h² in which a proportion r of the 1,000 SNPs was causal (see Online Methods). We applied our method to estimate heritability over a range of values of F_STC, θ, r, and h². For each parameter setting we estimated heritability from 2,000 independent simulated data sets. The results shown in Table 1 show that our heritability estimates are accurate across a range of parameter settings, confirming our analytical derivation. Results for additional parameter settings are shown in Supplementary Table 1.

The results also demonstrate the relationship between $h_{\gamma }^2$ and the parameters F_STC, θ, and h². For a fixed value of r, phenotypes with a larger h² will have larger genetic effects resulting in larger $h_{\gamma }^2$. When ancestral populations are genetically distant (larger F_STC), variants are more likely to have a different frequency in the ancestral populations resulting in a concomitant increase in $h_{\gamma }^2$. Increasing the variance of θ results in a larger standard error around the heritability estimates.

Simulations with Real Genotypes

We made several simplifying assumptions in the above simulations that do not hold in real data sets. These include a single SNP per ancestry block, no genotyping error, no local ancestry inference error, no LD, a normal or uniform distribution of ancestry proportion, continuous phenotypes, and that the effect size distribution of common and rare variants used in computing F_STC was identical. To address these complexities, we took the approach of using real genotypes and simulating phenotypes. We simulated continuous and case-control phenotypes over 5,129 individuals (excluding close relatives) from the CARe cohort (see Online Methods). Although phenotypes were generated from SNPs sampled across all genotyped SNPs, we only used local ancestry information from every 5^th SNP.

We tried a range of parameters for h². Instead of simulating phenotypes under an infinitesimal model, we sampled a proportion of causal variants r. We could not alter ancestry proportion θ, since this is fixed in the real data set. However, we altered the effect size distribution of SNPs according to their value of F_STC.

The data did not contain a sufficient number of genotyped variants that were rare in both populations to simulate rare versus common effects. Instead we examined SNPs common in both populations (common) vs. SNPs rare in at least one population (uncommon). Only common variants were used in constructing the kinship matrix, and so uncommon variants will only contribute to h_g² via LD. The common SNPs had an F_STC of 0.15, while the uncommon SNPs had an F_STC of 0.25. We simulated phenotypes with a different proportion of phenotypic variance from uncommon variants (α). When α is different from 0, the kinship matrix variant and causal variant frequencies are different. The results in Table 2 show that simulations involving a large proportion of causal variants not included in the kinship matrix (high α) had a lower value of h_g² than h² because the common variants did not completely capture the phenotypic variance driven by the uncommon variants. The parameter α also determines the study wide F_STC according to F_STC = (0.15(1-α) + 0.25α) (see Online Methods). The results shown in Table 2 use the correct value of α, and hence the estimates of h² are unbiased. However, if we incorrectly assume that α=0 when it does not, then h² will be biased by factor of (0.15(1- α) + 0.25α)/0.15. We describe this (and other potential sources of bias) in detail in the Discussion.

Setting individuals with the lowest P% of phenotypes as cases and all other as controls generated dichotomous phenotypes with prevalence P. The small number of individuals prevented simulation of case-control ascertainment, which may produce downward bias for low prevalence diseases in very large studies (see Supplementary Table 9 in [¹⁹]). Those biases are expected to be small in the prostate cancer data analyzed here because of the high prevalence of prostate cancer and moderate sample size. For large sample sizes, replacing mixed model based estimates with Haseman-Elston regression estimates will alleviate the issue of ascertainment bias²⁰.

The results in Table 2 also demonstrate that complexities such as genotyping error, LD, or errors in local ancestry inference in African Americans do not introduce bias into the heritability estimates when phenotypes are generated under a non-infinitesimal mixture model. This may not be the case for other admixed populations such as Latinos²¹ (see Discussion).

Application to WHI, CARe, and AAPC cohorts

We applied our method to 21,497 African-American individuals from the WHI, CARe, and AAPC cohorts over a total of 12 quantitative phenotypes and 1 case-control phenotype (see Online Methods). Local ancestry was inferred using the HAPMIX, SABER+, and RFMix methods, which are extremely accurate in African Americans (r²=0.98 or greater)^22,23,24. For each phenotype we estimated h_g², $h_{\gamma }^2$ and by extension h². For h_g² and $h_{\gamma }^2$ we used the GCTA software package applied to the genotypes and local ancestry at each SNP respectively¹⁷. For those phenotypes measured in both cohorts we compute the inverse variance-weighted mean and standard error. For each phenotype we also list previously published estimates of heritability from family studies using twins and African-American estimates where available ( $h_{\mathit{\text{pub}}}^2$) The results are shown in Table 3, and published African-American estimates are marked for reference. Estimates from European populations may not be directly comparable if the genetic or environmental bases for the phenotype differ substantially.

Several phenotypes, including height, BMI, HDL, TG, PC, and WBC (conditioned on ancestry at the Duffy antigen locus FY; see below), had h² estimates lower than family-based estimates. This could be due to the phenotype-specific effects of epistasis, gene environment interaction, and/or shared environmental factors that can inflate family based estimates^12,13. In our recent work using an extended genealogy inclusive of more distantly related individuals we also found height and BMI estimates lower than previous heritability estimates, providing further evidence of inflation¹¹. The lower estimates could also reflect a difference in the heritability between African Americans and the previous study populations. There were no statistically significant differences in h² estimates between the cohorts.

Yang et al. proposed an adjustment to account for the incomplete coverage of genotyping platforms¹⁰. We applied this approach in the CARe data (see Supplementary Table 3), and observed an increase in h_g² of less than 1% in all phenotypes. We include genome-wide ancestry proportion as a fixed effect in our mixed model. If there exists an environmental factor that affects phenotype and is correlated with ancestry, our heritability estimates will discount this environmental effect leading to higher estimates of heritability. Specifically, it will remove the variance of the environmental factor that can be explained by ancestry from the environmental component ( ${\sigma }_e^2$) in the denominator of the heritability estimate $h^2=\frac{{\sigma }_g^2}{{\sigma }_g^2+{\sigma }_e^2}$ (see Online Methods).

Differences between our heritability estimates and those of previous studies can also be due to differences between the value of F_STC we used in this study and the true value of F_STC for the phenotype in question. Based on recent evidence that rare variants unlikely to contribute to a large proportion of phenotypic variation^25,26, we computed an F_STC of 0.182 over the common variants (MAF > 5%) in African-Americans. However, this estimate drops to 0.165 for low-frequency variants (MAF < 5%) and 0.054 for rare variants (MAF < 1%). Estimates of heritability assuming a rare variant only phenotype model would be more than three times as large as from a common variant only phenotype model. Therefore, if rare variants contribute substantially to phenotypic variation or if balancing or negative selection constrained the genetic distance at causal variants, then our estimates of heritability will be biased downward (see Discussion).

Positive selection acting at causal variants could induce such a bias in F_STC, and we included WBC as a positive control for this type of bias. A SNP in the DARC gene (FY, Duffy null allele) is highly differentiated between CEU and YRI, likely due to its protective effect against vivax malaria²⁷. It is also a SNP of large effect size for WBC²⁸. Therefore, the average F_STC at causal variants for WBC, is much higher than the value 0.182 estimated from common variants (see Online Methods). The h² estimate of WBC is 3.42 due to the effect of this positive selective pressure. Ancestry at the FY locus accounts for ∼20% of the phenotypic variation in WBC²⁸. By including ancestry at FY as a fixed effect (WBC|FY) we obtain an h² estimate of 0.19, which is lower than the published estimate of 0.48.

We perform a sensitivity analysis to assess whether this type of bias is likely to be problematic. Since strong positive selection is unusual²⁹, we consider a single locus under positive selection. We estimate bias as a function of F_STC at the locus and the variance explained by the locus. The results in Supplementary Table 4 show that only for extreme values of both locus F_STC and heritability will there be significant bias in heritability due to positive selection. As an example we consider the 8q24 locus in prostate cancer, which contains causal SNPs that are highly differentiated SNPs between African and European ancestors, producing an admixture-mapping peak³⁰. However, because this locus explains less than 2% of the heritability of prostate cancer, even exceedingly strong population differentiation at this locus will not substantially bias our overall results.

Partitioning heritability across the genome

Our method is also capable of estimating the total narrow sense heritability attributable to a particular genomic region. This is accomplished by constructing the kinship matrix using just those ancestry segments in the region of interest and applying the variance component model to the phenotype of interest using the region-specific kinship matrix (see Online Methods). We partitioned the heritability for each of the phenotypes from the CARe data set across each of the chromosomes³¹. We applied weighted linear regression to determine the relationship between heritability and chromosome length (see Online Methods). The results for height are presented in Figure 2 and the full results are provided in Supplementary Table 2. We find a strong correlation between chromosome length and the heritability of height (Pearson correlation = 0.513, weighted p-value = 0.0028). logHDL, BMI, and SBP, also produced significant results (weighted p-value < 0.03, 0.02, 0.02 respectively). Other phenotypes had standard errors too large to produce meaningful results. To address this, we averaged the heritability from each chromosome across all phenotypes (using WBC|FY instead of WBC) and we observed a significant correlation between chromosome length and mean chromosomal heritability (Pearson correlation=0.686, weighted p-value <0.0002).

Definition of h²

Heritability is the ratio of genetic variance to the sum of genetic and environmental variance $h^2=\frac{{\sigma }_g^2}{{\sigma }_g^2+{\sigma }_e^2}$. In this case we are defining these elements with respect to an admixed population. For a given phenotype, both ${\sigma }_g^2$ and ${\sigma }_e^2$ can vary between the ancestral European and African populations. For example, ${\sigma }_g^2$ will vary with ancestry if the minor allele frequency at causal variants is systematically larger in one of the two populations. It is also possible for ancestry to be associated with environmental factors. In this case, by conditioning on genome-wide ancestry, our method will remove the environmental variance that can be explained by ancestry and estimate the heritability of the component of phenotype that cannot be predicted by genome-wide ancestry, thereby increasing the heritability estimate.

Estimation of ${\mathrm{h}}_{\gamma }^2$

We use a variance components approach to determine the phenotypic variance described by local ancestry $h_{\gamma }^2$ using θ as a fixed effect to prevent confounding from environmental factors association with ancestry. This method is equivalent to recent methods used to determine the phenotypic variance described by genotyped SNPs ( $h_g^2$), replacing genotypes with inferred local ancestry¹⁰.

Derivation of relationship between h² and ${\mathrm{h}}_{\gamma }^2$

Let i denote (diploid) individuals and s index SNPs. Individual i is assigned global ancestry proportion θ_i from some distribution F(.) with mean E[θ_i] = θ and variance σ²_θ. Given θ_i, an individual is assigned maternal and paternal local ancestries γ_i,s,M and γ_i,s,P at each SNP (0 or 1 copies of European ancestry), from Bernoulli distribution Ber(θ_i). Given local ancestries γ_i,s,M, γ_i,s,P and allele frequencies p_s,o, p_s,1 at SNP s in populations 0 and 1, individuals are assigned maternal genotypes g_i,s,M =γ_i,s,M Z_i,s,1 + (1-γ_i,s,M) Z_i,s,0 where Z_i,s,0∼Ber(p_s,0) and Z_i,s,1∼Ber(p_s,1), and similarly for paternal genotypes. The diploid genotype g_i,s= g_i,s,P+ g_i,s,M (0, 1 or 2), and the diploid local ancestry γ_i,s = γ_i,s,P+γ_i,s,M (0, 1 or 2).

We define E[g_i,s] = μ_g,s and Var[g_i,s] = σ²_g,s, and the normalized genotype ${\overline{g}}_{i,s}=\frac{g_{i,s}-{\mu }_{g,s}}{{\sigma }_{g,s}}$, where

Similarly, we define E[γ_i,s] = μ_γ and Var[γ_i,s] = σ²_γ, and the normalized local ancestry at each locus ${\overline{\gamma }}_{i,s}=\frac{{\gamma }_{i,s}-{\mu }_{\gamma }}{{\sigma }_{\gamma }}$, where

Although though equation (4) may not be strictly true (e.g. in a population where all individuals have 1 European parent and 1 African parent), it is approximately true for African Americans²². Furthermore, ${\sigma }_{\gamma }^2$ can be estimated empirically, and we do so in this work. We model the phenotype of individual i as

where ${\epsilon }_i~N(0,{\sigma }_{\epsilon }^2)$, Var[y_i]=1, E[y_i]=0, the effect size of SNP s is β_s, and $h^2=\sum _s{\beta }_s^2$. By substitution and algebra we get

Plugging into equation (5), we get

Note that δ_i does not depend on local ancestry, which allows us to compute the heritability due to local ancestry h²_γ as:

We define F_STC as a measure genetic distance between ancestral populations weighted by the square of effect size β_s:

This results in a final relationship

In practice we do not know the effect size of every SNP and must make simplifying assumptions about their distribution in order to estimate F_STC. First consider a simple phenotypic model in which genotypic effect size β_s is independent of p_s,o and p_s,1. Then

where N is the number of SNPs. Then equation (8) becomes

The F_STC in equation (12) is a genome-wide measure of genetic difference between the ancestral populations. This is related to the classic parameter F_ST when all variants are causal (i.e. the infinitesimal model).

Now consider a more complex model in which the effect size of SNPs can fall into one of L classes such that the effect size distribution is a function of the class L. These classes could be, for example, rare and common variants (used in this work). We defined the genetic distance between ancestral populations within each class as F_STL and the phenotypic variance explained by SNPs in this class as h²_L. Again substituting into equation (8) we have,

Therefore $F_{\mathit{\text{STC}}}=\sum _L\frac{h_L^2}{h^2}{F}_{\mathit{\text{STL}}}$ a weighted measure of genetic distance in each class.

To obtain an estimate of h² we must estimate θ, F_STC, and $h_{\gamma }^2$. The parameter θ is estimated from local ancestry inference. The parameter F_STC is estimated from assumptions about the variance explained by SNPs in each genotypic class combined with external reference panels^45,46.

Definition and Estimation of F_STC

As shown in the equations above we are defining F_ST to be the weighted average (across all SNPs s) of ratios $\frac{{(p_{s,1}-p_{s,0})}^2}{{\sigma }_{g,s}^2}$. While this is similar to standard versions of F_ST, a ratio of averages is recommended instead when the goal is to draw population genetic inferences⁴⁷. If the distribution of SNPs effect sizes is not a function F_ST then this would be the appropriate definition for our heritability estimation approach. However, recent work has shown that rare variants are unlikely to contribute to a large proportion of phenotypic variation^48,25. As has been reported previously⁴⁷, the average of ratios estimate will shrink when including many rare variants in the estimate. This is reflected in the 1000 Genomes based estimate of F_ST =0.07, which used an average of ratios⁴⁹. Therefore, F_ST will produce a biased estimate of heritability because for the variance explained by rare variants is different from the variance explained by common variants. To account for this we defined a parameter F_STC, which is a weighted measure of genetic distance between ancestral populations (equation 9).

In practice we defined F_STC as the average F_ST within each class L of SNPs (F_STL), weighted by the proportion of phenotypic variance explained by that class:

Consider a situation in which L contains two classes, rare and common SNPs, with F_ST 0.054 and 0.182 respectively. If rare variants explained 10% of the heritability and common variants explain 90% of the heritability, then F_STC=0.1692. We estimated F_STC over the HapMap3³⁷ data set by using CEU and YRI as proxies for the ancestral populations of African-Americans, using an admixture proportion of 18.3% European ancestry, and assuming distribution of causal variant frequencies. We estimated a value of 0.182 assuming causal variant MAF > 5% (which we used in this work), 0.165 assuming MAF < 5%, and 0.054 assuming MAF < 1%.

Simulations with Simulated Genotypes

In order to examine the properties of our approach, we first applied our method to data generated under a simple simulation framework for generating genotypes, local ancestries, and phenotypes of individuals from an admixed population. Allele frequencies p_A1, p_A2, …,p_AN of N SNPs from an ancestral population were drawn uniformly from [0.1-0.9]. Allele frequencies of SNPs from P₀ were drawn from a beta distribution with parameters p(1- F_STC)/F_STC and (1-p)(1- F_STC)/F_STC for each SNP s, and similarly for P₁. The parameter F_STC determines the genetic distance between the two populations. The global proportion of P₀ ancestry θ₁, θ₂, …θ_M for each of M individuals was drawn either uniformly from [0.4,0.6], from the normal distribution N(0.5,0.1), or fixed at 0.5. Local ancestry for individual i at SNP s (γ_is), was generated by two draws from binomial distribution with parameter θ_s. The genotypes from individual i at SNP s(g_is) were then generated by drawing from the binomial distributions with allele frequencies specified by the local ancestry for that individual at that SNP. That is, if the individual had two copies of ancestry from P₀ at SNP s then two draws from a binomial with parameter p_0s were used. To create a phenotype we first selected Nr causal variants where r is the proportion of causal variants. Effect sizes were drawn from the normal distribution N(0,h²/(Nr)) and the genetic element of the phenotype was generated by taking the inner product of the causal variants, normalized to have mean 0 and variance 1, and the effect sizes for the variants. Normally distributed random noise was added such that the total heritability in the population was h².

Simulations with Real Genotypes

We split the genotypes from 5,129 distantly related CARe individuals into two groups. The common group contained those SNPs with MAF > 5% in both CEU and YRI. The uncommon group contained all other SNPs (i.e. MAF < 5% in either or both of CEU and YRI). The genotype kinship matrix K was constructed over the common SNPs and the local ancestry kinship matrix K_γ was constructed using the local ancestry called at every 5^th common SNP.

We simulated a phenotype by first selecting a proportion r of causal variants at random from the common and uncommon SNPs, leaving N_c common causal and N_n uncommon causal SNPs. We then selected a fraction of phenotypic variance α explained by the uncommon SNPs. At α=0.0 uncommon variants had no effect and the genetic basis of the phenotype was entirely determined by common variants. We then chose effect sizes for each common and uncommon SNP by drawing from normal distributions N(0,(1-α)h²/(N_c)) and N(0,(α)h²/(N_n)) respectively. The genetic element of the phenotype was generated by taking the inner product of the causal variants, normalized to have mean 0 and variance 1 in the admixed population, and the effect sizes for the variants. Normally distributed random noise was added such that the total heritability in the population was h². The F_STC of the common and uncommon SNPs was 0.15 and 0.25 respectively. The study F_STC used to estimate heritability was the weighted mean 0.15(1-α) + 0.25α as described in the derivation above. Setting individuals with the lowest P% of phenotypes as cases and all other as controls generated dichotomous phenotypes with prevalence P.

Data set approvals

The CARe project has been approved by the Committee on the Use of Humans as Experimental Subjects (COUHES) of the Massachusetts Institute of Technology, and by the Institutional Review Boards of each of the nine parent cohorts.

The WHI project has been approved by the Human Subjects Committees at the WHI Clinical Coordinating Center (FHCRC) and at the 40 WHI Field Centers.

The AAPC project has been approved by the Institutional Review Board of the University of Southern California. The 11 studies contributing to the AAPC each received approval for the use of specimens from their patients.

CARe data set

Affymetrix 6.0 genotyping and QC filtering of African-American samples from the CARe cardiovascular consortium was performed as described previously⁵⁰. After QC filtering for each of ARIC, CARDIA, CFS, JHS and MESA cohorts and subsequent merging, 8,367 samples and 770,390 SNPs remained. To limit relatedness among samples we restricted all analyses to a subset of 5,129 samples in which all pairs have genome-wide relatedness of 0.05 or less and had between 5% and 45% European ancestry. We performed local ancestry inference using the HAPMIX software with the CEU and YRI HapMap populations as reference ancestral populations. We examined seven phenotypes from the CARe cohort, height, body mass index (BMI), log transformed high density lipoprotein cholesterol (logHDL), low density lipoprotein cholesterol (LDL), white blood cell count (WBC), diastolic blood pressure (DBP), and systolic blood pressure (SBP). For each phenotype we included age, sex, study center, proportion of European ancestry, and the top 5 principal components as fixed effects. A detailed description of the phenotypes can be found here⁵¹.

WHI Data Set

Affymetrix 6.0 genotyping and QC filtering of African-American samples from the Women's Health Initiative (WHI) SNP Health Association Resource (SHARe) was performed as described previously⁵². The dataset includes extensive phenotypic and genotypic data on 12,008 African American and Hispanic women aged 50-79 enrolled in one or more components of the WHI program. We included only African American samples and to limit relatedness among samples we restricted all analyses to a subset of 8,153 samples in which all pairs have genome-wide relatedness of 0.05 or less. We performed local ancestry inference using the SABER+²³ software with the CEU and YRI HapMap populations as reference ancestral populations. We examined 10 phenotypes from the WHI cohort (BMI), log transformed high density lipoprotein cholesterol (logHDL), low density lipoprotein cholesterol (LDL), white blood cell count (WBC), log transformed triglycerides (logTG), glucose, log transformed insulin (logInsulin), QT interval duration (QT-INTERVAL), C-reactive protein (CRP), diastolic blood pressure (DBP), and systolic blood pressure (SBP). For each phenotype we included age and proportion of European ancestry as fixed effects. A detailed description of the phenotypes can be found here⁵².

African American Prostate Cancer Data Set (AAPC)

IlluminaHuman1M-Duov3_B genotyping and QC filtering of African-American samples from the African American Prostate Cancer Study (AAPC) from a total of 11 participating studies was performed as described previously^55,53,54. The cleaned dataset includes 9,641 African American subjects and 1,001,899 autosomal SNPs. To limit relatedness among samples we restricted all analyses to a subset of 8,215 samples in which all pairs have genome-wide relatedness of 0.05 or less. We performed local ancestry inference using the RFMix²⁴ with the CEU and YRI HapMap populations as reference ancestral populations. We examined prostate cancer (PC) outcome for each subject. There were 4207 cases and 4008 controls after QC. Due to the admixture signal at the 8q24 locus⁵⁴, we also estimated heritability removing 8q24 from the SNPs used to estimate the kinship (PC|8q24). For each phenotype we included age and the top 10 principal components as fixed effects. For conversion to the liability scale we used a prevalence of 5%⁵⁴.

Partitioning Heritability across the genome

To estimate the heritability for a particular genomic segment we compute the genetic relatedness matrix as defined in Yang et al¹⁰, replacing genotypic for local ancestry calls, and restricting to just those SNPs contained in the region of interest. Given a partitioning of segments along the genome (in our case 22 segments), it is possible to fit them individually or jointly. We attempted both approaches, but found that the joint fit produced a numerical instability in the optimization algorithm preventing convergence. Thus all results reported for the single chromosome analyses are provided by individual and not joint estimates.

We performed both weighted and standard linear regression to assess the relationship between the heritability explained by a chromosome and the length of the chromosome. The weighted version accounts for the differences in number of SNPs contained in longer and shorter chromosomes and the weighting factor we used was the length of the chromosome in centimorgans.