2.1 Ancestral heterogeneity in WHI SHARe, JHS, and COPDGene African Americans

We focus our analyses on three examples of ancestrally heterogeneous admixed populations. Specifically, we consider samples of unrelated African American individuals from the WHI SHARe, JHS, and COPDGene studies. (Note that these studies also include related individuals, Hispanic Americans, and/or European Americans. To facilitate comparisons, we primarily focus our attention on the subset of mutually unrelated African Americans within each study. Methods to identify these subsets can be found in Section 4.1.) Before investigating the performance of PCA in these samples, we first explored the extent of ancestral heterogeneity present in each. In each sample, we estimated genome-wide admixture proportions using model-based genetic ancestry inference approaches. Inferred admixture proportions for these three samples of African American individuals are presented in Fig 1.

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Fig 1. Estimated admixture proportions in three samples of unrelated African American individuals.

Each individual is represented by a narrow vertical bar in the plot, and the individuals are sorted by their estimated proportion of African ancestry. The admixture proportions shown here were estimated using RFMix in WHI SHARe (A) and an unsupervised ADMIXTURE analysis in TOPMed JHS (B) and COPDGene (C).

https://doi.org/10.1371/journal.pgen.1011242.g001

In WHI SHARe, we compared admixture proportion estimates from three techniques. Fig 1A presents admixture proportions estimated as genome-wide average local ancestry, using local ancestry calls from RFMix with two ancestral populations. The reference panel for local ancestry inference consisted of Utah residents with Northern and Western European ancestry (denoted as EUR in Fig 1) and Yoruba in Ibadan, Nigeria (AFR in Fig 1) from the International HapMap Project [66]). These local ancestry based admixture proportion estimates were highly correlated (Pearson correlation > 0.998) with estimated admixture proportions from both supervised and unsupervised ADMIXTURE analyses with two ancestral populations.

In TOPMed samples, we performed unsupervised ADMIXTURE analyses with two ancestral populations. Fig 1B and 1C present results for unrelated African American individuals in JHS and COPDGene, respectively. As these analyses were unsupervised, we cannot be certain which ancestral populations are represented and thus we denote the two populations simply as “Pop 1” and “Pop 2” in Fig 1. However, in comparison to the distribution of admixture proportions seen in WHI SHARe African Americans (Fig 1A) and results from prior studies of admixture in African Americans, we hypothesize that “Pop 1” may correspond to European ancestry and “Pop 2” to African ancestry.

In all three samples, we observed considerable variability in the relative proportions of African and European ancestry across individuals. Thus, these three samples exemplify contexts in which careful adjustment for global ancestry will be necessary when performing GWAS.

2.2 Top PC correlates strongly with global ancestry

In addition to model-based global ancestry inference, we also performed PCA in the three admixed samples and compared results from the two techniques. We considered sets of PCs computed on the full set of variants that passed initial quality control filters, as well as smaller subsets of variants that remained after LD-based exclusions and/or pruning. Methodological details are provided in Section 4.2.2.

In an African American population, we might expect that only one PC is needed to capture ancestral heterogeneity, at least with respect to differences in the relative proportion of African and European continental ancestry. Comparing estimated admixture proportions to PCs in WHI SHARe, JHS, and COPDGene confirmed this hypothesis. In all three samples of African Americans, the first PC was highly correlated with the inferred proportion of African ancestry (Pearson correlation: 0.993–0.998). Later PCs, however, showed very little correlation with genome-wide continental ancestry (Pearson correlation: -0.015–0.034). We observed similar patterns of correlation between PCs and inferred admixture proportions regardless of the type of LD filtering, or lack thereof, performed prior to running PCA (Section D in S1 Text).

2.3 Multiple local genomic features captured by later PCs

In these three African American samples, the first PC was highly correlated with global ancestry, whereas later PCs were not. It is possible that these higher-order PCs may reflect sub-continental structure that is not captured by the estimated admixture proportions. However, in many cases we saw that these later PCs captured local genomic features rather than genome-wide ancestry. This was evident from inspection of SNP loadings—which represent the contribution of each variant to each PC—or, similarly, from investigation of the correlation between PC scores and genotypes.

Fig 2 presents the correlation between the top four PCs and genotypes in JHS and COPDGene African Americans when PCs were generated without any prior LD-based pruning or filtering. (Results for PCs 5–20 can be found in Section A in S1 Text.) We see that variants across the genome contributed relatively equally to the first PC, whereas the second, third, and fourth PCs had out-sized contributions from variants on a select number of chromosomes. In JHS, for example, the second PC was particularly highly correlated with variants on chromosomes 6 and 8, and to a lesser extent with variants on chromosomes 2, 3, and 11. We saw similar patterns, although with peaks on different combinations of chromosomes, in COPDGene (Fig 2B) and WHI SHARe African Americans (Fig 3A). The peaks in these genotype-PC correlation plots indicate that those PCs primarily captured variation at these positions rather than genome-wide global ancestry.

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Fig 2. Correlation between naively generated PCs (i.e., PCs that were constructed without any prior LD-based filtering or exclusions) and genotypes in JHS and COPDGene African Americans.

Each panel plots the absolute value of the correlation between PCs and genotypes (on the y-axis) versus the position along the genome (x-axis). Panels are organized vertically according to which PC is being investigated (1, 2, 3, 4) and horizontally according to the sample (A: JHS, B: COPDGene). Peaks in this plot indicate that a variant has a larger loading, i.e., a larger contribution to that PC.

https://doi.org/10.1371/journal.pgen.1011242.g002

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Fig 3. Correlation between PCs and genotypes in WHI SHARe African Americans with different choices of pre-processing.

Each panel plots the absolute value of the correlation between PCs and genotypes (on the y-axis) versus the position along the genome (x-axis). Panels are organized vertically according to which PC is being investigated (1, 2, 3, 4) and horizontally according to the level of filtering that was applied prior to running PCA (none: all SNPs, exclude: after excluding regions in Table 1, prune: after LD pruning with an r² threshold of 0.1 and window size of 0.5 Mb, and both: after both exclusions and LD pruning).

https://doi.org/10.1371/journal.pgen.1011242.g003

To compare the patterns we saw in these admixed samples to a European population, we also ran PCA in a subset of European American individuals from the COPDGene study. SNP loadings for the first four PCs are presented in Fig C.1 in S1 Text. We see pronounced peaks in the loadings plots particularly for the second and third PC. In both cases, the PCs were highly correlated with variants on a single chromosome (chromosome 11). This pattern of observing a single, pronounced peak in the SNP loadings for a given PC is in agreement with that mentioned in previous studies [37, 44, 49, 53–55, 58–61], but stands in contrast to the pattern of multiple peaks per PC that we observed in the JHS, COPDGene, and WHI SHARe African American samples (Figs 2 and 3A). As we will discuss below, this difference in terms of how many regions of the genome are captured by each PC has distinct consequences for GWAS.

2.4 Impact of LD pruning on PCA

Previous authors have suggested that this phenomenon of PCs capturing local genomic features arises due to high or otherwise unusual patterns of LD among variants; as a result, they recommend that variants in high LD with one another be removed prior to running PCA. Following these recommendations, we compared the set of PCs based on all variants to PCs generated after first removing regions of the genome known to have high LD (Table 1), performing LD pruning, or both.

Table 2 presents the number of variants that remained after each set of pre-processing steps. Removing previously-identified high LD regions cut the number of variants used for PCA by only about 3% across the three samples we considered. LD pruning, with or without additional Table 1 exclusions, reduced the number of variants more dramatically, removing over 90% of the original variants. Despite the smaller number of variants used for PCA, initial PCs were still very highly correlated with model-based estimates of global ancestry (Fig D.2 in S1 Text). The greatly reduced set of variants also offered computational advantages for running PCA.

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Table 2. Number of autosomal SNPs remaining after different combinations of pre-processing steps were applied prior to running PCA.

https://doi.org/10.1371/journal.pgen.1011242.t002

Fig 3 illustrates the impact of these pre-processing steps on the correlation between genotypes and PCs in WHI SHARe African Americans. For the first PC, we see very similar patterns in this SNP-PC correlation regardless of the pre-processing steps used: the four sets of results presented in Panels A–D are very highly correlated (pairwise Pearson correlation > 0.9999). However, we see notable differences across pre-processing choices (i.e., panels A–D) when we consider latter PCs. Panel A presents results for PCs that were generated without any prior LD-based filtering or pruning (i.e., using the entire set of 551,025 variants): In this setting, we see that PCs 2–4 captured local genomic features rather than genome-wide ancestry. Panel B shows what happened when we excluded the ≈ 15,000 variants located in the previously-identified high LD regions reported in Table 1: although the pattern of which SNPs were driving PCs 2–4 changes, the overarching issue of PCs capturing local genomic features was not resolved. However, after LD pruning with an r² threshold of 0.1 and a window size of 0.5 Mb we saw similar patterns with PCs 2–4 as with the first PC: in Panel C, the peaks are gone, with all variants contributing relatively equally to each PC. When we removed previously-identified high LD regions in addition to performing LD pruning (Panel D), we saw very similar patterns of correlation between PCs and genotypes as we did with LD pruning alone (Panel C): the correlation between these two sets of results ranges between 0.99428 (PC 4) and 0.99999 (PC 1).

Although LD pruning proved effective in preventing top PCs from capturing local genomic features in the WHI SHARe dataset, careful consideration was need to identify the optimal the choice of LD pruning parameters. When performing LD pruning using a program such as PLINK [63] or SNPRelate [41], two parameters must be selected: r² threshold and window size. The r² threshold we used in WHI SHARe (r² < 0.1) is stricter than the default for many software programs and the threshold used in many studies of European populations (r² < 0.2). When we used the larger r² threshold of 0.2 in WHI African Americans, we saw improvement for the second and third PCs, but the fourth continued to capture local genomic features (Fig B.1 in S1 Text). In contrast, LD pruning with the default settings (r² < 0.2, 0.5 Mb windows) did alleviate issues in the TOPMed COPDGene European American sample (Fig C.3 in S1 Text). In TOPMed JHS African Americans, both the stricter r² threshold of 0.1 and a wider window size of 10 Mb were needed (Fig B.5 in S1 Text). Yet, even these strict pre-processing settings were not sufficient in TOPMed COPDGene African Americans: peaks still remained in the PC-genotype correlation plots for the second, third, and fourth PCs after removing regions listed in Table 1 and LD pruning with an r² threshold of 0.1 and window size of 10 Mb (Fig B.7 in S1 Text). An even smaller r² threshold did not alleviate these issues (Fig B.8 in S1 Text).

2.5 Adjusting for PCs that capture local genomic features can induce spurious associations

As we have shown above, top PCs in admixed populations may capture multiple local genomic features, and this can occur even when PCs were generated after first removing previously identified high LD regions (Table 1) and/or performing LD pruning, depending on the parameter settings used. If pre-processing steps are not performed, or if default settings are used and PCs are not carefully checked, GWAS models will then be adjusting for PCs that capture these local genomic features instead of genome-wide ancestry. It remains to be fully understood what the downstream implications would be of adjusting for these PCs in GWAS. We derived expected effect size estimates and conducted simulation studies to investigate these implications further.

Fig 4 presents Manhattan plots from one replicate of our simulation study using real genotype data from WHI SHARe African Americans and simulated traits (see Section 4.3.3 for simulation set-up details). In this particular setting, there was a single causal variant on chromosome 4. We compared results from GWAS models using different ancestral heterogeneity adjustment approaches. As expected, we saw extreme inflation, i.e., statistically significant associations on every chromosome, when we did not make any adjustment for ancestral heterogeneity (Fig 4A). When we inferred and adjusted for ancestral heterogeneity using either PCA or estimated admixture proportions, we saw a single peak in our Manhattan plot on chromosome 4—as hoped, given that is where the causal variant was located—with one notable exception. When we adjusted for the first four PCs (as had been done in previous GWAS in WHI SHARe [24, 25]), where those PCs were generated without any prior LD-based pruning or filtering, then we saw a spurious association on chromosome 6 (Fig 4C). However, this spurious association disappeared if we only adjusted for the first of these PCs (Fig 4B). Likewise, no spurious association arose when we adjusted for estimated admixture proportions (Fig 4D) or when we used PCs that were generated after strict LD pruning and Table 1 exclusions (Fig 4E and 4F). On their own, LD pruning proved effective in preventing the spurious association but excluding regions from Table 1 did not (Fig F.2 in S1 Text.). Note that the causal variant, on chromosome 4, and the spurious signal, on chromosome 6, are both located in regions of the genome that are highly correlated with the PCs that were generated without any prior LD pruning (Fig 3). We observed a similar phenomenon in our TOPMed simulations: when GWAS models adjusted for an extraneous, artificially-generated PC that was constructed in such a way that it was highly correlated with the causal variant as well as a variant on a distinct chromosome, a spurious association appeared at that second variant (Section F.2 in S1 Text).

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Fig 4. Manhattan plots from GWAS in WHI SHARe African Americans using different approaches to adjust for ancestral heterogeneity.

In this example, the simulated trait depends only on the genotype at a single variant on chromosome 4. Panels present results using different adjustment approaches: (A) no adjustment; (B) one PC, with PCs calculated using all variants; (C) four PCs, with PCs calculated using all variants; (D) estimated admixture proportions; (E) one PC, with PCs calculated after LD pruning (r² < 0.1, window size = 0.5 Mb) and Table 1 exclusions; and (F) four PCs, with PCs calculated after LD pruning and exclusions. The horizontal dashed line in all panels represents the genome-wide significance threshold of 5 × 10⁻⁸. Note that we have restricted the range of the y-axis to [0, 12.5] to facilitate comparisons across panels and focus attention near the 5 × 10⁻⁸ threshold. There are some SNPs (exclusively located on chromosome 4, in the case of panels B–F) with small p-values that extend beyond the range of the plot.

https://doi.org/10.1371/journal.pgen.1011242.g004

These results are not unique to the choice of causal variant. In Fig 5, we see that adjusting for PCs that captured local genomic features led to higher numbers of spurious associations, on average, across all settings of our simulation using WHI SHARe data. Comparing models that made some sort of adjustment for ancestral heterogeneity, we observed the largest number of spurious associations when GWAS models adjusted for four PCs without any prior LD-based pruning or exclusions (represented by the orange solid line with circles in Fig 5). Excluding the high LD regions from Table 1 prior to running PCA (the orange solid line with triangles) reduced the number of observed spurious associations slightly, but not to the levels of the other approaches. Given what we saw in Fig 3, this is not surprising: even with these exclusions, PCs 2–4 still captured local genomic features—unless those exclusions were also combined with strict LD pruning. On the other hand, when models only included PCs that did not capture local genomic features, the rate of spurious associations dropped. This includes models that adjusted for just the first PC (the green lines in Fig 5) and models that included four PCs, but only after strict LD pruning (the orange dashed lines). Models that adjusted for estimated admixture proportions (the purple solid line) performed nearly identically to models that adjusted for the first PC. This, again, is not surprising, given the high correlation between admixture proportions and the first PC observed in this sample (Figs D.1 and D.2 in S1 Text).

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Fig 5. Comparison of the number of spurious associations in GWAS in WHI SHARe African Americans using different approaches to adjust for ancestral heterogeneity.

Panel (A) displays the average number of spurious associations that were observed across all simulation settings. Remaining panels focus on the subset of simulation settings in which the causal variant has (B) a small difference in ancestral allele frequencies, (C) low SNP loadings for each of the first four PCs, (D) a high SNP loading for at least one of the first four PCs, or (E) the highest SNP loading on its chromosome for one of the first four PCs. Within each panel, we compare the number of spurious associations when GWAS models adjust for estimated admixture proportions, 1 PC (with or without LD pruning and/or Table 1 exclusions), or 4 PCs (with or without LD pruning and/or Table 1 exclusions). Results shown here are for simulated traits with a single causal variant with effect size (β) ranging from 0 to 4.

https://doi.org/10.1371/journal.pgen.1011242.g005

2.6 Factors that influence the rate of spurious associations

Our simulation results highlight various factors that influence when, and how many, spurious associations arise when adjusting for PCs that capture local genomic features. First, we note that there were very few spurious associations, regardless of the adjustment approach, when there were small differences in ancestral allele frequencies at the causal variant (Fig 5B). Even models that made no adjustment for ancestry whatsoever performed similarly in this setting (Fig F.1B in S1 Text). This is to be expected: in this scenario, the causal variant was not associated with global ancestry, so global ancestry was not a confounding variable and thus adjustment was not needed. Considering other simulation settings in which the causal variant had a larger difference in ancestral allele frequencies (Fig 5C, 5D and 5E), and thus adjusting for ancestral heterogeneity was needed, the number of observed spurious associations remained low for models that adjusted for admixture proportions, a single PC (regardless of pre-processing), or four PCs—if those PCs were generated after strict LD pruning. For the two models that adjusted for PCs capturing local genomic features (i.e., the models that adjusted for 4 PCs that were generated with or without Table 1 exclusions, but no LD pruning), however, we saw a higher rate of spurious associations, particularly when the causal variant was highly correlated with one of those PCs. Notably, as the size of the causal variant’s SNP loading increased from low (Fig 5C), to high (Fig 5D), to the highest on its chromosome (Fig 5E), we saw an increasing number of spurious associations for these two approaches. This mirrors the pattern we saw in Fig 4, where a spurious association arose when we adjusted for PCs that were highly correlated with variants in several regions across the genome, and both the causal variant and spurious signal were located in one of those regions. Finally, we note that these problems worsened as the effect size of the causal variant increased.

2.6.1 Connecting simulation results to theory.

The patterns observed in our simulation studies are in agreement with our work to derive the expected effect size estimates for GWAS models using different techniques to adjust for ancestral heterogeneity. The setup, assumptions and derivations for this theoretical work are presented in Section 4.3.2 and Section E in S1 Text. In brief, we derived the expected effect size estimate () for each model and compared this to the true effect size (β). These derivations shed light on the situations in which GWAS models yield biased estimates and the factors that impact the magnitude of this bias.

Fig 6 summarizes our findings about GWAS models that do not make any adjustment for ancestral heterogeneity. We see that these unadjusted models yield a biased estimate of the effect size of the causal variant () unless there is no ancestral heterogeneity (i.e., Var(π) = 0), global ancestry does not have a direct effect on the trait (i.e., β_π = 0), or the causal variant has the same allele frequencies in the two ancestral populations (i.e., p₁₁ = p₁₀). We see, also, that the unadjusted model can yield a biased effect size at the unlinked neutral variant () even if global ancestry does not have a direct effect on the trait, provided that there is ancestral heterogeneity (Var(π)>0) and both the causal variant and the variant being tested have allele frequencies that differ between the two ancestral populations (i.e., p₁₁ ≠ p₁₀ and p₂₁ ≠ p₂₀). These biased effect size estimates at neutral variants will translate into spurious associations as sample sizes increase, just as we saw in our simulations when we used models that did not adjust for ancestral heterogeneity in any way (Fig 4A and Fig F.1 in S1 Text).

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Fig 6. Directed acyclic graphs (DAGs) summarizing the conditions for confounding by global ancestry in GWAS.

Each circle represents a variable: y is the trait; π is global ancestry; g₁ is the genotype of Variant 1, the causal variant; g₂ is the genotype of Variant 2, a second variant that is not associated with the trait and is unlinked with Variant 1. Dashed and solid lines represent relationships between variables, with arrows indicating the direction of the relationship. We used a solid line to indicate a relationship we assume to be present and a dashed line to indicate a relationship that may be present depending on whether certain conditions are met; those conditions are indicated by the text above the line. The question marks indicate the relationships we are interested in testing. On the left, we see that global ancestry confounds the association at the causal variant (Variant 1) if there is ancestral heterogeneity in the population (V_π > 0), the causal variant has different allele frequencies in the ancestral populations (p₁₁ ≠ p₁₀), and global ancestry has a direct effect on the trait (β_π ≠ 0). On the right, we see that global ancestry can confound the association at an unlinked neutral variant (Variant 2) even if global ancestry does not have a direct effect on the trait (β_π = 0), provided that there is ancestral heterogeneity (V_π > 0) and both the causal variant and the variant being tested have different allele frequencies in the ancestral population (p₁₁ ≠ p₁₀, p₂₁ ≠ p₂₀).

https://doi.org/10.1371/journal.pgen.1011242.g006

In contrast, our theoretical work shows that models that appropriately adjust for ancestral heterogeneity will yield unbiased estimates of the effect size at the causal and unlinked neutral variants (Equations E.3 and E.4 in S1 Text). At the causal variant, the same holds true even when models adjust for an extraneous PC (Equation E.5 in S1 Text). However, at the unlinked neutral variant these same models will see effect size estimates that are biased away from zero when there is ancestral heterogeneity (i.e., Var(π) > 0) and the model includes an extraneous PC that is correlated with both the causal variant and the variant being tested (i.e., Cov(g₁, z ∣ π) ≠ 0 and Cov(g₂, z ∣ π) ≠ 0)) rather than global ancestry. In other words, these results indicate that if a model adjusts for a PC that is correlated with the causal variant as well as a second variant that is not associated with the trait, then spurious associations will arise at that second neutral variant in large enough samples. This is exactly what we observed in our simulations (Figs 4C, 5D and 5E). However, if the extra PC is not correlated with the causal variant, then spurious associations will not arise; again, this is in agreement with our simulation results (Figs 4F and 5C).

3.1 Importance of adjusting for ancestral heterogeneity

We observed considerable variability in global ancestry proportions across all three admixed populations studied in this paper: the Women’s Health Initiative SNP Health Association Resource (WHI SHARe), Trans-Omics for Precision Medicine Jackson Heart Study (TOPMed JHS), and TOPMed Genetic Epidemiology of Chronic Obstructive Pulmonary Disease Study (COPDGene) African Americans. It is widely understood that adjusting for this ancestral heterogeneity in GWAS is needed in order to control for potential confounding by global ancestry and the spurious associations that can arise as a result. As we showed above, this confounding can occur even when global ancestry does not have a direct effect on the trait itself, provided that there is a causal variant elsewhere in the genome that has different allele frequencies across the ancestral populations of interest. Although this fact has been recognized previously (e.g., [67]), it is sometimes overlooked. Our theoretical work (Equations E.1 and E.2 in S1 Text) illustrates the factors that impact the magnitude of the bias incurred by GWAS models that fail to adjust for global ancestry. We hope that our results will serve as a reminder to researchers of the various ways in which global ancestry can confound genetic studies in admixed populations and the importance of ensuring that GWAS models appropriately adjust for ancestral heterogeneity.

3.2 PCs can induce collider bias

A common approach for adjusting for ancestral heterogeneity in GWAS involves including global ancestry as a covariate in marginal regression models, with global ancestry estimated using either model-based approaches or PCA. In WHI SHARe, TOPMed JHS, and TOPMed COPDGene African Americans, the first PC was highly correlated with estimates of the genome-wide proportion of African ancestry and models adjusting for either performed similarly. Later PCs, however, did not correlate with global ancestry and models that adjusted for these PCs anyway—a common practice in the literature—yielded elevated rates of spurious associations, particularly when those PCs captured local genomic features rather than genome-wide ancestry. Prior work (e.g., [53, 55]) has shown that PCs can detect regions with high, extensive, or otherwise unusual patterns of LD (Table 1), but the patterns observed in those studies—primarily involving individuals of European ancestry—differ from what we saw here. In particular, those studies typically saw PCs that were driven by variants in a single high-LD region, just as we saw in TOPMed COPDGene European Americans (Fig C.1 in S1 Text). However, in all three of the admixed samples that we investigated, we saw instead that PCs were correlated with variants in multiple regions, across multiple chromosomes. This has important downstream implications.

Our work shows that adjusting for PCs that capture multiple local genomic features can induce spurious associations. This issue can be explained by the concept of collider bias (Fig 7). If a PC is correlated with the genotype of multiple variants, and at least one of those variants is associated with the trait, then the PC becomes a collider variable when testing the association between the other variants and the trait. Adjusting for collider variables can induce a spurious association between variables that are otherwise unlinked [68]. This is precisely what we saw above. GWAS models adjusting for an extraneous PC will yield biased estimates of variant effect sizes, with the magnitude of that bias increasing with the effect size of the causal variant, the strength of the correlation between the PC and the variants it captures, and the amount of ancestral heterogeneity within the population (Equation E.6 in S1 Text). When that bias is large enough, it can lead to a spurious association—as our simulations show. Given the similarities in terms of which regions of the genome tend to be correlated with PCs across different admixed populations (see Figs 2 and 3, for example), it is possible that these spurious associations may even replicate across studies.

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Fig 7. Collider bias in GWAS.

Suppose that, instead of genome-wide global ancestry (π), a PC (z) captures the genotype of two variants (g₁, g₂). If one of those variants (g₁) is associated with the trait (y), then the PC will be a collider variable—rather than a confounding variable—when testing the association between the other variant (g₂) and the trait. Adjusting for the PC can then induce a spurious association between the trait and the neutral variant as a result of collider bias.

https://doi.org/10.1371/journal.pgen.1011242.g007

The focus in the literature has typically been on demonstrating issues that can arise when adjusting for too few PCs [7, 18, 47, 48, 69]. Prior discussion of the impact of adjusting for PCs capturing local genomic features have focused on persistent inflation due to sub-optimal ascertainment of population structure [44, 46, 49, 53, 54, 59, 60] and decreased power when a causal variant is located in one of the regions highly correlated with a PC [44, 46, 48, 53–55, 61, 62]. In studying the impact on type I error, many of these studies focus on the inflation factor λ [44, 49, 59, 61]. Although useful for detecting widespread inflation, our simulations suggest that inflation factors are not sensitive to the presence of spurious associations arising from collider bias (Section F.2 in S1 Text). This could partially explain why the issue has not been detected previously. Furthermore, the vast majority of these studies focused on European populations, where PCs are typically correlated with variants on just a single chromosome (e.g., Fig C.1 in S1 Text) and thus collider bias is not a concern. Those studies that did see multiple peaks in SNP loading plots (e.g., [53, 55, 59]) did not make any mention of its implications. We did identify one prior study [48] that investigated the performance of PCA in a simulated admixed population; they saw slightly higher type I error rates when using 10 PCs instead of 2 (consistent with our findings), but the authors did not provide any discussion of what those later PCs might have captured or what may have caused the elevated type I error rate.

In recent years, increasing attention has been paid to the issue of collider bias in genetic association studies [70–73], but to the best of our knowledge this is the first paper to fully demonstrate the concerns related to including extraneous PCs in GWAS models. Most closely related to our work, Dahl et al. [74] showed that adjusting for PCs can lead to replicating spurious associations in gene expression studies due to collider bias. As a short aside, the authors mentioned that collider bias could arise in GWAS, but they implied that the magnitude of this bias would be small—and thus not of practical concern—given that “causal SNPs have much lower leverage on genetic PCs” (i.e., the correlation between PCs and causal variants is small). Our findings are consistent with their conclusion that the magnitude of collider bias depends on the strength of the correlation between the PCs and the variants they capture. However, we found that this correlation between PCs and genetic variants can be non-trivial in admixed populations (without careful pre-processing prior to running PCA), implying that collider bias is of more concern in this setting. We also showed that the amount of bias depends on the variance of admixture proportions across a population (denoted V_π in Equation E.6 in S1 Text), which could explain why work that has focused on more ancestrally homogenous populations may not have identified this issue previously. That said, it is important to raise the question of the magnitude of bias that could be expected in a “typical” GWAS. The effect size of the simulated trait underlying the analyses in Fig 4 is useful for illustrative purposes but larger than would be typical for a real, complex trait/disease. Across other simulation settings, we observe that the smaller the effect size of the causal variant, the smaller the number of spurious associations that we observe (Fig 5). However, in these settings we would also expect that there are multiple causal variants—not just a single causal variant, as we assumed for the sake of simplicity in our simulations and theoretical work—and then these effects may be aggregated [75]. It is also important to acknowledge that the impact of including PCs that capture local genomic features extends beyond spurious associations. Our theoretical results (particularly, Equations E.6 and E.8 in S1 Text) illustrate the expected bias of effect size estimates from GWAS models adjusting for extraneous PCs. If the causal variant has a smaller effect size, this bias may not be large enough to yield a spurious statistically significant hypothesis testing result, but it will still have adverse consequences on downstream analyses such as designing appropriately-powered replication studies (akin to concerns about “winner’s curse” [76, 77]) and constructing polygenic risk scores [37]. Of course, the collider bias we are highlighting here is not the only potential source of bias for genome-wide association studies. Just as work has been done to draw attention to, prevent, and correct these other sources of bias, we believe it is likewise important to consider what steps can be taken to reduce or eliminate concerns about potential collider bias.

3.3 Impact of LD-based pre-processing

In our analysis of genotype and sequence data from unrelated WHI SHARe, TOPMed JHS, and TOPMed COPDGene African Americans, we found that all but the first PC were correlated with small regions of the genome—and thus have the potential to be collider variables—unless careful pre-processing of genotype data was performed prior to running PCA. As mentioned earlier, previous studies have found that PCs can be driven primarily by small regions of the genome, and as a result have suggested that these regions be excluded (Table 1) and/or that LD pruning be performed prior to running PCA. However, the motivation for this LD-based filtering has typically been framed in terms of the ability of the PCs to capture global ancestry, as well as the computational complexity of running PCA, rather than the downstream implications on association testing that we have highlighted here. Furthermore, we found that excluding the regions listed in Table 1, without also performing LD pruning, did not solve these issues in admixed populations: we still saw peaks in the PC-genotype correlation plots and models adjusting for these PCs continued to show elevated rates of spurious associations. A more tedious iterative approach of identifying and removing potentially problematic regions based on our own data (similar to that proposed by [60]) also did not prevent PCs from capturing local genomic features within a reasonable number of iterations (Fig B.3 in S1 Text). LD pruning, on the other hand, proved to be more successful, at least when looking at the first four PCs in WHI SHARe African Americans. Our results highlight that a stricter threshold (e.g., r² = 0.1) may be needed for LD pruning in admixed populations than the r² = 0.2 threshold that is often suggested in the literature: see Section B in S1 Text for a comparison. Given that LD patterns differ between admixed populations and the European populations upon which much of this prior work was based, it is not surprising that different LD-based filtering techniques are required here.

It is important to acknowledge, however, that LD pruning with the parameters we used in WHI SHARe African Americans (r² = 0.1, window size = 0.5 Mb) may not be a universal solution. For example, in TOPMed JHS African Americans, LD pruning with these parameters partially, but not completely, resolved the issue of PCs being highly correlated with variants on multiple chromosomes. Comparing Fig 2A to Fig B.4 in S1 Text, we saw improvement for the second and third PCs, but there continued to be some peaks (albeit shorter) in the PC-genotype correlation plot for the fourth PC. It was not until we used a wider window size of 10 Mb that the peaks disappeared (Fig B.5 in S1 Text) for all of the top four PCs. In TOPMed COPDGene African Americans, even this wider window size was not sufficient in preventing PCs from capturing local genomic features (Fig B.7 in S1 Text).

3.4 Generalizability of findings

Our analyses in this paper have focused on three samples of African American individuals, but we believe that our findings generalize more broadly. We have, in fact, observed similar patterns of PCs capturing multiple local genomic features through our involvement in a study including African American, African Caribbean, and Hispanic/Latino TOPMed participants [78], as well as a study focused on individuals of Mexican descent [79]. For example, in [79] a larger number of PCs (three) reflected population structure than we observed in WHI SHARe, JHS, and COPDGene African Americans. After this third PC, however, we see similar patterns in terms of higher order PCs capturing multiple local genomic features (see Supplementary Figs 6–8 in [79]). Pre-processing alleviated some of these concerns: with an r² threshold of 0.005, peaks disappeared from the SNP loadings plots for the 4th–7th PCs, although latter PCs continued to exhibit evidence of capturing local genomic features. Our theoretical work suggests that GWAS including any of these higher order PCs would be susceptible to collider bias. That said, we have not conducted an exhaustive study of all admixed populations. It is also important to acknowledge that patterns, implications, and ideal solutions may differ in other non-European populations or in trans-ancestry analyses.

Results may also be impacted by the choice of PCA software. We investigated the impact of algorithm choice when using the snpgdsPCA() function in the SNPRelate package [41]—specifically comparing the default “randomized” approach of [58] versus a more traditional “exact” approach—and saw very little difference in our results (Pearson correlation between comparable sets of results > 0.97). However, we have not performed an exhaustive investigation of all PCA software. Results may also be affected by the choice of which, if any, reference samples are included in the PCA analysis. We focus here on unsupervised PCA analyses (i.e., PCA was run on admixed individuals only, without any reference samples) as this is commonly done throughout the literature and presents one advantage of PCA over other (supervised) ancestry inference approaches. However, prior work has shown that PC scores can be sensitive to the choice of reference populations included in the analysis (see Fig 13 in [80]), and thus is yet another aspect of PCA that should be carefully considered by investigators. Thus in any sample (admixed or otherwise) and for any PCA software (SNPRelate or otherwise) and associated analysis choices (pre-processing, inclusion of reference samples, etc.) we strongly recommend that investigators carefully investigate PCs before including them in GWAS models. If multiple variants are associated with a PC and sample size is sufficiently large, spurious associations can arise. In some samples, it may be necessary to consider stricter LD pruning (e.g., smaller r² threshold, larger window size), to include fewer PCs, to reevaluate the choice of included reference samples (or lack thereof), or to consider an alternate approach altogether (e.g., adjusting for estimated admixture proportions).

3.5 Future work

This work raises a number of questions that warrant further exploration. We, like others before us, have clearly documented that PCs can capture local genomic features. The reason for this phenomenon has previously been attributed to LD [52, 53, 55, 60]. We have strong reason to suspect the same is true for our results: the peaks we see in the PC-genotype correlation plots for WHI SHARe, JHS, and COPDGene African Americans frequently overlap with known high LD regions (Table 1) and LD pruning proves largely effective in alleviating this concern. However, it would be interesting to further investigate the underlying biological and statistical mechanisms for these patterns. For example: why, in admixed populations, do we more frequently see that a single PC captures local genomic features on multiple chromosomes? There is further work that could be done here to more deeply understand these patterns and, in turn, develop alternate strategies for dealing with them besides (or in addition to) LD pruning. In addition, the sensitivity of mixed model approaches to long-range LD and other artifacts or biological features causing the issues we see here would be interesting to investigate further. The work of Day et al. [71] show that using a mixed model in and of itself does not eliminate concerns about collider bias. However, if a mixed model uses only random effects to capture distant relatedness (i.e., shared ancestry) between samples without also including PCs (i.e., potential collider variables) as fixed effects, could issues still arise? Further work is needed to answer this question.

3.6 Conclusions and recommendations

Ultimately, our work demonstrates the challenges that can arise in appropriately adjusting for ancestral heterogeneity in admixed populations. Compared to prior work in European populations, we see that the patterns of which and how many local genomic features are captured by PCs differs, leading to distinct downstream implications for GWAS, namely, collider bias. The previously recommended pre-processing step of removing known high LD regions (Table 1) shows limited utility in admixed populations, and the effectiveness of LD pruning is sensitive to the choice of parameters. These results are particularly concerning given the wide-spread use of PCA in the GWAS literature, and the inconsistent use and reporting of pre-processing steps.

Inspired by our findings above, we conclude with the following reminders and recommendations for genetic association studies in ancestrally heterogeneous populations.

1. Association models must carefully account for ancestrally heterogeneity. Even if global ancestry does not have a direct effect on the trait, it can still confound genetic association studies.
2. For populations where we have a good idea of the number of ancestral populations of interest and relevant reference panel data is readily available, GWAS models adjusting for estimated global ancestry proportions (rather than PCs) can perform well.
3. PCA can offer advantages over model-based ancestry inference methods. However, careful consideration must be given to how many PCs should be included and what those PCs are capturing. Specifically:
   1. (a). More PCs is not always better. In the African American samples studied here, for example, a single PC was sufficient for controlling spurious associations induced by population structure. In some simulation settings, we did see a small drop in the number of spurious associations when we included three additional PCs, but only after strict LD pruning.
   2. (b). Investigators must carefully check PCs before including them in GWAS models. We recommend calculating and plotting SNP loadings or the correlation between PCs and genotypes to assess whether PCs are capturing local genomic features rather than genome-wide ancestry. Inflation factors cannot reliably detect these issues.
   3. (c). Careful pre-processing of data, with a particular attention to LD, is critical. This may include a combination of LD pruning, excluding known high LD regions (Table 1), or other techniques that account for LD [53, 60]. In admixed populations, more stringent r² thresholds and wider window sizes may be required compared to the default settings for many software packages that implement LD pruning. If issues persist, it may be necessary to consider including fewer PCs or using an alternate approach altogether.
4. Publications, presentations, and other reporting of association study results must transparently describe and justify all statistical methods used for inferring and adjusting for ancestral heterogeneity. This should include the choice of which and how many PCs to include, as well as the steps taken to ensure that those PCs are not causing the very problem—spurious associations—that these techniques aim to solve.

4.1 Data and quality control

Our analyses focus on array-based and whole genome sequence-based genotype data from three samples of African American individuals. In particular, we consider genotype data from the Women’s Health Initiative SNP Health Association Resource (WHI SHARe), as well as whole genome sequencing data from two contributing studies to the Trans-Omics for Precision Medicine (TOPMed) Whole Genome Sequencing Project: the Jackson Heart Study (JHS) and the Genetic Epidemiology of Chronic Obstructive Pulmonary Disease Study (COPDGene). We performed quality control and identified subsets of unrelated African American individuals prior to running any further analyses—details are provided below. All analyses are restricted to the autosomes.

4.2 Genetic ancestry inference

We consider two approaches to inferring genetic ancestry in these admixed samples: model-based approaches and PCA.

4.3 Investigation of downstream implications for GWAS

To explore the impact of adjusting for PCs that capture local genomic features, as well as to further evaluate the utility of the proposed solutions of LD pruning and removing known high LD regions (Table 1) prior to running PCA, we derived expected effect size estimates and conducted simulation studies comparing the rate of spurious associations for GWAS models adjusting for different sets of covariates.