3.1 Ancestry affects the correlation in state within a locus

Imagine a collection of individuals who differ in their ancestry. We can think of this collection as a population, but we do not mean to assert anything in particular about breeding within this collection. Pick an individual $i$ at random from this collection. Individual $i$ has two alleles at our diploid locus $v$. Call those alleles $A_{v_i}$ and $A^*_{v_i}$. Let $q_{v_i}$ be the frequency of the $A_{v_1}$ allele among individual $i$'s ancestors. Let $C_{v_i}$ be the covariance in allelic state for our chosen individual $i$. Consider $C_v$ to be the covariance in allelic state in the entire collection of individuals from whom we picked individual $i$. To find this, we randomize over the choice of individual $i$ and use the law of total covariance in equation . In this collection of individuals,

Thus, in this collection of individuals, the covariance in allelic state is the average covariance in allelic state within individuals, plus the variance in allele frequency across the ancestries of the individuals in the collection. Notice that we now think of $q_{v_i}$ as the frequency of the $A_{v_1}$ allele in the ancestors of $i$, a personalized allele frequency for individual $i$, if you will [13], and we can sensibly discuss $\mathrm{Var}[q_{v_i}$] as the variation in this personalized frequency across individuals. Similarly we think of $C_{v_i}$ as a property of $i$'s immediate ancestry. Are the parents of $i$ related to one another? Is there some form of assortative mating occurring between $i$'s parents? Any sort of correlation between genotype and the identity of $i$'s parents (inbreeding / assortative mating) will lead to $\mathrm{E}[C_{v_i}$] being non-zero. Modeling these factors' effect on phenotype might conveniently be done here. If we assume that within individuals there is no covariance in allelic state, i.e., that individuals are not inbred or the result of assortative mating, in a formal sense assuming that the two alleles are independent within an individual, then we arrive at the simplified

Since variances are non-negative, we see that absent assortative mating (or some other factor that would create negative covariance in allelic state within individuals), this collection of individuals who vary in their ancestry will have a positive $C_v$, and a deficit of heterozygotes relative to HWE. This is essentially a restatement of the Wahlund effect [14], albeit without explicit reference to discrete populations. A somewhat more modern and familiar quantification of this concept is as Wright's $F_{ST}$ [15],

where $\mathrm{E}[q_{v_i}$] is the average frequency of the $A_{v_1}$ allele in the ancestors of this collection of individuals. In the Wrightian conception, we might identify $E[C_{v_i}$] as proportional to $F_{IS}$. There is a long history in human genetics of quantifying the variation in allele frequency across ancestry in terms of $F_{ST}$ [16-18], and it is well established that in humans, when ancestry varies across different continents, $F_{ST} \approx 0.1-0.2$. When ancestry varies widely within a single continent, $F_{ST} \approx 0.05-0.1$, and it tends to be less across smaller geographic regions. In the population genetics context, it is widely understood that variation in allele frequency due to ancestry leads to departures from HWE. In the quantitative genetics context, it is less widely appreciated that variation in allele frequency due to ancestry (Figure 1) can play havoc with everything, including creating dominance variance and making the additive variance larger than the total genetic variance, because negative covariance is created between the additive and dominance contributions. This is the first, albeit minor, reason all quantities in human genetics should be estimated with some procedure (linear mixed model / inclusion of PCs as covariates, etc.) that attempts to condition on the ancestry of the studied individuals. Variation in allele frequency within a single locus does not, though, create "false positive" association. From equation above, we have $\beta = \tilde{\beta}( 1 + F_{ST})$. If we interpret effects in the Kempthorne sense, we would say the effect size differs between a population in HWE versus one out of it. In the Falconer sense, where we might think of $\tilde{\beta}$ as the "true" $\beta$, we would say that departures from HWE inflate our estimate of $\tilde{\beta}$. Of course, if $\tilde{\beta}$ is zero, this has no effect on association test statistics. However, variation in ancestry has effects on estimates of $\beta$ that go far beyond those of departures from HWE.

3.2 Ancestry affects the correlation in state between loci

All effect sizes in human genetics should be estimated via a method that accounts for individuals' ancestry - because variation in ancestry creates correlation in state between every site in the genome. This correlation in state between loci caused by variation in ancestry among study participants causes substantial false positive association in genetic association studies if uncorrected. To understand why this is, consider two diploid loci, $v$ and $w$. Let $\mathrm{I}_{A_v}$ and $\mathrm{I}_{A_w}$ be the indicators that a randomly picked allele is $A_{v_1}$ and $A_{w_1}$, respectively. Recall [6] that the standard measures of linkage disequilibrium $D$ and $r^2$ can be written as

Now imagine a collection of individuals whose ancestry differs from one another. Each individual $i$ in this collection has some ancestry where $q_{v_i}$ and $q_{w_i}$ are the frequency of the $A_{v_1}$ and $A_{w_1}$ alleles among $i$'s ancestors. Let $D_i$ be the standard measure of linkage disequilibrium between the loci found in individual $i$, a personalized linkage disequilibrium for $i$.

Note that equations and follow from the law of total expectation, equation uses the definition of variance ( $\mathrm{Var}[X] = \mathrm{E}[X^2] - \mathrm{E}[X]^2$), and equation uses the law of total covariance. Thus, we arrive at the result that $D$ in the collection of individuals is the average $D$ within individuals, plus the covariance in allele frequencies over ancestries [19]. If loci $v$ and $w$ are not physically linked (loci on different chromosomes, say), then we expect that $E[D_i] = 0$, and this simplifies to

We conclude that when a group of individuals varies in ancestry, there will appear to be linkage disequilibrium (covariance in allelic state between loci) at any pair of loci that vary in frequency over ancestry, even if those loci are physically unlinked. From the fact that all correlations are bound between $-1$ and $1$, we know that

where $F_{{ST}_v}$ and $F_{{ST}_w}$ are Wright's measures of $F_{ST}$ at locus $v$ and $w$ respectively. Thus, $r^2$ between unlinked loci is bounded by the product of $F_{ST}$ at the individual loci. If in this collection of individuals, there are only two different ancestries with any appreciable difference in allele frequency (the precise situation treated in [19]), then equality necessarily holds and $D = Cov[q_{v_i},q_{w_i}] = \pm \sqrt{Var[q_{v_i}]Var[q_{w_i}]}$ regardless of the proportions of those ancestries (e.g., the respective percentages of European and African ancestry, say). To see this, call $q_{v_1}, q_{w_1}$ and $q_{v_2}, q_{w_2}$ the allele frequencies in the two different ancestries, and let $c_1$ and $c_2$, with $c_1 = 1 - c_2$, be the fraction of individuals in this collection whose ancestries are from each type. Let $\delta_{q_v} = q_{v_1} - q_{v_2}$ and $\delta_{q_w} = q_{w_1} - q_{w_2}$.

Recall that we assumed $v$ and $w$ are independent loci (unlinked). Thus, in any collection of individuals, when there are only two ancestries significantly contributing variance in allele frequency, $D$ between every pair of sites in the genome will be approximately plus-or-minus the square root of the product of the variance in allele frequency at each site. These variances are defined relative to the ancestry of the individuals in the collection under consideration. Differing individuals with differing ancestry necessarily have differing variance in their allele frequency.

Human population structure can be a major cause of "false positive" association between genetic variation and phenotype whenever there is allele frequency variation in the ancestors of the individuals examined. This has been known for decades, motivating the development of a large range of complementary methods to correct for these effects (e.g., TDT, genomic control, structure, principal component analysis, linear mixed models) [20-24]. To understand more quantitatively exactly what is occurring, imagine attempting to estimate the contribution of locus $v$ to phenotype, and describing that effect as $\beta_v$. Let $\tilde{\beta}_v$ be the effect of this locus if there were no variation in ancestry among the studied individuals. Thus, $\tilde{\beta}_v$ is the effect of this locus in an idealized collection of individuals without variation in ancestry. $\beta_v$ is the effect in the real collection under study (the true effect is $\tilde{\beta}_v$, versus the estimated effect, $\beta_v$, in the Falconer sense). Finally assume that $\tilde{\beta}_v = 0$. So, in the absence of variation in ancestry, this locus has no effect on phenotype. In the first paper in this series [6], we showed that for any pair of additive loci $v$ and $w$, in the absence of any interactions other than correlation in allelic state

where $D_{v,w}$ is the standard linkage disequilibrium measure between loci $v$ and $w$. Now imagine the collection of all loci $w_k$ contributing an additive effect, $\tilde{\beta}_{w_k}\ne 0$, in the idealized population, but in this case we are imagining that $\tilde{\beta}_v = 0$. The total additive variance in the idealized population is

Now consider the combined effects of all loci $w_k$ on $\beta_v$ in a population with variation in ancestry.

when only a small number of ancestries contribute significantly to variation in allele frequency. This value is necessarily bigger than zero, if there is any variation in $q_v$ over ancestries. Recall that a natural method to test for association between locus $v$ and phenotype would be to asses a null model where $2Np_vq_v\beta_v^2$ is approximately $\chi^2_1$ distributed, where $N$ is the number of individuals studied [25].

We thus arrive at the full intuition for the need to account for ancestry in human genetic studies. A naive $\chi^2$ test for association between locus $v$ and phenotype will not have an expectation centered at $1$, as the test assumes, but instead centered around a larger value that is a complex function of the covariance in allele frequencies and $V_A$. This inflation will occur at every site in the genome whose allele frequency varies over ancestry, i.e., potentially every site tested. While nearly all human geneticists understand that tests for association must account for ancestry, the reason for this is often only incompletely understood. Many discussions of this revolve around the idea that the mean phenotype (or disease prevalence for dichotomous traits) may differ between ancestries, and this difference in means causes correlation between ancestry and phenotype. Other discussions might suggest a "mechanism" for a correlation between phenotype and ancestry involving differing environmental states across ancestries, perhaps differing diets, rates of smoking, or other social determinants of health. While correlations between genes and mean phenotype or between genetic and environmental state can cause inflation of test statistics, they are in no sense required for the inflation to be pronounced. If there is additive variance for the trait, and if any of the alleles contributing to that variance differ in frequency among the ancestors of the study participants, then there will be test statistic inflation, and that inflation will increase with increasing sample size, $N$, and with increasing variance in allele frequency among the participants, $F_{ST}$. When effects are defined as the expectation of phenotype given the context in which they occur (i.e., in a Kempthorne-inspired presentation), this is not "inflation" per se, but real association between marker and phenotype, though the cause of the association is not the direct effects of the tested marker. Instead, the detected association is caused by the correlation in state between the tested marker and other variants scattered throughout the genome, i.e., the association is positive indication of the interaction—the correlation in state—between this marker and other unlinked sites. Given that the goal of association studies often is to identify loci that putatively have causal influence on phenotype, this phenomenon of association caused by interactions due to correlation in allelic state is nearly universally referred to as an issue of test statistic inflation, not one of "true" interaction, and we shall do the same.

Patterns common in disease genetics and human demographic history accentuate this inflation. The key aspect to consider is the sign of $\mathrm{Cov}[q_v,q_{w_k}] \beta_{w_k}$. Since the signs of both $\mathrm{Cov}[q_v,q_{w_k}$] and $\beta_{w_k}$ could be positive or negative, one might naively assume (hope) that the terms in the sum above will alternate signs sufficiently frequently that the mean over all SNPs might approach $0$. This is unlikely to be true for disease traits. Recall we have oriented $\beta$ so that it is the effect of substituting the rarer allele for the more common allele. If $\beta > 0$ then the rare allele makes the trait bigger, or in the case of a disease trait, increases the risk of disease relative to the common allele. For non-disease phenotypes there may be little reason to doubt that the sign of $\beta$ is "random". However, for disease traits there is good reason to believe that on average $\beta$ will be positive [13], because selection is likely to keep alleles rare if they increase the risk of a disease. This is especially likely if the allele has a strong effect on disease. Human demographic history suggests that $\mathrm{Cov}[q_v,q_{w_k}$] will also be positive most of the time. In general, a positive covariance between two variables implies that larger values at both variables occur more often than expected by chance. In this context it means the ancestry with the higher minor allele frequency at one locus tends to have the higher minor allele frequency at the other. In the second paper in this series [13], we saw that human demographic history tends to cause this exact pattern of correlation (conditional on alleles being present in two ancestries, one ancestry has systematically higher minor allele frequencies than the other). In that paper, the correlation in minor allele frequencies across unlinked sites appears to be the likely explanation for the most perplexing observations when applying polygenic risk scores across populations with differing ancestry. Here, the correlation in minor allele frequency induced by human demographic history is seen as the primary driver of test statistic inflation.

In this context, we can best understand how and why various procedures designed to account for this inflation work. Perhaps the most intuitively simple, but complex algorithmically, is the STRUCTURE / STRAT approach developed by Jonathan Pritchard and colleagues [22,26]. The idea is simple, variation in ancestry creates deviations from HWE within loci as well as correlation in allele frequency between loci, therefore, in principle, one can infer ancestral groups from the data [22] and then conditional on the inferred ancestry, perform association studies within ancestry groups in HWE [26]. In the structured association approach developed by Pritchard and colleagues, individuals are assigned to a subpopulation (possibly accounting for admixture with fractional cluster membership) using a model-based clustering program (STRUCTURE) [22] and association statistics are computed stratifying by subpopulation (STRAT) [26]. The approach is natural, but extremely computationally challenging for large sample sizes, and a bit sensitive to a correct a priori determination of the number of "meaningful" ancestral groups contributing to the collection.

At the other end of the computational spectrum is genomic control, pioneered by Devlin and Roeder [20]. Here the approach is incredibly simple, calculate the mean (median) $\chi^2$ test statistic, usually called $\lambda$, from a genome-wide study and divide each site's $\chi^2$ by $\lambda$. The intuition comes from the fact that the number of SNPs contributing to additive variance is likely very large. If so, it is not unreasonable to believe that some sort of strong law of large numbers holds, and the distribution across SNPs of $\beta_v$ may therefore approximate a normal distribution, and the association test statistic will approach a non-central $\chi^2$ with $\lambda \sim \mathrm{E}[\beta^2_v$]. That genomic control robustly controls for type-1 error in genetic studies in virtually all cases [27-28] is strong evidence in support of this intuition. While undoubtedly a robust tool, genomic control is also thought to be something of a blunt instrument, destroying "signal" in the process of achieving well calibrated type-1 error rates. By ignoring the magnitude of $\mathrm{Cov}[q_v,q_{w_k}$] and correcting for mean behavior alone, this technique penalizes all sites equally, even when an individual site $v$ has little or no allelic covariance with any other site in the genome, perhaps because $v$ has no substantial variation in allele frequency across ancestries.

Since the fundamental problem being solved is caused by covariance in allele frequency and additive variance for the trait, the most natural solution to this problem is to include both in a linear mixed model [24]. Here, the tested model takes the form

where $\vec{P}$ is the vector of phenotypes, $\vec{S_v}$ is a vector of minor allele counts at locus $v$, $\mathbf{R}$ is a matrix describing the covariance in allelic state between each pair of individuals (much more on this in ), and $V_A$ is the total additive variance. If both $P$ and $S_v$ have been normalized to have mean $0$, to the extent that $\mathbf{R}$ fully captures the covariance in allelic state between each pair of individuals and no other interactions exist, this should produce an estimate of $\tilde{\beta}_v$, the effect of substituting the rare allele for the common allele in a collection of individuals without ancestral variation in allele frequency - the correct $\beta$ in the Falconer sense, not inflated by population structure.

Linear mixed models can be quite computationally burdensome when compared to a linear model, particularly with thousands of individuals. Note that to perform certain statistical analyses within a linear mixed model framework, such as calculating individual genetic effects, the inverse of the genetic relatedness matrix $\mathbf{R}$ is needed. Calculating the inverse of a large relatedness matrix can be computationally demanding, especially with large sample sizes. To dramatically reduce this burden, many newer linear mixed model approaches (e.g., [29-30]) for large datasets implement algorithms that approximate the degree of relatedness among individuals in the sample without requiring a pre-computed genetic relationship matrix i.e., without explicitly calculating the covariance in allelic state between each pair of individuals. For example, a linear mixed model can be approximated with a linear model by replacing the $\mathbf{R}$ matrix with a subset of its eigenvectors. In this way, we can view a linear model with inclusion of principle components (PCs) derived from $\mathbf{R}$ [21] as a natural approximation to the more computationally burdensome linear mixed model. These approximate approaches to linear mixed models may have comparable computational efficiency to that of a linear model.

Of course, to a geneticist, the most natural way to remove the effects of variation in allele frequency over ancestries is to ensure that the test for association being performed is not affected by correlation in allelic state at unlinked markers. This can this be achieved by focusing only on alleles transmitted from parents known to be heterozygotes [23]. The transmission-disequilibrium test (TDT) counts the number of transmissions from parents known to be heterozygotes to children with some given phenotype, generally a disease state. Because Mendelian segregation is strongly regulated during meiosis to have nearly 50-50 transmission rates, regardless of the identity of any particular allele at a given site, the observed transmission rate from heterozygous parents is a function of the penetrance of the allele for the conditioned phenotype, and this transmission rate is fundamentally independent of any correlations in state at other unlinked sites [31-32]. While this approach is virtually guaranteed to solve issues associated with ancestry, it is often far harder to collect sample sets that include both parents and offspring, and undetected genotyping error can have particularly challenging effects. Computation of the TDT statistic on trios in which one parent is missing marker genotype data increases the type-1 error rate of the statistic [33] as does genotyping error, which can cause apparent over-transmission of common alleles [34-35].

3.3 Phenotypic covariance, relatedness, and the covariance in allelic state

The first paper in this series [6] derived the covariance between individuals in a generalized form. Here we focus on a very specialized case where there is potentially correlation in genotypic state, but no other non-additive interactions of any kind - no dominance, additivity between all loci, and no correlation in state of non-genetic factors. Imagine a collection of individuals that might be a single population in HWE or could be collection of individuals with differing and complex ancestry. We are interested in two randomly picked individuals from this collection; call them individuals $1$ and $2$ and their phenotypes $P_1$ and $P_2$. At some locus $v$ contributing to this phenotype, let $S_{v_1}$ and $S_{v_2}$ be the counts of minor alleles, $A_{v_1}$, in individual $1$ and $2$, respectively. Let $\vec{S_1}$ and $\vec{S_2}$ be the vector of these counts over all loci contributing to the phenotype. Let $\alpha_{v_{0_1}}$ and $\alpha_{v_{0_2}}$ be the average phenotype of an individual with the common allele at locus $v$ in individuals with ancestral minor allele frequencies $q_{v_1}$ and $q_{v_2}$ respectively. While the allelic effect $\alpha$ will be a function of allele frequency, assume the difference in allelic effects $\tilde{\beta} = \alpha_1 - \alpha_0$ [13] is the same in all individuals regardless of allele frequency.

Line 77 uses the lack of dominance or any other interactions extensively. Thus, for an additive trait, the covariance between individuals is the sum over all contributing loci of $\tilde{\beta}^2$ multiplied by the covariance in minor allele counts, plus the sum of the product of two different $\beta$'s, and their covariance in allelic state (LD) for pairs of sites across the genome.