Several groups have described the relationship of statistical power at a marker site in linkage disequilibrium with a causal site. In 1999, using the coalescent process to investigate the density of markers necessary for adequate coverage across the genome to detect disease-associated regions, Kruglyak presented the outline of an argument that the sample size necessary to detect association at a marker locus in linkage disequilibrium with a causal site is approximately S/d², where S is the number of samples required to detect disease association at the causal site with a given level of power and d² = [q(1 − q)p⁻¹(1 − p)⁻¹]r², such that r² is the standard measure of linkage disequilibrium between the causal site and the marker site and q and p are the allele frequencies at the marker and causal sites, respectively (Kruglyak, 1999). Later, Pritchard and Pzreworski performed a derivation showing a similar result, also with regard to power (Pritchard and Przeworski, 2001). Under the Pritchard- Przeworski derivation, the power to detect disease association at a causal site and marker site were found to be approximately the same if the sample size at a marker site is increased by a factor of (r²)⁻¹ over that used in interrogating the causal site. While certainly an intriguing relationship between sample sizes, as it is, the finding may not always have utility in fine mapping applications as most association studies use the same number samples at all sites interrogated. That said, this relationship can be used to motivate related and illuminating properties regarding how fast the disease association signal can be expected to decay as a function of declining linkage disequilibrium from a causal site. Equating the power at the disease-predisposing site to that at the marker site, it follows that,

where Z_D and Z_M are the normally-distributed Z-scores for testing disease-association at the causal site and marker site, respectively; and α is the significance level. Taking the inverse functions and squaring yields the provocative approximation,

where χD2 and χM2 are the Chi-Square statistics for disease association at the disease and marker sites, respectively. An interesting parallel was described by Luo, Thompson, and Wooliams in the context of marker-assisted selection of quantitative traits where the authors showed that the proportion of the additive variance of a trait due to a marker in linkage disequilibrium with a causal quantitative trait locus, σM2/σA2, is equal to r² (Luo et al., 1997).

Plotting the Equation (2) approximation with the χ² disease-association statistic on the ordinate and 1 − r² on the abscissa is a simple method of displaying the expected linear decay in the χ² -values as the linkage disequilibrium with a causal site declines at different marker sites. Figure 1 shows this relationship. This decay pattern was first used empirically in 2007–2008 to fine map the TRAF1 region in rheumatoid arthritis (Schrodi et al., 2007a) and the IL23R region in psoriasis (Garcia et al., 2008) and has, in an analogous form, subsequently been used in other applications (Farh et al., 2015). Although this approximation is very useful in understanding the decay of disease association with declining linkage disequilibrium from a causal site, several simplifying assumptions were made in the original Pritchard-Przeworski derivation. While the impact of these assumptions have been explored to some extent in previous work (Hu et al., 2004), it is not known how violations of the original assumptions might produce departures from Equation (2) nor what the effect of sampling haplotypes does to the relationship. Hence, an exact relationship between disease association statistics and r² -values with a causal site would aid in clarifying this relationship and motivate statistical approaches to harnessing this pattern for the purpose of fine-mapping functional alleles. Further, Monte Carlo simulations can be used to explore the how treating haplotype counts as random variables generates stochastic variation around this central relationship.

In this section, we will show the algebraic relationship between the Chi-Square-test statistics at a causal site and marker site, without any assumptions regarding the probabilistic properties (or whether they are fixed parameters) of the allele frequencies or haplotype frequencies of which the statistics are composed. Note that in the Monte Carlo Simulations Section we will treat the haplotype counts as random variables; and hence the Chi-Squared statistics and r² will each carry stochastic properties and we investigate these properties in that section.

Defining the Chi-Square-test statistics for a disease-causing site (χD2) and a marker closely linked to the disease site (χM2) following the Pritchard-Przeworski derivation,

where a two-site model is considered (site A segregating alleles A₁ and A₂, and site B segregating alleles B₁ and B₂), p, p_D, and p_C are the frequencies of the A₁ allele in the combined population, disease-affected population, and the control population, respectively, and where q, q_D, and q_C are the frequencies of the B₁ allele in the combined population, disease-affected population, and the control population, respectively. n_D and n_C are the sample sizes for diploid cases and controls, respectively, and n = n_D + n_C. For this work, haplotype and allele probabilities conditional on disease status (i.e., within cases or within controls) are derived. For the haplotype and allele probabilities in the general population, we weighted the disease status conditional probabilities by the probability of disease or healthy control attributable to the causal site, in accordance with the law of total probability. Note, that the form of these Chi-Square statistics in Equations (3) and (4) is twice the value of traditionally-defined Chi-Square statistic. However, this scalar inflation factor cancels out in the subsequent derivation.

Noting that

where K is the P(Case) attributable to the causal site, we can substitute pC=p-KpD1-K and qC=q-KqD1-K into Equation (5), resulting in

This treatment of the allele frequencies using the law of total probability holds for all populations in which each individual is either a case or control (e.g., cohort studies or case/control study designs). The next aim in the derivation is to substitute quantities for the allele frequencies in the affected population at both sites in terms of penetrances, disease prevalence, and general population allele frequencies. The allele frequencies at both the causal and marker sites have been previously described for two-locus systems under general disease models (Schrodi et al., 2007b):

where f₁₁, f₁₂, and f₂₂ are the prevalences of the A₁A₁, A₁A₂, and A₂A₂ genotypes, respectively, such that f_ij = P(Case|A_iA_j); which, under this monogenic model and assuming Hardy-Weinberg Equilibrium in the general population and using the law of total probability we can express the disease prevalence as, K=f11p2 + 2f12p(1-p)+ f22(1-p)2; and haplotype frequencies P₁₁ = P(A₁B₁), and P₂₁ = P(A₂B₁). Applied to complex diseases, it may be useful to think of this disease model as the subset of individuals with a common disease that is primarily driven by a particular locus. With the substitution into Equation (6),

In Equation (9), the R.H.S. numerator can be simplified to

Noting that P₂₁ = q − P₁₁ and substituting f11p2+2f12p(1-p)+f22(1-p)2=K, the numerator becomes

whereas, the denominator in Equation (9) can be simplified to

Hence, Equation (9) can be written as

where D = P₁₁P₂₂ − P₁₂P₂₁ = P₁₁ − pq.

Again substituting K=f11p2+2f12p(1-p)+f22(1-p)2,

Therefore, we have shown the exact relationship under our model,

Not only is this relationship an exact result under the model employed, but it is universal in that there is no dependence on the penetrances. Thus, we may expect that from a true disease-susceptibility site, that there should be a linear decay in the Chi-square statistics for disease association with declining r² -values with the causal site. Figure 1 shows the expected disease association decay with declining linkage disequilibrium from the causal site for additive, multiplicative, recessive, and dominant sets of models. The patterns arising from various relative risks are presented. Similarly, Figure 2 presents the patterns expected as a function of sample sizes. Aside from Equation (13) illuminating a central aspect of disease genetics, we suspect that it carries utility in fine mapping applications—we hypothesize that identifying this type of pattern in fine mapping data will better enable the pinpointing of truly causal sites through harnessing correlated data.

Consider the situation where there is a disease-susceptibility site and other sites in differing levels of linkage disequilibrium with the disease-susceptibility site. From large-scale genotyping or sequencing studies, we often know the matrix of pairwise r² -values, and allele frequencies at each site in the general population, broadly defined. An interesting question arises: If one has genotyped a marker site in a case/control sample set and calculated χM2 testing for disease association, can we infer the expected effect size at a non-interrogated causal site? Using Equation (13), and substituting allele frequencies at the causal site,

where ne=4nDnCnD+nC, the effective total number of independent diploid samples. Defining a traditional allelic odds ratio, R, calculated at the causal site as

the allele frequency in the cases can be solved: pD=RpC1-pC+RpC.

Therefore,

To simplify the derivation, we will assume that the disease studied is not very common such that the allele frequency in controls is well-approximated by the allele frequency in the general population, p_C ≅ p. This is also true if samples drawn from the general population are serving as the controls. Hence,

Solving for R,

To illustrate the use and implications of Equation (17), suppose that we have genotyped a site in 500 diploid cases and 500 diploid controls and calculated the test statistic χ² = 20, corresponding to p = 1.57E-03 (recall that half the Pritchard-Przeworski statistic is Chi-Square distributed with one degree of freedom). Further assume that this region has previously been subjected to next-generation sequencing in individuals derived from the same source population as the cases and controls which has yielded the discovery of numerous additional variants closely linked to the genotyped site, allele frequencies at those variants, and an array of pairwise linkage disequilibrium values across the region of interest. Under that scenario, one would typically have access to good estimates of the general population allele frequencies and r² -values at sites neighboring the genotyped site that produced the original finding. Suppose that one of these adjacent sites has a general population allele frequency p = 0.03 and a linkage disequilibrium value with the genotyped site of r² = 0.2. Under the two-site model, we would therefore estimate the odds ratio at the putative, non-genotyped, causal site to be 5.17. Put another way, the putative causal site, having the general population allele frequency and linkage disequilibrium values above, would have to have an odds ratio of 5.17 in order to generate twice a standard Chi-Square statistic value at the genotyped site of 20 given 500 cases and 500 controls. Indirect inference of the properties of non-interrogated causal sites can be helpful in subsequent experimental efforts to identify disease-predisposing sites in a fine-mapped region. Figure 3 displays the relationship between the inferred odds ratio at the causal site from disease association data at the marker site as a function of linkage disequilibrium between the two sites. Graphs for various p-values at marker site are shown. Additional work under a stochastic model would enable the calculation of the posterior probabilities of properties of non-interrogated causal sites given genetic data at linked markers.

The results detailed in Equations (1–17) do not treat any of the parameters, such as haplotype frequencies, as random variables. Clearly, haplotype counts in cases and controls should be treated with sampling processes from a larger population. To address this issue, we have constructed a Monte Carlo simulation program to generate haplotypes under a probabilistic model. Under this program we are able to explore the variation around Equation (13) generated by sampling haplotypes and to observe effects that may be produced by different sets of parameters.

In an effort to understand the variation in the patterns of disease association decay as a function of linkage disequilibrium with a causative site, we constructed a Monte Carlo simulation using a generalized disease model (penetrances for each of the three genotypes at the causal site are parameterized) and treating the haplotype counts in cases and controls as random variables. Recombination was introduced between a causal site and a closely linked marker as a realistic method of generating different sets of 2-site haplotypes for the general population (Hartl and Clark, 1989). For a rate of recombination, c, and generation time t, we used the following set of recursions (Haldane model of recombination):

Hence, for the general population, we can express r² as a function of generation time using the recursions in Equations (18–21):

Assuming Hardy-Weinberg equilibrium in the general population at both sites, the proportion of individuals affected by the disease attributable to this locus, is calculated through the previously-described formula for disease prevalence. To calculate the expected haplotype frequencies in cases, we used Bayes theorem. Hence, the expected frequency of the A₁B₁ haplotype in cases is

In an analogous manner, the remaining haplotype frequencies in cases, where the subscript indicates the haplotype, are

The haplotype frequencies in controls are simply

Sampling of the case and control haplotypes from the expected frequencies is accomplished through two independent multinomial variates such that the joint densities are given by

Hence, the sample frequency of the causal allele in cases and controls, respectively, are

and

We employed an additive model for the penetrances at the causal site and a design using 10,000 cases and 40,000 controls. As the time parameter is increased, the number of recombination events between the casual site and the marker site increases and there is a corresponding reduction in the linkage disequilibrium between the two sites. Figure 4 shows the distribution of the association statistic at the marker site (Equation 4) plotted against the product of the association statistic at the causal site (Equation 3) and the rt2 -value between the two sites. Four different time points were evaluated in the simulation, each with 10,000 replicates generated. The patterns show the general linear trend of how the association statistics scale with linkage disequilibrium and the variation around this pattern. For fixed properties at a causal site, Figure 5 displays the mean value and 95% confidence interval of the association statistic at the marker site as the rt2 -value declines.

All indicated earlier, there are several uses of the theorem presented here. The pattern of linear decay of association (as measured by the test statistic) with declining linkage disequilibrium can be used to support various markers as causal sites. Conversely, significant departure from the expected pattern can indicate multiple causal sites segregating at the disease locus. And additionally, understanding this fine mapping theorem can be used to infer properties of non-interrogated causal sites. To illustrate the application of the relationship described in Equation (13) to experimental data, we used GWAS data around the well-established obesity locus, FTO, generated by a recent large study of extreme BMI (Berndt et al., 2013). The FTO gene encodes for an alpha-ketoglutarate-dependent dioxygenase (Gerken et al., 2007), playing a role in growth and development (Boissel et al., 2009; Daoud et al., 2016), and has been reliably associated with the related conditions of type 2 diabetes, BMI, adiposity and other obesity-related traits (Scott et al., 2007; Zeggini et al., 2007; Lindgren et al., 2009; Thorleifsson et al., 2009; Fox et al., 2012; DIAGRAM Consortium et al., 2014; Wood et al., 2016). Within the FTO gene region, the study found that rs11075990 exhibited the strongest association with extreme BMI with a reported p-value of 9.3E-33. From this study, we identified 752 SNPs residing within a ~1 Mb region surrounding FTO, having linkage disequilibrium data from the 1000 Genomes project (The 1000 Genomes Project Consortium et al., 2015). Figure 6 displays the positional association of these data, showing a substantial peak localized on chr16q over the FTO gene. Plotting these association results as a function of pairwise linkage disequilibrium (as measured by r²) with rs11075990, there is a general decay of the Chi-Square association statistics with declining r² -values (Figure 7). Pearson's correlation is 0.979 and the p-value for this relationship (testing Spearman's rho under the null model of no correlation) is 2.87E-29. For this example, there are some immediate findings by visual inspection. The general pattern following the theorem is present. In addition, there appear to be some SNPs with extreme BMI associations that substantially exceed the level of association expected to be driven through linkage disequilibrium with rs11075990. That is, the theoretical model of one causal site (rs11075990) driving the extreme BMI association patterns in the FTO gene region may not explain the association statistics at some SNPs, such as rs2058908, where the theory only predicts a Chi-Square-value of 12.36 (r² with rs11075990 is 0.087) and yet the observed Chi-Square statistic is 73.98. Hence, the genetic information at rs2058908 may be driven by a causal signal independent of rs11075990 (rs2058908 is denoted with a green circle in Figure 7). A test of conditional association could be used to verify these types of hypotheses. Since the residuals obtains from the fitted line (the line that passes through the origin and the Chi-Square value associated to the causal site) and the observed Chi-Square-values are not normally-distributed, we used a resampling approach to obtain a 95% confidence band (dashed lines in Figure 7). In this approach, we treat the fitted Chi-Square-values to be the expected response for the bootstrap samples, and by resampling the original residuals, we obtain bootstrap replicates for the fixed covariate (r²) (Fox and Weisberg, 2012). Here, we resampled the original residuals 100,000 times in the R programming language (R Core Team, 2014) and used the 0.025 and 0.975 quantiles of the resampled fits to achieve the 95% confidence band in Figure 7.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.