The sibling relative risk (λ_s) attributable to a gene reflects the extent to which a particular genetic locus contributes to familial aggregation. The calculation of attributable λ_s values was first described in an early paper of James 4. If the frequency of the high-risk allele is denoted by RAF, the relative risk of RA for high-risk allele homozygotes compared to low-risk allele homozygotes by GRR_hom, the relative risk of heterozygotes compared to low-risk allele homozygotes by GRR_het and the disease prevalence among low-risk allele homozygotes by κ₀, the attributable λ_s is given by: λ_s = 1+(1/2V_a+1/4V_d)/K², where V_a (additive genetic variance divided by κ₀²) equals 2RAF(1-RAF)[(1-RAF)(1-GRR_het)-RAF(GRR_hom-GRR_het)]², V_d (dominance genetic variance divided by κ₀²) equals RAF²(1-RAF)²[1+GRR_hom-2GRR_het]² and K (population prevalence divided by κ₀) equals RAF²GRR_Hom+2RAF(1-RAF)GRR_het+ (1-RAF)². Note that if the overall disease prevalence is 50%, λ_s cannot exceed two by definition. However, an important property of the attributable λ_s is that it only depends on RAF, GRR_hom, and GRR_het. That is, the attributable λ_s is independent of the disease prevalence among low-risk homozygotes κ₀.

The BF statistic is the ratio of the probability of the observed data under the assumption that there is a true association to its probability under the null hypothesis (absence of association). A small BF provides evidence in favor of a true association. To investigate the relationship between attributable λ_s and the BF, we calculated the expected distribution of genotypes in 1000 cases and 1000 controls when GRR_hom = 1.5, 2, 3, 4, 5 and 10. Calculations were carried out for the dominant model, which assumes that GRR_het = GRR_hom, the recessive model (GRR_het = 1), and for the additive model (GRR_het = (1+GRR_hom)/2). Under Hardy-Weinberg equilibrium, the expected distribution of genotypes in controls (D = 0) is: Pr(G = aa|D = 0) = (1-RAF)²; Pr(G = Aa|D = 0) = 2RAF(1-RAF); Pr(G = AA|D = 0) = RAF² The expected distribution of genotypes in cases (D = 1) satisfies:

GRR_hom = (Pr(D = 1 | G = AA)/Pr(D = 0 | G = AA))/(Pr(D = 1 | G = aa)/Pr(D = 0 | G = aa)) and GRR_het = (Pr(D = 1 | G = Aa)/Pr(D = 0 | G = Aa))/(Pr(D = 1 | G = aa)/Pr(D = 0 | G = aa)). For example, if RAF = 0.2 and GRR_hom = GRR_het = 2 (dominance), the expected distribution is (aa = 640, Aa = 320, AA = 40) in controls and (aa = 471, Aa = 471, AA = 59) in cases.

The expected distributions of genotypes were investigated by Bayesian logistic regression. We considered a general three-genotype model of association. Calculation of BFs requires assumptions about effect sizes. We assumed N(0,1) priors on the mean and the two genetic effects. The function MCMClogit (R package MCMCpack 5) was used to generate a 10,000 sample dataset from the posterior distribution using a random walk Metropolis algorithm. The BayesFactor function in the same package was used to compute log marginal likelihoods for a model 'with' compared to a model 'without' genotypic information using a Laplace approximation.

This study investigated regions around the 12 signals detected in the WTCCC study with associated probability values of less than 10^-5 (Table 1). The left side of Table 1 represents the 12 selected regions from the WTCCC study. For example, the most associated marker on 1p13 was rs6679677. We retrieved NARAC data from 1000-kb regions centred on the positions of the 12 signals from the WTCCC study. For example, our first region comprised the chromosomal interval (10,901,850::11,901,850) around rs6679677. The WTCCC study reported two SNPs in 6p21 that resulted in two overlapping intervals which were independently investigated.

SNPs in the 12 selected intervals were extracted from NARAC data, and the association between RA and the retrieved SNPs was represented by 1) the base 10 logarithm of the BF (log₁₀BF) and 2) the attributable λ_s. RA status (affected or unaffected) was modelled by logistic regression, taking into account the covariates gender, number of HLA-DRB1 shared-epitope alleles (NN = 0, SN = 1 or SS = 2) and genotype (three levels). Individuals with missing information were excluded from the calculations. Only SNPs with the three genotypes represented in both cases and controls were considered.

As already mentioned, calculation of BFs requires assumptions on effect sizes. We calculated frequentist estimates of logistic regression coefficients over the corresponding entire chromosomes using NARAC data. Density plots and tests of goodness of fit of frequentist estimates of log₁₀GRR_Hom and log₁₀GRR_Het for the seven investigated chromosomes indicated that GRR variation was better represented by the median absolute deviation (MAD) than by the standard deviation (data not shown). Therefore, we assumed N(0, MAD²) priors on the logarithms of genetic effects. MCMClogit and BayesFactor 5 were used to calculate BFs for each SNP in the selected regions based on NARAC data. The number of individuals with complete data and the observed allele frequencies in controls were used to draw 10,000 samples from a binomial distribution, which were combined with the posterior estimates of genetic effects to simulate the distribution of attributable λ_s for each SNP.

A bagplot is a bivariate generalization of the non-parametric univariate boxplot 6. In a bagplot, the region which contains 50% of the observations with greatest bivariate depth is denominated the bag. Observations outside the bag expanded three times are considered outliers, they are too far away from the data's central bulk. These outliers are indicated by closed dots in univariate boxplots. Bagplots have been shown to be equivariant for linear transformations and not limited to elliptical distributions. Due to these properties and the characteristics of our data (unknown shape of the underlying distributions), we used non-parametric bagplots to identify deviating SNPs. Bagplots were determined using the function compute.bagplot in the R package aplpack 7.

Table 1 shows the RAFs at which the maximum log₁₀BF and λ_s were reached assuming several GRR_hom values under the dominant, recessive and additive modes of inheritance. For example, under GRR_hom = 2 and dominance, the maximum log₁₀BF (25.3) was reached when RAF = 0.22 and the maximum λ_s (1.06) for RAF = 0.17. Alternative sample-sizes/priors would result in different log₁₀BFs, but the RAF to reach the maximum log₁₀BF would not change. The representation of log₁₀BF versus attributable λ_s (data not shown) indicated that the two parameters were correlated, but they did not correspond one-to-one. In the three models, the RAF corresponding to the highest attributable λ_s and the RAF corresponding to the highest log₁₀BF decreased with increasing GRR_hom values. The RAF at maximum log₁₀BF was always higher than the RAF at maximum λ_s, i.e., balanced designs resulted in stronger association evidences. The difference between RAF at maximum log₁₀BF and RAF at maximum attributable λ_s increased with increasing GRR_hom, for example (0.24-0.21) = 0.03 if GRR_hom = 1.5 versus (0.14-0.05) = 0.09 when GRR_hom = 10 in the dominant model.

The analysis of complete chromosomes by frequentist logistic regression resulted in 2(GRR_hom, GRR_het) × 8(chromosomes 1, 4, 6, 7, 10, 13, 21, 22) sets of genetic effects. Table 1 shows the MADs of log₁₀GRR_hom and log₁₀GRR_het used as scale parameters in the normal prior distributions. As expected, the a priori variance of genetic effects was highest for chromosome 6 (MHC region).

Figure 1 represents the bivariate distribution of log₁₀BF by attributable λ_s based on NARAC data for individual SNPs in the 12 explored regions. Gray points show SNPs within the bulk of data, outlying observations (observations outside the bag expanded three times) are represented by black points. The right part of Table 1 shows SNP numbers and close genes for the most extreme outliers. For example, rs2476601 in the 1p13 region: based on NARAC data, the log₁₀BF was 3.91, the high-risk allele frequency in controls is 0.084, and the estimated attributable λ_s with 95% confidence interval was 1.048 (1.017 to 1.098) for this SNP. The other two regions on chromosome 1 did not reveal strong associations with RA. The investigation of the two overlapping intervals in chromosome 6p21 showed practically identical results; the highest attributable λ_s values were found for SNPs rs7765379 (1.175) and rs2395175 (1.164). Many SNPs in this region showed a strong association with RA. The investigation of the 6q23, 7q32, and 10p15 regions resulted in log₁₀BF values below three. The log₁₀BF and the attributable λ_s for SNP rs1407961 in the 13q12 region clearly departed from the majority of data. Two extreme outlier SNPs were identified in the 21q22 region and three outliers in the 22q13 region.