Notation and assumptions

The study sample contains n₁ cases and n₀ controls (n₁ + n₀ = n) with corresponding allele frequencies p₁ and p₀ and common frequency p under the null hypothesis of no association. There are m cases from sibships with at least two sibs and n₁ - m independent cases. At the candidate locus, each individual has two alleles, X_i1 and X_i2 (i = 1,..,n) coded as 0/1. Usually only the genotype X_i. = X_i1 + X_i2 is known. For all individuals the affection status y _i = 0/1 is given. The cases from families comprise k = 1,...,K sibships of size m _k , and z _i denotes the sibship of individual i. For the cases, the X _ij values have a Bernoulli(p₁) distribution. Cases from different sibships are assumed to be independent, cases from the same sibship are not independent. To describe the correlation structure between sibs we use a model from population genetics that considers a population consisting of different subpopulations based on the coefficient F _ST and the inbreeding coefficient F _IT . Sibships are regarded as small subpopulations and F _ST denotes the correlation between two randomly chosen alleles of two individuals from the same sibship. Under the assumption of no population structure, correlations within sibships only arise from IBD sharing between sibs and F _ST equals the expected kinship coefficient between two siblings. F _IT measures the correlation of the two alleles within an individual and equals 0 under assumption of random mating and no further population structure.

The test statistic

The inflation γ for the allelic χ²-test Var_γ=1(T) is defined as λ = Var _γ T/Var_γ=1T.

Strategies to determine the correlations F _ST and F _IT

To estimate Var(T), different strategies for determining F _ST and F _IT were investigated. In strategy I ("no linkage") F _ST is directly calculated under the assumptions of no linkage and F _IT = 0. Here F _ST corresponds to the prior kinship coefficient of a sib pair. F _ST = 0.25, since 2F _ST is the probability that two alleles from the same parent of a sib pair are IBD. In the two other strategies F _ST is estimated to account for regions of linkage where the true F _ST is larger than 0.25.

In strategy II ("ANOVA") F _ST and F _IT are estimated by analysis of variance based on the marker data of the affected sibships at the candidate locus [7]. This strategy has no further assumptions and is based on a partitioning of the total sum of squares into three sums of squares: within individuals, within sibships, and between sibships. Each of them describes the additional variance compared to the lower level in the given order. Because F _ST and F _IT can be expressed as ratios of variance components, estimates for F _ST and F _IT can be derived as functions of the sums of squares.

Strategy III ("MULTI") uses a multipoint F _ST estimate assuming F _IT = 0, requiring genotype information at adjacent markers, e.g., for cases previously analyzed for linkage with these markers. F _ST can be directly estimated from the estimated mean number Y of alleles IBD within the affected sib pairs. The expectation of Y can be expressed as E(Y) = 2N·2F _ST , where $2N={\displaystyle \sum {}_{k=1}^K{m}_k(m_k-1)}$ is the total number of allelic pairs considered and 2F _ST , is the probability that such an allele pair is IBD. The estimated number Y of alleles IBD has to be calculated from individual IBD estimates. If there are only affected sib pairs in the data (N = K), Y can be derived from the nonparametric linkage-score (NPL- or Z-score), which is then equivalent to the classical mean test statistic $Z=(Y-K)/\sqrt{K/2}$. Here the same IBD measure is used as in linkage analysis.

To evaluate the strategies we implemented the test statistic in the computer program R. For strategy I F _ST = 0.25, for strategy II F _ST was estimated in the ANOVA framework implemented in R, and for strategy III we calculated NPL-scores with Merlin.

Application to data from a candidate gene study for rheumatoid arthritis

The proposed methods were applied to case-control data from 20 candidate genes for rheumatoid arthritis previously analyzed by Plenge et al. [6]. The 839 cases were from the North American Rheumatoid Arthritis Consortium (NARAC) and include 717 cases from affected sibships and 122 unrelated cases. The 855 unrelated controls were selected from healthy individuals who were enrolled in the New York Cancer Project (NYCP). Because we have to include additional data for strategy III, we only investigated the introduced test statistics based on strategy I, II, and the traditional allelic χ²-test based on allele frequencies ignoring familial correlations. We compared our results to Plenge et al. [6] who analyzed the same sample with only a few additional individuals.

Application to the simulated data

Additionally, the SNP genome data from Replicate 1 of the simulated Genetic Analysis Workshop 15 data were analyzed knowing the solutions. The data contain 1500 families of two parents and an affected sib pair and 2000 controls. We calculated our test statistics based on strategies I-III for all 9187 SNPs of the genome scan comparing 3000 cases to 2000 controls. Subsequently, in order to remove true associations, we excluded SNPs in a region around ±3 cM of simulated disease loci to analyze data simulated under the null hypothesis of no association but allowing for linkage. For the remaining SNPs we verified the type I error rate of the test statistics. We also analyzed chromosome 6 containing the major disease locus to concentrate the analysis on a region of known linkage.

Results for the candidate gene study for rheumatoid arthritis

Results for the simulated data

Figure 1 shows the estimated F _ST values for chromosome 6. As expected, the multipoint F _ST estimation is more stable than the single-point ANOVA method. However, even with the single-point method, the F _ST estimate is in most cases larger than 0.25, thus accounting for linkage correctly. For the simulated data an F _ST value of 0.25 leads to an inflation factor of 1.2, whereas an F _ST = 0.3 corresponds to λ = 1.24. Because of this small difference between the inflation factors, the method to determine F _ST is expected to have only a minor impact on the test statistic. After excluding regions of true associations, 9055 SNPs remained, including 627 out of 674 SNPs on chromosome 6. Figure 2 shows the observed type I error rate for the different test statistics. The results for the entire genome indicate that the allelic χ²-test is far too liberal. In contrast, the observed type I error rates for the test statistics accounting for familial correlations are all very close to each other within the expected range for all significance levels up to 0.1. The separate analysis of chromosome 6 confirms that even in a region of known linkage there is only a minor difference between the three strategies, with the "no-linkage" being the most liberal.