Case-control studies with affected sibships.

Related cases may be included in case-control association studies if correlations between related individuals due to identity-by-descent (IBD) sharing are taken into account. We derived a framework to test for association in a case-control design including affected sibships and unrelated controls. First, a corrected variance for the allele frequency difference between cases and controls was directly calculated or estimated in two ways on the basis of the fixation index FST and the inbreeding coefficient. Then the correlation-corrected association test including controls and affected sibs was carried out. We applied the three strategies to 20 candidate genes on the Genetic Analysis Workshop 15 rheumatoid arthritis data and to 9187 single-nucleotide polymorphisms of replicate one of the Genetic Analysis Workshop 15 simulated data with knowledge of the "answers". The three strategies used to correct for correlation give only minor differences in the variance estimates and yield an almost correct type I error rate for the association tests. Thus, all strategies considered to correct the variance performed quite well.


Background
It is desirable to include related cases in case-control studies because pedigrees of multiple affected individuals have a higher expected frequency of susceptibility allele(s), leading to increased power [1]. Several methods have been proposed to test for association in case-control designs that take correlations due to IBD sharing into account [1][2][3][4]. Most of these determine correlations of related individuals based on prior kinship coefficients assuming no linkage under the hypothesis of no association. Only Slager and Schaid [4] incorporate individual identity-by-descent (IBD) estimates from previous linkage analyses. A comparison of the two strategies with respect to their power has been presented by Bourgain [5]. To integrate both strategies in one model we derive a unified framework to test for association including affected sibships and unrelated controls and apply the introduced test statistics to the candidate gene data set of Plenge et al. [6] as well as a replicate of the simulated single-nucleotide polymorphism (SNP) genome data.

Notation and assumptions
The study sample contains n 1 cases and n 0 controls (n 1 + n 0 = n) with corresponding allele frequencies p 1 and p 0 and common frequency p under the null hypothesis of no association. There are m cases from sibships with at least two sibs and n 1 -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(p 1 ) 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. Under the null hypothesis of no association, its variance can be derived by dividing the sum of alleles within cases into two parts: one for affected sib pairs and one for independent cases, leading to

The test statistic
The inflation γ for the allelic χ 2 -test Var γ=1 (T) is defined as

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 where 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 .
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 NPLscores 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].
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 χ 2 -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.  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.

Conclusion
If related cases are included in a case-control study, the allelic χ 2 -test can lead to an increased rate of false positives, as indicated by the simulations and the real data analysis. All strategies to correct the variance perform quite well and lead to an almost correct type I error rate on  Observed type I error rates in the simulated data excluding regions of true associations (expected values and 95% confidence bounds in gray) Figure 2 Observed type I error rates in the simulated data excluding regions of true associations (expected values and 95% confidence bounds in gray).