Our finding that about 17% of Illumina controls have signatures with a >70% chance of exceptional longevity (Figure S14) suggests that a substantial proportion of this group has a genetic predisposition to exceptional longevity. If this observation is replicated in more representative samples of the population, it could partly explain why centenarians are the fastest-growing age370 samples were analyzed together, including 102 from the NECS study, 176 from centenarians, and 92 from the ELIXIR study. The samples underwent sample-based QC checks, and sample call rates (CR) were computed using QC statistics. Samples with CR below 96.5% were eliminated, and re-clustering was performed. The excluded samples with CR above 93% were included using the new cluster file. The allele coding in other arrays was tested for agreement by comparing allele frequencies, which ruled out general bias between arrays and removed SNPs with reversed alleles. An additional sample of 60 centenarians included 39 subjects genotyped in September 2010 using the 610-Quad array. A principal components analysis was run on the NECS and Illumina SNP data to control for population stratification, and clustering was used to group individuals with similar ancestry into the same cluster. The Bayes Factor (BF) was used to score each SNP association by computing the posterior odds for association between the SNP and phenotype. The genomic control factor was estimated to determine the expected number of false positive associations of the decision rule that selects a significant association when the BF of at least one of the four models is greater than the threshold. For genotype association, two ORs for exceptional longevity (EL) were estimated based on gender effect. Significant SNPs in the discovery set were tested for modification of association due to gender effect.text does not affect the meaning. Therefore, no correction is needed for this part of the text.

When a gender-SNP interaction model was used, we used the retrospective likelihood to test whether the distribution of each selected SNP is independent of the phenotype once we condition on gender. Accepting the null hypothesis implies that the association between SNPs and phenotype is explained away by gender, and none of the associations could be explained away by gender. 

The probability of association ranges from 10 for multiple dependencies to 100 for two-way SNPxSNP interaction. The methodology based on Bayesian networks is described in detail in [62,90]. 

We estimated the conditional probabilities of genotypes as shown in Table 6 (Notation of allele frequencies in the allelic model) and Table 4 (Notation of genotype frequencies). 

For each SNP set, S, the Bayesian classification rule calculates k the posterior probability of EL as: 

p(ELjS)~i~1 i k p(EL)Pk p(SNPjEL)zp(AL)Pk p(SNPjAL)i~1 i i~1 i 

The rule is to classify a subject as predisposed to exceptional longevity if p(ELjS)wp(ALjS). 

We found 1214 SNPs in the Illumina array that we used for the GWAS of EL with acceptable quality. We calculated the number of disease alleles carried by centenarians versus all Caucasian controls included in our analysis. 

We choose K = 500 that corresponds to testing SNPs with approximately a posterior probability of an association .0.95 and removed from this set 100 SNPs that are highly correlated. To this end, we built a Bayesian network to capture mutual dependencies between SNPs that represent either strong linkage disequilibrium or strong SNP-SNP associations and removed those SNPs that are conditionally independent of the phenotype given more significant SNPs. We used a threshold on the posterior. 

We estimated the conditional probabilities p(SNPjphenotype) i using conjugate Bayesian analysis as described earlier. 

To validate the genotyping of the SNPs included in the model, 30 SNPs (.10% of the SNPs in the model) were selected to re-genotype using the TaqMan platform (Applied Biosystems, Carlsbad, CA). This genotyping was performed at Yale University. The samples included 688 centenarians and 221 controls from the NECS that were included in the discovery set and for whom we had available DNA. 

This can be written as 

p(SNP p(EL)Pk i~1 p(SNPijEL)zp(AL)Pk i~1 p(SNP p(SNPkz1jEL)|p(EL)Pk i~1 p(SNPijEL)zp(SNPkz1jAL)p(AL)Pk i~1 p(SNPijAL)