assumed to be missing at random [87]. The formulas are given below.

Table 5. Notation of the hyper-parameters in the Dirichlet prior distributions.

We assume that genotypes frequencies in cases and controls follow independent multinomial distributions with parameters that follow Dirichlet distributions with uniform prior hyper- parameters. This is the standard parameterization for conjugate Bayesian analysis of a contingency table when we condition on one dimension of the table. See for example the supplement material of the review article of Balding [86]. In our case, we condition on the phenotype (case/control status) so that we use the retrospective likelihood that is appropriate in a case-control design.

Prior Hyper-parameters MM Mm mm Total Cases (Y=1) a a a a . 11 12 13 1 Controls(Y=0) a a a a . 01 02 03 0 Total a. a. a. a 1 2 3

The table defines the mathematical notation for the hyper-parameters of the Dirichlet distribution used in the methods. doi:10.1371/journal.pone.0029848.t005

Then the marginal likelihood of the data, given a genotype association, is the formula:

and the marginal likelihood of the data, assuming no association between SNP and phenotype, is the formula:

C(a) C(a zn ) p(D

)~ P .k .k ; independence C(azn) k C(a ) .k X X a ~ a a~ a .k jk .k j j

The formulas are similar for the other genetic models. We estimated the genomic control factor as described in [88].

where the genotype frequencies n and hyper-parameters a of the ij ij Dirichlet distribution are defined in Table 4 and 5. The Bayes factor is the ratio between the two marginal likelihoods:

to compute 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 each allele frequency p(a)=0.05,0.10,0.15,…,0.5, we simulated 100,000 data sets with no associations with 1750 subjects that we randomly split into 800 cases and 950 controls, to mimic the sample size of the discovery set. We used thresholds varying between 10 and 1,800 and, in each simulated data set, we computed the BF for the 4 models of association as described above, and determined the SNP as significantly associated if the BF of at least one of the 4 models was greater than the threshold. The simulations are summarized in Figure S2 and show how to interpret different thresholds for the

The Bayes factors for the other models are calculated using the same formulas, after the genotype frequencies are converted into allele frequencies (Table 6), or frequencies for dominant alleles (Table 7), and recessive alleles (Table 8).

For genotype association, we estimated the two ORs for exceptional longevity (EL) as:

Gender effect. For the significant SNPs in the discovery set, we tested whether the associations are substantially modified when a gender-SNP interaction model was used. We used the retrospective likelihood and tested 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.

p(Mm

L)p(MM

L) OR(Mm v MM)~ and p(Mm

L)p(MM

L) p(mm

L)p(MM

L) OR(mm

M)~ p(mm

L)p(MM

L)

And we estimated the conditional probabilities of genotypes as:

Table 6. Notation of allele frequencies in the allelic model.

Table 4. Notation of genotype frequencies.

Genotype Frequencies MM Mm mm Total Cases (Y=1) n n n n . 11 12 13 1 Controls(Y=0) n n n n . 01 02 03 0 Total n. n. n. N 1 2 3

Allele Frequencies M M Total Cases (Y=1) 2n +n n +2n 2n . 11 12 12 13 1 Controls(Y=0) 2n +n n +2n 2n . 01 02 02 03 0 Total 2n. +n. n. +2n. 2N 1 2 2 3

The table defines the mathematical notation for the allele frequencies used the methods. doi:10.1371/journal.pone.0029848.t006

The table defines the mathematical notation for the genotype frequencies used in the methods. doi:10.1371/journal.pone.0029848.t004

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

Table 7. Notation of allele frequencies in the dominant model.

Allele Frequencies MM/Mm Mm Total Cases (Y=1) n 11+n 12 n 13 n 1. Controls(Y=0) n 01+n 02 n 03 n 0. Total n. +n. n. N 1 2 3

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(SNP jEL)zp(AL)Pk p(SNP jAL) i~1 i i~1 i

where p(EL) and p(AL) =1-p(EL) are the prior probabilities of exceptional and average longevity. The conditional probabilities p(SNP jEL) and p(SNP jAL) represent the distribution of the ith i i SNP genotype in cases (EL) and controls (AL). The rule is to classify a subject as predisposed to exceptional longevity if p(ELjS )wp(ALjS ). k k

The table defines the mathematical notation for the dominant model for the M allele in the methods. doi:10.1371/journal.pone.0029848.t007

Association with known disease alleles. We

62,339 unique SNPs that were associated with a variety of diseases and traits in several GWASs from the catalogue of published genome wide association studies at http://www.genome.gov/ 26525384 [89], and the Human Gene Mutation Database (HGMD). We found 1214 of these 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 used the prior Pr(EL) =Pr(AL) =0.5 as described in the caption of Figure 4. This choice of a prior probability 0.5 for both EL and AL means that the classification becomes independent of the prior because, by Bayes’ theorem, the rule becomes ‘‘assign EL’’ if

p(ELjS )wp(ALjS )u P p(SNP jEL)w P SNP jAL) k k i i i i