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

and hence when the probability of the data given EL is greater than the probability of the data given AL.

Nested Bayesian Models. We define k nested SNP sets (k = 1,…,K), starting from the most significant SNP

S 1~½rs2075650(cid:2)

are the joint probabilities of the SNPs in the set S that are k estimated from the cases and controls. The rationale of this formula is that the SNPs are modeled as conditionally independent given the phenotype so that the probability distribution of a SNP set, given the phenotype, has the product form

and then we increment the set by adding one SNP at a time in order of maximum Bayesian factor (MBF). The latter is the maximum Bayes Factor among the 4 genetic models that we tested for each SNP in the GWAS. Therefore, the (k+1)th SNP set is defined as

p(SNP , (cid:3)(cid:3)(cid:3),SNP jphenotype)~P p(SNP jphenotype) 1 k i i

The product form is equivalent to assuming that the SNPs have a multiplicative effect, as in an additive logistic regression model. Compared to logistic regression, the Bayesian classification rule uses the retrospective likelihood to update the prior probabilities of EL and AL into the posterior probabilities. Also, the product form in the retrospective likelihood has the advantage that the genetic effect of each SNP can be estimated independently of the other SNPs and so there is virtually no upper limit on the number of SNPs that we can include in the SNP set.

where SNP is the SNP with the (k+1)th Bayesian significance. k+1 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 build 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 estimate the conditional probabilities p(SNP jphenotype) i using conjugate Bayesian analysis as described earlier.

Evaluation of sensitivity and specificity. Sensitivity (how many centenarians are predicted as centenarians) and specificity (how many controls are predicted as controls) of each SNP set were estimated as:

Table 8. Notation of allele frequencies in the recessive model.

Allele Frequencies M M 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. 1 n. 2+n. 3 N

Sensitivity= proportion of centenarians in the discovery set for whom p(EL | S )$p(AL | S ); k k Specificity= proportion of controls in the discovery set for whom p(EL | S ),p(AL | S ); k k

Resampling Method.

Resampling Method. repeatedly split the discovery set into non overlapping 2/3 training and 1/3 test sets that were respectively used to estimate the nested genetic risk models and to evaluate their predictive value. We

The table defines the mathematical notation for the recessive model for the M allele in the methods. doi:10.1371/journal.pone.0029848.t008



repeated this random procedure 1000 times for each SNP set, and summarize the sensitivity and specificity into the average values (See Figure S4). We evaluated the growth of sensitivity and specificity in the 1000 resampled sets. The mean number of SNPs in which the absolute difference between sensitivity and specificity was ,0.02 and accuracy was .85% was 281.

Prediction in independent tests.

the ensemble of 281 genetic risk models trained in the discovery set and computed the posterior probability of AL and EL in cases and controls of the two sets. We assumed uniform prior probabilities (P(AL) =P(EL) = 0.5), and classify a subjects as EL if the posterior probability of EL given the genotype of 281 SNPs was.posterior probability of AL. We assessed sensitivity and specificity by the number of centenarians classified as EL and the number of controls classified as AL.

ordering heuristics to see whether different orderings may lead to better risk prediction models. We conducted two types of tests. In the first test, we randomly permuted the order of the top 281 SNPs and repeated the heuristic of building nested genetic risk models by adding one SNP at a time from the randomized list of SNPs. In each test, we examined the effect of changing SNP order on the sensitivity and specificity in the discovery set, and also in the bootstrap procedure. The results of these analyses are shown in

Genetic Signatures

Clustering of genetic risk profiles.

model-based clustering procedure implemented in the program CAGED [91] to cluster the genetic risk profiles of centenarians and controls, independently, in the discovery and replication sets. The method in CAGED is designed to cluster row profiles of a two dimensional array by preserving the column ordering and it uses a Bayesian search strategy to identify the number of clusters by maximizing a Bayesian score [92]. We organized the genetic risk profiles into a N x 281 array, with rows that represent subjects and the jth column that represents the genetic risk calculated from the jth SNP set. We used polynomial models up to order 4 to capture a variety of patterns [93] and then used hierarchical clustering of the profiles to check whether similar clusters could be further merged. The signatures in the merged replication sets were generated by cluster analysis of the predicted profiles calculated using the 281 genetic risk models trained in the discovery set.

Interpretation of genetic risk profiles.