Rubio et al. – supplementary data
Estimation of prior probabilites for the most associated IMSGC candidate genes
Methods
In the replication dataset R, let yR denote the observed phenotype data and genotype data
for a particular SNP, and let ˆR denote the estimated log odds ratio for that SNP. Then,
given a null hypothesis H0 and an alternative hypothesis H1, replace the Bayes factor
Pr(yR | H0) / Pr(yR | H1) by an Approximate Bayes Factor Pr( ˆR | H0) / Pr( ˆR | H1) (30).
The null hypothesis H0 is that the true value of the log odds ratio  is 0. Take the
alternative hypothesis H1 to be that the true value of  comes from a normal distribution
N (ˆI ,  I ) , where ˆI is the estimated log odds ratio from the combined IMSGC study,
and  I is its standard error. Then the Approximate Bayes Factor (ABF) is
ABF =
1
R
 ˆR 



 R
 
1
 R2   I2
 ˆ  ˆ
R
I
  2  2
R
I



,


where  R is the standard error of the estimate ˆR and
  x2 

 ( x)  exp 
2


is a Gaussian function. An ABF less than 1 implies that the replication dataset has
increased the evidence for ‘true’ association, whereas an ABF greater than 1 implies that
our data has reduced the evidence for ‘true’ association. Given a probability 1 of
association prior to this replication study (incorporating the evidence from the IMSGC
study and previous replication studies), the posterior probability of association,
incorporating the evidence from this replication study, is
1
 1  ABF (1   1 )
Rubio et al. – supplementary data
This is equal to one minus the Bayesian False Discovery Probability (BFDP) (30). For the
seven most associated non-HLA SNPs (six genes) in the IMSGC study, posterior
probabilities were calculated for prior probabilities 1 of 0.99, 0.95, 0.9, 0.7, 0.5 and 0.3,
respectively.
Results
A summary of the results is presented in the text of the manuscript. Below is a more in
depth description of the data for each of seven SNPs represented in Table 2.
For KIAA0350 (rs6498169), the ABF was 0.008, and if, for example, the prior probability
of this association were 0.5 before our data were added (i.e. an evens, or 1:1, chance), the
posterior probability of this association being ‘true’ now might be 0.992 (or odds of about
100:1 in favour), where the odds of association = probability of association/(1 probability of association).
For IL2RA, there is more prior evidence that this association is ‘true’, and thus the current
prior probability for this locus might be greater than for KIAA0350 i.e. 0.9 or odds of 9:1
in favour. We found that the ABF for rs12722489 (a SNP that did not reach our p≤0.05
significance threshold) was 0.714, and thus if this scenario were true, we would still have
contributed to an increase in the posterior probability from 0.9 to 0.927. In contrast, the
more associated IL2RA SNP (rs2104286), with an ABF of 0.119, would have contributed
much more to the posterior probability of this association being ‘true’ (now 0.987, or
odds of nearly 100:1).
If we assume that the less significant p-values achieved by the IMSGC for RPL5 and
CD58 meant that the prior probabilities for these loci being truly associated with risk
were less than for the other more highly ‘ranked’ genes (e.g. prior probabilities of 0.3, or
Rubio et al. – supplementary data
odds of around 1:2 against them being truly associated) then our data has increased the
posterior probabilities to 0.724 and 0.721, respectively, now 3:1 in favour of these
associations being ‘true’.
Our failure to replicate the association with the IL7R SNP (rs6897932) and its
correspondingly high ABF of 33.54 diminish the posterior probability for this
association. However, there is more supporting evidence for IL7R being a ‘true’ MS
susceptibility locus than for any other non-HLA gene (6-8, 10), so the prior probability for
this locus might have been 0.99 prior to this study. If this were the case, then the posterior
probability of this association would now be 0.747, still about 3:1 in favour of this locus
being a bone fide MS susceptibility locus.