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closeCorrection to conjunction FDR definitiion
Posted by wesleyt on 16 Jun 2015 at 02:36 GMT
<<Correction for Page 13-14 of Main Text>>
Conjunction statistics—test of association with both phenotypes
In order to identify which of the SNPs were associated with schizophrenia and bipolar disorder we used a conjunction FDR
procedure similar to that described for p-value statistics in Nichols et al. [45]. This minimizes the effect of a single phenotype
driving the common association signal. Conjunction FDR is defined as the posterior probability that a given SNP is null for
either phenotype or both phenotypes simultaneously when the p-values for both phenotypes are as small or smaller than the
observed p-values. Formally, conjunction FDR is given by
FDRSCZ&BD(p1, p2) = π0 F0(p1, p2)/ F(p1, p2) + π1 F1(p1, p2)/ F(p1, p2) + π2 F2(p1, p2)/ F(p1, p2), [6]
where π0 is the a priori proportion of SNPs null for both SCZ and BD simultaneously and F0(p1, p2) is the joint null cdf, π1 is
the a priori proportion of SNPs non-null for SCZ and null for BD with F1(p1, p2) the joint cdf of these SNPs, and π2 is the a priori
proportion of SNPs non-null for BD and null for SCZ, with joint cdf F2(p1, p2). F(p1, p2) is the joint overall mixture cdf for
all SCZ and BD SNPs. Conditional empirical cdfs provide a model-free method to obtain conservative estimates of Eq. [6].
This can be seen as follows. Estimate the conjunction FDR by
FDRSCZ&BD = max{FDRSCZ|BD,FDRBD| SCZ} [7]
where FDRSCZ|BD and FDRBD| SCZ (the estimated conditional FDRs described above) are conservative (upwardly biased)
estimates of Eq. [5]. Thus, Eq. [7] is a conservative estimate of max{p1/F(p1| p2), p2/F(p2|p1)} = max{p1F2(p2)/F(p1, p2), p2F1(p1)/F(p1, p2)},
with F1(p1) and F2(p2) the marginal non-null cdfs of SNPs for SCZ and BD, respectively. For enriched samples, p-values will tend to be
smaller than predicted from the uniform distribution, so that F1(p1) ≥ p1 and F2(p2) ≥ p2. Then
max{p1F2(p2)/F(p1, p2), p2F1(p1)/F(p1, p2)}
≥ [π0 + π1+ π2] max{p1F2(p2)/F(p1, p2), p2F1(p1)/F(p1, p2)}
≥ [π0 p1p2 + π1p2F1(p1)+ π2p1F2(p2)]/F(p1, p2).
Under the assumption that SNPs are independent if one or both are null, reasonable for disjoint samples, this last quantity is precisely
the conjunction FDR given in Eq. [6]. Thus, Eq. [7] is a conservative model-free estimate of the conjunction FDR. We present a complementary
model-based approach to estimating conjunction FDR in the Text S1.
<<Correction for Page 4 of Text S1 (changes from published text in red)
We can also compute the conjunctional fdr of both phenotypes as
fdr(z1, z2) = [f(z1, z2| z1 null, z2 null) Pr(z1 null, z2 null)
+ f(z1, z2| z1 non-null, z2 null) Pr(z1 non-null, z2 null)
+ f(z1, z2| z1 null, z2 non-null) Pr(z1 null, z2 non-null)]/ f(z1, z2)
[S8]
With densities given in Eq. [S5], this becomes
fdr(z1, z2) = [π0 f0(z1, z2)+ π1 f1(z1, z2)+ π2 f2(z1, z2)]/ f(z1, z2) [S9]