Incorrect mathematical definition of the conjunction FDR in the section ‘Conjunction statistics—test of association with both phenotypes’

Under the sub-heading of ‘Conjunction statistics—test of association with both phenotypes’ in the ‘Materials and Methods’ section of the manuscript, there are errors in the mathematical definition of the conjunction FDR. The authors have provided an updated version here with corrections to the text in bold:

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 (6) where π₀ is the a priori proportion of SNPs null for both SCZ and BD simultaneously and F₀(p_1, p₂) is the joint null cdf, π₁ is the a priori proportion of SNPs non-null for SCZ and null for BD with F₁(p_1, 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(p_1, p₂). F(p_1, 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 (7) where FDR_SCZ|BD and FDR_{BD| 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

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 S1 Text.

We assigned the conjunction FDR values by interpolation into a bi-directional two-dimensional look-up table (S3 Fig). All SNPs with conjunction FDR<0.05 (−log₁₀(FDR)>1.3) with schizophrenia and bipolar disorder considered jointly are listed in Table 3 (after pruning), together with the corresponding z-scores and minor alleles. The z-scores were calculated from the p-values and the direction of effect was determined by the risk allele.

Incorrect mathematical definition of the conjunction FDR in the ‘Conditional and Conjunction Local False Discovery Rate’ section of the S1 Text.

There are also errors under the sub-heading ‘Conditional and Conjunction Local False Discovery Rate’ in the S1 Text. Please view the correct S1 Text here, with updates to the text in red.