the function batch(⋅) is a function mapping each sample to a batch (plate) ∈ {1,2,3}. The κi are independently Gaussian distributed with mean zero and variance σ2 κ, for subject i representing inter individual variability. εij is the residual error term incorporating technical (aliquot) variability, assumed to be independently Gaussian distributed with mean zero and variance σ2 ε. The model was fitted in R, using the lme4 package[57] and the lmer() function, using maximum-likelihood. A likelihood ratio test was applied to assess the significance of the case-control effect (η). The p-value of the case-control effect in each model was calculated using a likelihood ratio (LR) test with the D = −2*log(LR) as the test statistic, which can be approximated by a Chi-square distribution with one degree of freedom. The p-values were adjusted for multiple testing by FDR correction [58]. In the replication study the batch effect was omitted from the model due to change of Illumina platform from 96-array format to 12-array format with no apparent batch differences present. Results were considered to be replicating if the fixed effect of interest (η) had a p-value<0.05 and consistent direction of the effect.
