Model

We assume the following model for the observed data x_ijg:

Here i is the index for the observation, j the index for the batch and g the index for the variable. The term aijTβg parametrizes the effect of experimental conditions or, in general, any factors of interest a_ij on the measurements of variable g. In this paper, a_ij is a dummy variable representing the binary variable of interest y_ij, with a_ij=1 if y_ij=2 and a_ij=0 if y_ij=1, respectively. The term ε_ijg represents random noise, unaffected by batch effects. The term γ_jg corresponds to the mean shift in location of variable g in the j-th batch compared to the unobserved—hypothetical—data xijg∗ unaffected by batch effects. The term δ_jg corresponds to the scale shift of the residuals for variable g in the j-th batch. As in the SVA model (Appendix A.2, Additional file 1), Z_ijl are random latent factors. In contrast to the latter model, in our model the distribution of the latent factors is independent of the individual observation. However, since the loadings b_jgl of the latent factors are batch-specific, the latter induce batch effects in our model as well. More precisely, they lead to varying correlation structures in the batches. In the SVA model, by contrast, all batch effects are induced by the latent factors. Without the summand ∑l=1mjbjglZijl model (1) would equal the model underlying the ComBat-method, see Appendix A.1 (Additional file 1).

The unobserved data xijg∗ not affected by batch effects is assumed to have the form

The remaining batch effect adjustment methods considered in this paper are described in Appendix A.3 (Additional file 1).