Response to Sul & Eskin

We thank Sul & Eskin for carefully examining and confirming the limitation of standard mixed model association methods that we identified in our review, and for developing an interesting new way to address it.

In Price et al. 2010¹, we investigated the limits of mixed model methods by considering an extreme simulation in which most markers had low population differentiation (F_ST=0.01) but a small fraction of markers were unusually differentiated (allele frequency difference = 0.6). We found that standard mixed model methods² did not fully correct for population structure, but mixed models with PC covariates³ did fully correct for population structure. We stated that “population structure is a fixed effect, and spurious associations might result if it is modeled as a random effect based on overall covariance”.

Sul & Eskin⁴ have confirmed that in this extreme simulation, standard mixed model methods do not fully correct for population structure, and that mixed models with PC covariates do fully correct for population structure. They also investigated a new approach, which is to employ a mixed model using two kinship matrices, one computed using unusually differentiated markers identified by their SPA method⁵ and one computed using the remaining markers. They reported that this approach also fully corrects for population structure in this simulation. Thus, population stratification (a fixed effect in this simulation) can be addressed using random effects, in a way that we had not previously considered—our review only considered mixed models with a single random effect based on overall covariance^{2–3, 6–8}, but did not consider mixed models with multiple random effects⁴.

Another possibility, very similar to the Sul & Eskin approach, is to employ a mixed model using two kinship matrices, one computed from PC1 and one computed using the remaining PCs, based on the natural decomposition of a kinship matrix into its PCs⁹. This would also fully correct for population structure in this extreme simulation, since Sul & Eskin showed that using a single kinship matrix computed from PC1 fully corrects for population structure.

A broader question is whether the limitation of standard mixed model methods that arises in this extreme simulation is a major concern in empirical studies. In Price et al. 2010¹, we stated that standard mixed model methods are an appealing and simple approach and are sufficient to correct for stratification in many settings. Sul & Eskin indicated that the limitation we described did not arise in the Finnish and UK data sets that they analyzed. We agree that mixed models with a single random effect based on overall covariance will likely be sufficient to fully correct for population structure in most settings.

We finally note that recent work has raised additional points about mixed model methods, including inclusion vs. exclusion of the candidate marker in the kinship matrix, use of only a small subset of markers in computing the kinship matrix, and effects of case-control ascertainment^10–13. We believe that these are important points that merit further investigation, which is however outside the scope of the current Correspondence.