To the editor:
In the study of complex disease, separating causal from confounded factors is a challenge for genetic epidemiologists. One tool useful for separating these factors is Genomic Control (GC). In this communication we clarify how and when to use GC. We also describe a refined approach to GC, which should be used when GC is applied to extreme settings.
Population-based studies, such as case-control studies, are common designs used to determine the genetic and environmental bases of disease. To avoid false positive associations, design and analysis of population-based studies should account for population stratification, which can inflate association test statistics. One analytic method used to control the false positive rate is GC. In our original paper1, we investigated two scenarios with two corresponding analytic methods. GC is the version similar to the typical approach to hypothesis testing, and GCB is the version that uses Bayesian inference. GC is suitable when a modest number of candidate genes are assessed and L supplementary loci are included for control. The supplementary loci, called null loci, are used to correct any inflation, λ, in association test statistic(s) by estimating λ from the null test statistics. GC produces average rejection rates close to the targeted 0.05 significance level2,3,4.
We also considered population-based studies when large numbers of markers are tested1. GCB is designed for this scenario. Rather than preselecting null loci, GCB delineates loci associated with disease as 'outliers' relative to most of the loci tested.
In our original papers1,2, we argued that a population-based study should attempt to remove the effect of stratification by experimental design and analysis, such as by matching cases and controls for ethnicity and environmental covariates. GC then adjusts for the residual effects of stratification. Careful study design and implementation pay off in statistical power5,6; even small stratification can have considerable consequences for large samples1,2,3,4,5.
Marchini et al.7 explored the efficacy of GC for population-based studies under less ideal conditions, using subjects that originate from different populations and including environmental effects that induce geographically distinct prevalences; both of these possibilities were ignored in the design and analysis. Because they genotyped a large number of loci, they required an extremely small significance level (α) for P values. They found that GC could be anticonservative when α is small. Their results are sensible because GC treats λ as a known constant8. For small values of α, variability in the estimate of λ matters. The population-based studies explored by Marchini et al.7 can produce highly inflated test statistics (Fig. 1), and, because these population-based studies involve a large number of candidate loci, they are more appropriately analyzed by using GCB rather than GC.
Because λ is determined by sample size, stratification and differential prevalence1, we can generate and compactly represent the general findings of Marchini et al.7 (Fig. 1). Four features stand out from our results: (i) GC works well in situations for which it was originally intended, namely larger values of α and/or smaller values of λ (refs. 1,
Is there a way to adjust the procedure if a researcher wishes to apply the logic of GC and use an extremely small α value? Correcting the bias in GC is straightforward by simple modification of the test statistic (GCF). For GCF, estimate λ using the mean (λm) of the null test statistics and account for the variability of λm by using an F test to determine the P values. Notably, GCF is accurate throughout the parameter space, even for only 30 null loci (Fig. 2).
Our means of validating the results of Marchini et al.7 and our own results use a shortcut method that Marchini et al.7 did not use. Our results are also supported by using the simulation methods of Marchini et al.7. When we used their methods and analyzed the data using GCF, we again found that GCF yielded an excellent approximation for small values of α (Table 1), even when λ is inflated substantially by large sample size or geographically distinct prevalences.
In summary, when a large number of candidate loci are genotyped or when α is small, application of GC produces misleading results (Fig. 1), as Marchini et al.7 show. Because GCF corrects this bias for small values of α, and does so in a range of settings (Fig. 2 and Table 1), we conclude that the bias is largely due to the uncertainty in λ. GCF accounts for this uncertainty in its degrees of freedom. Thus, GCF provides a simple alternative to recently suggested methods based on the confidence interval for λ (ref. 9).
As we have pointed out before1,10, for experiments involving a large number of tests of genetic markers, one should analyze the entire distribution of test statistics. In this setting different statistical paradigms should be considered, such as methods based on the false discovery rate principle11, which has great promise for this setting (refs. 10,12 and S.-A.B., B.D., K.R. and L. Wasserman, unpublished data).
Note: Supplementary information is available on the Nature Genetics website.