We show in the Supplementary Note that LD Score regression is robust to oversampling of cases in case/control studies, modulo transformation observed and liability scale heritability and genetic covariance. Oversampling of cases is the most common form of biased sampling, but there are many other forms of biased sampling. For example, consider case/control/covariate ascertainment, where the sampling of cases and controls takes into account a covariate. As as concrete example, we know that high BMI is a major risk factor for T2D. If we wish to discover genetic variants that influence risk for T2D via mechanisms other than BMI, we may wish to perform a case/control study for T2D where we compare BMI-matched cases and controls. If we were to use such a T2D study and a random population study of BMI to compute the genetic correlation between BMI and T2D, the result would be substantially attenuated relative to the population genetic correlation between T2D and BMI. (Note that this example holds irrespective of whether there is sample overlap and applies to all genetic correlation estimators, not just LD Score).
