This has two practical implications: first, in studies with biased sampling schemes and sample overlap, if one wishes to constrain the intercept, one should use the sample correlation between phenotypes ρ̂ rather than the population correlation ρ. Under biased sampling, plimN→∞ρ̂=E[yi1yi2|si=1], which is typically not equal to ρ. Second, even if there is no sample verlap, biased sampling can affect the genetic correlation estimate. If the biased sampling mechanism (i.e., the function f(Ci):=P[si=1|Ci]) is known, then it may be possible to explicitly model the biased sampling and derive a function for converting genetic correlation estimates from the biased sample to population genetic correlations (similar to the derivations in sections 1.3 and 1.4 of the Supplementary Note). If the biased sampling mechanism can only be described qualitatively, then it should at least be possible to guess the magnitude and direction of the bias by reasoning about E[yi1yi2|si=1] and E[ai1ai2|si=1].
