Table 1 Inflation of heritability estimates and their corrected values for select UK Biobank traits.

Phenotype	\(r\)	\(n\)	\({\hat{h}}_{{{{{{\rm{HE}}}}}}}^{2}\) [s.e.]	\({\hat{h}}_{{{{{{\rm{REML}}}}}}}^{2}\) [s.e.]	\({\hat{h}}_{{{{{{\rm{\infty }}}}}}}^{2}\) [s.e.]	\({\hat{h}}_{0}^{2}\) [s.e.]	\({\hat{h}}_{{{{{{\rm{HE}}}}}}}^{2}/{\hat{h}}_{{{{{{\rm{\infty }}}}}}}^{2}\) [s.e.]
Body mass index^a	0.228	334,429	0.257 [3.15e−3]	0.256 [2.29e−3]	–	–	–
Bone mineral density^a,b	–	191,330	0.277 [4.37e−3]	0.295 [3.28e−3]	–	–	–
Height	0.240	334,798	0.567 [4.68e−3]	0.525 [2.22e−3]	0.499 [4.23e−3]	0.467 [3.37e−3]	1.14 [8.70e−4]
Years of education	0.412	332,198	0.174 [2.64e−3]	0.155 [2.01e−3]	0.163 [2.46e−3]	0.153 [2.06e−3]	1.07 [9.46e−4]

Spousal correlations (\(r\)) as previously reported in British cohorts^2,4 and heritability estimates for selected UK Biobank phenotypes (\(n\) denotes sample size). Height and years of education were selected a priori on the basis of previous evidence for primary phenotypic AM whereas body mass index and bone mineral density were selected a priori as negative controls. Assuming equilibrium, the equilibrium (\({\hat{h}}_{{{{{{\rm{\infty }}}}}}}^{2}\)) and panmictic (\({\hat{h}}_{0}^{2}\)) heritability estimates (defined in Eq. 3) provide unbiased estimates of the present day and panmictic heritabilites, respectively. Under disequilibrium, they respectively provide probabilistic lower bounds, with the HE regression (\({\hat{h}}_{{{{{{\rm{HE}}}}}}}^{2}\)) and REML (\({\hat{h}}_{{{{{{\rm{REML}}}}}}}^{2}\)) estimates providing complementary upper bounds. The ratio \({\hat{h}}_{{{{{{\rm{HE}}}}}}}^{2}/{\hat{h}}_{{{{{{\rm{\infty }}}}}}}^{2}\) reflects the extent to which HE regression overestimates the true heritability under the assumption of equilibrium.
^aAdjusted equilibrium and panmictic heritability estimates omitted for negative control traits.
^bNo estimates of the phenotypic mating correlation for bone mineral density were available.

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