Phantom heritability under the limiting pathway model. (*A*) Quantitative trait model *LP*(*k*, *h*^{2}_{pathway}, *c*_{R}). For various parameters, curves show apparent heritability *h*^{2}_{pop}(*ACE*) and phantom heritability π_{phantom}. Curves connect points with various values of *k* (1, 2, 3, 4, 5, 6, 7, 10, with tip of arrow at *k* = 10), and for specific values of *h*^{2}_{pathway} (10, 30, 50, 70, and 90%, indicated by color of the curve) and *c*_{R} (0% filled circles and arrows, 50% open boxes and arrows). The red asterisk indicates example *P** referred to in the text. (The quantity *h*^{2}_{pop} can exceed 100%, as seen for some models with high heritability and for some real traits. In such cases, we set *h*^{2}_{pop} = 100%.) Raw data are in *SI Appendix*, Table 6. (*B*) Disease model *LP*_{Δ}(*k*, *h*^{2}_{pathway}, 0%, μ). For various parameters, curves show value of λ_{MZ} and phantom heritability π_{phantom}. Values of *k* and *h*^{2}_{pathway} are as in *A*. Values of prevalence μ are 0.1% (solid), 1% (dotted), and 10% (dashed). The red asterisk indicates example Δ* referred to in the text. In both *A* and *B*, as *k* increases, the trait becomes more nonlinear; phantom heritability increases to 50% and beyond. Raw data are in *SI Appendix*, Table 7.

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