Table 1 Type I error rate of $${\tilde{Z}}_C$$ and $${TDT}_{\mathrm{BRV}}$$

$${\tilde{Z}}_C$$ $${TDT}_{\mathrm{BRV}}$$
α = 0.05 α = 0.001 α = 0.05 α = 0.001
Sample size (N) Sample size (N) Sample size (N) Sample size (N)
Homogeneous population N0 + $$\frac{1}{10}$$ N0 N0 + $$\frac{1}{5}$$ N0 N0 + $$\frac{1}{2}$$ N0 N0 + $$\frac{1}{10}$$ N0 N0 + $$\frac{1}{5}$$ N0 N0 + $$\frac{1}{2}$$ N0 N0 + $$\frac{1}{10}$$ N0 N0 + $$\frac{1}{5}$$ N0 N0 + $$\frac{1}{2}$$N0 N0 + $$\frac{1}{10}$$ N0 N0 + $$\frac{1}{5}$$ N0 N0 + $$\frac{1}{2}$$ N0
Ω0 + Ι 0.057 0.045 0.052 0.0012 0.0014 0.0009 0.053 0.046 0.051 0.0011 0.0012 0.0010
Ω0 + ΙΙ 0.055 0.055 0.050 0.0011 0.0008 0.0014 0.054 0.055 0.052 0.0012 0.0009 0.0014
Ω0 + Ι + ΙΙ 0.048 0.049 0.046 0.0009 0.0013 0.0012 0.048 0.049 0.049 0.0009 0.0013 0.0013
Population stratification
Ω0 + Ι 0.055 0.051 0.050 0.0013 0.0012 0.0010 0.055 0.047 0.054 0.0013 0.0012 0.0012
Ω0 + ΙΙ 0.053 0.056 0.053 0.0014 0.0009 0.0014 0.056 0.054 0.051 0.0012 0.0009 0.0011
Ω0 + Ι + ΙΙ 0.049 0.049 0.051 0.0009 0.0013 0.0014 0.048 0.050 0.047 0.0009 0.0010 0.0014
1. Note: The sample size N = N0 + NI, N = N0 + NII, N = N0 + NI + NII for Ω0 + Ι, Ω0 + ΙΙ, and Ω0+Ι+ΙΙ, respectively, with N0 (=500) complete case-parents trios (Ω0). There are NI case-parents trios of ΩI forΩ0 + Ιand NII case-parents trios of ΩII for Ω0 + ΙΙ with NI, NII=$$\frac{1}{10}$$N0, $$\frac{1}{5}$$N0, and $$\frac{1}{2}$$N0, respectively. For Ω0+Ι+ΙΙ, NI = NII, and NI + NII = $$\frac{1}{10}$$N0, $$\frac{1}{5}$$N0, and $$\frac{1}{2}$$N0, respectively