. Author manuscript; available in PMC: 2023 Feb 1.

Published in final edited form as: Genet Epidemiol. 2021 Nov 15;46(1):73–86. doi: 10.1002/gepi.22438

TABLE 2.

Type 1 error rates of ZIP-b and ZIP-k

	Parameters	w₀ = 1	w _p	$w_{\hat{p}}$	w _B	$w_{\hat{B}}$
ZIP-b	γ_π	0.051	0.056	0.053	0.050	0.051
	γ_λ	0.045	0.042	0.043	0.044	0.044
	Fisher-p	0.046	0.055	0.052	0.045	0.045
	Cauchy-p	0.046	0.054	0.051	0.047	0.046
	Min-p	0.092	0.101	0.094	0.089	0.089

ZIP-k	γ_π	0.051	0.001	0.004	0.049	0.049
	γ_λ	0.040	0.0004	0.003	0.037	0.037
	Fisher-p	0.048	0.0003	0.002	0.045	0.046
	Cauchy-p	0.050	0.001	0.002	0.047	0.048
	Min-p	0.050	0.002	0.005	0.088	0.089

Note: w₀ = 1 no weight; $w_{p} = 1 / \sqrt{(p (1 - p)}$ , and $w_{\hat{p}} = 1 / \sqrt{(\hat{p} (1 - \hat{p})}$ proposed by Madsen and Browning; w_B = Beta(p, 1, 25) and $w_{\hat{B}} = B e t a (\hat{p}, 1, 25)$ based on Beta density function, where p is the population MAF from COSI and $\hat{p}$ is the sample MAF estimate. All p-values were estimated based on 10, 000 replicates of 2000 samples. Parameters setting: proportion of structured zero of 20% and average counts at 4, the coefficients for covariates: α_π1 = α_λ1 = 0.25 and α_π2 = α_λ2 = 0.50. ZIP-b: ZIP burden test; ZIP-k: ZIP kernel.