Table 2.

The mean of $\widehat{\theta }$, mean of estimated standard error of $\widehat{\theta }$, and standard deviation of $\widehat{\theta }$ across simulation repetitions for the set-valued (SV) and logistic regression (LG and oLG) methods based on 1000 simulations^*.

			LG					oLG					SV
θ	SM	DM	$\overline{\widehat{\theta }}$	$\overline{\widehat{se}\left(\widehat{\theta }\right)}$	$sd\left(\widehat{\theta }\right)$	$\frac{\overline{\widehat{\theta }}}{\overline{\overline{\widehat{se}\left(\widehat{\theta }\right)}}}$	$\frac{\overline{\widehat{\theta }}}{sd\left(\widehat{\theta }\right)}$	$\overline{\widehat{\theta }}$	$\overline{\widehat{se}\left(\widehat{\theta }\right)}$	$sd\left(\widehat{\theta }\right)$	$\frac{\overline{\widehat{\theta }}}{\overline{\overline{\widehat{se}\left(\widehat{\theta }\right)}}}$	$\frac{\overline{\widehat{\theta }}}{sd\left(\widehat{\theta }\right)}$	$\overline{\widehat{\theta }}$	$\overline{\widehat{se}\left(\widehat{\theta }\right)}$	$sd\left(\widehat{\theta }\right)$	$\frac{\overline{\widehat{\theta }}}{\overline{\overline{\widehat{se}\left(\widehat{\theta }\right)}}}$	$\frac{\overline{\widehat{\theta }}}{sd\left(\widehat{\theta }\right)}$
Rand, pA=0.0075
0	LGsimu	H0	0.02	0.391	0.409	0.051	0.049	0.005	0.367	0.387	0.013	0.012	0.0017	0.219	0.231	0.008	0.007
0.5	LGsimu	ADD	0.535	0.398	0.402	1.344	1.329	0.514	0.353	0.354	1.456	1.452	0.3039	0.212	0.213	1.43	1.426
2	LGsimu	ADD	2.229	2.711	1.336	0.822	1.669	2.009	0.366	0.374	5.49	5.374	1.1942	0.218	0.222	5.478	5.373
0	SVsimu	H0	−0.02	0.409	0.408	−0.05	−0.05	−0.026	0.372	0.386	−0.07	−0.07	−0.016	0.221	0.229	−0.073	−0.071
0.5	SVsimu	ADD	0.84	0.44	0.47	1.91	1.789	0.807	0.361	0.377	2.236	2.139	0.4834	0.217	0.226	2.231	2.139
2	SVsimu	ADD	6.953	83.61	5.785	0.083	1.202	3.442	0.711	0.648	4.841	5.31	2.0333	0.284	0.303	7.15	6.709
Rand, pA=0.2
0	LGsimu	H0	−0.003	0.082	0.082	−0.04	−0.04	−0.004	0.078	0.078	−0.06	−0.06	−0.003	0.047	0.047	−0.064	−0.064
0.5	LGsimu	ADD	0.502	0.085	0.087	5.941	5.798	0.5	0.076	0.077	6.566	6.477	0.2976	0.045	0.046	6.555	6.475
2	LGsimu	ADD	2.003	0.124	0.128	16.2	15.66	1.995	0.093	0.093	21.43	21.35	1.179	0.051	0.053	23.3	22.13
0	SVsimu	H0	1E-03	0.086	0.086	0.011	0.011	1E-06	0.079	0.078	1E-4	1E-4	−2E-4	0.047	0.046	−0.004	−0.004
0.5	SVsimu	ADD	0.833	0.095	0.095	8.781	8.779	0.837	0.079	0.08	10.57	10.52	0.5012	0.047	0.047	10.76	10.57
2	SVsimu	ADD	3.581	0.221	0.214	16.19	16.73	3.418	0.132	0.129	25.84	26.59	2.0031	0.068	0.072	29.58	27.97
Same, pA=0.0075
0	LGsimu	H0	0.05	0.42	0.433	0.12	0.116	0.017	0.348	0.355	0.05	0.049	0.0103	0.211	0.216	0.049	0.048
0.5	LGsimu	ADD	0.632	0.717	0.631	0.882	1.002	0.506	0.329	0.332	1.538	1.523	0.3091	0.2	0.202	1.543	1.532
2	LGsimu	ADD	3.042	17.62	3.062	0.173	0.993	1.939	0.338	0.333	5.74	5.816	1.174	0.195	0.193	6.035	6.091
0	SVsimu	H0	0.004	0.442	0.46	0.009	0.008	−0.002	0.362	0.369	−5E-3	−4E-3	−8E-4	0.217	0.221	−0.004	−0.004
0.5	SVsimu	ADD	0.946	1.116	0.806	0.848	1.174	0.819	0.345	0.353	2.371	2.317	0.4936	0.206	0.21	2.392	2.348
2	SVsimu	ADD	8.881	140.8	6.375	0.063	1.393	3.368	0.493	0.86	6.836	3.917	1.9605	0.263	0.29	7.454	6.759
Same, pA=0.2
0	LGsimu	H0	−0.003	0.087	0.089	−0.04	−0.04	−0.005	0.074	0.074	−0.07	−0.07	−0.003	0.045	0.045	−0.071	−0.071
0.5	LGsimu	ADD	0.548	0.091	0.089	6.011	6.169	0.5	0.073	0.072	6.853	6.92	0.3035	0.044	0.044	6.895	6.963
2	LGsimu	ADD	2.042	0.131	0.128	15.65	15.93	1.979	0.092	0.092	21.46	21.56	1.1699	0.05	0.052	23.37	22.39
0	SVsimu	H0	−0.002	0.092	0.093	−0.03	−0.03	−1E-03	0.076	0.077	−0.01	−0.01	−6E-04	0.046	0.046	−0.014	−0.014
0.5	SVsimu	ADD	0.871	0.101	0.098	8.615	8.866	0.833	0.078	0.075	10.68	11.04	0.4999	0.046	0.045	10.91	11.2
2	SVsimu	ADD	3.337	0.22	0.205	15.14	16.31	3.273	0.127	0.119	25.73	27.4	1.9147	0.064	0.066	29.7	28.93

^*n=2000 and 1998 for Rand and Same sampling schema.

$\overline{\widehat{\theta }}$: The mean of $\overline{\widehat{\theta }}$ for 1000 replicates; $\overline{\widehat{se}\left(\widehat{\theta }\right)}$: The mean of estimated standard error of $\widehat{\theta }$ for 1000 replicates; $sd\widehat{\theta })$: The empirical standard deviation of $\widehat{\theta }$ for 1000 replicates; p_A is minor allele frequency of SNP; θ is the association coefficient of SNP with outcome; SM is simulation model; DM is disease model representing the underlying genetic disease model. LG stands for logistic regression model on the regrouped binary outcome (recoding as 0 or greater than 0); SV stands for the set-valued method; oLG stands for ordered logistic regression method; oPRB is the usual probit model with IRWLS estimation algorithm.