Table 2.

Impact of main effects on type I error and analytical bias for the case-only design

		Disease prevalence
${\mathit{\beta }}_{\mathit{G}}$	${\mathit{\beta }}_{\mathit{E}}$	1.0%	1.5%	2.0%	2.5%	3.0%	3.5%	4.0%	4.5%	5.0%	10.0%	15.0%	20.0%
(a) Type I error
$ln(3.846)$	$ln(5)$	0.141321	0.252332	0.396712	0.551859	0.695653	0.810552	0.891836	0.942837	0.972389	0.999998	1.000000	1.000000
$ln(3.846)$	$ln(2)$	0.056403	0.063479	0.072751	0.085891	0.101210	0.119065	0.138098	0.160377	0.183840	0.480285	0.735506	0.873881
$ln(1.2)$	$ln(2)$	0.050132	0.050603	0.049854	0.050842	0.050614	0.051195	0.051387	0.052100	0.052221	0.057378	0.063536	0.070049
$ln(1.05)$	$ln(1.1)$	0.049890	0.049729	0.049665	0.049657	0.049644	0.049992	0.049896	0.049923	0.050019	0.050016	0.049992	0.049666
$ln(0.952)$	$ln(0.909)$	0.049816	0.049991	0.049856	0.049738	0.049618	0.050020	0.050054	0.049950	0.049860	0.049651	0.049931	0.050423
$ln(0.833)$	$ln(0.5)$	0.050030	0.049658	0.049459	0.049875	0.049607	0.049515	0.049339	0.049723	0.049573	0.049940	0.051270	0.052579
$ln(0.26)$	$ln(0.5)$	0.048962	0.048521	0.048793	0.048931	0.049388	0.049655	0.050008	0.050319	0.051327	0.063964	0.088547	0.125843
$ln(0.26)$	$ln(0.2)$	0.047237	0.047101	0.046880	0.047278	0.047025	0.047459	0.047611	0.048081	0.048999	0.063687	0.096078	0.152817
(b) Analytical bias
$ln(3.846)$	$ln(5)$	−0.039119	−0.057894	−0.076150	−0.093890	−0.111119	−0.127843	−0.144066	−0.159797	−0.175042	−0.302652	−0.391307	−0.449137
$ln(3.846)$	$ln(2)$	−0.012563	−0.018684	−0.024697	−0.030605	−0.036407	−0.042105	−0.047698	−0.053188	−0.058576	−0.106964	−0.145864	−0.175993
$ln(1.2)$	$ln(2)$	−0.001667	−0.002479	−0.003277	−0.004061	−0.004831	−0.005588	−0.006330	−0.007060	−0.007775	−0.014227	−0.019476	−0.023631
$ln(1.05)$	$ln(1.1)$	.−0.000048	−0.000072	−0.000095	−0.000118	−0.000141	−0.000164	−0.000186	−0.000208	−0.000230	−0.000434	−0.000612	−0.000764
$ln(0.952)$	$ln(0.909)$	−0.000044	−0.000066	−0.000087	−0.000109	−0.000130	−0.000151	−0.000171	−0.000192	−0.000212	−0.000404	−0.000574	−0.000724
$ln(0.833)$	$ln(0.5)$	−0.000928	−0.001389	−0.001847	−0.002303	−0.002756	−0.003207	−0.003655	−0.004101	−0.004544	−0.008831	−0.012836	−0.016536
$ln(0.26)$	$ln(0.5)$	−0.005304	−0.007952	−0.010597	−0.013240	−0.015878	−0.018514	−0.021146	−0.023774	−0.026399	−0.052388	−0.077767	−0.102296
$ln(0.26)$	$ln(0.2)$	−0.008789	−0.013196	−0.017611	−0.022035	−0.026467	−0.030907	−0.035354	−0.039810	−0.044273	−0.089276	−0.134820	−0.180639

Data (N = 10,000 cases) were generated under the null with no interaction (${\beta }_{G\times E}=0$). The MAF for the genetic variant is 0.2 and the frequency of the environmental exposure is 0.1. Type I error is calculated as the proportion of instances that the null hypothesis is rejected among all 1,000,000 replicates ($\alpha =0.05)$. Analytical bias was calculated using formula (4). Type I errors exceeding 5% of the alpha level are highlighted in bold in the table