Table 2. Type I error rate $\left(\text{\%}\right)$ of each test under the null hypothesis of no interaction effect, i.e., $\gamma =0$.

		Type I error rate $\left(\text{\%}\right)$
		Probit			Logit			Log penetrance
Tests	d.f.	$p=10\text{\%}$	$p=1\text{\%}$	$p=0.1\text{\%}$	$p=10\text{\%}$	$p=1\text{\%}$	$p=0.1\text{\%}$	$p=10\text{\%}$	$p=1\text{\%}$	$p=0.1\text{\%}$
A. ${\beta }_1={\beta }_2=0$
SI	4	4.96	5.28	5.07	5.22	4.96	5.03	5.39	4.95	5.34
ImAP	4	3.06	3.28	3.16	3.10	3.32	3.09	3.36	3.21	3.38
$\mathrm{ImA}{\mathrm{P}}_2$	4	2.49	2.80	2.59	2.54	2.82	2.52	2.90	2.72	2.89
$\mathrm{ImA}{\mathrm{P}}_3$	4	12.34	12.40	12.27	12.32	12.34	12.39	12.68	12.13	12.19
$\mathrm{CLRP}{\mathrm{C}}_1$	4	4.97	5.25	5.01	5.14	5.23	5.04	5.44	5.05	5.29
SC	1	5.29	5.19	4.83	4.77	4.83	4.98	5.00	5.10	5.43
$\mathrm{CLRP}{\mathrm{C}}_2$	1	5.42	5.17	5.00	4.57	4.71	5.01	5.06	4.98	5.34
TC	1	5.28	5.13	4.84	4.78	4.79	5.04	4.90	5.08	5.42
B. ${\beta }_1={\beta }_2=0.2$
SI	4	5.10	5.01	4.73	5.05	4.92	5.39	5.13	4.80	4.67
ImAP	4	8.45	26.05	60.20	8.03	10.57	10.68	11.13	10.81	10.69
$\mathrm{ImA}{\mathrm{P}}_2$	4	2.56	2.60	2.51	2.87	2.62	2.80	2.82	2.58	2.48
$\mathrm{ImA}{\mathrm{P}}_3$	4	79.41	99.42	100.00	77.02	88.38	89.14	89.19	89.08	88.92
$\mathrm{CLRP}{\mathrm{C}}_1$	4	4.93	5.13	5.07	5.40	4.81	5.08	5.15	4.70	5.06
SC	1	5.43	4.97	5.21	5.66	5.04	5.28	5.03	4.91	4.81
$\mathrm{CLRP}{\mathrm{C}}_2$	1	4.98	4.87	4.76	5.52	4.98	5.43	5.38	4.71	5.29
TC	1	5.03	4.94	5.77	5.44	5.30	5.44	5.24	5.06	5.17

Each entry represents an average over 10,000 simulated data sets. p, disease prevalence in the population.