Table 3. Permutation-based type I error rate of all tests under the null hypothesis of no interaction effect, i.e., $\gamma =0$.

		Type I error rate $\left(\text{\%}\right)$
Tests	d.f.	$p=10\text{\%}$	$p=1\text{\%}$	$p=0.1\text{\%}$
A. ${\beta }_1={\beta }_2=0$
SI	4	4.72	5.14	5.58
ImAP	4	5.22	5.81	5.48
$\mathrm{ImA}{\mathrm{P}}_2$	4	5.02	5.70	5.68
$\mathrm{ImA}{\mathrm{P}}_3$	4	4.82	5.25	5.89
$\mathrm{CLRP}{\mathrm{C}}_1$	4	5.33	6.60	5.19
SC	1	6.02	6.37	5.68
$\mathrm{CLRP}{\mathrm{C}}_2$	1	6.43	6.37	5.89
TC	1	6.02	6.26	5.58
B. ${\beta }_1={\beta }_2=0.2$
SI	4	4.91	4.97	4.10
ImAP	4	11.54	37.99	73.77
$\mathrm{ImA}{\mathrm{P}}_2$	4	4.70	5.40	4.71
$\mathrm{ImA}{\mathrm{P}}_3$	4	61.11	97.78	100.00
$\mathrm{CLRP}{\mathrm{C}}_1$	4	4.23	5.88	4.17
SC	1	5.02	6.14	5.43
$\mathrm{CLRP}{\mathrm{C}}_2$	1	5.24	6.03	4.92
TC	1	5.24	6.67	6.15

Each entry is based on 1000 permuted data sets. p, disease prevalence in the population.