Table 1.

Simulation of causal inference for ordered multi-categorical exposure.

Simulation	IVa	β	$\widehat{\beta }$	95% Confidence interval (CI)
				Normal	Basic	Percentile
Group A	8	1.5	2.503	2.314–2.783	2.315–2.784	2.221–2.690
Group B	6	1.5	2.395	2.067–2.817	2.003–2.781	2.010–2.787
Group C	5	1.5	2.227	1.913–2.577	1.907–2.556	1.898–2.547
Group D	4	2.7	4.419	3.912–5.054	3.848–5.014	3.823–4.990
Group E	6	2.7	4.419	4.022–4.938	4.005–4.916	3.922–4.833
Group F	6	− 2.7	− 4.068	− 4.551–− 3.672	− 4.525–− 3.580	− 4.556–− 3.611
Group G	6	− 1.5	− 2.223	− 2.495–− 2.014	− 2.491–− 1.992	− 2.453–− 1.954
Group H	5	0	− 0.027	− 0.112–0.063	− 0.111–0.064	− 0.118–0.057
Group I	8	0	− 0.008	− 0.073–0.062	− 0.071–0.060	− 0.075–0.056
Group J	3	0	0.048	− 0.067–0.171	− 0.064–0.180	− 0.084–0.160

"a" represents the number of instrumental variables, "Normal" means the CI was calculated by the normal approximation method, "Basic" indicates the CI was calculated by the basic bootstrap method and "Percentile" stands for the CI was calculated by the bootstrap percentage method. The ten data sets were randomly generated with a preset $\sigma$ of 1.5.