Table 2.

Simulation results for the piecewise linear method

			Decile of the IV‐free exposure distribution										Heuristic
Model	β	Parameter	1	2	3	4	5	6	7	8	9	10	FP	PL
Linear	0.5	Correct	0.201	0.357	0.460	0.550	0.641	0.743	0.870	1.040	1.302	1.995
		Mean	0.197	0.352	0.454	0.543	0.633	0.736	0.862	1.032	1.296	1.961	1.12	1.20
		Coverage	0.958	0.964	0.962	0.962	0.958	0.952	0.940	0.944	0.950	0.952	(0.74)	(0.68)
Quadratic	0.5	Correct	0.891	1.697	2.281	2.826	3.409	4.107	5.027	6.360	8.644	16.002
		Mean	1.011	1.831	2.426	2.982	3.572	4.281	5.214	6.568	8.891	16.617	1.07	2.94
		Coverage	0.680	0.784	0.786	0.786	0.772	0.758	0.764	0.762	0.750	0.922	(0.77)	(1.43)
Threshold	0.5	Correct	0.000	0.000	0.000	0.000	0.000	0.000	0.054	0.224	0.486	1.179
		Mean	0.001	0.000	0.001	0.003	0.005	0.020	0.088	0.239	0.499	1.171	1.35	1.25
		Coverage	0.960	0.964	0.952	0.950	0.942	0.930	0.934	0.930	0.940	0.946	(0.46)	(0.73)
Threshold	1	Correct	0.000	0.000	0.000	0.000	0.000	0.000	0.108	0.448	0.971	2.359
		Mean	0.008	0.012	0.013	0.012	0.014	0.041	0.181	0.493	1.017	2.393	2.31	1.33
		Coverage	0.946	0.940	0.936	0.922	0.916	0.904	0.904	0.916	0.916	0.958	(0.42)	(0.79)

Notes. β is the effect parameter. Mean is the mean value of the outcome at the mean value of the exposure in the deciles of the IV‐free distribution. Coverage refers to the number of replications where the correct value of the outcome at the mean value of the exposure in the decile of the IV‐free distribution was contained within the corresponding 95% prediction interval. The heuristic statistic (mean (SD) across simulations) is the sum of the absolute values of the predicted value of the outcome minus the correct value of the outcome at the mean value of the exposure in deciles of the IV‐free distribution. FP, fractional polynomial; PL, piecewise linear; IV, instrumental variable.