Table 4.

Mean Rate Ratios, Empirical Variance, and Percent Coverage of 95% Confidence Intervals From 1,000 Simulated Data Sets in the Presence of Unmeasured Confounding Due to “Treatment Withheld” Without or With Asymmetric Trimming of the Propensity Score According to Analytic Strategy (True RR = 2.0)

PS Implementation and Trimming Range	Treatment Prevalence = 0.2				Treatment Prevalence = 0.05
	RR	Variancea	MSEb	Coveragec	RR	Variancea	MSEb	Coveragec
Crude	2.85	0.007	0.130	0.1	2.93	0.016	0.157	4.4
True outcome model	2.00	0.003	0.003	94.0	2.00	0.007	0.007	94.1
Outcome model (X1–X6)d	1.30	0.006	0.193	0.0	1.34	0.013	0.182	1.1
PS analysesd
PS continuous
Unrestricted	1.31	0.007	0.191	0.0	1.38	0.023	0.171	5.9
Restricted
0–100	1.31	0.007	0.190	0.0	1.41	0.019	0.146	5.9
1–99	1.53	0.006	0.078	1.7	1.50	0.014	0.099	14.0
2.5–97.5	1.75	0.006	0.004	38.5	1.63	0.014	0.059	39.3
5–95	1.95	0.005	0.006	91.0	1.75	0.014	0.033	71.9
PS quintiles (Mantel-Haenszel)
Unrestricted	1.52	0.008	0.086	2.1	1.61	0.016	0.066	23.9
Restricted
0–100	1.50	0.007	0.092	1.5	1.58	0.015	0.073	19.6
1–99	1.65	0.006	0.045	11.7	1.63	0.012	0.056	33.8
2.5–97.5	1.82	0.005	0.015	59.7	1.71	0.013	0.040	56.6
5–95	1.98	0.005	0.005	91.8	1.80	0.014	0.027	77.5
PS quintiles (standardized mortality ratio)
Unrestricted	1.46	0.009	0.109	1.3	1.60	0.016	0.070	22.4
Restricted
0–100	1.45	0.008	0.116	1.1	1.57	0.015	0.077	19.0
1–99	1.59	0.006	0.060	6.7	1.62	0.013	0.059	30.9
2.5–97.5	1.78	0.006	0.020	49.4	1.70	0.013	0.042	53.6
5–95	1.97	0.005	0.006	91.3	1.79	0.014	0.028	76.7
PS matching
Unrestricted	1.22	0.008	0.255	0.0	1.28	0.033	0.250	9.5
Restricted
0–100	1.22	0.008	0.254	0.0	1.28	0.033	0.249	9.4
1–99	1.45	0.008	0.113	1.4	1.49	0.030	0.127	33.3
2.5–97.5	1.69	0.008	0.037	38.5	1.63	0.031	0.080	60.7
5–95	1.94	0.008	0.009	89.2	1.77	0.034	0.054	81.7
Inverse probability of treatment weighting
Unrestricted	1.31	0.007	0.192	0.0	1.57	0.014	0.076	50.2
Restricted
0–100	1.29	0.007	0.203	0.0	1.52	0.013	0.093	38.6
1–99	1.62	0.005	0.052	19.3	1.69	0.013	0.045	70.4
2.5–97.5	1.82	0.005	0.014	75.2	1.77	0.013	0.029	83.2
5–95	1.99	0.005	0.005	94.5	1.85	0.014	0.022	91.1

Abbreviations: MSE, mean squared error; PS, propensity score; RR, rate ratio.

^aVariance of treatment effect estimates [log(RR)] over 1,000 simulated data sets.

^bMean of [log(RR) – log(2.0)]² over 1,000 simulated data sets.

^cPercentage of simulated studies in which the 95% confidence interval includes the true value (RR = 2.0).

^dOutcome and PS models including all measured covariates X₁–X₆ but not including unmeasured covariate X₇.