Table 2 Open in new tab Simulation...

		Performance measures for estimated log(HR) for exposure
1	1.0 [0]	0.005	N/A	0.172	0.172	0.957^j	−0.023^b	N/A	0.177	0.178	0.967	N/A	1.03	52.2
2	1.5 [0.405]	−0.110^b	−27.1	0.167	0.200	0.910	−0.164^b	−40.5	0.162	0.231	0.872	1.49	1.16	57.3
3	2.0 [0.693]	−0.180^b	−26.0	0.149	0.234	0.802	−0.257^b	−37.0	0.152	0.298	0.649	1.43	1.27	68.3
4	2.5 [0.916]	−0.227^b	−24.8	0.149	0.271	0.675	−0.315^b	−34.4	0.147	0.348	0.452	1.39	1.28	71.7

		Performance measures for estimated log(HR) for exposure
1	1.0 [0]	0.005	N/A	0.172	0.172	0.957^j	−0.023^b	N/A	0.177	0.178	0.967	N/A	1.03	52.2
2	1.5 [0.405]	−0.110^b	−27.1	0.167	0.200	0.910	−0.164^b	−40.5	0.162	0.231	0.872	1.49	1.16	57.3
3	2.0 [0.693]	−0.180^b	−26.0	0.149	0.234	0.802	−0.257^b	−37.0	0.152	0.298	0.649	1.43	1.27	68.3
4	2.5 [0.916]	−0.227^b	−24.8	0.149	0.271	0.675	−0.315^b	−34.4	0.147	0.348	0.452	1.39	1.28	71.7

Table 2

Simulation results for Example 2: comparison of estimates for time-varying recent benzodiazepine use (exposure) from two strategies for imputing event times.

		Performance measures for estimated log(HR) for exposure
		Model 2: event at interval midpoint					Model 3: event at interval endpoint					Comparison of model 2 and model 3
Scenario	True HR for exposure [log(HR)]	Bias^a^,^b	Relative bias^c, %	Empirical SE^d	RMSE^e	Coverage rate 95% CI^f	Bias^a^,^b	Relative bias^c, %	Empirical SE^d	RMSE^e	Coverage rate 95% CI^f	Ratio bias model 3/model 2^g	Ratio RMSE model 3/model 2^h	% repetitions model 2 closer to truth than model 3ⁱ, %
Col. 1	Col. 2	Col. 3	Col. 4	Col. 5	Col. 6	Col. 7	Col. 8	Col. 9	Col. 10	Col. 11	Col. 12	Col. 13	Col. 14	Col. 15
1	1.0 [0]	0.005	N/A	0.172	0.172	0.957^j	−0.023^b	N/A	0.177	0.178	0.967	N/A	1.03	52.2
2	1.5 [0.405]	−0.110^b	−27.1	0.167	0.200	0.910	−0.164^b	−40.5	0.162	0.231	0.872	1.49	1.16	57.3
3	2.0 [0.693]	−0.180^b	−26.0	0.149	0.234	0.802	−0.257^b	−37.0	0.152	0.298	0.649	1.43	1.27	68.3
4	2.5 [0.916]	−0.227^b	−24.8	0.149	0.271	0.675	−0.315^b	−34.4	0.147	0.348	0.452	1.39	1.28	71.7

		Performance measures for estimated log(HR) for exposure
		Model 2: event at interval midpoint					Model 3: event at interval endpoint					Comparison of model 2 and model 3
Scenario	True HR for exposure [log(HR)]	Bias^a^,^b	Relative bias^c, %	Empirical SE^d	RMSE^e	Coverage rate 95% CI^f	Bias^a^,^b	Relative bias^c, %	Empirical SE^d	RMSE^e	Coverage rate 95% CI^f	Ratio bias model 3/model 2^g	Ratio RMSE model 3/model 2^h	% repetitions model 2 closer to truth than model 3ⁱ, %
Col. 1	Col. 2	Col. 3	Col. 4	Col. 5	Col. 6	Col. 7	Col. 8	Col. 9	Col. 10	Col. 11	Col. 12	Col. 13	Col. 14	Col. 15
1	1.0 [0]	0.005	N/A	0.172	0.172	0.957^j	−0.023^b	N/A	0.177	0.178	0.967	N/A	1.03	52.2
2	1.5 [0.405]	−0.110^b	−27.1	0.167	0.200	0.910	−0.164^b	−40.5	0.162	0.231	0.872	1.49	1.16	57.3
3	2.0 [0.693]	−0.180^b	−26.0	0.149	0.234	0.802	−0.257^b	−37.0	0.152	0.298	0.649	1.43	1.27	68.3
4	2.5 [0.916]	−0.227^b	−24.8	0.149	0.271	0.675	−0.315^b	−34.4	0.147	0.348	0.452	1.39	1.28	71.7

Abbreviations: Col., column; HR, hazard ratio; N/A, not applicable; RMSE, root mean squared error; SE, standard error.

^a Mean of the 1000 estimates of log(HR) for benzodiazepine exposure minus true log(HR) shown in column 2.

^b Indicates that the 95% CI for bias excludes 0.

^c Bias over true log(HR) value, presented as a percentage (not applicable if true log(HR) = 0, ie, in scenario 1).

^d Empirical SE of the log(HR) estimates was calculated as the standard deviation of the 1000 log(HR) estimates.

^e Root mean squared error of the 1000 log(HR) estimates, calculated as the square root of the sum of squared bias and variance, ie, |$\sqrt{{\mathrm{bias}}^2+\mathrm{Var}\left(\log \left(\mathrm{HR}\right)\right)}$|⁠, with lower values indicating better overall accuracy of estimates.

^f Proportion of the 1000 samples where the 95% CI included the true log(HR) (ideally should be very close to 0.95).

^g Ratio of bias for model 3 over bias for model 2, with respectively interval endpoint and midpoint event imputation.

^h Ratio of RMSE for model 3 over RMSE for model 2, with respectively interval endpoint and midpoint event imputation.

ⁱ Percentage among the 1000 repetitions for which the exposure estimate from model 2 (with events imputed at midpoint of intervals) is closer to the true log(HR) (shown in column 2) than the estimate from model 3 (events imputed at endpoint of intervals).

^j Indicates number mentioned in the text.

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