Ordinary competing-risk longitudinal TMLE

This study validates ordinary cumulative-incidence estimation for two competing causes. It covers two horizons, monotone censoring, static plans, and one dynamic plan. The comparison uses R lmtp 1.5.4 with the other cause supplied through compete=.

The study reports each unique parameter once. The dynamic plan equals the always-treated plan at the first horizon. A structural test checks this identity outside repeated sampling.

What was compared

setting	cleverly	R lmtp
datasets	1,600 panels from the same finite-support law	the identical rows
causes	relapse and death	each cause fitted as the event, with the other cause in compete=
horizons	cause-specific cumulative incidence at times one and two	a binary mean at time one and a survival fit at time two
plans	never treat, always treat, and continue when L2 equals one	the same unique plan and horizon combinations
mechanisms	exact treatment and censoring probabilities	the same exact per-node density ratios
sequential regressions	intercept-only working regressions	SL.mean working regressions on the same histories
learner path	scikit-learn estimators, called directly	lmtp’s internal run_ensemble replaced by the same single-learner fit, without SuperLearner’s inner cross-validation. The adapter checks the two agree to 1e-10 before every run
intervals	pointwise 95% Wald intervals from the influence curve	the same, after transforming one minus the incidence back to incidence

The outcome regressions are misspecified in this comparison. Therefore, targeting stays nonzero and the competing-event risk mask affects the result. Exact mechanisms isolate those two steps without making the comparison an oracle-law duplicate.

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
two-time-point, two-cause competing-risk law with monotone censoring	ate_regimen[always vs never, death @ t=1]	difference in cumulative incidence of death between the plans “treat at both times” against “treat at neither time” at horizon t = 1	cleverly competing-risk LTMLE	-0.0017 to 0.000777	0.9381	0.9815	pass
two-time-point, two-cause competing-risk law with monotone censoring	ate_regimen[always vs never, death @ t=1]	difference in cumulative incidence of death between the plans “treat at both times” against “treat at neither time” at horizon t = 1	R lmtp	-0.0017 to 0.000777	0.9381	0.9815	pass
two-time-point, two-cause competing-risk law with monotone censoring	ate_regimen[always vs never, death @ t=2]	difference in cumulative incidence of death between the plans “treat at both times” against “treat at neither time” at horizon t = 2	cleverly competing-risk LTMLE	-0.0032 to 0.000929	0.9519	0.9833	pass
two-time-point, two-cause competing-risk law with monotone censoring	ate_regimen[always vs never, death @ t=2]	difference in cumulative incidence of death between the plans “treat at both times” against “treat at neither time” at horizon t = 2	R lmtp	-0.0032 to 0.000929	0.9519	0.9833	pass
two-time-point, two-cause competing-risk law with monotone censoring	ate_regimen[always vs never, relapse @ t=1]	difference in cumulative incidence of relapse between the plans “treat at both times” against “treat at neither time” at horizon t = 1	cleverly competing-risk LTMLE	-0.000670 to 0.0018	0.9519	0.9919	pass
two-time-point, two-cause competing-risk law with monotone censoring	ate_regimen[always vs never, relapse @ t=1]	difference in cumulative incidence of relapse between the plans “treat at both times” against “treat at neither time” at horizon t = 1	R lmtp	-0.000670 to 0.0018	0.9519	0.9919	pass
two-time-point, two-cause competing-risk law with monotone censoring	ate_regimen[always vs never, relapse @ t=2]	difference in cumulative incidence of relapse between the plans “treat at both times” against “treat at neither time” at horizon t = 2	cleverly competing-risk LTMLE	-0.000818 to 0.0032	0.9375	0.9784	pass
two-time-point, two-cause competing-risk law with monotone censoring	ate_regimen[always vs never, relapse @ t=2]	difference in cumulative incidence of relapse between the plans “treat at both times” against “treat at neither time” at horizon t = 2	R lmtp	-0.000818 to 0.0032	0.9375	0.9784	pass
two-time-point, two-cause competing-risk law with monotone censoring	ate_regimen[continue_if_l2 vs never, death @ t=2]	difference in cumulative incidence of death between the plans “treat first, then continue if L2 equals one” against “treat at neither time” at horizon t = 2	cleverly competing-risk LTMLE	-0.0031 to 0.000878	0.9394	0.9632	pass
two-time-point, two-cause competing-risk law with monotone censoring	ate_regimen[continue_if_l2 vs never, death @ t=2]	difference in cumulative incidence of death between the plans “treat first, then continue if L2 equals one” against “treat at neither time” at horizon t = 2	R lmtp	-0.0031 to 0.000878	0.9394	0.9632	pass
two-time-point, two-cause competing-risk law with monotone censoring	ate_regimen[continue_if_l2 vs never, relapse @ t=2]	difference in cumulative incidence of relapse between the plans “treat first, then continue if L2 equals one” against “treat at neither time” at horizon t = 2	cleverly competing-risk LTMLE	-0.000864 to 0.0032	0.9425	0.9599	pass
two-time-point, two-cause competing-risk law with monotone censoring	ate_regimen[continue_if_l2 vs never, relapse @ t=2]	difference in cumulative incidence of relapse between the plans “treat first, then continue if L2 equals one” against “treat at neither time” at horizon t = 2	R lmtp	-0.000864 to 0.0032	0.9425	0.9599	pass
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[always, death @ t=1]	cumulative incidence of death under the plan treat at both times at horizon t = 1	cleverly competing-risk LTMLE	-0.000993 to 0.0010	0.9413	0.9872	pass
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[always, death @ t=1]	cumulative incidence of death under the plan treat at both times at horizon t = 1	R lmtp	-0.000993 to 0.0010	0.9413	0.9872	pass
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[always, death @ t=2]	cumulative incidence of death under the plan treat at both times at horizon t = 2	cleverly competing-risk LTMLE	-0.0022 to 0.0013	0.9406	0.9867	pass
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[always, death @ t=2]	cumulative incidence of death under the plan treat at both times at horizon t = 2	R lmtp	-0.0022 to 0.0013	0.9406	0.9867	pass
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[always, relapse @ t=1]	cumulative incidence of relapse under the plan treat at both times at horizon t = 1	cleverly competing-risk LTMLE	-0.000540 to 0.0016	0.9481	0.9952	pass
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[always, relapse @ t=1]	cumulative incidence of relapse under the plan treat at both times at horizon t = 1	R lmtp	-0.000540 to 0.0016	0.9481	0.9952	pass
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[always, relapse @ t=2]	cumulative incidence of relapse under the plan treat at both times at horizon t = 2	cleverly competing-risk LTMLE	-0.000574 to 0.0029	0.9369	0.9866	pass
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[always, relapse @ t=2]	cumulative incidence of relapse under the plan treat at both times at horizon t = 2	R lmtp	-0.000574 to 0.0029	0.9369	0.9866	pass
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[continue_if_l2, death @ t=2]	cumulative incidence of death under the plan treat first, then continue if L2 equals one at horizon t = 2	cleverly competing-risk LTMLE	-0.0022 to 0.0013	0.9325	0.9594	pass
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[continue_if_l2, death @ t=2]	cumulative incidence of death under the plan treat first, then continue if L2 equals one at horizon t = 2	R lmtp	-0.0022 to 0.0013	0.9325	0.9594	pass
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[continue_if_l2, relapse @ t=2]	cumulative incidence of relapse under the plan treat first, then continue if L2 equals one at horizon t = 2	cleverly competing-risk LTMLE	-0.000615 to 0.0029	0.9369	0.9675	pass
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[continue_if_l2, relapse @ t=2]	cumulative incidence of relapse under the plan treat first, then continue if L2 equals one at horizon t = 2	R lmtp	-0.000615 to 0.0029	0.9369	0.9675	pass
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[never, death @ t=1]	cumulative incidence of death under the plan treat at neither time at horizon t = 1	cleverly competing-risk LTMLE	-0.000249 to 0.0012	0.9494	0.9935	pass
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[never, death @ t=1]	cumulative incidence of death under the plan treat at neither time at horizon t = 1	R lmtp	-0.000249 to 0.0012	0.9494	0.9935	pass
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[never, death @ t=2]	cumulative incidence of death under the plan treat at neither time at horizon t = 2	cleverly competing-risk LTMLE	-0.000333 to 0.0017	0.9363	0.9813	pass
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[never, death @ t=2]	cumulative incidence of death under the plan treat at neither time at horizon t = 2	R lmtp	-0.000333 to 0.0017	0.9363	0.9813	pass
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[never, relapse @ t=1]	cumulative incidence of relapse under the plan treat at neither time at horizon t = 1	cleverly competing-risk LTMLE	-0.000673 to 0.000580	0.9500	0.9956	pass
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[never, relapse @ t=1]	cumulative incidence of relapse under the plan treat at neither time at horizon t = 1	R lmtp	-0.000673 to 0.000580	0.9500	0.9956	pass
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[never, relapse @ t=2]	cumulative incidence of relapse under the plan treat at neither time at horizon t = 2	cleverly competing-risk LTMLE	-0.000946 to 0.000931	0.9387	0.9809	pass
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[never, relapse @ t=2]	cumulative incidence of relapse under the plan treat at neither time at horizon t = 2	R lmtp	-0.000946 to 0.000931	0.9387	0.9809	pass

Agreement with the canonical implementation

Comparator-backed: a paired comparison with lmtp on the same samples. Each row reports estimate equivalence within the declared margin, the RMSE ratio and the coverage difference, and names its comparison kind.

law	estimand	what was compared	paired difference	share of margin used	RMSE ratio bound	coverage difference	comparison kind	calibration resolution	result
two-time-point, two-cause competing-risk law with monotone censoring	ate_regimen[always vs never, death @ t=1]	difference in cumulative incidence of death between the plans “treat at both times” against “treat at neither time” at horizon t = 1	1.253e-10	4.346e-08	1.0000	0	matching implementation	4.917e-09 vs 0.0500	equivalent
two-time-point, two-cause competing-risk law with monotone censoring	ate_regimen[always vs never, death @ t=2]	difference in cumulative incidence of death between the plans “treat at both times” against “treat at neither time” at horizon t = 2	-8.519e-11	1.783e-08	1.0000	0	matching implementation	4.289e-10 vs 0.0500	equivalent
two-time-point, two-cause competing-risk law with monotone censoring	ate_regimen[always vs never, relapse @ t=1]	difference in cumulative incidence of relapse between the plans “treat at both times” against “treat at neither time” at horizon t = 1	1.364e-09	4.781e-07	1.0000	0	matching implementation	1.690e-09 vs 0.0500	equivalent
two-time-point, two-cause competing-risk law with monotone censoring	ate_regimen[always vs never, relapse @ t=2]	difference in cumulative incidence of relapse between the plans “treat at both times” against “treat at neither time” at horizon t = 2	2.470e-11	5.336e-09	1.0000	0	matching implementation	6.692e-10 vs 0.0500	equivalent
two-time-point, two-cause competing-risk law with monotone censoring	ate_regimen[continue_if_l2 vs never, death @ t=2]	difference in cumulative incidence of death between the plans “treat first, then continue if L2 equals one” against “treat at neither time” at horizon t = 2	-1.918e-10	4.141e-08	1.0000	0	matching implementation	6.677e-10 vs 0.0500	equivalent
two-time-point, two-cause competing-risk law with monotone censoring	ate_regimen[continue_if_l2 vs never, relapse @ t=2]	difference in cumulative incidence of relapse between the plans “treat first, then continue if L2 equals one” against “treat at neither time” at horizon t = 2	5.143e-11	1.092e-08	1.0000	0	matching implementation	5.014e-10 vs 0.0500	equivalent
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[always, death @ t=1]	cumulative incidence of death under the plan treat at both times at horizon t = 1	-4.066e-12	1.750e-09	1.0000	0	matching implementation	3.228e-11 vs 0.0500	equivalent
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[always, death @ t=2]	cumulative incidence of death under the plan treat at both times at horizon t = 2	-1.084e-10	2.607e-08	1.0000	0	matching implementation	4.629e-10 vs 0.0500	equivalent
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[always, relapse @ t=1]	cumulative incidence of relapse under the plan treat at both times at horizon t = 1	-8.801e-11	3.605e-08	1.0000	0	matching implementation	2.009e-08 vs 0.0500	equivalent
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[always, relapse @ t=2]	cumulative incidence of relapse under the plan treat at both times at horizon t = 2	-8.995e-11	2.225e-08	1.0000	0	matching implementation	6.593e-10 vs 0.0500	equivalent
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[continue_if_l2, death @ t=2]	cumulative incidence of death under the plan treat first, then continue if L2 equals one at horizon t = 2	-2.150e-10	5.387e-08	1.0000	0	matching implementation	7.877e-10 vs 0.0500	equivalent
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[continue_if_l2, relapse @ t=2]	cumulative incidence of relapse under the plan treat first, then continue if L2 equals one at horizon t = 2	-6.323e-11	1.538e-08	1.0000	0	matching implementation	3.586e-10 vs 0.0500	equivalent
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[never, death @ t=1]	cumulative incidence of death under the plan treat at neither time at horizon t = 1	-1.293e-10	7.763e-08	1.0000	0	matching implementation	1.505e-08 vs 0.0500	equivalent
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[never, death @ t=2]	cumulative incidence of death under the plan treat at neither time at horizon t = 2	-2.319e-11	9.968e-09	1.0000	0	matching implementation	1.702e-09 vs 0.0500	equivalent
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[never, relapse @ t=1]	cumulative incidence of relapse under the plan treat at neither time at horizon t = 1	-1.452e-09	9.960e-07	1.0000	0	matching implementation	1.031e-09 vs 0.0500	equivalent
two-time-point, two-cause competing-risk law with monotone censoring	cif_regimen[never, relapse @ t=2]	cumulative incidence of relapse under the plan treat at neither time at horizon t = 2	-1.147e-10	5.250e-08	1.0000	0	matching implementation	7.086e-10 vs 0.0500	equivalent

Theory properties

property	cell	role	what was tested	what must hold	measured	result
competing_risk_recursion_necessity	death_always_t2__all_cause	positive	always-treat death incidence at horizon two: the estimator removes every first-node event from the later risk set	bias interval inside the equivalence margin	bias -0.0032 to 0.0010, margin 0.0070	pass
competing_risk_recursion_necessity	death_always_t2__cause_specific_control	control	always-treat death incidence at horizon two: the same recursion wrongly lets the competing cause remain at risk	bias interval must fall entirely outside the margin	bias 0.1488 to 0.1581, margin 0.0156	pass
competing_risk_recursion_necessity	relapse_always_t2__all_cause	positive	always-treat relapse incidence at horizon two: the estimator removes every first-node event from the later risk set	bias interval inside the equivalence margin	bias -0.000904 to 0.0031, margin 0.0067	pass
competing_risk_recursion_necessity	relapse_always_t2__cause_specific_control	control	always-treat relapse incidence at horizon two: the same recursion wrongly lets the competing cause remain at risk	bias interval must fall entirely outside the margin	bias 0.0608 to 0.0673, margin 0.0109	pass
double_robustness	death_static_t2__both_correct	positive	static death contrast at horizon two: both the outcome regression and the treatment mechanism are correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.0046 to -0.000077, margin 0.0077, SE ratio 0.9953	pass
double_robustness	death_static_t2__both_wrong	control	static death contrast at horizon two: both nuisances are misspecified	bias interval must fall entirely outside the margin, with the reported standard error still on the scale of the empirical spread	bias -0.0175 to -0.0129, margin 0.0077, SE ratio 0.7070	pass
double_robustness	death_static_t2__mechanism_correct	positive	static death contrast at horizon two: only the treatment and censoring mechanisms are correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.0026 to 0.0022, margin 0.0080, SE ratio 0.9778	pass
double_robustness	death_static_t2__outcome_correct	positive	static death contrast at horizon two: only the outcome regression is correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.0028 to 0.0019, margin 0.0079, SE ratio 0.6760	pass
double_robustness	relapse_dynamic_t2__both_correct	positive	dynamic relapse contrast at horizon two: both the outcome regression and the treatment mechanism are correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.0014 to 0.0031, margin 0.0076, SE ratio 0.9738	pass
double_robustness	relapse_dynamic_t2__both_wrong	control	dynamic relapse contrast at horizon two: both nuisances are misspecified	bias interval must fall entirely outside the margin, with the reported standard error still on the scale of the empirical spread	bias 0.0260 to 0.0302, margin 0.0071, SE ratio 0.7865	pass
double_robustness	relapse_dynamic_t2__mechanism_correct	positive	dynamic relapse contrast at horizon two: only the treatment and censoring mechanisms are correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.0020 to 0.0025, margin 0.0076, SE ratio 0.9916	pass
double_robustness	relapse_dynamic_t2__outcome_correct	positive	dynamic relapse contrast at horizon two: only the outcome regression is correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.0012 to 0.0033, margin 0.0076, SE ratio 0.7296	pass
interval_calibration	death_static_t2__correctly_specified	positive	static death contrast at horizon two: both nuisances are correctly specified with an independently computed efficiency bound	SE ratio and coverage intervals both inside their calibration bands, with both efficiency-ratio intervals inside their bands	coverage 0.9395 to 0.9515, SE ratio 0.9700 to 1.0066, empirical efficiency ratio 0.9913 to 1.0289, reported efficiency ratio 0.9975 to 0.9984	pass
interval_calibration	death_static_t2__noise_control	control	static death contrast at horizon two: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8207 to 0.8405, SE ratio 0.6896 to 0.7160, empirical efficiency ratio 1.3935 to 1.4470, reported efficiency ratio 0.9975 to 0.9984	pass
interval_calibration	death_static_t2__shrunken_se_control	control	static death contrast at horizon two: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8168 to 0.8368, SE ratio 0.6794 to 0.7045, empirical efficiency ratio 0.9915 to 1.0281, reported efficiency ratio 0.6982 to 0.6989	pass
interval_calibration	relapse_dynamic_t2__correctly_specified	positive	dynamic relapse contrast at horizon two: both nuisances are correctly specified with an independently computed efficiency bound	SE ratio and coverage intervals both inside their calibration bands, with both efficiency-ratio intervals inside their bands	coverage 0.9421 to 0.9539, SE ratio 0.9732 to 1.0098, empirical efficiency ratio 0.9888 to 1.0261, reported efficiency ratio 0.9981 to 0.9991	pass
interval_calibration	relapse_dynamic_t2__noise_control	control	dynamic relapse contrast at horizon two: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8198 to 0.8397, SE ratio 0.6908 to 0.7167, empirical efficiency ratio 1.3933 to 1.4455, reported efficiency ratio 0.9982 to 0.9991	pass
interval_calibration	relapse_dynamic_t2__shrunken_se_control	control	dynamic relapse contrast at horizon two: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8133 to 0.8335, SE ratio 0.6818 to 0.7073, empirical efficiency ratio 0.9882 to 1.0253, reported efficiency ratio 0.6987 to 0.6994	pass
power	death_static_t2__alternative	positive	static death contrast at horizon two: the same test applied to a law with a real effect	rejection lower bound clears the minimum power	rejection 0.9969, 0.9912 to 0.9993	pass
power	relapse_dynamic_t2__alternative	positive	dynamic relapse contrast at horizon two: the same test applied to a law with a real effect	rejection lower bound clears the minimum power	rejection 0.9869, 0.9776 to 0.9931	pass
root_n_and_efficiency	death_static_t2__n_32000	positive	static death contrast at horizon two: bias, coverage and SE calibration at n = 32,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.000294, coverage 0.9471 to 0.9726, SE ratio 1.0323	pass
root_n_and_efficiency	death_static_t2__n_4000	positive	static death contrast at horizon two: bias, coverage and SE calibration at n = 4,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias 0.000180, coverage 0.9147 to 0.9477, SE ratio 0.9633	pass
root_n_and_efficiency	death_static_t2__n_8000	positive	static death contrast at horizon two: bias, coverage and SE calibration at n = 8,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.000635, coverage 0.9216 to 0.9532, SE ratio 0.9755	pass
root_n_and_efficiency	relapse_dynamic_t2__n_32000	positive	dynamic relapse contrast at horizon two: bias, coverage and SE calibration at n = 32,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.000220, coverage 0.9307 to 0.9603, SE ratio 1.0189	pass
root_n_and_efficiency	relapse_dynamic_t2__n_4000	positive	dynamic relapse contrast at horizon two: bias, coverage and SE calibration at n = 4,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.000270, coverage 0.9140 to 0.9471, SE ratio 0.9388	pass
root_n_and_efficiency	relapse_dynamic_t2__n_8000	positive	dynamic relapse contrast at horizon two: bias, coverage and SE calibration at n = 8,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.000841, coverage 0.9300 to 0.9597, SE ratio 0.9871	pass
root_n_rate	death_static_t2__empirical_sd	positive	static death contrast at horizon two: log empirical spread of the estimates regressed on log n across three sizes	slope interval inside the root-n band and excluding -1/4	slope -0.5570 to -0.4995	pass
root_n_rate	death_static_t2__reported_se	positive	static death contrast at horizon two: the same regression applied to the mean reported standard error	slope interval inside the root-n band and excluding -1/4	slope -0.4954 to -0.4924	pass
root_n_rate	relapse_dynamic_t2__empirical_sd	positive	dynamic relapse contrast at horizon two: log empirical spread of the estimates regressed on log n across three sizes	slope interval inside the root-n band and excluding -1/4	slope -0.5614 to -0.5036	pass
root_n_rate	relapse_dynamic_t2__reported_se	positive	dynamic relapse contrast at horizon two: the same regression applied to the mean reported standard error	slope interval inside the root-n band and excluding -1/4	slope -0.4972 to -0.4942	pass
simultaneous_coverage	all_reported__pointwise_joint_control	control	every parameter the calibration fit reports: the pointwise 95% intervals of the same fits, read jointly	joint coverage upper endpoint must fall below the nominal rate	joint coverage 0.7416 to 0.7644	pass
simultaneous_coverage	all_reported__simultaneous_band	positive	every parameter the calibration fit reports: the max-t multiplier band over every estimand the law reports, from the same-row centered influence curves	joint coverage interval inside the calibration coverage band	joint coverage 0.9405 to 0.9524	pass
simultaneous_coverage	curve_always__pointwise_joint_control	control	the always plan alone, at every declared horizon: the pointwise 95% intervals of the same fits, read jointly	joint coverage upper endpoint must fall below the nominal rate	joint coverage 0.8159 to 0.8552	pass
simultaneous_coverage	curve_always__simultaneous_band	positive	the always plan alone, at every declared horizon: the max-t multiplier band over every estimand the law reports, from the same-row centered influence curves	joint coverage interval inside the calibration coverage band	joint coverage 0.9365 to 0.9600	pass
targeting_necessity	death_static_t2__targeted	positive	static death contrast at horizon two: the estimator fluctuates a misspecified outcome model, so targeting does all the adjusting	bias interval inside the equivalence margin	bias -0.0043 to 0.000449, margin 0.0080	pass
targeting_necessity	death_static_t2__untargeted	control	static death contrast at horizon two: the identical fit with every fluctuation step removed	bias interval must fall entirely outside the margin	bias -0.0193 to -0.0147, margin 0.0078	pass
targeting_necessity	relapse_dynamic_t2__targeted	positive	dynamic relapse contrast at horizon two: the estimator fluctuates a misspecified outcome model, so targeting does all the adjusting	bias interval inside the equivalence margin	bias -0.0018 to 0.0027, margin 0.0075	pass
targeting_necessity	relapse_dynamic_t2__untargeted	control	dynamic relapse contrast at horizon two: the identical fit with every fluctuation step removed	bias interval must fall entirely outside the margin	bias 0.0264 to 0.0309, margin 0.0074	pass
type_i_error	death_static_t2__sharp_null	positive	static death contrast at horizon two: a confounded law whose true contrast is exactly zero	one-sided rejection bound stays under the declared type-I ceiling	rejection 0.0600, 0.0457 to 0.0770	pass
type_i_error	relapse_dynamic_t2__sharp_null	positive	dynamic relapse contrast at horizon two: a confounded law whose true contrast is exactly zero	one-sided rejection bound stays under the declared type-I ceiling	rejection 0.0531, 0.0397 to 0.0693	pass

The property study checks one relapse contrast and one death contrast. It measures double robustness, root-n behavior, exact-EIF efficiency, calibration, type-I error, and power. Separate controls remove targeting and replace all-cause survival with target-cause survival.

At the second horizon a baseline-only standardisation misses the null by -0.0077 for death and 0.0091 for relapse. The null is therefore one an estimator has to be longitudinal to find. At the first horizon that same analysis returns exactly zero, because no time-varying node precedes the first event node. A crude comparison of arms is biased at both horizons, so the first-horizon cells still test baseline and censoring adjustment.

The simultaneous_coverage family reads the default band, 2000 Rademacher draws seeded by random_state=0. The all_reported cells read the calibration fits at \(n = 32{,}000\), whose band covers six incidences and four contrasts at horizon two. The curve_always cells fit the always plan alone at both horizons in a dedicated batch of 2,400 samples of the same size, so the band covers both causes at both horizons.

Measured values

Names beginning margin: are thresholds declared before the run. The other values come from the committed results.

quantity	value	source
replicates	1600	paired replications
n	4000	observations per paired replication
independent_tests_total	32	implementation-estimand truth tests
independent_tests_passed	32	truth tests passing
paired_tests_total	16	paired estimand comparisons
paired_tests_passed	16	paired comparisons passing
property_cells_total	40	independent property cells
property_cells_passed	40	property cells passing
properties[simultaneous_coverage/curve_always__simultaneous_band]:coverage	0.9492	joint coverage of the band over both causes at both horizons
properties[simultaneous_coverage/curve_always__pointwise_joint_control]:coverage	0.8363	joint coverage of the pointwise intervals of the same fits
properties[simultaneous_coverage/all_reported__simultaneous_band]:coverage	0.9467	joint coverage of the band over the ten horizon-two parameters
properties[simultaneous_coverage/all_reported__pointwise_joint_control]:coverage	0.7531	joint coverage of the pointwise intervals of the same fits
max_standardized_bias	0.0432	largest primary standardized bias
min_coverage	0.9325	lowest primary coverage
min_coverage_ci_lower	0.9147	lowest primary coverage lower endpoint
min_se_ratio_ci_lower	0.9172	lowest primary SE-ratio endpoint
max_se_ratio_ci_upper	1.0427	highest primary SE-ratio endpoint
properties[targeting_necessity/relapse_dynamic_t2__targeted]:targeting_displacement	0.4698	least paired targeting displacement
properties[competing_risk_recursion_necessity/relapse_always_t2__all_cause]:recursion_displacement	2.3623	least competing-risk recursion displacement
margin:confidence_level	0.9900	Monte Carlo confidence level
margin:alpha	0.0500	test size
margin:nominal_coverage	0.9500	nominal interval coverage
margin:bootstrap_replicates	10000	bootstrap replications
margin:standardized_bias	0.2500	standardized-bias margin
margin:coverage_floor	0.9000	primary coverage floor
margin:over_coverage_ceiling	0.9900	descriptive overcoverage threshold
margin:se_ratio_sanity_lower	0.8000	primary SE-ratio lower screen
margin:se_ratio_sanity_upper	1.2000	primary SE-ratio upper screen
margin:calibration_se_ratio_lower	0.9300	calibration SE-ratio lower bound
margin:calibration_se_ratio_upper	1.0700	calibration SE-ratio upper bound
margin:calibration_coverage_lower	0.9200	calibration coverage lower bound
margin:calibration_coverage_upper	0.9800	calibration coverage upper bound
margin:type_i_ceiling	0.1000	type-I upper bound
margin:paired_difference	0.1500	paired similarity margin
margin:rmse_noninferiority	1.1000	RMSE-ratio noninferiority bound
margin:coverage_noninferiority	-0.0250	coverage-difference noninferiority bound
margin:calibration_noninferiority	0.0500	calibration-excess noninferiority bound
margin:minimum_power	0.8000	power lower bound
margin:root_n_slope	-0.5000	expected root-n slope
margin:root_n_slope_lower	-0.6250	accepted slope lower bound
margin:root_n_slope_upper	-0.3750	accepted slope upper bound
margin:excluded_slope	-0.2500	rate the interval must exclude
margin:union_model_se_lower	0.1000	union-model SE-ratio screen, lower limit
margin:union_model_se_upper	10	union-model SE-ratio screen, upper limit
margin:efficiency_ratio_lower	0.9000	exact-EIF ratio lower bound
margin:efficiency_ratio_upper	1.1000	exact-EIF ratio upper bound
margin:shrunken_se_factor	0.7000	deliberate SE mutation factor
margin:targeting_displacement	0.2500	least required targeting displacement
margin:recursion_displacement	0.2500	least required recursion displacement

Limitations

limitation	what it means for use
The comparison covers two causes and two horizons	Longer event processes and more causes need separate evidence
The first-horizon null is not a longitudinal null	No time-varying node precedes the first event node, so a baseline-only standardisation recovers that null exactly. The first-horizon type-I cells test baseline and censoring adjustment only
The bands are measured at one law	The curve_always band covers both causes at both horizons for one plan. The all_reported band covers the ten parameters of horizon two. A band over more plans and horizons together is not measured
The mechanisms are supplied	The comparison does not test learned-mechanism parity or active truncation
The fit is ordinary	Flexible learning and cross-fitting belong to the separate cross-fitted row
Competing events remain natural	The row does not validate an estimand that eliminates a competing event
The data are independent and unweighted	The row does not validate observation weights, clustering, or time-respecting splits

The causal interpretation requires consistency, sequential exchangeability, longitudinal positivity, conditionally independent censoring, and correct competing-event coding.

Reproduction

The fixture README gives the commands. The manifest records the seeds, configuration, source digests, and artifact hashes.