Ordinary survival-curve longitudinal TMLE

This study validates ordinary, non-cross-fitted cumulative risk estimation under absorbing failure. It covers two horizons, monotone censoring, static plans, and a dynamic plan. The canonical comparison uses R ltmle 1.3-0 with survivalOutcome=TRUE.

The study reports each unique parameter once. At the first horizon, the dynamic plan equals the always-treated plan by construction. A structural test checks that identity instead of counting the duplicate as repeated-sampling evidence.

What was compared

setting	cleverly	R ltmle
datasets	1,600 censored survival panels generated in Python	the identical rows
horizons	cumulative risk by times one and two	one survivalOutcome=TRUE fit for each data prefix
plans	never treat, always treat, and continue after initial treatment when L2 is positive	the same unique plan and horizon combinations
contrasts	always-minus-never at both horizons and dynamic-minus-never at time two	differences of rowwise influence curves, preserving covariance
mechanisms	the generating treatment and censoring probabilities	the same fixed probabilities
sequential regressions	follower-stratified quasibinomial	the same formulas and link
cumulative-g bounds	nonbinding	nonbinding
intervals	pointwise 95% identity-scale Wald, influence-curve variance	the same, with variance.method="ic"

R fits each requested prefix because ltmle returns the terminal cumulative risk for that input. The Python implementation performs one backward pass per horizon. Comparing the prefixes tests the same horizon-specific parameters without treating a terminal outcome as a survival curve.

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
two-time-point absorbing-event law with monotone censoring	ate_regimen[always vs never @ t=1]	difference in cumulative risk between the plans “treat at both times” against “treat at neither time” at horizon t = 1	cleverly	-0.0022 to 0.000243	0.9494	1.0064	pass
two-time-point absorbing-event law with monotone censoring	ate_regimen[always vs never @ t=1]	difference in cumulative risk between the plans “treat at both times” against “treat at neither time” at horizon t = 1	R ltmle	-0.0022 to 0.000243	0.9494	1.0064	pass
two-time-point absorbing-event law with monotone censoring	ate_regimen[always vs never @ t=2]	difference in cumulative risk between the plans “treat at both times” against “treat at neither time” at horizon t = 2	cleverly	-0.0036 to 0.000104	0.9406	0.9938	pass
two-time-point absorbing-event law with monotone censoring	ate_regimen[always vs never @ t=2]	difference in cumulative risk between the plans “treat at both times” against “treat at neither time” at horizon t = 2	R ltmle	-0.0036 to 0.000104	0.9406	0.9938	pass
two-time-point absorbing-event law with monotone censoring	ate_regimen[treat then continue if l2 positive vs never @ t=2]	difference in cumulative risk between the plans “treat, then continue only if L2 is positive” against “treat at neither time” at horizon t = 2	cleverly	-0.0038 to 0.000020	0.9387	0.9923	pass
two-time-point absorbing-event law with monotone censoring	ate_regimen[treat then continue if l2 positive vs never @ t=2]	difference in cumulative risk between the plans “treat, then continue only if L2 is positive” against “treat at neither time” at horizon t = 2	R ltmle	-0.0038 to 0.000020	0.9387	0.9923	pass
two-time-point absorbing-event law with monotone censoring	risk_regimen[always @ t=1]	cumulative risk under the plan treat at both times at horizon t = 1	cleverly	-0.0012 to 0.000324	0.9444	1.0076	pass
two-time-point absorbing-event law with monotone censoring	risk_regimen[always @ t=1]	cumulative risk under the plan treat at both times at horizon t = 1	R ltmle	-0.0012 to 0.000324	0.9444	1.0076	pass
two-time-point absorbing-event law with monotone censoring	risk_regimen[always @ t=2]	cumulative risk under the plan treat at both times at horizon t = 2	cleverly	-0.0018 to 0.000412	0.9544	1.0105	pass
two-time-point absorbing-event law with monotone censoring	risk_regimen[always @ t=2]	cumulative risk under the plan treat at both times at horizon t = 2	R ltmle	-0.0018 to 0.000412	0.9544	1.0105	pass
two-time-point absorbing-event law with monotone censoring	risk_regimen[never @ t=1]	cumulative risk under the plan treat at neither time at horizon t = 1	cleverly	-0.000434 to 0.0015	0.9519	0.9976	pass
two-time-point absorbing-event law with monotone censoring	risk_regimen[never @ t=1]	cumulative risk under the plan treat at neither time at horizon t = 1	R ltmle	-0.000434 to 0.0015	0.9519	0.9976	pass
two-time-point absorbing-event law with monotone censoring	risk_regimen[never @ t=2]	cumulative risk under the plan treat at neither time at horizon t = 2	cleverly	-0.000431 to 0.0025	0.9425	0.9999	pass
two-time-point absorbing-event law with monotone censoring	risk_regimen[never @ t=2]	cumulative risk under the plan treat at neither time at horizon t = 2	R ltmle	-0.000431 to 0.0025	0.9425	0.9999	pass
two-time-point absorbing-event law with monotone censoring	risk_regimen[treat then continue if l2 positive @ t=2]	cumulative risk under the plan treat, then continue only if L2 is positive at horizon t = 2	cleverly	-0.0020 to 0.000348	0.9537	1.0104	pass
two-time-point absorbing-event law with monotone censoring	risk_regimen[treat then continue if l2 positive @ t=2]	cumulative risk under the plan treat, then continue only if L2 is positive at horizon t = 2	R ltmle	-0.0020 to 0.000348	0.9537	1.0104	pass

Agreement with the canonical implementation

Comparator-backed: a paired comparison with ltmle 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 absorbing-event law with monotone censoring	ate_regimen[always vs never @ t=1]	difference in cumulative risk between the plans “treat at both times” against “treat at neither time” at horizon t = 1	3.324e-11	1.160e-08	1.0000	0	matching implementation	1.353e-09 vs 0.0500	equivalent
two-time-point absorbing-event law with monotone censoring	ate_regimen[always vs never @ t=2]	difference in cumulative risk between the plans “treat at both times” against “treat at neither time” at horizon t = 2	-3.360e-10	7.845e-08	1.0000	0	matching implementation	4.322e-08 vs 0.0500	equivalent
two-time-point absorbing-event law with monotone censoring	ate_regimen[treat then continue if l2 positive vs never @ t=2]	difference in cumulative risk between the plans “treat, then continue only if L2 is positive” against “treat at neither time” at horizon t = 2	-3.903e-10	8.898e-08	1.0000	0	matching implementation	4.072e-08 vs 0.0500	equivalent
two-time-point absorbing-event law with monotone censoring	risk_regimen[always @ t=1]	cumulative risk under the plan treat at both times at horizon t = 1	-8.728e-11	4.910e-08	1.0000	0	matching implementation	2.193e-09 vs 0.0500	equivalent
two-time-point absorbing-event law with monotone censoring	risk_regimen[always @ t=2]	cumulative risk under the plan treat at both times at horizon t = 2	-2.473e-10	9.571e-08	1.0000	0	matching implementation	1.359e-09 vs 0.0500	equivalent
two-time-point absorbing-event law with monotone censoring	risk_regimen[never @ t=1]	cumulative risk under the plan treat at neither time at horizon t = 1	-1.205e-10	5.271e-08	1.0000	0	matching implementation	1.342e-09 vs 0.0500	equivalent
two-time-point absorbing-event law with monotone censoring	risk_regimen[never @ t=2]	cumulative risk under the plan treat at neither time at horizon t = 2	8.867e-11	2.594e-08	1.0000	0	matching implementation	5.747e-08 vs 0.0500	equivalent
two-time-point absorbing-event law with monotone censoring	risk_regimen[treat then continue if l2 positive @ t=2]	cumulative risk under the plan treat, then continue only if L2 is positive at horizon t = 2	-3.016e-10	1.104e-07	1.0000	0	matching implementation	1.204e-09 vs 0.0500	equivalent

Theory properties

property	cell	role	what was tested	what must hold	measured	result
double_robustness	dynamic_t2__both_correct	positive	dynamic plan 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 -0.0023 to 0.0042, margin 0.0109, SE ratio 0.9756	pass
double_robustness	dynamic_t2__both_wrong	control	dynamic plan 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 -0.0447 to -0.0382, margin 0.0109, SE ratio 0.7604	pass
double_robustness	dynamic_t2__mechanism_correct	positive	dynamic plan 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 -0.0029 to 0.0037, margin 0.0111, SE ratio 1.0338	pass
double_robustness	dynamic_t2__outcome_correct	positive	dynamic plan 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 -0.0010 to 0.0055, margin 0.0109, SE ratio 0.7158	pass
double_robustness	static_t1__both_correct	positive	static plan at horizon one: 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 -0.0022 to 0.0018, margin 0.0067, SE ratio 1.0378	pass
double_robustness	static_t1__both_wrong	control	static plan at horizon one: 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 -0.0449 to -0.0408, margin 0.0068, SE ratio 0.9436	pass
double_robustness	static_t1__mechanism_correct	positive	static plan at horizon one: 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 -0.0024 to 0.0016, margin 0.0068, SE ratio 1.0530	pass
double_robustness	static_t1__outcome_correct	positive	static plan at horizon one: 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 -0.000895 to 0.0033, margin 0.0071, SE ratio 0.8840	pass
double_robustness	static_t2__both_correct	positive	static plan 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 -0.0025 to 0.0043, margin 0.0114, SE ratio 0.9587	pass
double_robustness	static_t2__both_wrong	control	static plan 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 -0.0368 to -0.0301, margin 0.0113, SE ratio 0.6847	pass
double_robustness	static_t2__mechanism_correct	positive	static plan 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 0.0042, margin 0.0113, SE ratio 1.0324	pass
double_robustness	static_t2__outcome_correct	positive	static plan 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 -0.000743 to 0.0059, margin 0.0112, SE ratio 0.6544	pass
interval_calibration	dynamic_t2__correctly_specified	positive	dynamic plan 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 0.9300 to 0.9429, SE ratio 0.9483 to 0.9849, empirical efficiency ratio 1.0094 to 1.0482, reported efficiency ratio 0.9922 to 0.9961	pass
interval_calibration	dynamic_t2__noise_control	control	dynamic plan 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 0.8113 to 0.8315, SE ratio 0.6760 to 0.7028, empirical efficiency ratio 1.4154 to 1.4707, reported efficiency ratio 0.9922 to 0.9961	pass
interval_calibration	dynamic_t2__shrunken_se_control	control	dynamic plan 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 0.8042 to 0.8247, SE ratio 0.6636 to 0.6894, empirical efficiency ratio 1.0096 to 1.0484, reported efficiency ratio 0.6945 to 0.6972	pass
interval_calibration	static_t1__correctly_specified	positive	static plan at horizon one: 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 0.9393 to 0.9513, SE ratio 0.9700 to 1.0068, empirical efficiency ratio 0.9930 to 1.0306, reported efficiency ratio 0.9993 to 1.0003	pass
interval_calibration	static_t1__noise_control	control	static plan at horizon one: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8175 to 0.8374, SE ratio 0.6827 to 0.7091, empirical efficiency ratio 1.4098 to 1.4644, reported efficiency ratio 0.9993 to 1.0003	pass
interval_calibration	static_t1__shrunken_se_control	control	static plan at horizon one: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8177 to 0.8376, SE ratio 0.6788 to 0.7047, empirical efficiency ratio 0.9932 to 1.0310, reported efficiency ratio 0.6995 to 0.7002	pass
interval_calibration	static_t2__correctly_specified	positive	static plan 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 0.9276 to 0.9407, SE ratio 0.9411 to 0.9774, empirical efficiency ratio 1.0122 to 1.0515, reported efficiency ratio 0.9874 to 0.9914	pass
interval_calibration	static_t2__noise_control	control	static plan 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 0.8060 to 0.8264, SE ratio 0.6757 to 0.7008, empirical efficiency ratio 1.4119 to 1.4641, reported efficiency ratio 0.9874 to 0.9915	pass
interval_calibration	static_t2__shrunken_se_control	control	static plan 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 0.8001 to 0.8208, SE ratio 0.6585 to 0.6842, empirical efficiency ratio 1.0125 to 1.0519, reported efficiency ratio 0.6911 to 0.6941	pass
power	dynamic_t2__alternative	positive	dynamic plan at horizon two: the same test applied to a law with a real effect	rejection lower bound clears the minimum power	rejection 1, 0.9934 to 1	pass
power	static_t1__alternative	positive	static plan at horizon one: the same test applied to a law with a real effect	rejection lower bound clears the minimum power	rejection 1, 0.9934 to 1	pass
power	static_t2__alternative	positive	static plan at horizon two: the same test applied to a law with a real effect	rejection lower bound clears the minimum power	rejection 1, 0.9934 to 1	pass
root_n_and_efficiency	dynamic_t2__n_1000	control	dynamic plan at horizon two: bias, coverage and SE calibration at n = 1,000	coverage interval lies below nominal or clears the declared floor	bias 0.0020, coverage 0.8766 to 0.9309, SE ratio 0.8690	pass
root_n_and_efficiency	dynamic_t2__n_2000	positive	dynamic plan at horizon two: bias, coverage and SE calibration at n = 2,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.0016, coverage 0.9136 to 0.9585, SE ratio 0.9810	pass
root_n_and_efficiency	dynamic_t2__n_8000	positive	dynamic plan 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 -0.000974, coverage 0.9150 to 0.9596, SE ratio 0.9872	pass
root_n_and_efficiency	static_t1__n_1000	control	static plan at horizon one: bias, coverage and SE calibration at n = 1,000	coverage interval lies below nominal or clears the declared floor	bias 0.000396, coverage 0.9107 to 0.9565, SE ratio 0.9596	pass
root_n_and_efficiency	static_t1__n_2000	positive	static plan at horizon one: bias, coverage and SE calibration at n = 2,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.0012, coverage 0.9341 to 0.9727, SE ratio 1.0450	pass
root_n_and_efficiency	static_t1__n_8000	positive	static plan at horizon one: bias, coverage and SE calibration at n = 8,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.000241, coverage 0.9179 to 0.9616, SE ratio 0.9838	pass
root_n_and_efficiency	static_t2__n_1000	control	static plan at horizon two: bias, coverage and SE calibration at n = 1,000	coverage interval lies below nominal or clears the declared floor	bias -0.000599, coverage 0.8696 to 0.9255, SE ratio 0.8669	pass
root_n_and_efficiency	static_t2__n_2000	positive	static plan at horizon two: bias, coverage and SE calibration at n = 2,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.0024, coverage 0.9208 to 0.9637, SE ratio 0.9893	pass
root_n_and_efficiency	static_t2__n_8000	positive	static plan 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 -0.000698, coverage 0.9121 to 0.9575, SE ratio 0.9733	pass
root_n_rate	dynamic_t2__empirical_sd	positive	dynamic plan 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 -0.5866 to -0.4928	pass
root_n_rate	dynamic_t2__reported_se	positive	dynamic plan 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 -0.4906 to -0.4814	pass
root_n_rate	static_t1__empirical_sd	positive	static plan at horizon one: 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 -0.5463 to -0.4630	pass
root_n_rate	static_t1__reported_se	positive	static plan at horizon one: the same regression applied to the mean reported standard error	slope interval inside the root-n band and excluding -1/4	slope -0.5008 to -0.4986	pass
root_n_rate	static_t2__empirical_sd	positive	static plan 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 -0.5701 to -0.4752	pass
root_n_rate	static_t2__reported_se	positive	static plan 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 -0.4811 to -0.4709	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 0.7110 to 0.7346	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 0.9140 to 0.9283	fail
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 0.8760 to 0.8929	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 0.9288 to 0.9418	pass
survival_recursion_necessity	always_t2__survival	positive	always-treat risk at horizon two: the survival estimator keeps failures in their event node and removes them afterward	bias interval inside the equivalence margin	bias -0.0038 to 0.0022, margin 0.0100	pass
survival_recursion_necessity	always_t2__survivor_only	control	always-treat risk at horizon two: the same horizon-two outcome analyzed only among first-node survivors	bias interval must fall entirely outside the margin	bias -0.1523 to -0.1430, margin 0.0155	pass
targeting_necessity	dynamic_t2__targeted	positive	dynamic plan at horizon two: the estimator fluctuates a misspecified outcome model, so targeting does all the adjusting	bias interval inside the equivalence margin	bias -0.0035 to 0.0029, margin 0.0108	pass
targeting_necessity	dynamic_t2__untargeted	control	dynamic plan at horizon two: the identical fit with every fluctuation step removed	bias interval must fall entirely outside the margin	bias -0.0444 to -0.0379, margin 0.0108	pass
targeting_necessity	static_t1__targeted	positive	static plan at horizon one: the estimator fluctuates a misspecified outcome model, so targeting does all the adjusting	bias interval inside the equivalence margin	bias -0.0019 to 0.0021, margin 0.0068	pass
targeting_necessity	static_t1__untargeted	control	static plan at horizon one: the identical fit with every fluctuation step removed	bias interval must fall entirely outside the margin	bias -0.0448 to -0.0408, margin 0.0067	pass
targeting_necessity	static_t2__targeted	positive	static plan at horizon two: the estimator fluctuates a misspecified outcome model, so targeting does all the adjusting	bias interval inside the equivalence margin	bias -0.0033 to 0.0034, margin 0.0113	pass
targeting_necessity	static_t2__untargeted	control	static plan at horizon two: the identical fit with every fluctuation step removed	bias interval must fall entirely outside the margin	bias -0.0366 to -0.0299, margin 0.0114	pass
type_i_error	dynamic_t2__sharp_null	positive	dynamic plan at horizon two: a confounded law whose true contrast is exactly zero	one-sided rejection bound stays under the declared type-I ceiling	rejection 0.0537, 0.0353 to 0.0777	pass
type_i_error	static_t1__sharp_null	positive	static plan at horizon one: a confounded law whose true contrast is exactly zero	one-sided rejection bound stays under the declared type-I ceiling	rejection 0.0500, 0.0323 to 0.0733	pass
type_i_error	static_t2__sharp_null	positive	static plan at horizon two: a confounded law whose true contrast is exactly zero	one-sided rejection bound stays under the declared type-I ceiling	rejection 0.0575, 0.0384 to 0.0821	pass

The property study samples an exact binary survival law. It checks static risks at both horizons and the dynamic risk at time two. Its independent instruments include double robustness, exact-EIF efficiency, root-n rates, calibration, null size, power, and targeting controls.

The survival-recursion control is separate from targeting. It replaces cumulative risk with a survivor-only terminal hazard while leaving the rest of the procedure unchanged. The control must miss the exact time-two risk, while the correct recursion must recover it.

The two horizons do not behave alike. The study measures that difference on one set of draws, so the comparison is internal to the calibration cell rather than across studies.

The first horizon is calibrated. Its SE ratio and its coverage both surround their nominal values. At the second horizon the whole SE-ratio interval sits below one and the whole coverage interval sits below nominal. The empirical spread runs above the exact efficiency bound there, while the reported standard error sits on it. So the reported standard error understates the sampling spread at the second horizon by a few percent. The measured-values table below carries all four endpoints.

Both intervals stay inside the declared calibration bands, so the cell passes. The second backward regression is the source, and it is why this cell carries four times the shared calibration budget. A reader who needs calibrated horizon-two inference at this sample size should treat the shortfall as measured rather than as absent.

The sharp null keeps the treatment, censoring, and L2 mechanisms of the law and replaces the two hazards. Every replacement value is a multiple of one quarter, so an N-row sample realises the null law exactly, as it does the law it derives from. All three contrasts are exactly zero under it.

At the second horizon a baseline-only standardisation returns 0.0349 rather than zero, so the null is one an estimator has to be longitudinal to find. At the first horizon it returns exactly zero, because no time-varying node precedes the first event node. The first-horizon type-I cells therefore test baseline and censoring adjustment and not longitudinal adjustment. A crude comparison of arms is biased by -0.05 under the null at that horizon, so those cells are not vacuous either.

The power alternative keeps the law’s own first hazard and replaces the second. It gives the two horizon-two contrasts different values, so the two horizon-two power cells report two parameters rather than one number twice.

The simultaneous_coverage family reads the default band, 2000 Rademacher draws seeded by random_state=0. The curve_always cells fit the always plan alone in a dedicated batch of 9,600 samples, so the band covers that plan’s risk at both declared horizons. The all_reported cells read the calibration fits, whose band covers all ten reported parameters. The always and continue_if_l2 plans agree at the first node, so their first-horizon risks and contrasts are two exact duplicate pairs and the family has eight distinct parameters.

The all_reported band is the one red verdict, and the study publishes under the reporting policy. Its joint coverage is 0.9214, with a 99% interval from 0.9140 to 0.9283 that crosses the 0.92 edge. The declared diagnostic read the same rows at the design critical value of 2.622. That oracle band covers 0.9241, so the multiplier is not the main cause. The package critical value averages 2.605, which accounts for the 0.0027 between the two coverages. The family inherits the horizon-two shortfall of the pointwise intervals above. tests/unit/test_band_shortfall_reading.py rebuilds this reading from the committed rows.

The default band stays on, and band-finite-sample owns the cell. The one-step fit of hazard-based survival targeting is a different construction that supplies a band over the whole curve.

Measured values

Names beginning margin: are thresholds declared before the run. Everything else is measured from the committed results and checked at the precision printed.

quantity	value	source
replicates	1600	paired replications
n	2000	observations per paired replication
independent_tests_total	16	implementation-estimand truth tests
independent_tests_passed	16	truth tests passing
paired_tests_total	8	paired estimand comparisons
paired_tests_passed	8	paired comparisons passing
property_cells_total	54	independent property cells
property_cells_passed	53	property cells passing
properties[simultaneous_coverage/curve_always__simultaneous_band]:coverage	0.9355	joint coverage of the band over the always risk curve
properties[simultaneous_coverage/curve_always__pointwise_joint_control]:coverage	0.8847	joint coverage of the pointwise intervals of the same fits
properties[simultaneous_coverage/all_reported__simultaneous_band]:coverage	0.9214	joint coverage of the band over all ten parameters
properties[simultaneous_coverage/all_reported__simultaneous_band]:coverage_ci_lower	0.9140	its 99% lower endpoint
properties[simultaneous_coverage/all_reported__simultaneous_band]:coverage_ci_upper	0.9283	its 99% upper endpoint
properties[simultaneous_coverage/all_reported__pointwise_joint_control]:coverage	0.7229	joint coverage of the pointwise intervals of the same fits
max_standardized_bias	0.0638	largest primary standardized bias
min_coverage	0.9387	lowest primary coverage
min_coverage_ci_lower	0.9216	lowest primary coverage lower endpoint
min_se_ratio_ci_lower	0.9491	lowest primary SE-ratio endpoint
max_se_ratio_ci_upper	1.0615	highest primary SE-ratio endpoint
max_margin_utilization	1.104e-07	largest share of paired similarity margin used
max_rmse_ratio_upper	1.0000	largest paired RMSE-ratio bound
min_coverage_difference_lower	0	smallest paired coverage-difference bound
max_calibration_excess_upper	8.055e-08	largest paired calibration-excess bound
max_targeting_displacement	0.3560	largest final-fluctuation move, in standard errors
median_targeting_displacement	0.0343	median final-fluctuation move, in standard errors
properties[targeting_necessity/static_t1__targeted]:targeting_displacement	0.7387	least-displaced contrast, in targeted standard deviations
properties[survival_recursion_necessity/always_t2__survival]:recursion_displacement	3.6840	survivor-only control’s distance, in recursion standard deviations
properties[interval_calibration/static_t1__correctly_specified]:se_ratio	0.9880	horizon-one reported SE over empirical spread
properties[interval_calibration/static_t1__correctly_specified]:coverage	0.9455	horizon-one calibration coverage
properties[interval_calibration/static_t2__correctly_specified]:se_ratio_ci_upper	0.9774	highest horizon-two SE ratio the draws support
properties[interval_calibration/static_t2__correctly_specified]:coverage_ci_upper	0.9407	highest horizon-two coverage the draws support
properties[interval_calibration/static_t2__correctly_specified]:replicates	9600	replications the horizon-two calibration cell required
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 in pooled SDs
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 the fluctuation must move the estimate
margin:recursion_displacement	0.2500	least the survivor-only control must move the estimate

Limitations

limitation	what it means for use
This is the ordinary row	Nuisances are fitted on the analysis sample. The separate cross-fitted row below validates held-out nuisance fitting and fold-specific recursion
The band over the curve is measured at two horizons	The law has two horizons. The one-plan band passes, and the band over all ten parameters covers 0.9214, a red cell that band-finite-sample owns. Each horizon is targeted in its own backward pass, so the reported curve is not constrained to increase
Horizon-two inference is mildly anticonservative at n = 2,000	The reported standard error sits a few percent below the sampling spread, and coverage sits below nominal. Both endpoints stay inside the declared calibration bands. The first horizon does not show this on the same draws
The event process has two horizons	Both backward prefixes are exercised. Longer follow-up may compound the horizon-two shortfall above, and this study cannot say by how much
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
initial_estimate measures the final fluctuation only	The earlier node’s regression is of the already targeted later node. R’s fit$Q[[1]] regresses the updated Q.kplus1 and cleverly’s first step does the same. So max_targeting_displacement and median_targeting_displacement measure the last fluctuation rather than the whole targeting step. They are not the quantity margin:targeting_displacement bounds, which is the targeted-against-unfluctuated distance in the property study
The paired study fixes the mechanisms	It isolates survival recursion, sequential regression, targeting, and inference. It does not claim parity for learned treatment or censoring models
Positivity is comfortable	No primary cell validates active truncation or near-positivity behavior
The size ladder starts at n = 1,000	Every other registered study starts at 500. An absorbing event thins the risk set, so at 500 some samples of the property law leave the horizon-two parameter unestimable and the estimator refuses. The rate cells therefore say nothing below 1,000
Failure is absorbing and has one cause	Competing-risk cumulative incidence remains a different parameter with no R parity claim

The causal interpretation requires consistency, sequential exchangeability, longitudinal positivity, and conditionally independent censoring. The event process must also be coded so a failure remains in the cumulative risk at later horizons.

Reproduction

The fixture README gives the regeneration commands. The manifest records the pinned R package, source commit, image digest, formulas, seeds, and artifact hashes. The replications, paired verdicts, and property results carry every published row.