Longitudinal incremental propensity-score interventions under dropout

This study validates IPSI in cleverly.longitudinal with a censoring mechanism. IPSI is the one-step estimator of Kim, Kennedy and Naimi (2021), Algorithm 1 and Theorem 5.1. The incremental interventions over time section states the estimator, its conditions and its exact-law evidence.

No maintained package ships this construction with dropout. So no canonical implementation is compared. The study rests on exact truths and on repeated-sampling properties. The complete-data study pairs the same estimator with R npcausal, where that package applies. Its comparator runs are pending.

What was fitted

scenario	law and estimands	folds
dropout_folds_1	law C: three nodes with a continuous end-of-study outcome and dropout after each arm. The three plan means and their two contrasts with the natural course	one
dropout_folds_5	law C, the same estimands	five
survival_folds_1	law D: three nodes with a binary absorbing event and the same dropout. The risk of each plan by horizons 1, 2 and 3, and the two contrasts at each horizon	one
survival_folds_5	law D, the same estimands	five

Three plans tilt every node by 0.5, 1 (the natural course) and 2. The laws are in tests/discrete_law_longitudinal_incremental.py. Law C is law A of the complete-data study with dropout at random given the history. The dropout probabilities \(\omega_t\) lie in [0.85, 0.97]. Law D has event hazards in [0.05, 0.30]. Every learner is a saturated cell mean.

The one-fold scenarios are primary rows. A one-fold fit with saturated cell means is the plug-in g-formula on the empirical cell frequencies. Its tilt is nonlinear in \(\hat\pi\), so it carries a bias of order (cells)/n. The conditions for the one-fold interval are in the table below.

source	what it states
Kennedy (2019), Algorithm 2 (Appendix 8.3)	the one-fold interval, under Donsker conditions
Kim et al. (2021), Theorem 5.1	the dropout result, with sample splitting only
the argument of this study	saturated cell means on a finite history set form a finite-dimensional class. The remainder is a product of \(O_P(n^{-1/2})\) errors. The class is Donsker and the fitted influence function converges in \(L_2\), so the empirical-process term is \(o_P(n^{-1/2})\). The argument therefore extends the dropout result to one fold

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
three-node law with dropout after each arm, one fold	ate_regimen[odds x0.5 vs natural course]	difference in mean outcome between the plans “multiply the treatment odds by 0.5” against “leave the observed treatment mechanism unchanged”	cleverly IPSI with a censoring mechanism	-0.000084 to 0.000487	0.9400	0.9614	pass
three-node law with dropout after each arm, one fold	ate_regimen[odds x2 vs natural course]	difference in mean outcome between the plans “multiply the treatment odds by two” against “leave the observed treatment mechanism unchanged”	cleverly IPSI with a censoring mechanism	-0.000472 to 0.000114	0.9510	0.9871	pass
three-node law with dropout after each arm, one fold	ey_regimen[natural course]	mean outcome under the plan leave the observed treatment mechanism unchanged	cleverly IPSI with a censoring mechanism	-0.000933 to 0.000541	0.9490	0.9874	pass
three-node law with dropout after each arm, one fold	ey_regimen[odds x0.5]	mean outcome under the plan multiply the treatment odds by 0.5	cleverly IPSI with a censoring mechanism	-0.000717 to 0.000727	0.9540	0.9974	pass
three-node law with dropout after each arm, one fold	ey_regimen[odds x2]	mean outcome under the plan multiply the treatment odds by two	cleverly IPSI with a censoring mechanism	-0.0011 to 0.000356	0.9440	0.9807	pass
three-node law with dropout after each arm, five folds	ate_regimen[odds x0.5 vs natural course]	difference in mean outcome between the plans “multiply the treatment odds by 0.5” against “leave the observed treatment mechanism unchanged”	cleverly IPSI with a censoring mechanism	-0.000522 to 0.000020	0.9530	1.0133	pass
three-node law with dropout after each arm, five folds	ate_regimen[odds x2 vs natural course]	difference in mean outcome between the plans “multiply the treatment odds by two” against “leave the observed treatment mechanism unchanged”	cleverly IPSI with a censoring mechanism	0.000009 to 0.000579	0.9420	1.0174	pass
three-node law with dropout after each arm, five folds	ey_regimen[natural course]	mean outcome under the plan leave the observed treatment mechanism unchanged	cleverly IPSI with a censoring mechanism	-0.000723 to 0.000738	0.9500	0.9964	pass
three-node law with dropout after each arm, five folds	ey_regimen[odds x0.5]	mean outcome under the plan multiply the treatment odds by 0.5	cleverly IPSI with a censoring mechanism	-0.000970 to 0.000483	0.9450	0.9911	pass
three-node law with dropout after each arm, five folds	ey_regimen[odds x2]	mean outcome under the plan multiply the treatment odds by two	cleverly IPSI with a censoring mechanism	-0.000415 to 0.0010	0.9490	1.0014	pass
three-node survival law with dropout, horizons 1 to 3, one fold	ate_regimen[odds x0.5 vs natural course @ t=1]	difference in cumulative risk between the plans “multiply the treatment odds by 0.5” against “leave the observed treatment mechanism unchanged” at horizon t = 1	cleverly IPSI with a censoring mechanism	-0.000030 to 0.000029	0.9430	0.9916	pass
three-node survival law with dropout, horizons 1 to 3, one fold	ate_regimen[odds x0.5 vs natural course @ t=2]	difference in cumulative risk between the plans “multiply the treatment odds by 0.5” against “leave the observed treatment mechanism unchanged” at horizon t = 2	cleverly IPSI with a censoring mechanism	-0.000071 to 0.000043	0.9350	0.9669	pass
three-node survival law with dropout, horizons 1 to 3, one fold	ate_regimen[odds x0.5 vs natural course @ t=3]	difference in cumulative risk between the plans “multiply the treatment odds by 0.5” against “leave the observed treatment mechanism unchanged” at horizon t = 3	cleverly IPSI with a censoring mechanism	-0.000096 to 0.000054	0.9440	0.9815	pass
three-node survival law with dropout, horizons 1 to 3, one fold	ate_regimen[odds x2 vs natural course @ t=1]	difference in cumulative risk between the plans “multiply the treatment odds by two” against “leave the observed treatment mechanism unchanged” at horizon t = 1	cleverly IPSI with a censoring mechanism	-0.000036 to 0.000039	0.9420	0.9895	pass
three-node survival law with dropout, horizons 1 to 3, one fold	ate_regimen[odds x2 vs natural course @ t=2]	difference in cumulative risk between the plans “multiply the treatment odds by two” against “leave the observed treatment mechanism unchanged” at horizon t = 2	cleverly IPSI with a censoring mechanism	-0.000037 to 0.000092	0.9440	0.9643	pass
three-node survival law with dropout, horizons 1 to 3, one fold	ate_regimen[odds x2 vs natural course @ t=3]	difference in cumulative risk between the plans “multiply the treatment odds by two” against “leave the observed treatment mechanism unchanged” at horizon t = 3	cleverly IPSI with a censoring mechanism	-0.000051 to 0.000111	0.9430	0.9678	pass
three-node survival law with dropout, horizons 1 to 3, one fold	risk_regimen[natural course @ t=1]	cumulative risk under the plan leave the observed treatment mechanism unchanged at horizon t = 1	cleverly IPSI with a censoring mechanism	-0.000094 to 0.000134	0.9500	0.9753	pass
three-node survival law with dropout, horizons 1 to 3, one fold	risk_regimen[natural course @ t=2]	cumulative risk under the plan leave the observed treatment mechanism unchanged at horizon t = 2	cleverly IPSI with a censoring mechanism	-0.000127 to 0.000158	0.9480	1.0236	pass
three-node survival law with dropout, horizons 1 to 3, one fold	risk_regimen[natural course @ t=3]	cumulative risk under the plan leave the observed treatment mechanism unchanged at horizon t = 3	cleverly IPSI with a censoring mechanism	-0.000141 to 0.000172	0.9550	0.9977	pass
three-node survival law with dropout, horizons 1 to 3, one fold	risk_regimen[odds x0.5 @ t=1]	cumulative risk under the plan multiply the treatment odds by 0.5 at horizon t = 1	cleverly IPSI with a censoring mechanism	-0.000105 to 0.000143	0.9440	0.9568	pass
three-node survival law with dropout, horizons 1 to 3, one fold	risk_regimen[odds x0.5 @ t=2]	cumulative risk under the plan multiply the treatment odds by 0.5 at horizon t = 2	cleverly IPSI with a censoring mechanism	-0.000156 to 0.000160	0.9510	1.0062	pass
three-node survival law with dropout, horizons 1 to 3, one fold	risk_regimen[odds x0.5 @ t=3]	cumulative risk under the plan multiply the treatment odds by 0.5 at horizon t = 3	cleverly IPSI with a censoring mechanism	-0.000182 to 0.000172	0.9500	0.9876	pass
three-node survival law with dropout, horizons 1 to 3, one fold	risk_regimen[odds x2 @ t=1]	cumulative risk under the plan multiply the treatment odds by two at horizon t = 1	cleverly IPSI with a censoring mechanism	-0.000090 to 0.000133	0.9590	1.0065	pass
three-node survival law with dropout, horizons 1 to 3, one fold	risk_regimen[odds x2 @ t=2]	cumulative risk under the plan multiply the treatment odds by two at horizon t = 2	cleverly IPSI with a censoring mechanism	-0.000108 to 0.000195	0.9550	1.0232	pass
three-node survival law with dropout, horizons 1 to 3, one fold	risk_regimen[odds x2 @ t=3]	cumulative risk under the plan multiply the treatment odds by two at horizon t = 3	cleverly IPSI with a censoring mechanism	-0.000126 to 0.000218	0.9510	1.0023	pass
three-node survival law with dropout, horizons 1 to 3, five folds	ate_regimen[odds x0.5 vs natural course @ t=1]	difference in cumulative risk between the plans “multiply the treatment odds by 0.5” against “leave the observed treatment mechanism unchanged” at horizon t = 1	cleverly IPSI with a censoring mechanism	-0.000043 to 0.000016	0.9500	0.9960	pass
three-node survival law with dropout, horizons 1 to 3, five folds	ate_regimen[odds x0.5 vs natural course @ t=2]	difference in cumulative risk between the plans “multiply the treatment odds by 0.5” against “leave the observed treatment mechanism unchanged” at horizon t = 2	cleverly IPSI with a censoring mechanism	-0.000068 to 0.000046	0.9420	0.9661	pass
three-node survival law with dropout, horizons 1 to 3, five folds	ate_regimen[odds x0.5 vs natural course @ t=3]	difference in cumulative risk between the plans “multiply the treatment odds by 0.5” against “leave the observed treatment mechanism unchanged” at horizon t = 3	cleverly IPSI with a censoring mechanism	-0.000059 to 0.000092	0.9430	0.9732	pass
three-node survival law with dropout, horizons 1 to 3, five folds	ate_regimen[odds x2 vs natural course @ t=1]	difference in cumulative risk between the plans “multiply the treatment odds by two” against “leave the observed treatment mechanism unchanged” at horizon t = 1	cleverly IPSI with a censoring mechanism	-0.000019 to 0.000055	0.9490	0.9975	pass
three-node survival law with dropout, horizons 1 to 3, five folds	ate_regimen[odds x2 vs natural course @ t=2]	difference in cumulative risk between the plans “multiply the treatment odds by two” against “leave the observed treatment mechanism unchanged” at horizon t = 2	cleverly IPSI with a censoring mechanism	-0.000056 to 0.000071	0.9420	0.9848	pass
three-node survival law with dropout, horizons 1 to 3, five folds	ate_regimen[odds x2 vs natural course @ t=3]	difference in cumulative risk between the plans “multiply the treatment odds by two” against “leave the observed treatment mechanism unchanged” at horizon t = 3	cleverly IPSI with a censoring mechanism	-0.000079 to 0.000080	0.9440	0.9871	pass
three-node survival law with dropout, horizons 1 to 3, five folds	risk_regimen[natural course @ t=1]	cumulative risk under the plan leave the observed treatment mechanism unchanged at horizon t = 1	cleverly IPSI with a censoring mechanism	-0.000065 to 0.000155	0.9480	1.0087	pass
three-node survival law with dropout, horizons 1 to 3, five folds	risk_regimen[natural course @ t=2]	cumulative risk under the plan leave the observed treatment mechanism unchanged at horizon t = 2	cleverly IPSI with a censoring mechanism	-0.000139 to 0.000157	0.9480	0.9873	pass
three-node survival law with dropout, horizons 1 to 3, five folds	risk_regimen[natural course @ t=3]	cumulative risk under the plan leave the observed treatment mechanism unchanged at horizon t = 3	cleverly IPSI with a censoring mechanism	-0.000170 to 0.000146	0.9490	0.9900	pass
three-node survival law with dropout, horizons 1 to 3, five folds	risk_regimen[odds x0.5 @ t=1]	cumulative risk under the plan multiply the treatment odds by 0.5 at horizon t = 1	cleverly IPSI with a censoring mechanism	-0.000085 to 0.000148	0.9490	1.0181	pass
three-node survival law with dropout, horizons 1 to 3, five folds	risk_regimen[odds x0.5 @ t=2]	cumulative risk under the plan multiply the treatment odds by 0.5 at horizon t = 2	cleverly IPSI with a censoring mechanism	-0.000163 to 0.000158	0.9530	0.9927	pass
three-node survival law with dropout, horizons 1 to 3, five folds	risk_regimen[odds x0.5 @ t=3]	cumulative risk under the plan multiply the treatment odds by 0.5 at horizon t = 3	cleverly IPSI with a censoring mechanism	-0.000175 to 0.000184	0.9500	0.9731	pass
three-node survival law with dropout, horizons 1 to 3, five folds	risk_regimen[odds x2 @ t=1]	cumulative risk under the plan multiply the treatment odds by two at horizon t = 1	cleverly IPSI with a censoring mechanism	-0.000050 to 0.000176	0.9480	0.9933	pass
three-node survival law with dropout, horizons 1 to 3, five folds	risk_regimen[odds x2 @ t=2]	cumulative risk under the plan multiply the treatment odds by two at horizon t = 2	cleverly IPSI with a censoring mechanism	-0.000143 to 0.000175	0.9440	0.9753	pass
three-node survival law with dropout, horizons 1 to 3, five folds	risk_regimen[odds x2 @ t=3]	cumulative risk under the plan multiply the treatment odds by two at horizon t = 3	cleverly IPSI with a censoring mechanism	-0.000185 to 0.000161	0.9500	0.9963	pass

Theory properties

property	cell	role	what was tested	what must hold	measured	result
censoring_complete_case	mean_x2__complete_case	diagnostic	odds-x2 mean at every node, three-node law: a fit to the rows that never drop out, without a censoring model, beside the exact-law limit	reported only	bias 0.0705 to 0.0716, margin 0.0029, coverage 0.0000	reported
clustered_inference	mean_x2__cluster_robust	positive	odds-x2 mean at every node, three-node law: five-fold point-treatment TMLE with cluster-robust ATE inference	SE-ratio and coverage intervals both stay inside their calibration bands	coverage 0.9451 to 0.9648, SE ratio 0.9809 to 1.0465, paired coverage gain 0.1530 to 0.1887	pass
clustered_inference	mean_x2__iid_control	control	odds-x2 mean at every node, three-node law: the identical rows, point estimates, and influence curves treated as independent	the SE-ratio upper endpoint must not exceed the declared IID-control ceiling	coverage 0.7654 to 0.8043, SE ratio 0.6212 to 0.6625, paired coverage gain 0.1530 to 0.1887	pass
interval_calibration	competing_contrast_x2_h3__correctly_specified	positive	odds-x2 contrast in the cause-1 incidence by t = 3: 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 0.9491 to 0.9681, SE ratio 1.0124 to 1.0830, empirical efficiency ratio 0.9239 to 0.9884, reported efficiency ratio 1.0005 to 1.0007	fail
interval_calibration	competing_contrast_x2_h3__noise_control	control	odds-x2 contrast in the cause-1 incidence by t = 3: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8328 to 0.8667, SE ratio 0.6925 to 0.7421, empirical efficiency ratio 1.3483 to 1.4449, reported efficiency ratio 1.0005 to 1.0007	pass
interval_calibration	competing_contrast_x2_h3__shrunken_se_control	control	odds-x2 contrast in the cause-1 incidence by t = 3: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8259 to 0.8604, SE ratio 0.7091 to 0.7584, empirical efficiency ratio 0.9235 to 0.9877, reported efficiency ratio 0.7003 to 0.7005	pass
interval_calibration	dropout_contrast_x2__correctly_specified	positive	odds-x2 contrast at the end of study, with dropout: 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 0.9407 to 0.9612, SE ratio 0.9876 to 1.0548, empirical efficiency ratio 0.9489 to 1.0135, reported efficiency ratio 1.0007 to 1.0012	pass
interval_calibration	dropout_contrast_x2__noise_control	control	odds-x2 contrast at the end of study, with dropout: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8141 to 0.8495, SE ratio 0.6886 to 0.7342, empirical efficiency ratio 1.3634 to 1.4534, reported efficiency ratio 1.0007 to 1.0012	pass
interval_calibration	dropout_contrast_x2__shrunken_se_control	control	odds-x2 contrast at the end of study, with dropout: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8210 to 0.8559, SE ratio 0.6914 to 0.7379, empirical efficiency ratio 0.9496 to 1.0133, reported efficiency ratio 0.7005 to 0.7008	pass
interval_calibration	dropout_contrast_x2_in_sample__correctly_specified	positive	odds-x2 contrast at the end of study, with dropout, one fold: 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 0.9388 to 0.9597, SE ratio 0.9877 to 1.0566, empirical efficiency ratio 0.9467 to 1.0126, reported efficiency ratio 0.9999 to 1.0004	pass
interval_calibration	dropout_contrast_x2_in_sample__noise_control	control	odds-x2 contrast at the end of study, with dropout, one fold: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8182 to 0.8534, SE ratio 0.6880 to 0.7354, empirical efficiency ratio 1.3600 to 1.4537, reported efficiency ratio 0.9999 to 1.0004	pass
interval_calibration	dropout_contrast_x2_in_sample__shrunken_se_control	control	odds-x2 contrast at the end of study, with dropout, one fold: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8186 to 0.8537, SE ratio 0.6923 to 0.7392, empirical efficiency ratio 0.9471 to 1.0112, reported efficiency ratio 0.7000 to 0.7003	pass
interval_calibration	msm_log_delta__correctly_specified	positive	slope of the identity-link working model in log odds multiplier: 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 0.9451 to 0.9648, SE ratio 0.9792 to 1.0457, empirical efficiency ratio 0.9573 to 1.0222, reported efficiency ratio 1.0007 to 1.0012	pass
interval_calibration	msm_log_delta__noise_control	control	slope of the identity-link working model in log odds multiplier: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8269 to 0.8613, SE ratio 0.6966 to 0.7467, empirical efficiency ratio 1.3405 to 1.4368, reported efficiency ratio 1.0007 to 1.0012	pass
interval_calibration	msm_log_delta__shrunken_se_control	control	slope of the identity-link working model in log odds multiplier: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8207 to 0.8556, SE ratio 0.6848 to 0.7325, empirical efficiency ratio 0.9566 to 1.0232, reported efficiency ratio 0.7005 to 0.7009	pass
interval_calibration	survival_contrast_x2_h3__correctly_specified	positive	odds-x2 contrast in the risk by t = 3, with dropout: 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 0.9429 to 0.9630, SE ratio 0.9881 to 1.0552, empirical efficiency ratio 0.9486 to 1.0131, reported efficiency ratio 1.0009 to 1.0012	pass
interval_calibration	survival_contrast_x2_h3__noise_control	control	odds-x2 contrast in the risk by t = 3, with dropout: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8127 to 0.8483, SE ratio 0.6969 to 0.7437, empirical efficiency ratio 1.3461 to 1.4366, reported efficiency ratio 1.0009 to 1.0012	pass
interval_calibration	survival_contrast_x2_h3__shrunken_se_control	control	odds-x2 contrast in the risk by t = 3, with dropout: 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 0.8521, SE ratio 0.6918 to 0.7384, empirical efficiency ratio 0.9490 to 1.0128, reported efficiency ratio 0.7006 to 0.7008	pass
interval_calibration	survival_contrast_x2_h3_in_sample__correctly_specified	positive	odds-x2 contrast in the risk by t = 3, with dropout, one fold: 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 0.9436 to 0.9636, SE ratio 0.9869 to 1.0539, empirical efficiency ratio 0.9484 to 1.0128, reported efficiency ratio 0.9993 to 0.9996	pass
interval_calibration	survival_contrast_x2_h3_in_sample__noise_control	control	odds-x2 contrast in the risk by t = 3, with dropout, one fold: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8103 to 0.8460, SE ratio 0.6819 to 0.7277, empirical efficiency ratio 1.3735 to 1.4659, reported efficiency ratio 0.9993 to 0.9996	pass
interval_calibration	survival_contrast_x2_h3_in_sample__shrunken_se_control	control	odds-x2 contrast in the risk by t = 3, with dropout, one fold: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8137 to 0.8492, SE ratio 0.6916 to 0.7377, empirical efficiency ratio 0.9483 to 1.0117, reported efficiency ratio 0.6995 to 0.6997	pass
interval_calibration	weighted_contrast_x0_5__correctly_specified	positive	odds-x0.5 contrast on the law weighted by 1 + A1 + W/2: 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 0.9425 to 0.9627, SE ratio 0.9666 to 1.0318, empirical efficiency ratio 0.9699 to 1.0352, reported efficiency ratio 1.0005 to 1.0009	pass
interval_calibration	weighted_contrast_x0_5__noise_control	control	odds-x0.5 contrast on the law weighted by 1 + A1 + W/2: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8186 to 0.8537, SE ratio 0.6963 to 0.7436, empirical efficiency ratio 1.3456 to 1.4369, reported efficiency ratio 1.0004 to 1.0009	pass
interval_calibration	weighted_contrast_x0_5__shrunken_se_control	control	odds-x0.5 contrast on the law weighted by 1 + A1 + W/2: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8127 to 0.8483, SE ratio 0.6774 to 0.7230, empirical efficiency ratio 0.9688 to 1.0340, reported efficiency ratio 0.7003 to 0.7007	pass
simultaneous_coverage	survival_grid__pointwise_joint_control	control	three plans by three horizons, the risks and six contrasts: the pointwise 95% intervals of the same fits, read jointly	joint coverage upper endpoint must fall below the nominal rate	joint coverage 0.6746 to 0.7181	pass
simultaneous_coverage	survival_grid__simultaneous_band	positive	three plans by three horizons, the risks and six contrasts: 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 0.9370 to 0.9582	pass

One cell is red. interval_calibration/competing_contrast_x2_h3__correctly_specified fails its SE-ratio band on the conservative side: the interval 1.0124 to 1.0830 crosses the upper bound of 1.07. Its coverage interval, 0.9491 to 0.9681, stays inside the band. The reported efficiency ratio is 1.0006, so the reported standard error matches the exact efficient influence-function standard deviation. The empirical spread sits 4.4% below that bound. A regular estimator does not stay below the bound as n grows, so the diagnosis reads a finite-sample excursion and finds no defect. The design probes on other draws read SE ratios of 0.948 at n = 32,000 and 0.980 at n = 64,000 (tests/diagnostics/x19_sizing/reading.csv).

By the declared route the study moved to reporting. The roadmap row X19-competing-calibration owns the cell. No fit of this cell was repeated: regenerate.py --reuse-subject --skip-properties summarized the committed rows again under the new policy. No property fit failed.

Every property fit uses five folds, except the two one-fold labels, saturated cell means at every slot and 3,000 draws. The default size is n = 48,000. x2 is the plan that doubles the odds of treatment at every node. contrast_x2 is its contrast with the natural course. tests/studies/longitudinal_incremental_dropout_properties.py declares every cell.

family	cells and fits	what must hold
interval_calibration	seven labels with three cells each: the correctly specified fit and two derived controls	SE ratio and coverage intervals inside the calibration bands, and both efficiency-ratio intervals inside their band. The shrunken-SE control falls below the band. The noise control rises above the efficiency-ratio band
simultaneous_coverage	the band over the risks of three plans at horizons 1 to 3 and their six contrasts, on law D at n = 48,000, and its pointwise control	the band’s joint coverage interval lies inside the calibration coverage band. The pointwise control’s joint coverage interval lies below the nominal rate
clustered_inference	ey_regimen[odds x2] on law Ck in 800 clusters of 10 (n = 8,000): the cluster-robust standard error, and the standard error that treats each unit as independent (the control)	the cluster-robust arm is calibrated. The control’s SE-ratio interval lies below 0.80. The coverage gain of the pair is at least 0.03
censoring_complete_case	ey_regimen[odds x2] on its own draws of law C, fitted to the rows that never drop out, with no censoring model. 3,000 draws at n = 48,000	no verdict. The cell prints its bias beside the exact complete-case limit (complete_case_limit)

The calibration labels:

label	law and estimand	n
dropout_contrast_x2	C, contrast_x2 at the end of study	48,000
survival_contrast_x2_h3	D, contrast_x2 of the risk by horizon 3, read off the band fit	48,000
dropout_contrast_x2_in_sample	C, contrast_x2, one fold (Kennedy 2019, Algorithm 2)	48,000
survival_contrast_x2_h3_in_sample	D, contrast_x2 of the risk by horizon 3, one fold, on the band draws	48,000
competing_contrast_x2_h3	Dc, contrast_x2 of the cause-1 incidence by horizon 3	96,000
msm_log_delta	C, the slope of an identity-link working model linear in \(\log\delta\) over the three plans	48,000
weighted_contrast_x0_5	C with the informative weight \(1 + A_1 + W/2\), odds x0.5 vs natural course. The truth is the weighted law’s	48,000

Law Dc adds a second, competing cause at each node. Law Ck is law C with a shared binary cluster effect. The one-fold cells are gated like the cross-fitted ones. The censoring_complete_case cell shows why the weights matter: a fit that ignores dropout sits away from the target, and the cell reports how far.

Sizing

The five-fold contrasts carry a second-order bias of order 1/n, so their standardized bias falls as \(1/\sqrt n\). The sizing probe in tests/diagnostics/x19_sizing/ measured the bias at n = 8,000. The size rule requires that every primary row passes the bias rule with probability at least 0.95 at the projected bias. It requires 0.90 for each calibration cell.

sizing input	value	source
standardized bias at n = 8,000, dropout contrasts	0.31	tests/studies/canonical_longitudinal_incremental_dropout.py
one-fold standardized bias at n = 8,000	about 0.2	the same
primary n	80,000	the same
calibration replications	3,000. At 2,000 an ideal cell passes with probability 0.935, and at 3,000 with 0.996	tests/studies/longitudinal_incremental_dropout_properties.py
calibration n	48,000. The calibration cells pass jointly with probability 0.93	the same, with tests/unit/test_longitudinal_incremental_design.py
competing-risk n	96,000. At n = 8,000 a training fold can hold one or two treated units of a rare history, and the censoring fit reaches its lower bound. The SE ratio reads 0.948 at 32,000 and 0.980 at 64,000	tests/studies/longitudinal_incremental_dropout_properties.py
clustered cell	n = 8,000 and 3,000 draws. The calibration rule cannot pass at 1,000	the same
weighted truth	the informative weight moves the truth by 9.5 standard errors at n = 48,000	tests/unit/test_longitudinal_incremental_design.py
independent-unit control	the exact SE ratio is 0.60 and the exact coverage is 0.76	the same
complete-case limit	6.2 standard errors from the truth	the same
band control	the pointwise intervals cover jointly with exact probability 0.695. The control clears the control-power rule at that value plus 0.005	the same

The fallback-read bias bound holds far below these sizes. The study fixed every size, replication count, margin, law, learner, fold count and seed before any run. After the run, the censoring_complete_case diagnostic moved to its own sample stream, because it reused the law C draws of interval_calibration. It was refitted at the declared size and spliced into the committed rows. targeted-refit.json in the study directory records the baseline, the selection, the replacement fits and the identical carried rows. The cell states no verdict.

Red-cell route

event	route, declared before any run
a red primary truth row	blocks the merge and gets a diagnosis
a red property cell that reads as a finite-sample limit	is re-registered reporting with an owner row before any re-run
a replication that raises	is never redrawn. The failing cell drops to its owner with its failure count stated as a page limit, and the run repeats without it
any verdict	moves no budget, margin, law, learner, fold count or seed. Controls carry the verdict their family states

Measured values and declared margins

Names beginning margin: are thresholds declared before the run. Names beginning bound: are exact standard errors of the efficient influence curve at the primary n. Everything else is measured from the committed results and checked at the precision printed.

quantity	value	source
replicates	1000	primary replications
n	80000	observations per primary replication
independent_tests_passed	40	truth tests passing
independent_tests_total	40	truth tests reported
property_cells_passed	24	property cells passing
property_cells_total	25	property cells reported
max_standardized_bias	0.0843	largest primary standardized bias
min_coverage	0.9350	lowest primary coverage
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. No cell of this study reads it
margin:paired_difference	0.1500	paired similarity margin in pooled SDs. No cell of this study reads it
margin:rmse_noninferiority	1.1000	RMSE-ratio noninferiority bound. No cell of this study reads it
margin:coverage_noninferiority	-0.0250	coverage-difference noninferiority bound. No cell of this study reads it
margin:calibration_noninferiority	0.0500	calibration-excess noninferiority bound. No cell of this study reads it
margin:minimum_power	0.8000	minimum power lower bound. No cell of this study reads it
margin:root_n_slope	-0.5000	expected root-n slope. No cell of this study reads it
margin:root_n_slope_lower	-0.6250	root-n slope lower bound. No cell of this study reads it
margin:root_n_slope_upper	-0.3750	root-n slope upper bound. No cell of this study reads it
margin:excluded_slope	-0.2500	slower rate the interval must exclude. No cell of this study reads it
margin:iid_control_se_ceiling	0.8000	independent-unit control SE-ratio ceiling
margin:clustered_coverage_gain	0.0300	paired coverage-gain floor
margin:shrunken_se_factor	0.7000	negative-control SE multiplier
margin:subject_failure_bound	0.0100	declared share of replications a cell may lose to a recordable fitting failure
margin:subject_failures_allowed	10	the same bound, in primary replications
margin:efficiency_ratio_lower	0.9000	exact-EIF ratio lower bound
margin:efficiency_ratio_upper	1.1000	exact-EIF ratio upper bound
bound:ate_regimen[odds x0.5 vs natural course @ t=1]_standard_error	0.000359	exact influence-curve standard error of this estimand at the primary n = 80,000
bound:ate_regimen[odds x0.5 vs natural course @ t=2]_standard_error	0.000675	exact influence-curve standard error of this estimand at the primary n = 80,000
bound:ate_regimen[odds x0.5 vs natural course @ t=3]_standard_error	0.000901	exact influence-curve standard error of this estimand at the primary n = 80,000
bound:ate_regimen[odds x0.5 vs natural course]_standard_error	0.0034	exact influence-curve standard error of this estimand at the primary n = 80,000
bound:ate_regimen[odds x2 vs natural course @ t=1]_standard_error	0.000451	exact influence-curve standard error of this estimand at the primary n = 80,000
bound:ate_regimen[odds x2 vs natural course @ t=2]_standard_error	0.000762	exact influence-curve standard error of this estimand at the primary n = 80,000
bound:ate_regimen[odds x2 vs natural course @ t=3]_standard_error	0.000962	exact influence-curve standard error of this estimand at the primary n = 80,000
bound:ate_regimen[odds x2 vs natural course]_standard_error	0.0035	exact influence-curve standard error of this estimand at the primary n = 80,000
bound:ey_regimen[natural course]_standard_error	0.0089	exact influence-curve standard error of this estimand at the primary n = 80,000
bound:ey_regimen[odds x0.5]_standard_error	0.0088	exact influence-curve standard error of this estimand at the primary n = 80,000
bound:ey_regimen[odds x2]_standard_error	0.0088	exact influence-curve standard error of this estimand at the primary n = 80,000
bound:risk_regimen[natural course @ t=1]_standard_error	0.0014	exact influence-curve standard error of this estimand at the primary n = 80,000
bound:risk_regimen[natural course @ t=2]_standard_error	0.0018	exact influence-curve standard error of this estimand at the primary n = 80,000
bound:risk_regimen[natural course @ t=3]_standard_error	0.0019	exact influence-curve standard error of this estimand at the primary n = 80,000
bound:risk_regimen[odds x0.5 @ t=1]_standard_error	0.0015	exact influence-curve standard error of this estimand at the primary n = 80,000
bound:risk_regimen[odds x0.5 @ t=2]_standard_error	0.0020	exact influence-curve standard error of this estimand at the primary n = 80,000
bound:risk_regimen[odds x0.5 @ t=3]_standard_error	0.0021	exact influence-curve standard error of this estimand at the primary n = 80,000
bound:risk_regimen[odds x2 @ t=1]_standard_error	0.0014	exact influence-curve standard error of this estimand at the primary n = 80,000
bound:risk_regimen[odds x2 @ t=2]_standard_error	0.0019	exact influence-curve standard error of this estimand at the primary n = 80,000
bound:risk_regimen[odds x2 @ t=3]_standard_error	0.0021	exact influence-curve standard error of this estimand at the primary n = 80,000

Limits

limit	what it means for use
dropout at random given the history, after each arm	the laws set \(\omega_t\) in [0.85, 0.97]. Heavier dropout and informative dropout are outside this evidence
saturated cell means at every slot	the study does not test a flexible learner. The one-fold claim rests on the finite-dimensional class
one-fold primary rows rest on an argument	Kim et al. (2021) state the dropout result with sample splitting only. The Donsker argument above is the extension, and no source states it
finite-sample bias is sized, not removed	at n = 8,000 the standardized bias reads 0.31 on the dropout contrasts. The rows pass at the sizes above, and smaller sizes are not claimed
three nodes and horizons 1 to 3	longer panels and later horizons are not claimed
no comparator	no pinned implementation fits this design, so the study has no paired verdict
the clustered cell reads one mean	ey_regimen[odds x2] on law Ck in 800 clusters of 10. Fewer clusters are unmeasured
the complete-case cell states no verdict	it reports a bias beside the exact limit

Reproduction

tests/canonical/longitudinal_incremental_dropout/regenerate.py regenerates the study. It runs no reference process, because the study has no comparator.