Repeated-outcome longitudinal IPSI#

This study validates IPSI with outcome=Repeated(["Y1", "Y2", "Y3"]). A repeated outcome is measured at every node and is not absorbing. The parameter is the mean \(E[Y_h^{q}]\) at each horizon \(h\) under each odds tilt \(\delta\) of the treatment.

Each horizon is the end-of-study fit of the panel truncated after node \(h\). The functional is Theorem 3.1 of Kim, Kennedy and Naimi (2021) at \(t = h\). The influence function is their Theorem 4.1 on the prefix. The section repeated outcomes states the estimator and its exact-law evidence.

The evidence is Python property evidence. No canonical implementation is compared, because no maintained implementation fits this parameter. The npcausal package reads an end-of-study outcome, has no dropout and pools its treatment fit over (id, time). A dropout-free pairing at each horizon would repeat the pairing of the incremental interventions study. The dropout claim rests on the exact witnesses E5, E6 and the mutation N9 in tests/unit/test_influence_gateaux_repeated.py.

What was fitted#

setting	value
datasets	2,400 censored panels of 8,000 rows per scenario from `make_longitudinal_repeated`, with three nodes
fit	one `IPSI` fit with `outcome=Repeated(["Y1", "Y2", "Y3"])`
scenarios	`repeated_folds_5` (Algorithm 1 of Kim et al.: five folds) and `repeated_folds_1` (in sample)
plans	the odds tilts \(\delta\) = 0.5, 1 (the natural course, the reference) and 2
reported parameters	the mean of each plan at \(t = 1, 2, 3\), and the contrasts of the two other plans against the natural course. That is nine means and six contrasts
outcome regressions	cell means with a backoff (`CellMeansBackoff`)
mechanisms	main-terms quasibinomial GLMs for treatment and censoring, correctly specified on this law. The study declares no mechanism
intervals	pointwise 95% Wald from the influence curve
initial estimate	not published. `IPSI` fits no fluctuation, and its one-step estimate is the only estimate it reports

Each truth is the forward composition of the law’s conditionals under the tilt (RepeatedOutcomeLaw.tilt_counterfactual in tests/studies/ltmle_repeated_law.py). A fast test shows that the observed-data functional of Theorem 3.1 gives the same numbers to 1e-14.

With correct mechanisms, Kim’s fourth condition holds for any consistent outcome regression. The sample size follows from that. The mechanisms are parametric, so the remainder \((\hat\pi - \pi)^2\) is of order \(p / n\) with \(p\) near 6. The primary pilot set \(n\) = 8,000 and 2,400 replications without a doubling of \(n\).

Accuracy against known truth#

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
three-time-point law with a repeated binary outcome and monotone censoring, in sample	`ate_regimen[odds x0.5 vs natural course @ t=1]`	difference in the mean of the repeated outcome between the plans “multiply the treatment odds by 0.5” against “leave the observed treatment mechanism unchanged” at horizon t = 1	`cleverly`	-0.000089 to 0.000075	0.9404	0.9866	pass
three-time-point law with a repeated binary outcome and monotone censoring, in sample	`ate_regimen[odds x0.5 vs natural course @ t=2]`	difference in the mean of the repeated outcome between the plans “multiply the treatment odds by 0.5” against “leave the observed treatment mechanism unchanged” at horizon t = 2	`cleverly`	-0.000100 to 0.000138	0.9421	0.9764	pass
three-time-point law with a repeated binary outcome and monotone censoring, in sample	`ate_regimen[odds x0.5 vs natural course @ t=3]`	difference in the mean of the repeated outcome between the plans “multiply the treatment odds by 0.5” against “leave the observed treatment mechanism unchanged” at horizon t = 3	`cleverly`	-0.000216 to 0.000100	0.9367	0.9805	pass
three-time-point law with a repeated binary outcome and monotone censoring, in sample	`ate_regimen[odds x2 vs natural course @ t=1]`	difference in the mean of the repeated outcome between the plans “multiply the treatment odds by two” against “leave the observed treatment mechanism unchanged” at horizon t = 1	`cleverly`	-0.000066 to 0.000077	0.9413	0.9869	pass
three-time-point law with a repeated binary outcome and monotone censoring, in sample	`ate_regimen[odds x2 vs natural course @ t=2]`	difference in the mean of the repeated outcome between the plans “multiply the treatment odds by two” against “leave the observed treatment mechanism unchanged” at horizon t = 2	`cleverly`	-0.000099 to 0.000122	0.9358	0.9797	pass
three-time-point law with a repeated binary outcome and monotone censoring, in sample	`ate_regimen[odds x2 vs natural course @ t=3]`	difference in the mean of the repeated outcome between the plans “multiply the treatment odds by two” against “leave the observed treatment mechanism unchanged” at horizon t = 3	`cleverly`	-0.000110 to 0.000185	0.9504	0.9952	pass
three-time-point law with a repeated binary outcome and monotone censoring, in sample	`ey_regimen[natural course @ t=1]`	mean of the repeated outcome under the plan leave the observed treatment mechanism unchanged at horizon t = 1	`cleverly`	-0.000289 to 0.000165	0.9492	0.9871	pass
three-time-point law with a repeated binary outcome and monotone censoring, in sample	`ey_regimen[natural course @ t=2]`	mean of the repeated outcome under the plan leave the observed treatment mechanism unchanged at horizon t = 2	`cleverly`	-0.000274 to 0.000212	0.9537	1.0212	pass
three-time-point law with a repeated binary outcome and monotone censoring, in sample	`ey_regimen[natural course @ t=3]`	mean of the repeated outcome under the plan leave the observed treatment mechanism unchanged at horizon t = 3	`cleverly`	-0.000347 to 0.000221	0.9492	0.9836	pass
three-time-point law with a repeated binary outcome and monotone censoring, in sample	`ey_regimen[odds x0.5 @ t=1]`	mean of the repeated outcome under the plan multiply the treatment odds by 0.5 at horizon t = 1	`cleverly`	-0.000327 to 0.000189	0.9479	0.9836	pass
three-time-point law with a repeated binary outcome and monotone censoring, in sample	`ey_regimen[odds x0.5 @ t=2]`	mean of the repeated outcome under the plan multiply the treatment odds by 0.5 at horizon t = 2	`cleverly`	-0.000300 to 0.000277	0.9546	1.0144	pass
three-time-point law with a repeated binary outcome and monotone censoring, in sample	`ey_regimen[odds x0.5 @ t=3]`	mean of the repeated outcome under the plan multiply the treatment odds by 0.5 at horizon t = 3	`cleverly`	-0.000462 to 0.000221	0.9471	0.9834	pass
three-time-point law with a repeated binary outcome and monotone censoring, in sample	`ey_regimen[odds x2 @ t=1]`	mean of the repeated outcome under the plan multiply the treatment odds by two at horizon t = 1	`cleverly`	-0.000278 to 0.000165	0.9542	0.9917	pass
three-time-point law with a repeated binary outcome and monotone censoring, in sample	`ey_regimen[odds x2 @ t=2]`	mean of the repeated outcome under the plan multiply the treatment odds by two at horizon t = 2	`cleverly`	-0.000263 to 0.000225	0.9533	1.0105	pass
three-time-point law with a repeated binary outcome and monotone censoring, in sample	`ey_regimen[odds x2 @ t=3]`	mean of the repeated outcome under the plan multiply the treatment odds by two at horizon t = 3	`cleverly`	-0.000323 to 0.000273	0.9558	0.9823	pass
three-time-point law with a repeated binary outcome and monotone censoring, five folds	`ate_regimen[odds x0.5 vs natural course @ t=1]`	difference in the mean of the repeated outcome between the plans “multiply the treatment odds by 0.5” against “leave the observed treatment mechanism unchanged” at horizon t = 1	`cleverly`	-0.000045 to 0.000118	0.9529	0.9968	pass
three-time-point law with a repeated binary outcome and monotone censoring, five folds	`ate_regimen[odds x0.5 vs natural course @ t=2]`	difference in the mean of the repeated outcome between the plans “multiply the treatment odds by 0.5” against “leave the observed treatment mechanism unchanged” at horizon t = 2	`cleverly`	-0.000136 to 0.000102	0.9437	0.9781	pass
three-time-point law with a repeated binary outcome and monotone censoring, five folds	`ate_regimen[odds x0.5 vs natural course @ t=3]`	difference in the mean of the repeated outcome between the plans “multiply the treatment odds by 0.5” against “leave the observed treatment mechanism unchanged” at horizon t = 3	`cleverly`	-0.000153 to 0.000152	0.9558	1.0361	pass
three-time-point law with a repeated binary outcome and monotone censoring, five folds	`ate_regimen[odds x2 vs natural course @ t=1]`	difference in the mean of the repeated outcome between the plans “multiply the treatment odds by two” against “leave the observed treatment mechanism unchanged” at horizon t = 1	`cleverly`	-0.000103 to 0.000039	0.9517	0.9971	pass
three-time-point law with a repeated binary outcome and monotone censoring, five folds	`ate_regimen[odds x2 vs natural course @ t=2]`	difference in the mean of the repeated outcome between the plans “multiply the treatment odds by two” against “leave the observed treatment mechanism unchanged” at horizon t = 2	`cleverly`	-0.000089 to 0.000134	0.9479	0.9736	pass
three-time-point law with a repeated binary outcome and monotone censoring, five folds	`ate_regimen[odds x2 vs natural course @ t=3]`	difference in the mean of the repeated outcome between the plans “multiply the treatment odds by two” against “leave the observed treatment mechanism unchanged” at horizon t = 3	`cleverly`	-0.000113 to 0.000178	0.9600	1.0220	pass
three-time-point law with a repeated binary outcome and monotone censoring, five folds	`ey_regimen[natural course @ t=1]`	mean of the repeated outcome under the plan leave the observed treatment mechanism unchanged at horizon t = 1	`cleverly`	0.000039 to 0.000493	0.9442	0.9882	pass
three-time-point law with a repeated binary outcome and monotone censoring, five folds	`ey_regimen[natural course @ t=2]`	mean of the repeated outcome under the plan leave the observed treatment mechanism unchanged at horizon t = 2	`cleverly`	-0.000375 to 0.000111	0.9533	1.0213	pass
three-time-point law with a repeated binary outcome and monotone censoring, five folds	`ey_regimen[natural course @ t=3]`	mean of the repeated outcome under the plan leave the observed treatment mechanism unchanged at horizon t = 3	`cleverly`	-0.000077 to 0.000466	0.9517	1.0344	pass
three-time-point law with a repeated binary outcome and monotone censoring, five folds	`ey_regimen[odds x0.5 @ t=1]`	mean of the repeated outcome under the plan multiply the treatment odds by 0.5 at horizon t = 1	`cleverly`	0.000048 to 0.000557	0.9529	0.9974	pass
three-time-point law with a repeated binary outcome and monotone censoring, five folds	`ey_regimen[odds x0.5 @ t=2]`	mean of the repeated outcome under the plan multiply the treatment odds by 0.5 at horizon t = 2	`cleverly`	-0.000436 to 0.000138	0.9533	1.0185	pass
three-time-point law with a repeated binary outcome and monotone censoring, five folds	`ey_regimen[odds x0.5 @ t=3]`	mean of the repeated outcome under the plan multiply the treatment odds by 0.5 at horizon t = 3	`cleverly`	-0.000133 to 0.000521	0.9575	1.0369	pass
three-time-point law with a repeated binary outcome and monotone censoring, five folds	`ey_regimen[odds x2 @ t=1]`	mean of the repeated outcome under the plan multiply the treatment odds by two at horizon t = 1	`cleverly`	0.000009 to 0.000458	0.9450	0.9799	pass
three-time-point law with a repeated binary outcome and monotone censoring, five folds	`ey_regimen[odds x2 @ t=2]`	mean of the repeated outcome under the plan multiply the treatment odds by two at horizon t = 2	`cleverly`	-0.000354 to 0.000135	0.9492	1.0090	pass
three-time-point law with a repeated binary outcome and monotone censoring, five folds	`ey_regimen[odds x2 @ t=3]`	mean of the repeated outcome under the plan multiply the treatment odds by two at horizon t = 3	`cleverly`	-0.000059 to 0.000513	0.9529	1.0299	pass

Agreement with the canonical implementation#

No canonical implementation is compared. equivalence.csv records no row. The accuracy table above and the theory properties below carry the evidence of this study.

Theory properties#

property	cell	role	what was tested	what must hold	measured	result
`booster_calibration`	`ipsi_0.5_t3__crossfit_boosted`	diagnostic	mean under the odds tilt 0.5 at horizon t = 3, repeated binary outcome: five folds, a shallow regularized booster for both outcome learners, main-terms quasibinomial mechanisms, n = 8,000	reported, no verdict	SE ratio 0.9767 to 1.0606	reported
`booster_calibration`	`ipsi_0.5_vs_1.0_t3__crossfit_boosted`	diagnostic	contrast of the odds tilt 0.5 against the natural course at horizon t = 3, repeated binary outcome: five folds, a shallow regularized booster for both outcome learners, main-terms quasibinomial mechanisms, n = 8,000	reported, no verdict	SE ratio 0.9958 to 1.0822	reported
`booster_calibration`	`ipsi_1.0_t3__crossfit_boosted`	diagnostic	mean under the natural course at horizon t = 3, repeated binary outcome: five folds, a shallow regularized booster for both outcome learners, main-terms quasibinomial mechanisms, n = 8,000	reported, no verdict	SE ratio 0.9781 to 1.0596	reported
`booster_calibration`	`ipsi_2.0_t3__crossfit_boosted`	diagnostic	mean under the odds tilt 2 at horizon t = 3, repeated binary outcome: five folds, a shallow regularized booster for both outcome learners, main-terms quasibinomial mechanisms, n = 8,000	reported, no verdict	SE ratio 0.9870 to 1.0678	reported
`booster_calibration`	`ipsi_2.0_vs_1.0_t3__crossfit_boosted`	diagnostic	contrast of the odds tilt 2.0 against the natural course at horizon t = 3, repeated binary outcome: five folds, a shallow regularized booster for both outcome learners, main-terms quasibinomial mechanisms, n = 8,000	reported, no verdict	SE ratio 0.9889 to 1.0719	reported
`interval_calibration`	`ipsi_0.5_t1__in_sample`	positive	mean under the odds tilt 0.5 at horizon t = 1, repeated binary outcome: in sample, cell-mean outcome regressions with a backoff, main-terms quasibinomial treatment and censoring mechanisms, n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9439 to 0.9575, SE ratio 0.9815 to 1.0255	pass
`interval_calibration`	`ipsi_0.5_t1__noise_control`	control	mean under the odds tilt 0.5 at horizon t = 1, repeated binary outcome: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8201 to 0.8436, SE ratio 0.6896 to 0.7207	pass
`interval_calibration`	`ipsi_0.5_t1__shrunken_se_control`	control	mean under the odds tilt 0.5 at horizon t = 1, repeated binary outcome: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8165 to 0.8401, SE ratio 0.6870 to 0.7176	pass
`interval_calibration`	`ipsi_0.5_t2__in_sample`	positive	mean under the odds tilt 0.5 at horizon t = 2, repeated binary outcome: in sample, cell-mean outcome regressions with a backoff, main-terms quasibinomial treatment and censoring mechanisms, n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9420 to 0.9559, SE ratio 0.9727 to 1.0171	pass
`interval_calibration`	`ipsi_0.5_t2__noise_control`	control	mean under the odds tilt 0.5 at horizon t = 2, repeated binary outcome: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8177 to 0.8413, SE ratio 0.6835 to 0.7147	pass
`interval_calibration`	`ipsi_0.5_t2__shrunken_se_control`	control	mean under the odds tilt 0.5 at horizon t = 2, repeated binary outcome: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8202 to 0.8437, SE ratio 0.6805 to 0.7119	pass
`interval_calibration`	`ipsi_0.5_t3__continuous`	positive	mean under the odds tilt 0.5 at horizon t = 3, repeated binary outcome: the three-value outcome law, in sample, cell-mean outcome regressions with a backoff, main-terms quasibinomial mechanisms, outcome bounds (0, 1), n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9408 to 0.9548, SE ratio 0.9708 to 1.0150	pass
`interval_calibration`	`ipsi_0.5_t3__crossfit`	positive	mean under the odds tilt 0.5 at horizon t = 3, repeated binary outcome: five folds, cell-mean outcome regressions with a backoff, main-terms quasibinomial treatment and censoring mechanisms, n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9423 to 0.9562, SE ratio 0.9784 to 1.0232	pass
`interval_calibration`	`ipsi_0.5_t3__in_sample`	positive	mean under the odds tilt 0.5 at horizon t = 3, repeated binary outcome: in sample, cell-mean outcome regressions with a backoff, main-terms quasibinomial treatment and censoring mechanisms, n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9430 to 0.9567, SE ratio 0.9742 to 1.0179	pass
`interval_calibration`	`ipsi_0.5_t3__noise_control`	control	mean under the odds tilt 0.5 at horizon t = 3, repeated binary outcome: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8183 to 0.8419, SE ratio 0.6894 to 0.7209	pass
`interval_calibration`	`ipsi_0.5_t3__shrunken_se_control`	control	mean under the odds tilt 0.5 at horizon t = 3, repeated binary outcome: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8157 to 0.8394, SE ratio 0.6822 to 0.7118	pass
`interval_calibration`	`ipsi_0.5_vs_1.0_t1__in_sample`	positive	contrast of the odds tilt 0.5 against the natural course at horizon t = 1, repeated binary outcome: in sample, cell-mean outcome regressions with a backoff, main-terms quasibinomial treatment and censoring mechanisms, n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9480 to 0.9611, SE ratio 0.9966 to 1.0424	pass
`interval_calibration`	`ipsi_0.5_vs_1.0_t1__noise_control`	control	contrast of the odds tilt 0.5 against the natural course at horizon t = 1, repeated binary outcome: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8270 to 0.8501, SE ratio 0.7011 to 0.7323	pass
`interval_calibration`	`ipsi_0.5_vs_1.0_t1__shrunken_se_control`	control	contrast of the odds tilt 0.5 against the natural course at horizon t = 1, repeated binary outcome: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8269 to 0.8500, SE ratio 0.6976 to 0.7302	pass
`interval_calibration`	`ipsi_0.5_vs_1.0_t2__in_sample`	positive	contrast of the odds tilt 0.5 against the natural course at horizon t = 2, repeated binary outcome: in sample, cell-mean outcome regressions with a backoff, main-terms quasibinomial treatment and censoring mechanisms, n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9380 to 0.9523, SE ratio 0.9625 to 1.0050	pass
`interval_calibration`	`ipsi_0.5_vs_1.0_t2__noise_control`	control	contrast of the odds tilt 0.5 against the natural course at horizon t = 2, repeated binary outcome: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8198 to 0.8433, SE ratio 0.6950 to 0.7265	pass
`interval_calibration`	`ipsi_0.5_vs_1.0_t2__shrunken_se_control`	control	contrast of the odds tilt 0.5 against the natural course at horizon t = 2, repeated binary outcome: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8103 to 0.8343, SE ratio 0.6741 to 0.7038	pass
`interval_calibration`	`ipsi_0.5_vs_1.0_t3__continuous`	positive	contrast of the odds tilt 0.5 against the natural course at horizon t = 3, repeated binary outcome: the three-value outcome law, in sample, cell-mean outcome regressions with a backoff, main-terms quasibinomial mechanisms, outcome bounds (0, 1), n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9353 to 0.9499, SE ratio 0.9533 to 0.9958	pass
`interval_calibration`	`ipsi_0.5_vs_1.0_t3__crossfit`	positive	contrast of the odds tilt 0.5 against the natural course at horizon t = 3, repeated binary outcome: five folds, cell-mean outcome regressions with a backoff, main-terms quasibinomial treatment and censoring mechanisms, n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9425 to 0.9563, SE ratio 0.9804 to 1.0237	pass
`interval_calibration`	`ipsi_0.5_vs_1.0_t3__in_sample`	positive	contrast of the odds tilt 0.5 against the natural course at horizon t = 3, repeated binary outcome: in sample, cell-mean outcome regressions with a backoff, main-terms quasibinomial treatment and censoring mechanisms, n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9430 to 0.9567, SE ratio 0.9720 to 1.0155	pass
`interval_calibration`	`ipsi_0.5_vs_1.0_t3__noise_control`	control	contrast of the odds tilt 0.5 against the natural course at horizon t = 3, repeated binary outcome: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8219 to 0.8453, SE ratio 0.6883 to 0.7192	pass
`interval_calibration`	`ipsi_0.5_vs_1.0_t3__shrunken_se_control`	control	contrast of the odds tilt 0.5 against the natural course at horizon t = 3, repeated binary outcome: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8171 to 0.8407, SE ratio 0.6801 to 0.7113	pass
`interval_calibration`	`ipsi_1.0_t1__in_sample`	positive	mean under the natural course at horizon t = 1, repeated binary outcome: in sample, cell-mean outcome regressions with a backoff, main-terms quasibinomial treatment and censoring mechanisms, n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9442 to 0.9578, SE ratio 0.9801 to 1.0240	pass
`interval_calibration`	`ipsi_1.0_t1__noise_control`	control	mean under the natural course at horizon t = 1, repeated binary outcome: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8219 to 0.8453, SE ratio 0.6926 to 0.7240	pass
`interval_calibration`	`ipsi_1.0_t1__shrunken_se_control`	control	mean under the natural course at horizon t = 1, repeated binary outcome: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8180 to 0.8416, SE ratio 0.6866 to 0.7163	pass
`interval_calibration`	`ipsi_1.0_t2__in_sample`	positive	mean under the natural course at horizon t = 2, repeated binary outcome: in sample, cell-mean outcome regressions with a backoff, main-terms quasibinomial treatment and censoring mechanisms, n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9411 to 0.9551, SE ratio 0.9741 to 1.0177	pass
`interval_calibration`	`ipsi_1.0_t2__noise_control`	control	mean under the natural course at horizon t = 2, repeated binary outcome: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8246 to 0.8479, SE ratio 0.6973 to 0.7285	pass
`interval_calibration`	`ipsi_1.0_t2__shrunken_se_control`	control	mean under the natural course at horizon t = 2, repeated binary outcome: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8178 to 0.8414, SE ratio 0.6816 to 0.7122	pass
`interval_calibration`	`ipsi_1.0_t3__continuous`	positive	mean under the natural course at horizon t = 3, repeated binary outcome: the three-value outcome law, in sample, cell-mean outcome regressions with a backoff, main-terms quasibinomial mechanisms, outcome bounds (0, 1), n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9408 to 0.9548, SE ratio 0.9815 to 1.0263	pass
`interval_calibration`	`ipsi_1.0_t3__crossfit`	positive	mean under the natural course at horizon t = 3, repeated binary outcome: five folds, cell-mean outcome regressions with a backoff, main-terms quasibinomial treatment and censoring mechanisms, n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9440 to 0.9577, SE ratio 0.9834 to 1.0271	pass
`interval_calibration`	`ipsi_1.0_t3__in_sample`	positive	mean under the natural course at horizon t = 3, repeated binary outcome: in sample, cell-mean outcome regressions with a backoff, main-terms quasibinomial treatment and censoring mechanisms, n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9423 to 0.9562, SE ratio 0.9765 to 1.0197	pass
`interval_calibration`	`ipsi_1.0_t3__noise_control`	control	mean under the natural course at horizon t = 3, repeated binary outcome: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8310 to 0.8539, SE ratio 0.6994 to 0.7311	pass
`interval_calibration`	`ipsi_1.0_t3__shrunken_se_control`	control	mean under the natural course at horizon t = 3, repeated binary outcome: 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 0.8376, SE ratio 0.6837 to 0.7144	pass
`interval_calibration`	`ipsi_2.0_t1__in_sample`	positive	mean under the odds tilt 2 at horizon t = 1, repeated binary outcome: in sample, cell-mean outcome regressions with a backoff, main-terms quasibinomial treatment and censoring mechanisms, n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9445 to 0.9581, SE ratio 0.9817 to 1.0254	pass
`interval_calibration`	`ipsi_2.0_t1__noise_control`	control	mean under the odds tilt 2 at horizon t = 1, repeated binary outcome: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8222 to 0.8456, SE ratio 0.6927 to 0.7235	pass
`interval_calibration`	`ipsi_2.0_t1__shrunken_se_control`	control	mean under the odds tilt 2 at horizon t = 1, repeated binary outcome: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8187 to 0.8423, SE ratio 0.6873 to 0.7178	pass
`interval_calibration`	`ipsi_2.0_t2__in_sample`	positive	mean under the odds tilt 2 at horizon t = 2, repeated binary outcome: in sample, cell-mean outcome regressions with a backoff, main-terms quasibinomial treatment and censoring mechanisms, n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9392 to 0.9534, SE ratio 0.9700 to 1.0119	pass
`interval_calibration`	`ipsi_2.0_t2__noise_control`	control	mean under the odds tilt 2 at horizon t = 2, repeated binary outcome: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8181 to 0.8417, SE ratio 0.6854 to 0.7159	pass
`interval_calibration`	`ipsi_2.0_t2__shrunken_se_control`	control	mean under the odds tilt 2 at horizon t = 2, repeated binary outcome: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8112 to 0.8351, SE ratio 0.6788 to 0.7093	pass
`interval_calibration`	`ipsi_2.0_t3__continuous`	positive	mean under the odds tilt 2 at horizon t = 3, repeated binary outcome: the three-value outcome law, in sample, cell-mean outcome regressions with a backoff, main-terms quasibinomial mechanisms, outcome bounds (0, 1), n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9439 to 0.9575, SE ratio 0.9760 to 1.0192	pass
`interval_calibration`	`ipsi_2.0_t3__crossfit`	positive	mean under the odds tilt 2 at horizon t = 3, repeated binary outcome: five folds, cell-mean outcome regressions with a backoff, main-terms quasibinomial treatment and censoring mechanisms, n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9453 to 0.9588, SE ratio 0.9889 to 1.0324	pass
`interval_calibration`	`ipsi_2.0_t3__in_sample`	positive	mean under the odds tilt 2 at horizon t = 3, repeated binary outcome: in sample, cell-mean outcome regressions with a backoff, main-terms quasibinomial treatment and censoring mechanisms, n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9423 to 0.9562, SE ratio 0.9716 to 1.0134	pass
`interval_calibration`	`ipsi_2.0_t3__noise_control`	control	mean under the odds tilt 2 at horizon t = 3, repeated binary outcome: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8193 to 0.8429, SE ratio 0.6821 to 0.7129	pass
`interval_calibration`	`ipsi_2.0_t3__shrunken_se_control`	control	mean under the odds tilt 2 at horizon t = 3, repeated binary outcome: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8125 to 0.8364, SE ratio 0.6795 to 0.7097	pass
`interval_calibration`	`ipsi_2.0_vs_1.0_t1__in_sample`	positive	contrast of the odds tilt 2.0 against the natural course at horizon t = 1, repeated binary outcome: in sample, cell-mean outcome regressions with a backoff, main-terms quasibinomial treatment and censoring mechanisms, n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9473 to 0.9605, SE ratio 0.9984 to 1.0440	pass
`interval_calibration`	`ipsi_2.0_vs_1.0_t1__noise_control`	control	contrast of the odds tilt 2.0 against the natural course at horizon t = 1, repeated binary outcome: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8334 to 0.8561, SE ratio 0.7013 to 0.7326	pass
`interval_calibration`	`ipsi_2.0_vs_1.0_t1__shrunken_se_control`	control	contrast of the odds tilt 2.0 against the natural course at horizon t = 1, repeated binary outcome: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8310 to 0.8539, SE ratio 0.6985 to 0.7309	pass
`interval_calibration`	`ipsi_2.0_vs_1.0_t2__in_sample`	positive	contrast of the odds tilt 2.0 against the natural course at horizon t = 2, repeated binary outcome: in sample, cell-mean outcome regressions with a backoff, main-terms quasibinomial treatment and censoring mechanisms, n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9456 to 0.9590, SE ratio 0.9788 to 1.0223	pass
`interval_calibration`	`ipsi_2.0_vs_1.0_t2__noise_control`	control	contrast of the odds tilt 2.0 against the natural course at horizon t = 2, repeated binary outcome: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8216 to 0.8450, SE ratio 0.6844 to 0.7148	pass
`interval_calibration`	`ipsi_2.0_vs_1.0_t2__shrunken_se_control`	control	contrast of the odds tilt 2.0 against the natural course at horizon t = 2, repeated binary outcome: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8160 to 0.8397, SE ratio 0.6849 to 0.7159	pass
`interval_calibration`	`ipsi_2.0_vs_1.0_t3__continuous`	positive	contrast of the odds tilt 2.0 against the natural course at horizon t = 3, repeated binary outcome: the three-value outcome law, in sample, cell-mean outcome regressions with a backoff, main-terms quasibinomial mechanisms, outcome bounds (0, 1), n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9412 to 0.9552, SE ratio 0.9664 to 1.0099	pass
`interval_calibration`	`ipsi_2.0_vs_1.0_t3__crossfit`	positive	contrast of the odds tilt 2.0 against the natural course at horizon t = 3, repeated binary outcome: five folds, cell-mean outcome regressions with a backoff, main-terms quasibinomial treatment and censoring mechanisms, n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9420 to 0.9559, SE ratio 0.9778 to 1.0228	pass
`interval_calibration`	`ipsi_2.0_vs_1.0_t3__in_sample`	positive	contrast of the odds tilt 2.0 against the natural course at horizon t = 3, repeated binary outcome: in sample, cell-mean outcome regressions with a backoff, main-terms quasibinomial treatment and censoring mechanisms, n = 8,000	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9466 to 0.9599, SE ratio 0.9759 to 1.0201	pass
`interval_calibration`	`ipsi_2.0_vs_1.0_t3__noise_control`	control	contrast of the odds tilt 2.0 against the natural course at horizon t = 3, repeated binary outcome: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8211 to 0.8446, SE ratio 0.6837 to 0.7148	pass
`interval_calibration`	`ipsi_2.0_vs_1.0_t3__shrunken_se_control`	control	contrast of the odds tilt 2.0 against the natural course at horizon t = 3, repeated binary outcome: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8149 to 0.8387, SE ratio 0.6832 to 0.7139	pass
`outcome_history_necessity`	`ipsi_0.5_t3__history_free`	control	mean under the odds tilt 0.5 at horizon t = 3, repeated binary outcome: an end-of-study fit with no earlier outcome in any history, cell-mean outcome regressions and mechanisms on the coarse key	bias interval outside the margin, and inside it around the coarse tilt functional	bias -0.0122 to -0.0110, margin 0.0016, bias around its limit -0.000702 to 0.000501, margin 0.0016	pass
`outcome_history_necessity`	`ipsi_0.5_t3__repeated`	positive	mean under the odds tilt 0.5 at horizon t = 3, repeated binary outcome: the repeated-outcome fit, correctly specified mechanisms, cell-mean outcome regressions	bias interval inside the equivalence margin	bias -0.000655 to 0.000552, margin 0.0017	pass
`outcome_history_necessity`	`ipsi_2.0_t3__history_free`	control	mean under the odds tilt 2 at horizon t = 3, repeated binary outcome: an end-of-study fit with no earlier outcome in any history, cell-mean outcome regressions and mechanisms on the coarse key	bias interval outside the margin, and inside it around the coarse tilt functional	bias 0.0045 to 0.0056, margin 0.0014, bias around its limit -0.000420 to 0.000627, margin 0.0014	pass
`outcome_history_necessity`	`ipsi_2.0_t3__repeated`	positive	mean under the odds tilt 2 at horizon t = 3, repeated binary outcome: the repeated-outcome fit, correctly specified mechanisms, cell-mean outcome regressions	bias interval inside the equivalence margin	bias -0.000464 to 0.000557, margin 0.0014	pass
`repeated_outcome_necessity`	`ipsi_0.5_t2__absorbing_reading`	control	mean under the odds tilt 0.5 at horizon t = 2, repeated binary outcome: an end-of-study fit of the maximum of the outcomes through the horizon, with the earlier outcomes in the history	bias interval outside the margin, and inside it around the absorbing tilt functional	bias 0.1259 to 0.1269, margin 0.0015, bias around its limit -0.000680 to 0.000330, margin 0.0015	pass
`repeated_outcome_necessity`	`ipsi_0.5_t2__repeated`	positive	mean under the odds tilt 0.5 at horizon t = 2, repeated binary outcome: the repeated-outcome fit, correctly specified mechanisms, cell-mean outcome regressions	bias interval inside the equivalence margin	bias -0.000668 to 0.000250, margin 0.0014	pass
`repeated_outcome_necessity`	`ipsi_0.5_t3__absorbing_reading`	control	mean under the odds tilt 0.5 at horizon t = 3, repeated binary outcome: an end-of-study fit of the maximum of the outcomes through the horizon, with the earlier outcomes in the history	bias interval outside the margin, and inside it around the absorbing tilt functional	bias 0.1932 to 0.1943, margin 0.0017, bias around its limit -0.000644 to 0.000457, margin 0.0017	pass
`repeated_outcome_necessity`	`ipsi_0.5_t3__repeated`	positive	mean under the odds tilt 0.5 at horizon t = 3, repeated binary outcome: the repeated-outcome fit, correctly specified mechanisms, cell-mean outcome regressions	bias interval inside the equivalence margin	bias -0.000693 to 0.000369, margin 0.0016	pass
`repeated_outcome_necessity`	`ipsi_1.0_t2__absorbing_reading`	control	mean under the natural course at horizon t = 2, repeated binary outcome: an end-of-study fit of the maximum of the outcomes through the horizon, with the earlier outcomes in the history	bias interval outside the margin, and inside it around the absorbing tilt functional	bias 0.1213 to 0.1222, margin 0.0014, bias around its limit -0.000612 to 0.000278, margin 0.0014	pass
`repeated_outcome_necessity`	`ipsi_1.0_t2__repeated`	positive	mean under the natural course at horizon t = 2, repeated binary outcome: the repeated-outcome fit, correctly specified mechanisms, cell-mean outcome regressions	bias interval inside the equivalence margin	bias -0.000552 to 0.000211, margin 0.0012	pass
`repeated_outcome_necessity`	`ipsi_1.0_t3__absorbing_reading`	control	mean under the natural course at horizon t = 3, repeated binary outcome: an end-of-study fit of the maximum of the outcomes through the horizon, with the earlier outcomes in the history	bias interval outside the margin, and inside it around the absorbing tilt functional	bias 0.1916 to 0.1925, margin 0.0015, bias around its limit -0.000540 to 0.000417, margin 0.0015	pass
`repeated_outcome_necessity`	`ipsi_1.0_t3__repeated`	positive	mean under the natural course at horizon t = 3, repeated binary outcome: the repeated-outcome fit, correctly specified mechanisms, cell-mean outcome regressions	bias interval inside the equivalence margin	bias -0.000423 to 0.000444, margin 0.0013	pass
`repeated_outcome_necessity`	`ipsi_2.0_t2__absorbing_reading`	control	mean under the odds tilt 2 at horizon t = 2, repeated binary outcome: an end-of-study fit of the maximum of the outcomes through the horizon, with the earlier outcomes in the history	bias interval outside the margin, and inside it around the absorbing tilt functional	bias 0.1178 to 0.1187, margin 0.0014, bias around its limit -0.000620 to 0.000284, margin 0.0014	pass
`repeated_outcome_necessity`	`ipsi_2.0_t2__repeated`	positive	mean under the odds tilt 2 at horizon t = 2, repeated binary outcome: the repeated-outcome fit, correctly specified mechanisms, cell-mean outcome regressions	bias interval inside the equivalence margin	bias -0.000525 to 0.000221, margin 0.0011	pass
`repeated_outcome_necessity`	`ipsi_2.0_t3__absorbing_reading`	control	mean under the odds tilt 2 at horizon t = 3, repeated binary outcome: an end-of-study fit of the maximum of the outcomes through the horizon, with the earlier outcomes in the history	bias interval outside the margin, and inside it around the absorbing tilt functional	bias 0.1862 to 0.1872, margin 0.0016, bias around its limit -0.000517 to 0.000505, margin 0.0016	pass
`repeated_outcome_necessity`	`ipsi_2.0_t3__repeated`	positive	mean under the odds tilt 2 at horizon t = 3, repeated binary outcome: the repeated-outcome fit, correctly specified mechanisms, cell-mean outcome regressions	bias interval inside the equivalence margin	bias -0.000233 to 0.000659, margin 0.0014	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 0.6288 to 0.6589	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 0.9415 to 0.9555	pass
`single_robustness`	`ipsi_0.5_t3__one_step_necessity`	control	mean under the odds tilt 0.5 at horizon t = 3, repeated binary outcome: the tilt plug-in with the same wrong regression and the same fitted treatment mechanism, written without `cleverly`	bias interval outside the margin, and inside it around the plug-in’s exact limit	bias -0.0291 to -0.0270, margin 0.0029, coverage 0.4350	pass
`single_robustness`	`ipsi_0.5_t3__w_only_m`	positive	mean under the odds tilt 0.5 at horizon t = 3, repeated binary outcome: the outcome regression reads the baseline covariate and the current arm alone, which is wrong for the history, with the correctly specified mechanisms, n = 2,000	bias interval inside the equivalence margin	bias -0.0012 to 0.0012, margin 0.0033, coverage 0.9600	pass
`single_robustness`	`ipsi_0.5_vs_1.0_t3__one_step_necessity`	control	contrast of the odds tilt 0.5 against the natural course at horizon t = 3, repeated binary outcome: the tilt plug-in with the same wrong regression and the same fitted treatment mechanism, written without `cleverly`	bias interval outside the margin, and inside it around the plug-in’s exact limit	bias -0.0263 to -0.0257, margin 0.000847, coverage 0.0000	pass
`single_robustness`	`ipsi_0.5_vs_1.0_t3__w_only_m`	positive	contrast of the odds tilt 0.5 against the natural course at horizon t = 3, repeated binary outcome: the outcome regression reads the baseline covariate and the current arm alone, which is wrong for the history, with the correctly specified mechanisms, n = 2,000	bias interval inside the equivalence margin	bias -0.000858 to 0.000236, margin 0.0015, coverage 0.9625	pass
`single_robustness`	`ipsi_2.0_t3__one_step_necessity`	control	mean under the odds tilt 2 at horizon t = 3, repeated binary outcome: the tilt plug-in with the same wrong regression and the same fitted treatment mechanism, written without `cleverly`	bias interval outside the margin, and inside it around the plug-in’s exact limit	bias 0.0208 to 0.0228, margin 0.0028, coverage 0.5125	pass
`single_robustness`	`ipsi_2.0_t3__w_only_m`	positive	mean under the odds tilt 2 at horizon t = 3, repeated binary outcome: the outcome regression reads the baseline covariate and the current arm alone, which is wrong for the history, with the correctly specified mechanisms, n = 2,000	bias interval inside the equivalence margin	bias -0.000696 to 0.0013, margin 0.0028, coverage 0.9537	pass
`single_robustness`	`ipsi_2.0_vs_1.0_t3__one_step_necessity`	control	contrast of the odds tilt 2.0 against the natural course at horizon t = 3, repeated binary outcome: the tilt plug-in with the same wrong regression and the same fitted treatment mechanism, written without `cleverly`	bias interval outside the margin, and inside it around the plug-in’s exact limit	bias 0.0235 to 0.0242, margin 0.000948, coverage 0.0025	pass
`single_robustness`	`ipsi_2.0_vs_1.0_t3__w_only_m`	positive	contrast of the odds tilt 2.0 against the natural course at horizon t = 3, repeated binary outcome: the outcome regression reads the baseline covariate and the current arm alone, which is wrong for the history, with the correctly specified mechanisms, n = 2,000	bias interval inside the equivalence margin	bias -0.000502 to 0.000530, margin 0.0014, coverage 0.9600	pass
`small_sample_calibration`	`ipsi_0.5_t3__continuous`	diagnostic	mean under the odds tilt 0.5 at horizon t = 3, repeated binary outcome: the continuous calibration arm at n = 2,000	reported, no verdict	SE ratio 0.9169 to 0.9920	reported
`small_sample_calibration`	`ipsi_0.5_t3__in_sample`	diagnostic	mean under the odds tilt 0.5 at horizon t = 3, repeated binary outcome: the in-sample binary calibration arm at n = 2,000	reported, no verdict	SE ratio 0.9462 to 1.0256	reported
`small_sample_calibration`	`ipsi_0.5_vs_1.0_t3__continuous`	diagnostic	contrast of the odds tilt 0.5 against the natural course at horizon t = 3, repeated binary outcome: the continuous calibration arm at n = 2,000	reported, no verdict	SE ratio 0.8884 to 0.9612	reported
`small_sample_calibration`	`ipsi_0.5_vs_1.0_t3__in_sample`	diagnostic	contrast of the odds tilt 0.5 against the natural course at horizon t = 3, repeated binary outcome: the in-sample binary calibration arm at n = 2,000	reported, no verdict	SE ratio 0.9163 to 0.9898	reported
`small_sample_calibration`	`ipsi_1.0_t3__continuous`	diagnostic	mean under the natural course at horizon t = 3, repeated binary outcome: the continuous calibration arm at n = 2,000	reported, no verdict	SE ratio 0.9454 to 1.0250	reported
`small_sample_calibration`	`ipsi_1.0_t3__in_sample`	diagnostic	mean under the natural course at horizon t = 3, repeated binary outcome: the in-sample binary calibration arm at n = 2,000	reported, no verdict	SE ratio 0.9738 to 1.0568	reported
`small_sample_calibration`	`ipsi_2.0_t3__continuous`	diagnostic	mean under the odds tilt 2 at horizon t = 3, repeated binary outcome: the continuous calibration arm at n = 2,000	reported, no verdict	SE ratio 0.9388 to 1.0187	reported
`small_sample_calibration`	`ipsi_2.0_t3__in_sample`	diagnostic	mean under the odds tilt 2 at horizon t = 3, repeated binary outcome: the in-sample binary calibration arm at n = 2,000	reported, no verdict	SE ratio 0.9589 to 1.0406	reported
`small_sample_calibration`	`ipsi_2.0_vs_1.0_t3__continuous`	diagnostic	contrast of the odds tilt 2.0 against the natural course at horizon t = 3, repeated binary outcome: the continuous calibration arm at n = 2,000	reported, no verdict	SE ratio 0.9086 to 0.9863	reported
`small_sample_calibration`	`ipsi_2.0_vs_1.0_t3__in_sample`	diagnostic	contrast of the odds tilt 2.0 against the natural course at horizon t = 3, repeated binary outcome: the in-sample binary calibration arm at n = 2,000	reported, no verdict	SE ratio 0.9179 to 0.9956	reported

The property families draw from their own streams of make_longitudinal_repeated. The continuous arm draws the three-value law CONTINUOUS. The table above lists every cell. This section states what each family tests.

family	what it tests	size
`interval_calibration`	pointwise coverage and SE ratio inside the calibration bands. The cells are in sample for all 15 parameters, cross-fitted at five folds and continuous for the five parameters at \(t = 3\). A shrunken-SE control and a noise control read each in-sample cell	\(n\) = 8,000, 6,800 replications
`simultaneous_coverage`	the default band over all 15 reported parameters, against its pointwise control	\(n\) = 8,000, 6,800 replications
`booster_calibration`	the cross-fitted fit with a shallow regularized booster for both outcome learners. The role is `diagnostic`. The table reports these cells and no rule gates them	\(n\) = 8,000, 2,000 replications
`small_sample_calibration`	the in-sample and continuous arms refitted at \(n\) = 2,000. The role is `diagnostic`	\(n\) = 2,000, 2,000 replications
`single_robustness`	bias inside the margin when the outcome regression reads \(W\) and the current arm alone and the mechanisms are correct. The control is the tilt plug-in with the same regression and propensity. It must miss the truth and reach its exact limit	\(n\) = 2,000, 800 replications
`repeated_outcome_necessity`	an absorbing reading of the outcome, which fits \(\max_{s \le h} Y_s\) with the earlier outcomes in the history. The control must miss the truth and reach the absorbing tilt functional	\(n\) = 8,000, 1,000 replications
`outcome_history_necessity`	a fit that leaves every \(Y_s\) out of every history. Every nuisance is the coarse projection. The control must miss the truth and reach the coarse tilt functional	\(n\) = 8,000, 800 replications

There is no double-robustness family. IPSI is singly robust. It needs the treatment and censoring mechanisms. A wrong treatment mechanism with a correct outcome regression leaves a bias (witness E7).

In sample, the cell-means regression reproduces every seen cell. The outcome-residual correction of the one-step estimator is then zero on seen cells. The in-sample cells test the plug-in and the mechanism terms. Only the cross-fitted cells and single_robustness exercise the residual term with a nonzero residual. The tests E5, E6, N9 and W1 cover that term exactly.

Sizing#

Every size is a rule of tests/studies/repeated_sizing.py applied to a pilot table in tests/studies/ipsi_repeated_properties.py. The pilots draw from streams that no published cell reads. The pilot tables record the commands in tests/canonical/ipsi_repeated/README.md.

pilot	draws	sets
P4 primary, two scenarios	2,000 each	the primary \(n\) and the primary replications
P5 calibration arms	8,000 each	the calibration replications
P6 every property family	2,000 each	the size of each family and the \(n\) of the two necessity families

The study fixed every size, replication count, margin, law, learner, fold count and seed before the run.

Measured values and declared margins#

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`	2400	primary replications per scenario
`n`	8000	observations per primary replication
`independent_tests_total`	30	implementation-estimand tests against the truth
`independent_tests_passed`	30	of those, passing
`property_cells_total`	81	repeated-sampling property cells, reported cells excluded
`property_cells_passed`	81	cells whose own and family verdicts pass
`property_cells_reported`	15	cells reported and not gated, as the `small_sample_calibration` diagnostics
`max_standardized_bias`	0.0625	largest absolute primary bias in empirical standard deviations
`min_coverage`	0.9358	lowest measured primary-study coverage
`min_se_ratio_ci_lower`	0.9393	lowest bootstrap primary SE-ratio endpoint
`max_se_ratio_ci_upper`	1.0769	highest bootstrap primary SE-ratio endpoint
`margin:alpha`	0.0500	nominal size of the reported intervals, and the family level of the rule
`margin:bootstrap_replicates`	10000	resamples behind every bootstrap interval
`margin:calibration_coverage_lower`	0.9200	calibration-cell coverage band, lower limit
`margin:calibration_coverage_upper`	0.9800	calibration-cell coverage band, upper limit
`margin:calibration_noninferiority`	0.0500	largest external-comparison calibration excess bound
`margin:calibration_se_ratio_lower`	0.9300	calibration-cell SE-ratio band, lower limit
`margin:calibration_se_ratio_upper`	1.0700	calibration-cell SE-ratio band, upper limit
`margin:confidence_level`	0.9900	confidence level of every Monte Carlo interval
`margin:coverage_floor`	0.9000	validity floor the coverage lower endpoint must clear
`margin:coverage_noninferiority`	-0.0250	smallest external-comparison coverage difference bound
`margin:excluded_slope`	-0.2500	slower rate a root-n interval must exclude. No cell of this study reads it
`margin:minimum_power`	0.8000	rejection lower bound a power cell must clear. No cell of this study reads it
`margin:nominal_coverage`	0.9500	nominal coverage those intervals claim
`margin:over_coverage_ceiling`	0.9900	above this, coverage is conservative rather than invalid
`margin:paired_difference`	0.1500	paired similarity margin, in pooled empirical standard deviations
`margin:rmse_noninferiority`	1.1000	largest external-comparison RMSE ratio bound
`margin:root_n_slope`	-0.5000	contraction rate root-n asymptotics predict. No cell of this study reads it
`margin:root_n_slope_lower`	-0.6250	accepted root-n slope band, lower limit. No cell of this study reads it
`margin:root_n_slope_upper`	-0.3750	accepted root-n slope band, upper limit. No cell of this study reads it
`margin:se_ratio_sanity_lower`	0.8000	SE-ratio screen, lower limit
`margin:se_ratio_sanity_upper`	1.2000	SE-ratio screen, upper limit
`margin:standardized_bias`	0.2500	bias equivalence margin, in empirical standard deviations
`margin:type_i_ceiling`	0.1000	the rate a positive cell must bound and a control must exceed. No cell of this study reads it
`margin:shrunken_se_factor`	0.7000	factor the shrunken-SE control multiplies the standard error by
`margin:necessity_displacement`	0.2500	smallest paired displacement of a necessity control, in empirical standard deviations
`margin:subject_failure_bound`	0.0100	declared share of replications a cell may lose to a recordable fitting failure
`margin:subject_failures_allowed`	24	the same bound, in primary replications

Limits#

limit	what it means for use
one law with three nodes and a binary outcome	the continuous outcome has one calibration arm, and the rate in \(n\) is not measured
the study has no comparator	the claim rests on exact truths, exact-law witnesses and the controls. No other implementation fits this parameter
the in-sample interval runs below nominal at \(n\) = 2,000 for the continuous arm	the `small_sample_calibration` cells report the shortfall and no rule gates them. The gated calibration cells draw \(n\) = 8,000
cell-mean outcome regressions and one booster	the study does not cover a larger learner library or another fold count than five
the mechanisms are correctly specified	the study covers no wrong treatment or censoring mechanism. `IPSI` is singly robust, and witness `E7` shows the bias
the band covers the declared tilts and horizons	the study does not test a band over values of \(\delta\) between the declared ones
the absorbing and history-free controls read exact limits of this law	they show that the repeated reading and the earlier outcomes matter here. They do not bound the difference on another law
the second-order dependence acts at node 3 only	the law has three nodes, so the depth-two terms act at the last node. No cell reads a longer history
no weights, clusters, working model or survival outcome in this study	the dropout study measures them for the end-of-study and survival outcomes

The causal interpretation requires consistency, sequential exchangeability, longitudinal positivity, and conditionally independent censoring.

Reproduction#

The fixture README gives the regeneration and pilot commands. The manifest records the seeds, the pinned learners and the artifact hashes.