Categorical incremental policies#

This study validates the categorical odds tilt of Incremental({level: factor}). Family P fits point TMLE on a three-level treatment. Family L fits IPSI with two three-level tilt nodes. The point estimator and the longitudinal estimator state the construction, its remainder and its exact-law evidence.

No maintained package ships this construction with its influence curve. npcausal::ipsi fits two arms only. imtp 0.1.0 omits the derivative of the tilted propensity with respect to the mechanism. So no canonical implementation is compared. The study rests on exact truths and on repeated-sampling properties. It is Python property evidence.

What was fitted#

scenario	law and estimands	folds
`point_in_sample`	law P: four baseline values, three treatment levels, a continuous outcome. The three tilted means and the two contrasts with the natural course	one
`point_crossfit`	law P, the same estimands	five
`longitudinal_dropout_folds_5`	law L: a binary baseline covariate, a three-level treatment, dropout, a binary time-varying covariate, a second three-level treatment, dropout and a continuous outcome. The three plan means and the two contrasts	five
`longitudinal_folds_5`	law L0: law L without dropout, the same estimands	five

Plan T_a multiplies the odds of the second level by 2 and of the third by 0.5, against the first. Plan T_b multiplies them by 0.5 and 2. The natural course uses factor 1 at every level. Family L applies each plan at both treatment nodes. Every nuisance is a saturated cell frequency or a cell mean. The laws are exact finite sums in tests/discrete_law_categorical_ipsi_point.py and tests/discrete_law_categorical_ipsi_longitudinal.py.

law	key numbers
P	treatment propensities from 0.10 to 0.70. The contrast `T_a` has truth -0.8124 and efficient influence-curve SD 0.9468. The treatment-mechanism term holds 0.62 of the curve variance
P0	law P with the outcome mean replaced by its row mean (sharp null). Every tilted mean is equal
Pw	law P with the outcome contrast shrunk to 0.033 of its size. The exact power is 0.8724 at n = 4,000
L	propensities from 0.070 to 0.774 and retention 0.833. The contrast `T_a` has truth 0.9267 and SD 1.0172. With the treatment conditionals frozen at the truth the curve SD falls to 0.808 of this value
L0	law L without dropout, with the same truths and SD 0.9597

Two routes give each truth to \(10^{-12}\). The longitudinal second route is a g-formula written in loops from the conditional tables.

Accuracy against known truth#

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
two treatment nodes with three levels each, dropout after each node, five folds	`ate_regimen[T_a vs natural course]`	difference in mean outcome between the plans “multiply the odds of the second level by 2 and of the third by 0.5, against the first” against “leave the observed treatment mechanism unchanged”	`cleverly` TMLE and `IPSI` with a categorical odds tilt	-0.000580 to 0.0012	0.9610	1.0253	pass
two treatment nodes with three levels each, dropout after each node, five folds	`ate_regimen[T_b vs natural course]`	difference in mean outcome between the plans “multiply the odds of the second level by 0.5 and of the third by 2, against the first” against “leave the observed treatment mechanism unchanged”	`cleverly` TMLE and `IPSI` with a categorical odds tilt	-0.0013 to 0.000838	0.9580	1.0016	pass
two treatment nodes with three levels each, dropout after each node, five folds	`ey_regimen[T_a]`	mean outcome under the plan multiply the odds of the second level by 2 and of the third by 0.5, against the first	`cleverly` TMLE and `IPSI` with a categorical odds tilt	-0.0019 to 0.0020	0.9390	0.9633	pass
two treatment nodes with three levels each, dropout after each node, five folds	`ey_regimen[T_b]`	mean outcome under the plan multiply the odds of the second level by 0.5 and of the third by 2, against the first	`cleverly` TMLE and `IPSI` with a categorical odds tilt	-0.0023 to 0.0012	0.9390	0.9710	pass
two treatment nodes with three levels each, dropout after each node, five folds	`ey_regimen[natural course]`	mean outcome under the plan leave the observed treatment mechanism unchanged	`cleverly` TMLE and `IPSI` with a categorical odds tilt	-0.0021 to 0.0016	0.9350	0.9486	pass
two treatment nodes with three levels each, no dropout, five folds	`ate_regimen[T_a vs natural course]`	difference in mean outcome between the plans “multiply the odds of the second level by 2 and of the third by 0.5, against the first” against “leave the observed treatment mechanism unchanged”	`cleverly` TMLE and `IPSI` with a categorical odds tilt	0.000655 to 0.0024	0.9460	0.9962	pass
two treatment nodes with three levels each, no dropout, five folds	`ate_regimen[T_b vs natural course]`	difference in mean outcome between the plans “multiply the odds of the second level by 0.5 and of the third by 2, against the first” against “leave the observed treatment mechanism unchanged”	`cleverly` TMLE and `IPSI` with a categorical odds tilt	-0.0025 to -0.000496	0.9480	1.0017	pass
two treatment nodes with three levels each, no dropout, five folds	`ey_regimen[T_a]`	mean outcome under the plan multiply the odds of the second level by 2 and of the third by 0.5, against the first	`cleverly` TMLE and `IPSI` with a categorical odds tilt	-0.000922 to 0.0028	0.9480	0.9758	pass
two treatment nodes with three levels each, no dropout, five folds	`ey_regimen[T_b]`	mean outcome under the plan multiply the odds of the second level by 0.5 and of the third by 2, against the first	`cleverly` TMLE and `IPSI` with a categorical odds tilt	-0.0036 to -0.000559	0.9590	1.0288	pass
two treatment nodes with three levels each, no dropout, five folds	`ey_regimen[natural course]`	mean outcome under the plan leave the observed treatment mechanism unchanged	`cleverly` TMLE and `IPSI` with a categorical odds tilt	-0.0023 to 0.0011	0.9470	0.9924	pass
three-level treatment, four baseline values, five folds	`ate_ipsi[T_a vs natural course]`	difference in means under the incremental interventions “multiply the odds of the second level by 2 and of the third by 0.5, against the first” against “leave the observed treatment mechanism unchanged”	`cleverly` TMLE and `IPSI` with a categorical odds tilt	-0.0014 to 0.000299	0.9480	1.0145	pass
three-level treatment, four baseline values, five folds	`ate_ipsi[T_b vs natural course]`	difference in means under the incremental interventions “multiply the odds of the second level by 0.5 and of the third by 2, against the first” against “leave the observed treatment mechanism unchanged”	`cleverly` TMLE and `IPSI` with a categorical odds tilt	-0.000403 to 0.0014	0.9500	1.0068	pass
three-level treatment, four baseline values, five folds	`ey_ipsi[T_a]`	mean under the incremental intervention multiply the odds of the second level by 2 and of the third by 0.5, against the first	`cleverly` TMLE and `IPSI` with a categorical odds tilt	-0.0023 to 0.0019	0.9410	0.9898	pass
three-level treatment, four baseline values, five folds	`ey_ipsi[T_b]`	mean under the incremental intervention multiply the odds of the second level by 0.5 and of the third by 2, against the first	`cleverly` TMLE and `IPSI` with a categorical odds tilt	-0.0011 to 0.0028	0.9490	0.9964	pass
three-level treatment, four baseline values, five folds	`ey_ipsi[natural course]`	mean under the incremental intervention leave the observed treatment mechanism unchanged	`cleverly` TMLE and `IPSI` with a categorical odds tilt	-0.0017 to 0.0024	0.9430	0.9903	pass
three-level treatment, four baseline values, nuisances fitted in sample	`ate_ipsi[T_a vs natural course]`	difference in means under the incremental interventions “multiply the odds of the second level by 2 and of the third by 0.5, against the first” against “leave the observed treatment mechanism unchanged”	`cleverly` TMLE and `IPSI` with a categorical odds tilt	-0.000546 to 0.0012	0.9460	1.0124	pass
three-level treatment, four baseline values, nuisances fitted in sample	`ate_ipsi[T_b vs natural course]`	difference in means under the incremental interventions “multiply the odds of the second level by 0.5 and of the third by 2, against the first” against “leave the observed treatment mechanism unchanged”	`cleverly` TMLE and `IPSI` with a categorical odds tilt	-0.0013 to 0.000552	0.9570	1.0173	pass
three-level treatment, four baseline values, nuisances fitted in sample	`ey_ipsi[T_a]`	mean under the incremental intervention multiply the odds of the second level by 2 and of the third by 0.5, against the first	`cleverly` TMLE and `IPSI` with a categorical odds tilt	-0.0014 to 0.0026	0.9530	1.0295	pass
three-level treatment, four baseline values, nuisances fitted in sample	`ey_ipsi[T_b]`	mean under the incremental intervention multiply the odds of the second level by 0.5 and of the third by 2, against the first	`cleverly` TMLE and `IPSI` with a categorical odds tilt	-0.0019 to 0.0018	0.9530	1.0316	pass
three-level treatment, four baseline values, nuisances fitted in sample	`ey_ipsi[natural course]`	mean under the incremental intervention leave the observed treatment mechanism unchanged	`cleverly` TMLE and `IPSI` with a categorical odds tilt	-0.0016 to 0.0023	0.9510	1.0331	pass

Theory properties#

property	cell	role	what was tested	what must hold	measured	result
`censoring_complete_case`	`longitudinal_mean_t_a__complete_case`	diagnostic	two-node law L with dropout, mean under T_a: a fit to the rows that never drop out, without a censoring model, beside the exact-law limit	reported only	bias 0.0977 to 0.0996, margin 0.0042, coverage 0.0000	reported
`in_sample_fit`	`longitudinal_dropout_contrast_t_a__one_fold`	diagnostic	two-node law L with dropout and three-level treatment, T_a against the natural course: the same histories fitted with one fold, beside the exact-law truth	reported only	bias -0.000825 to -0.000110, margin 0.0021, coverage 0.9397	reported
`interval_calibration`	`longitudinal_contrast_t_a__correctly_specified`	positive	two-node law L0 without dropout and three-level treatment, T_a against the natural course: 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 0.9614, SE ratio 0.9819 to 1.0431, empirical efficiency ratio 0.9597 to 1.0194, reported efficiency ratio 1.0006 to 1.0014	pass
`interval_calibration`	`longitudinal_contrast_t_a__noise_control`	control	two-node law L0 without dropout and three-level treatment, T_a against the natural course: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8096 to 0.8423, SE ratio 0.6870 to 0.7287, empirical efficiency ratio 1.3736 to 1.4569, reported efficiency ratio 1.0006 to 1.0014	pass
`interval_calibration`	`longitudinal_contrast_t_a__shrunken_se_control`	control	two-node law L0 without dropout and three-level treatment, T_a against the natural course: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8199 to 0.8520, SE ratio 0.6865 to 0.7308, empirical efficiency ratio 0.9589 to 1.0207, reported efficiency ratio 0.7005 to 0.7010	pass
`interval_calibration`	`longitudinal_dropout_contrast_t_a__correctly_specified`	positive	two-node law L with dropout and three-level treatment, T_a against the natural course: 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 0.9360 to 0.9556, SE ratio 0.9637 to 1.0221, empirical efficiency ratio 0.9796 to 1.0389, reported efficiency ratio 1.0009 to 1.0016	pass
`interval_calibration`	`longitudinal_dropout_contrast_t_a__noise_control`	control	two-node law L with dropout and three-level treatment, T_a against the natural course: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8245 to 0.8562, SE ratio 0.6940 to 0.7372, empirical efficiency ratio 1.3581 to 1.4427, reported efficiency ratio 1.0009 to 1.0016	pass
`interval_calibration`	`longitudinal_dropout_contrast_t_a__shrunken_se_control`	control	two-node law L with dropout and three-level treatment, T_a against the natural course: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8090 to 0.8418, SE ratio 0.6754 to 0.7160, empirical efficiency ratio 0.9790 to 1.0376, reported efficiency ratio 0.7006 to 0.7011	pass
`interval_calibration`	`point_contrast_t_a__correctly_specified`	positive	three-level law P, T_a against the natural course: 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 0.9441 to 0.9624, SE ratio 0.9773 to 1.0383, empirical efficiency ratio 0.9640 to 1.0241, reported efficiency ratio 1.0001 to 1.0016	pass
`interval_calibration`	`point_contrast_t_a__noise_control`	control	three-level law P, T_a against the natural course: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8145 to 0.8469, SE ratio 0.6873 to 0.7279, empirical efficiency ratio 1.3750 to 1.4565, reported efficiency ratio 1.0001 to 1.0016	pass
`interval_calibration`	`point_contrast_t_a__shrunken_se_control`	control	three-level law P, T_a against the natural course: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8133 to 0.8458, SE ratio 0.6847 to 0.7262, empirical efficiency ratio 0.9649 to 1.0233, reported efficiency ratio 0.7000 to 0.7011	pass
`interval_calibration`	`point_contrast_t_b__correctly_specified`	positive	three-level law P, T_b against the natural course: 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 0.9390 to 0.9581, SE ratio 0.9759 to 1.0332, empirical efficiency ratio 0.9690 to 1.0259, reported efficiency ratio 1.0004 to 1.0018	pass
`interval_calibration`	`point_contrast_t_b__noise_control`	control	three-level law P, T_b against the natural course: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8159 to 0.8482, SE ratio 0.6847 to 0.7262, empirical efficiency ratio 1.3781 to 1.4619, reported efficiency ratio 1.0003 to 1.0018	pass
`interval_calibration`	`point_contrast_t_b__shrunken_se_control`	control	three-level law P, T_b against the natural course: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8101 to 0.8429, SE ratio 0.6833 to 0.7244, empirical efficiency ratio 0.9672 to 1.0256, reported efficiency ratio 0.7002 to 0.7012	pass
`interval_calibration`	`point_known_t_a__correctly_specified`	positive	law P with the treatment mechanism declared, T_a against the natural course: 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 0.9466 to 0.9644, SE ratio 0.9870 to 1.0490, empirical efficiency ratio 0.9546 to 1.0147, reported efficiency ratio 1.0005 to 1.0022	pass
`interval_calibration`	`point_known_t_a__noise_control`	control	law P with the treatment mechanism declared, T_a against the natural course: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8219 to 0.8538, SE ratio 0.7029 to 0.7461, empirical efficiency ratio 1.3421 to 1.4244, reported efficiency ratio 1.0005 to 1.0022	pass
`interval_calibration`	`point_known_t_a__shrunken_se_control`	control	law P with the treatment mechanism declared, T_a against the natural course: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8156 to 0.8480, SE ratio 0.6910 to 0.7346, empirical efficiency ratio 0.9542 to 1.0142, reported efficiency ratio 0.7004 to 0.7015	pass
`interval_calibration`	`point_stratified_t_a_v1__correctly_specified`	positive	law P in baseline stratum V = 1, T_a against the natural course: 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 0.9387 to 0.9579, SE ratio 0.9687 to 1.0295, empirical efficiency ratio 0.9705 to 1.0317, reported efficiency ratio 0.9980 to 1.0007	pass
`interval_calibration`	`point_stratified_t_a_v1__noise_control`	control	law P in baseline stratum V = 1, T_a against the natural course: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8165 to 0.8488, SE ratio 0.6956 to 0.7381, empirical efficiency ratio 1.3542 to 1.4367, reported efficiency ratio 0.9980 to 1.0007	pass
`interval_calibration`	`point_stratified_t_a_v1__shrunken_se_control`	control	law P in baseline stratum V = 1, T_a against the natural course: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8093 to 0.8421, SE ratio 0.6787 to 0.7205, empirical efficiency ratio 0.9712 to 1.0305, reported efficiency ratio 0.6986 to 0.7005	pass
`mechanism_targeting`	`point_contrast_t_a__mechanism_targeted`	positive	three-level law P, T_a against the natural course: the treatment mechanism is tilted along the T_a mechanism score by 2 / sqrt(n), then the full targeting runs	bias inside the margin and the SE-ratio interval inside the efficiency-ratio band	bias -0.0020 to -0.000300, margin 0.0038, SE ratio 0.9588 to 1.0371	pass
`mechanism_targeting`	`point_contrast_t_a__outcome_targeted_only`	control	three-level law P, T_a against the natural course: the same tilted mechanism with the outcome fluctuation alone	bias interval must fall above zero and entirely outside the margin	bias 0.0176 to 0.0193, margin 0.0037	pass
`mechanism_targeting`	`point_stratified_t_a_v1__mechanism_targeted`	positive	law P in baseline stratum V = 1, T_a against the natural course: the treatment mechanism is tilted along the T_a mechanism score by 2 / sqrt(n), then the full targeting runs	bias inside the margin and the SE-ratio interval inside the efficiency-ratio band	bias -0.0030 to -0.000502, margin 0.0054, SE ratio 0.9526 to 1.0349	pass
`mechanism_term_necessity`	`point_contrast_t_a__binary_blip`	control	three-level law P, T_a against the natural course: the same estimates with the mechanism term read from the first two levels only	the SE-ratio interval must fall below the calibration band	SE ratio 0.7789 to 0.8261	pass
`mechanism_term_necessity`	`point_contrast_t_a__full_eif`	positive	three-level law P, T_a against the natural course: the reported standard error from the full influence curve	the SE-ratio interval inside the efficiency-ratio band	SE ratio 0.9905 to 1.0511	pass
`mechanism_term_necessity`	`point_contrast_t_a__mechanism_term_dropped`	control	three-level law P, T_a against the natural course: the same estimates with the treatment-mechanism term, on the outcome’s own scale, removed from the curve	the SE-ratio interval must fall below the calibration band	SE ratio 0.6091 to 0.6468	pass
`power`	`point_contrast_t_a__alternative`	positive	three-level law P, T_a against the natural course: the same test applied to a law with a real effect	rejection lower bound clears the minimum power	rejection 0.8725, 0.8521 to 0.8911	pass
`single_robustness`	`point_contrast_t_a__mechanism_constant`	diagnostic	three-level law P, T_a against the natural course: constant treatment mechanism and cell-mean outcome regressions, beside the exact-law limit	reported only	bias -0.0653 to -0.0641, margin 0.0026, coverage 0.0000	reported
`single_robustness`	`point_contrast_t_a__mechanism_correct__n_16000`	positive	three-level law P, T_a against the natural course: cell-frequency treatment mechanism and a constant outcome regression, at n = 16,000	bias inside the margin at n = 8,000 and at n = 16,000	bias -0.000988 to -0.000125, margin 0.0019, coverage 0.9070	pass
`single_robustness`	`point_contrast_t_a__mechanism_correct__n_8000`	positive	three-level law P, T_a against the natural course: cell-frequency treatment mechanism and a constant outcome regression, at n = 8,000	bias inside the margin at n = 8,000 and at n = 16,000	bias -0.0014 to -0.000096, margin 0.0027, coverage 0.8940	pass
`type_i_error`	`point_contrast_t_a__sharp_null`	positive	three-level law P, T_a against the natural course: a confounded law whose true contrast is exactly zero	one-sided rejection bound stays under the declared type-I ceiling	rejection 0.0515, 0.0396 to 0.0656	pass

The calibration, necessity, type I, power, one-fold and complete-case fits use saturated learners. The mechanism_targeting fits use a mechanism tilted along its score. The single_robustness fits use a constant outcome regression or a constant mechanism. The tables above print one row for each cell. The families below state what each cell tests.

family	cells and fits	what must hold
`interval_calibration`	six labels with three cells each: the correctly specified fit and two derived controls. Four point labels at n = 4,000. Two longitudinal labels at n = 16,000	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
`mechanism_term_necessity`	the `T_a` contrast on law P: the full curve, the curve without the mechanism term, and the curve whose mechanism term reads the first two levels only (the binary blip)	the full curve is calibrated. Each control under-reports and its SE-ratio interval lies below the calibration band
`mechanism_targeting`	the `T_a` contrast on law P, with a mechanism tilted along its score by \(2/\sqrt n\). The TMLE, unstratified and in stratum V = 1, and the outcome-only targeted plug-in	each TMLE cell has bias inside the margin and an SE-ratio interval inside the efficiency band. The outcome-only control lies above the truth and outside the margin
`type_i_error` and `power`	the `T_a` contrast on law P0 and law Pw	the upper endpoint of the rejection interval on law P0 stays at or below the type-I ceiling. The lower endpoint on law Pw clears the minimum power
`single_robustness`	the mechanism saturated and the outcome regression constant, at n = 8,000 and 16,000. The mechanism constant with a saturated outcome regression is a diagnostic	bias inside the margin at both sizes. The diagnostic prints its bias beside the exact limit and states no verdict
`in_sample_fit`	the longitudinal contrast on law L with one fold, on its own sample stream	no verdict. The cell prints its bias and coverage beside the exact truth
`censoring_complete_case`	the mean under `T_a` on law L, fitted to the rows that never drop out, with no censoring model	no verdict. The cell prints its bias beside the exact complete-case limit

The mechanism_targeting family is the study-scale evidence for the softmax update. A saturated cross-fitted mechanism moves by \(O(1/n)\) only. The other cells therefore cannot see a defective update. The tilted mechanism has a first-order error, and only the mechanism fluctuation removes it. The outcome-only plug-in sits 1.18 standard errors (at n = 4,000) above the truth, which is its exact first-order limit.

tests/unit/test_categorical_incremental_design.py shows on the exact law that an uncentred mechanism covariate moves the TMLE by 0.37 standard errors at n = 4,000. A one-versus-rest update is a deliberate mutation. The fast-tier test tests/unit/test_categorical_incremental_mutations.py catches it.

The mechanism_term_necessity controls carry a verdict that asserts only the failure each control exists to show. The single_robustness cells gate bias only. A misspecified outcome regression leaves a first-order variance term, so their intervals are anti-conservative. A bias pass is not an interval claim.

Sizing#

The sizes follow the pilots in tests/unit/test_categorical_incremental_design.py. The study fixed every size, replication count, margin, law, learner and fold count before the first run. It fixed every seed before that run except one. After the first run, the in_sample_fit cell moved to its own seed stream (commit 2b5077d3), so that it shares no sample with another family. No verdict reads that cell. Every other committed row equals the first run.

run	commit	wall time	content
first	`7cb43c92`	2.7 hours on four cores (9,785 s)	every cell
resumed	`2b5077d3`	1.1 hours on four cores (3,870 s)	carried the primary and other property checkpoints. Fitted the law-L calibration cells and the `in_sample_fit` stream

The committed run.log and manifest.json record the resumed run.

sizing input	value	source
primary n	8,000 for all four scenarios. The plan’s rung rule gives 1,000 for family P and 2,000 for family L, where the twenty primary rows pass the bias gate with joint probability 0.25. At 8,000 the joint probability is 0.99 under a bias that falls as \(1/\sqrt n\)	`tests/studies/canonical_categorical_incremental.py`
rung pilots	400 draws per rung from n = 1,000 to 16,000. The largest standardized bias of a primary row is 0.164 at n = 1,000 and 0.054 at n = 16,000 for family P	the same
calibration n	4,000 for the point labels. 16,000 for the longitudinal labels, where the pilot of 1,600 draws read SE ratios 1.009 and 1.016	`tests/studies/categorical_incremental_properties.py`
calibration replications	3,600. The pilot predicts a family-wise pass probability of 0.966	the same
`mechanism_targeting` replications	2,000. The family pass probability is 0.995	the same
type I error and power	2,000 replications at n = 4,000, with exact power 0.8724	the same
single robustness	2,000 replications at n = 8,000 and 16,000	the same
complete-case cell	2,000 replications at n = 16,000. The exact limit sits 5.8 standard errors from the truth	the same

Red-cell route#

event	route, declared before any run
a red primary truth row	blocks the merge and gets a diagnosis
a red calibration row	is a finite-sample red only when its SE-ratio interval contains 1, its coverage is consistent with 0.95 and the diagnosis finds no defect. It then moves to `reporting` with an owner row before any re-run. Any other red row is a defect suspect. A defect is fixed and the study re-runs under a fresh declaration
a red `mechanism_targeting` positive cell	is a defect suspect in the softmax update
a control that does not fail	is a defect suspect in the instrument
any verdict	moves no budget, margin, law, learner, fold count or seed

The run found no red cell.

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`	8000	observations per primary replication
`independent_tests_passed`	20	truth tests passing
`independent_tests_total`	20	truth tests reported
`property_cells_passed`	28	property cells passing
`property_cells_total`	28	property cells reported
`max_standardized_bias`	0.1422	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	the rate the upper endpoint of the sharp-null rejection interval must not exceed
`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	rejection lower bound the power cell must clear
`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:shrunken_se_factor`	0.7000	negative-control SE multiplier
`margin:efficiency_ratio_lower`	0.9000	exact-EIF ratio lower bound
`margin:efficiency_ratio_upper`	1.1000	exact-EIF ratio upper bound
`bound:ate_ipsi[T_a vs natural course]_standard_error`	0.0106	exact influence-curve standard error of this estimand at the primary n = 8,000
`bound:ate_ipsi[T_b vs natural course]_standard_error`	0.0112	exact influence-curve standard error of this estimand at the primary n = 8,000
`bound:ey_ipsi[T_a]_standard_error`	0.0252	exact influence-curve standard error of this estimand at the primary n = 8,000
`bound:ey_ipsi[T_b]_standard_error`	0.0238	exact influence-curve standard error of this estimand at the primary n = 8,000
`bound:ey_ipsi[natural course]_standard_error`	0.0248	exact influence-curve standard error of this estimand at the primary n = 8,000
`bound:longitudinal_dropout_folds_5/ate_regimen[T_a vs natural course]_standard_error`	0.0114	exact influence-curve standard error of this estimand at the primary n = 8,000
`bound:longitudinal_dropout_folds_5/ate_regimen[T_b vs natural course]_standard_error`	0.0133	exact influence-curve standard error of this estimand at the primary n = 8,000
`bound:longitudinal_dropout_folds_5/ey_regimen[T_a]_standard_error`	0.0231	exact influence-curve standard error of this estimand at the primary n = 8,000
`bound:longitudinal_dropout_folds_5/ey_regimen[T_b]_standard_error`	0.0211	exact influence-curve standard error of this estimand at the primary n = 8,000
`bound:longitudinal_dropout_folds_5/ey_regimen[natural course]_standard_error`	0.0215	exact influence-curve standard error of this estimand at the primary n = 8,000
`bound:longitudinal_folds_5/ate_regimen[T_a vs natural course]_standard_error`	0.0107	exact influence-curve standard error of this estimand at the primary n = 8,000
`bound:longitudinal_folds_5/ate_regimen[T_b vs natural course]_standard_error`	0.0123	exact influence-curve standard error of this estimand at the primary n = 8,000
`bound:longitudinal_folds_5/ey_regimen[T_a]_standard_error`	0.0224	exact influence-curve standard error of this estimand at the primary n = 8,000
`bound:longitudinal_folds_5/ey_regimen[T_b]_standard_error`	0.0192	exact influence-curve standard error of this estimand at the primary n = 8,000
`bound:longitudinal_folds_5/ey_regimen[natural course]_standard_error`	0.0206	exact influence-curve standard error of this estimand at the primary n = 8,000
`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

Limits#

limit	what it means for use
three levels at each tilt node	the laws have three treatment levels. Four or more levels and the vector-treatment keys are outside this evidence
saturated learners	the study does not test a flexible learner. The cross-fitted mechanism moves by \(O(1/n)\), which the `mechanism_targeting` family compensates
two tilt nodes and one outcome	longer panels, survival and competing-risk outcomes are covered by exact laws and not by a registered study
dropout at random given the history	the retention is 0.833 on law L. Heavier and informative dropout are outside this evidence
finite-sample bias is sized, not removed	the primary rows pass at n = 8,000. Smaller sizes carry a larger standardized bias, which the rung pilots print
single-robustness intervals	the cells gate bias only. The intervals of a misspecified outcome regression are anti-conservative
the parametric g-formula	a tilt at a node with other than two levels is refused. This study does not cover it
no comparator	no maintained package ships the construction, so the study has no paired verdict

Reproduction#

tests/canonical/categorical_incremental/regenerate.py regenerates the study under the declared run form. It runs no reference process, because the study has no comparator. The first run took 2.7 hours on four cores, which is the time to reproduce the study from scratch. The resumed run described in Sizing took 1.1 hours.