Stochastic categorical longitudinal TMLE

This study validates ordinary longitudinal TMLE under known stochastic categorical policies. The law has two nodes and three treatment levels. A policy node draws each unit’s arm from a density that is fixed before the fit. The known stochastic policies section states the estimator, its conditions, and its exact-law evidence.

Pinned lmtp 1.5.4 takes one shifted value per unit. The adapter tests/canonical/lmtp_policy_adapter.R therefore realises each policy exactly by copying every unit four times. Copy \(c\) takes its shifted arm from row \(c\) of the node’s allocation table, and every policy probability is a multiple of one quarter.

What was compared

setting	cleverly	R lmtp
datasets	2,000 samples of 2,000 rows from the two-node three-level law	the identical rows
folds	one fold, in sample	one fold
plans	low at both nodes; mix, a policy at both nodes; taper, high then a policy	the same plans; mix and taper through four copies of each unit
mechanism	the generating probabilities	the same, as the exact per-node ratio \(q / g\) at the natural arm
node regressions	a quasibinomial GLM at the last node and least squares at the first	SL.glm: binomial at the last node and gaussian at the first
policy-node design	the covariate history, the current arm, and the arms of earlier policy nodes	the full history with the current arm replaced
intervals	pointwise 95% Wald from the influence curve	the unit mean of the per-copy eif

For mix the two designs span the same columns and the learners match, so the pair is an exactness witness. For low and taper, this package fits the label nodes on the units that follow the plan, while lmtp pools over every arm. For low and the two contrasts the mean absolute paired difference is about \(9 \times 10^{-3}\), and the largest is 0.040 to 0.042. For taper they are \(2.3 \times 10^{-3}\) and 0.017. For mix the mean paired difference is \(4.9 \times 10^{-11}\), and the largest is \(6.3 \times 10^{-10}\). The pairs are read under the default paired margins, as in the categorical study.

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
the three-level law with known stochastic policies at both nodes	ate_regimen[mix vs low]	difference in mean outcome between the plans “draw each arm from a known policy at both times” against “assign the low arm at both times”	cleverly ordinary LTMLE under known stochastic categorical policies	-0.0023 to 0.0018	0.9360	0.9691	pass
the three-level law with known stochastic policies at both nodes	ate_regimen[mix vs low]	difference in mean outcome between the plans “draw each arm from a known policy at both times” against “assign the low arm at both times”	R lmtp	-0.0018 to 0.0024	0.9385	0.9791	pass
the three-level law with known stochastic policies at both nodes	ate_regimen[taper vs low]	difference in mean outcome between the plans “assign the high arm first, then draw from a known policy” against “assign the low arm at both times”	cleverly ordinary LTMLE under known stochastic categorical policies	-0.0037 to 0.0011	0.9480	0.9759	pass
the three-level law with known stochastic policies at both nodes	ate_regimen[taper vs low]	difference in mean outcome between the plans “assign the high arm first, then draw from a known policy” against “assign the low arm at both times”	R lmtp	-0.0027 to 0.0023	0.9430	0.9825	pass
the three-level law with known stochastic policies at both nodes	ey_regimen[low]	mean outcome under the plan assign the low arm at both times	cleverly ordinary LTMLE under known stochastic categorical policies	-0.0021 to 0.0018	0.9355	0.9731	pass
the three-level law with known stochastic policies at both nodes	ey_regimen[low]	mean outcome under the plan assign the low arm at both times	R lmtp	-0.0028 to 0.0013	0.9385	0.9857	pass
the three-level law with known stochastic policies at both nodes	ey_regimen[mix]	mean outcome under the plan draw each arm from a known policy at both times	cleverly ordinary LTMLE under known stochastic categorical policies	-0.0012 to 0.000400	0.9520	0.9950	pass
the three-level law with known stochastic policies at both nodes	ey_regimen[mix]	mean outcome under the plan draw each arm from a known policy at both times	R lmtp	-0.0012 to 0.000400	0.9520	0.9950	pass
the three-level law with known stochastic policies at both nodes	ey_regimen[taper]	mean outcome under the plan assign the high arm first, then draw from a known policy	cleverly ordinary LTMLE under known stochastic categorical policies	-0.0028 to -0.000064	0.9450	0.9820	pass
the three-level law with known stochastic policies at both nodes	ey_regimen[taper]	mean outcome under the plan assign the high arm first, then draw from a known policy	R lmtp	-0.0023 to 0.000400	0.9500	0.9870	pass

Agreement with the canonical implementation

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

law	estimand	what was compared	paired difference	share of margin used	RMSE ratio bound	coverage difference	comparison kind	calibration resolution	result
the three-level law with known stochastic policies at both nodes	ate_regimen[mix vs low]	difference in mean outcome between the plans “draw each arm from a known policy at both times” against “assign the low arm at both times”	-0.000593	0.1109	0.9860	-0.0025	matching implementation	0.0144 vs 0.0500	equivalent
the three-level law with known stochastic policies at both nodes	ate_regimen[taper vs low]	difference in mean outcome between the plans “assign the high arm first, then draw from a known policy” against “assign the low arm at both times”	-0.0011	0.1691	0.9884	0.0050	matching implementation	0.0129 vs 0.0500	equivalent
the three-level law with known stochastic policies at both nodes	ey_regimen[low]	mean outcome under the plan assign the low arm at both times	0.000593	0.1136	0.9776	-0.0030	matching implementation	0.0148 vs 0.0500	equivalent
the three-level law with known stochastic policies at both nodes	ey_regimen[mix]	mean outcome under the plan draw each arm from a known policy at both times	-4.892e-11	2.285e-08	1.0000	0	matching implementation	3.839e-10 vs 0.0500	equivalent
the three-level law with known stochastic policies at both nodes	ey_regimen[taper]	mean outcome under the plan assign the high arm first, then draw from a known policy	-0.000478	0.1343	1.0195	-0.0050	matching implementation	0.0069 vs 0.0500	equivalent

Theory properties

property	cell	role	what was tested	what must hold	measured	result
crossfit_overfitting	cross_fitted_policy_ltmle	positive	five-fold policy LTMLE with fully grown outcome trees	SE ratio clears the overfitting floor and stays inside the sanity band	SE ratio 0.9764 to 0.9948	pass
crossfit_overfitting	in_sample_control	control	the same flexible learner fitted in sample, with no cross-fitting	SE ratio must fall below the overfitting ceiling	SE ratio 0.2722 to 0.2772	pass
double_robustness	mix__both_correct	positive	the known policy mix, as its contrast against low, and as its mean in randomizer_projection: both the outcome regression and the treatment mechanism are correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.0018 to 0.0032, margin 0.0084, SE ratio 0.9957	pass
double_robustness	mix__both_wrong	control	the known policy mix, as its contrast against low, and as its mean in randomizer_projection: both nuisances are misspecified	bias interval must fall entirely outside the margin, with the reported standard error still on the scale of the empirical spread	bias 0.0298 to 0.0349, margin 0.0084, SE ratio 0.9604	pass
double_robustness	mix__mechanism_correct	positive	the known policy mix, as its contrast against low, and as its mean in randomizer_projection: only the treatment and censoring mechanisms are correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.0030 to 0.0023, margin 0.0088, SE ratio 1.0065	pass
double_robustness	mix__outcome_correct	positive	the known policy mix, as its contrast against low, and as its mean in randomizer_projection: only the outcome regression is correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.0022 to 0.0029, margin 0.0086, SE ratio 0.9063	pass
interval_calibration	mix__correctly_specified	positive	the known policy mix, as its contrast against low, and as its mean in randomizer_projection: 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 0.9364 to 0.9550, SE ratio 0.9672 to 1.0281, empirical efficiency ratio 0.9630 to 1.0231, reported efficiency ratio 0.9879 to 0.9913	pass
interval_calibration	mix__noise_control	control	the known policy mix, as its contrast against low, and as its mean in randomizer_projection: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8173 to 0.8479, SE ratio 0.6828 to 0.7227, empirical efficiency ratio 1.3693 to 1.4490, reported efficiency ratio 0.9878 to 0.9913	pass
interval_calibration	mix__shrunken_se_control	control	the known policy mix, as its contrast against low, and as its mean in randomizer_projection: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8175 to 0.8482, SE ratio 0.6781 to 0.7193, empirical efficiency ratio 0.9632 to 1.0211, reported efficiency ratio 0.6915 to 0.6939	pass
interval_calibration	msm_policy__correctly_specified	positive	the dose coefficient of a working model over low, mix and taper: 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 0.9340 to 0.9530, SE ratio 0.9547 to 1.0127, empirical efficiency ratio 0.9780 to 1.0368, reported efficiency ratio 0.9889 to 0.9912	pass
interval_calibration	msm_policy__noise_control	control	the dose coefficient of a working model over low, mix and taper: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8106 to 0.8417, SE ratio 0.6750 to 0.7154, empirical efficiency ratio 1.3842 to 1.4663, reported efficiency ratio 0.9889 to 0.9912	pass
interval_calibration	msm_policy__shrunken_se_control	control	the dose coefficient of a working model over low, mix and taper: 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 0.8404, SE ratio 0.6686 to 0.7088, empirical efficiency ratio 0.9776 to 1.0362, reported efficiency ratio 0.6922 to 0.6938	pass
interval_calibration	policy_risk_h2__correctly_specified	positive	the cumulative risk at t = 2 of a known policy on the survival law: 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 0.9273 to 0.9472, SE ratio 0.9333 to 0.9871, empirical efficiency ratio 1.0029 to 1.0604, reported efficiency ratio 0.9876 to 0.9920	pass
interval_calibration	policy_risk_h2__noise_control	control	the cumulative risk at t = 2 of a known policy on the survival law: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.7930 to 0.8252, SE ratio 0.6658 to 0.7037, empirical efficiency ratio 1.4067 to 1.4864, reported efficiency ratio 0.9876 to 0.9920	pass
interval_calibration	policy_risk_h2__shrunken_se_control	control	the cumulative risk at t = 2 of a known policy on the survival law: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.7912 to 0.8235, SE ratio 0.6528 to 0.6913, empirical efficiency ratio 1.0026 to 1.0607, reported efficiency ratio 0.6913 to 0.6944	pass
policy_necessity	mix__declared_policy	positive	the known policy mix, as its contrast against low, and as its mean in randomizer_projection: the recursion integrates over the declared policy at each node	bias interval inside the equivalence margin	bias -0.0017 to 0.0033, margin 0.0084	pass
policy_necessity	mix__uniform_control	control	the known policy mix, as its contrast against low, and as its mean in randomizer_projection: the same fit replaces each declared policy with a uniform distribution	bias interval must fall entirely outside the margin	bias 0.0160 to 0.0208, margin 0.0079	pass
power	mix__alternative	positive	the known policy mix, as its contrast against low, and as its mean in randomizer_projection: the same test applied to a law with a real effect	rejection lower bound clears the minimum power	rejection 0.5575, 0.5114 to 0.6029	fail
randomizer_projection	mix__integrated	positive	the known policy mix, as its contrast against low, and as its mean in randomizer_projection: the integrated estimator, with the policy-weighted recursion and the ratio q / g	bias interval inside the equivalence margin; the sd ratio bound below one	bias -0.000562 to 0.0013, margin 0.0032, sd ratio 0.3343, 99% upper 0.3563	pass
randomizer_projection	mix__recorded_randomizer	positive	the known policy mix, as its contrast against low, and as its mean in randomizer_projection: the same sample with a recorded uniform randomizer per node and the rule that reads it	bias interval inside the equivalence margin	bias -0.0028 to 0.0028, margin 0.0095, sd ratio 0.3343, 99% upper 0.3563	pass
root_n_and_efficiency	mix__n_2000	positive	the known policy mix, as its contrast against low, and as its mean in randomizer_projection: bias, coverage and SE calibration at n = 2,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.0015, coverage 0.9282 to 0.9688, SE ratio 0.9981	pass
root_n_and_efficiency	mix__n_500	control	the known policy mix, as its contrast against low, and as its mean in randomizer_projection: bias, coverage and SE calibration at n = 500	coverage interval lies below nominal or clears the declared floor	bias -0.0030, coverage 0.9194 to 0.9627, SE ratio 0.9611	pass
root_n_and_efficiency	mix__n_8000	positive	the known policy mix, as its contrast against low, and as its mean in randomizer_projection: bias, coverage and SE calibration at n = 8,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.000889, coverage 0.9121 to 0.9575, SE ratio 0.9813	pass
root_n_rate	mix__empirical_sd	positive	the known policy mix, as its contrast against low, and as its mean in randomizer_projection: log empirical spread of the estimates regressed on log n across three sizes	slope interval inside the root-n band and excluding -1/4	slope -0.5297 to -0.4623	pass
root_n_rate	mix__reported_se	positive	the known policy mix, as its contrast against low, and as its mean in randomizer_projection: the same regression applied to the mean reported standard error	slope interval inside the root-n band and excluding -1/4	slope -0.4912 to -0.4855	pass
simultaneous_coverage	primary__pointwise_joint_control	control	every primary estimand of the study: the pointwise 95% intervals of the same fits, read jointly	joint coverage upper endpoint must fall below the nominal rate	joint coverage 0.8210 to 0.8634	pass
simultaneous_coverage	primary__simultaneous_band	positive	every primary estimand of the study: 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 0.9255 to 0.9533	pass
targeting_necessity	mix__targeted	positive	the known policy mix, as its contrast against low, and as its mean in randomizer_projection: the estimator fluctuates a misspecified outcome model, so targeting does all the adjusting	bias interval inside the equivalence margin	bias -0.0021 to 0.0032, margin 0.0088	pass
targeting_necessity	mix__untargeted	control	the known policy mix, as its contrast against low, and as its mean in randomizer_projection: the identical fit with every fluctuation step removed	bias interval must fall entirely outside the margin	bias 0.0640 to 0.0688, margin 0.0080	pass
type_i_error	mix__sharp_null	positive	the known policy mix, as its contrast against low, and as its mean in randomizer_projection: a confounded law whose true contrast is exactly zero	one-sided rejection bound stays under the declared type-I ceiling	rejection 0.0425, 0.0263 to 0.0644	pass

Every end-of-study family uses the three-level law with the declared policies. Its truths come from the finite-support g-formula in tests/discrete_law_longitudinal_policy.py. The survival calibration cell (policy_risk_h2) uses the binary censored survival law with a known policy at both nodes. The working-model cell (msm_policy) uses an identity-link projection over low, mix and taper, and its truth is the projection of the exact policy means.

policy_necessity scores a fit with uniform policies against the declared truth. randomizer_projection fits the same sample twice. The integrated estimator is the shipped one. The recorded estimator reads a fresh uniform randomizer at each node, which is Theorem 3 as stated. The one-sided 99% bootstrap upper bound of the ratio of their spreads must lie below one. These two cells read ey_regimen[mix]. Every other mix cell reads ate_regimen[mix vs low].

The power cell is red because it is underpowered at its declared size. Its n = 4,000 was copied from the categorical study, where the contrast is 0.125. Here the contrast is -0.0488, and the exact power of a two-sided 5% test is 0.5365. Exact power 0.80 needs n = 7,460. The cell measured 0.5575, and its 99% interval contains the exact power. Its bias is inside the margin and its coverage is 0.9475.

The study was declared gated. By its declared red-cell route it was re-registered under reporting, with the owner row F1-power-design, and the run repeated with no other change. The repeat reproduced every artifact of the first 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	2000	paired replications
n	2000	observations per paired replication
independent_tests_passed	10	truth tests passing
independent_tests_total	10	truth tests reported
paired_tests_passed	5	paired comparisons passing
paired_tests_total	5	paired comparisons reported
property_cells_passed	29	property cells passing
property_cells_total	30	property cells reported
max_standardized_bias	0.0603	largest primary standardized bias
min_coverage	0.9355	lowest primary coverage
max_margin_utilization	0.1691	largest paired similarity-margin share
properties[crossfit_overfitting/cross_fitted_policy_ltmle]:coverage	0.9404	cross-fitted tree coverage
properties[crossfit_overfitting/in_sample_control]:coverage	0.4041	in-sample tree coverage
properties[randomizer_projection/mix__integrated]:projection_sd_ratio	0.3343	spread ratio, integrated over recorded
properties[randomizer_projection/mix__integrated]:projection_sd_ratio_ci_upper	0.3563	one-sided 99% upper bound of the spread ratio
margin:confidence_level	0.9900	Monte Carlo confidence level
margin:standardized_bias	0.2500	standardized-bias margin
margin:coverage_floor	0.9000	primary coverage floor
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:paired_difference	0.1500	paired similarity margin in pooled SDs
margin:overfit_control_ceiling	0.7500	in-sample tree SE-ratio upper bound
margin:alpha	0.0500	test size
margin:bootstrap_replicates	10000	bootstrap replications
margin:nominal_coverage	0.9500	nominal interval coverage
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:coverage_noninferiority	-0.0250	coverage-difference noninferiority bound
margin:rmse_noninferiority	1.1000	RMSE-ratio noninferiority bound
margin:calibration_noninferiority	0.0500	calibration-excess noninferiority bound
margin:root_n_slope	-0.5000	expected root-n slope
margin:root_n_slope_lower	-0.6250	root-n slope lower bound
margin:root_n_slope_upper	-0.3750	root-n slope upper bound
margin:excluded_slope	-0.2500	slower rate the interval must exclude
margin:union_model_se_lower	0.1000	union-model SE-ratio screen, lower limit
margin:union_model_se_upper	10	union-model SE-ratio screen, upper limit
margin:efficiency_ratio_lower	0.9000	exact-EIF ratio lower bound
margin:efficiency_ratio_upper	1.1000	exact-EIF ratio upper bound
margin:shrunken_se_factor	0.7000	negative-control SE multiplier
margin:targeting_displacement	0.1000	minimum targeting displacement
margin:policy_displacement	0.1000	minimum policy displacement
margin:type_i_ceiling	0.1000	type-I upper bound
margin:minimum_power	0.8000	power lower bound
margin:overfit_se_floor	0.8500	cross-fitted tree SE-ratio lower bound
margin:overfit_coverage_gain	0.1500	minimum paired coverage gain

Limits

limit	what it means for use
one law, two nodes and three levels	the evidence covers one finite law with policies fixed before the run
the power cell is underpowered at its declared size	the size was copied from a study with a contrast 2.56 times larger, so the exact power is 0.5365. The F1-power-design owner holds it. The type_i_error cell, on the null variant of the law, passes
no study cell for weights, clusters, competing risks, a continuous outcome or fold repeats	these compositions have the exact-law and end-to-end evidence of the known stochastic policies section only
low, taper and both contrasts are not exactness pairs	the mean absolute paired difference is about \(9 \times 10^{-3}\) for low and the contrasts, with a largest of 0.040 to 0.042, and \(2.3 \times 10^{-3}\) for taper, with a largest of 0.017. lmtp pools label nodes over every arm, and this package fits them on the followers. Only ey_regimen[mix] is an exactness pair
the comparator copies each unit four times	the pairing needs every policy probability on a grid of one quarter. A policy off that grid has no exact lmtp pair
the mechanism is supplied	both implementations receive the generating probabilities. The double-robustness cells cover misspecified mechanisms
a policy must be known and fixed	a policy learned from the sample, a policy that reads the natural treatment, and a policy that depends on the law are outside this evidence
learner caution	the tree pair validates held-out prediction with flexible learners, not learner-library selection