Stochastic point-treatment regimes

This study validates an ordinary known stochastic regime over two treatment arms. It compares cleverly with pinned R lmtp 1.5.4 on identical binary-outcome samples and exact treatment probabilities.

The comparison runs through the shared point adapter rather than through lmtp’s shift interface. That interface takes a function of the natural treatment value, and this regime ignores that value. The adapter supplies the analytic density ratio and the shifted frame directly. The static regime uses mtp = FALSE. The tilt draws one treatment value per unit from the declared density and uses mtp = TRUE. Pinned tmle3 stays unusable for this parameter, because Param_TSM evaluates a counterfactual at one treatment value and integrates over no density.

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
binary-outcome law with a known stochastic treatment density	ate_regime[tilt vs never]	difference in means under the regimes “draw from the known stochastic tilt” against “assign no treatment”	cleverly	-0.000970 to 0.000901	0.9437	1.0142	pass
binary-outcome law with a known stochastic treatment density	ate_regime[tilt vs never]	difference in means under the regimes “draw from the known stochastic tilt” against “assign no treatment”	R lmtp	-0.000923 to 0.000980	0.9487	1.0295	pass
binary-outcome law with a known stochastic treatment density	ey_regime[never]	mean under the regime assign no treatment	cleverly	-0.0020 to 0.000657	0.9387	0.9948	pass
binary-outcome law with a known stochastic treatment density	ey_regime[never]	mean under the regime assign no treatment	R lmtp	-0.0020 to 0.000662	0.9375	0.9946	pass
binary-outcome law with a known stochastic treatment density	ey_regime[tilt]	mean under the regime draw from the known stochastic tilt	cleverly	-0.0018 to 0.000390	0.9375	0.9993	pass
binary-outcome law with a known stochastic treatment density	ey_regime[tilt]	mean under the regime draw from the known stochastic tilt	R lmtp	-0.0018 to 0.000482	0.9413	0.9976	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
binary-outcome law with a known stochastic treatment density	ate_regime[tilt vs never]	difference in means under the regimes “draw from the known stochastic tilt” against “assign no treatment”	-0.000063	0.0403	0.9997	-0.0050	different estimator	0.0387 vs 0.0500	equivalent
binary-outcome law with a known stochastic treatment density	ey_regime[never]	mean under the regime assign no treatment	-0.000001	0.000471	0.9981	0.0012	different estimator	0.0013 vs 0.0500	equivalent
binary-outcome law with a known stochastic treatment density	ey_regime[tilt]	mean under the regime draw from the known stochastic tilt	-0.000064	0.0348	0.9899	-0.0037	different estimator	0.0176 vs 0.0500	equivalent

Theory properties

property	cell	role	what was tested	what must hold	measured	result
density_necessity	tilt__declared	positive	known stochastic tilt: the estimator integrates over the declared covariate-dependent treatment density	bias interval inside the equivalence margin	bias -0.000592 to 0.000920, margin 0.0025	pass
density_necessity	tilt__uniform_control	control	known stochastic tilt: the same fit replaces the declared density with a uniform distribution	bias interval must fall entirely outside the margin	bias 0.0269 to 0.0285, margin 0.0026	pass
double_robustness	both_correct	positive	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 -0.000926 to 0.000563, margin 0.0025, SE ratio 0.9980	pass
double_robustness	both_wrong	control	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 -0.1967 to -0.1954, margin 0.0022, SE ratio 2.1881	pass
double_robustness	outcome_correct	positive	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 -0.000754 to 0.000361, margin 0.0019, SE ratio 2.3099	pass
double_robustness	treatment_correct	positive	only the treatment mechanism is correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.0011 to 0.000477, margin 0.0027, SE ratio 1.0028	pass
interval_calibration	tilt__correctly_specified	positive	known stochastic tilt: 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 0.9445 to 0.9619, SE ratio 0.9796 to 1.0372, empirical efficiency ratio 0.9625 to 1.0191, reported efficiency ratio 0.9971 to 0.9996	pass
interval_calibration	tilt__noise_control	control	known stochastic tilt: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8238 to 0.8539, SE ratio 0.6954 to 0.7353, empirical efficiency ratio 1.3579 to 1.4358, reported efficiency ratio 0.9971 to 0.9997	pass
interval_calibration	tilt__shrunken_se_control	control	known stochastic tilt: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8150 to 0.8457, SE ratio 0.6863 to 0.7265, empirical efficiency ratio 0.9620 to 1.0183, reported efficiency ratio 0.6979 to 0.6998	pass
power	alternative	positive	the same test applied to a law with a real effect	rejection lower bound clears the minimum power	rejection 1, 0.9934 to 1	pass
root_n_and_efficiency	n_2000	positive	bias, coverage and SE calibration at n = 2,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.000196, coverage 0.9223 to 0.9647, SE ratio 1.0015	pass
root_n_and_efficiency	n_500	control	bias, coverage and SE calibration at n = 500	coverage interval lies below nominal or clears the declared floor	bias -0.000643, coverage 0.9208 to 0.9637, SE ratio 0.9871	pass
root_n_and_efficiency	n_8000	positive	bias, coverage and SE calibration at n = 8,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.000148, coverage 0.9267 to 0.9677, SE ratio 0.9718	pass
root_n_rate	empirical_sd	positive	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 -0.5273 to -0.4583	pass
root_n_rate	reported_se	positive	the same regression applied to the mean reported standard error	slope interval inside the root-n band and excluding -1/4	slope -0.5019 to -0.4973	pass
targeting_necessity	tilt__targeted	positive	known stochastic tilt: the estimator fluctuates a misspecified outcome model, so targeting does all the adjusting	bias interval inside the equivalence margin	bias -0.000897 to 0.000709, margin 0.0027	pass
targeting_necessity	tilt__untargeted	control	known stochastic tilt: the identical fit with every fluctuation step removed	bias interval must fall entirely outside the margin	bias -0.2251 to -0.2250, margin 0.000208	pass
type_i_error	sharp_null	positive	a confounded law whose true contrast is exactly zero	one-sided rejection bound stays under the declared type-I ceiling	rejection 0.0512, 0.0333 to 0.0748	pass

The property study tests double robustness, exact-EIF calibration, a confounded sharp null, and targeting. A paired control replaces the declared density with a uniform density.

Measured values and declared margins

quantity	value	source
replicates	800	replications
n	2000	observations per replication
independent_tests_passed	6	truth tests passing
independent_tests_total	6	truth tests reported
paired_tests_passed	3	paired comparisons passing
paired_tests_total	3	paired comparisons reported
property_cells_passed	18	property cells passing
property_cells_total	18	property cells reported
max_standardized_bias	0.0588	largest primary standardized bias
min_coverage	0.9375	lowest primary coverage
max_margin_utilization	0.0403	largest paired similarity-margin share
margin:confidence_level	0.9900	Monte Carlo confidence level
margin:alpha	0.0500	test size
margin:nominal_coverage	0.9500	nominal interval coverage
margin:bootstrap_replicates	10000	bootstrap replications
margin:standardized_bias	0.2500	standardized-bias margin
margin:coverage_floor	0.9000	primary coverage floor
margin:over_coverage_ceiling	0.9900	descriptive overcoverage threshold
margin:se_ratio_sanity_lower	0.8000	primary SE-ratio lower screen
margin:se_ratio_sanity_upper	1.2000	primary SE-ratio upper screen
margin:calibration_se_ratio_lower	0.9300	calibration SE-ratio lower bound
margin:calibration_se_ratio_upper	1.0700	calibration SE-ratio upper bound
margin:calibration_coverage_lower	0.9200	calibration coverage lower bound
margin:calibration_coverage_upper	0.9800	calibration coverage upper bound
margin:type_i_ceiling	0.1000	type-I upper bound
margin:paired_difference	0.1500	paired-similarity margin
margin:rmse_noninferiority	1.1000	RMSE non-inferiority bound
margin:coverage_noninferiority	-0.0250	coverage non-inferiority bound
margin:calibration_noninferiority	0.0500	calibration non-inferiority bound
margin:minimum_power	0.8000	minimum power lower 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	efficiency-ratio lower bound
margin:efficiency_ratio_upper	1.1000	efficiency-ratio upper bound
margin:shrunken_se_factor	0.7000	negative-control SE multiplier
margin:targeting_displacement	0.2500	minimum targeting displacement
margin:necessity_displacement	0.2500	minimum density-control displacement

Limits

• The laws use binary treatment, binary outcome, and one three-level baseline covariate.

• The primary working outcome model is logistic and the natural treatment mechanism is supplied exactly.

• The intervention density is known, covariate dependent, and supported on both arms.

• The study covers ordinary, non-cross-fitted targeting and pointwise Wald intervals.

• It excludes missing outcomes, weights, clusters, simultaneous bands, and flexible learners. The default-band study measures the default band of this study’s subject fit.

• It does not validate continuous-dose, incremental, or longitudinal stochastic interventions.

• The reference evaluates its shifted regression at one draw from the declared density. It does not integrate over that density, so it carries a Monte Carlo variance that cleverly does not.

• Both implementations fit a logistic outcome regression on the covariate as one numeric column. The paired comparison is therefore symmetric in its outcome model. The calibration claim for a correctly specified pair belongs to the interval_calibration cells, which use exact nuisances.