Incremental propensity interventions

This study validates three incremental treatment-odds multipliers, including the natural course at one. It compares cleverly with R npcausal at commit 56a5ac1 on identical samples. Kennedy (2019) publishes npcausal, and its influence values carry the derivative through the treatment mechanism. The comparison therefore gates inference as well as the point curve.

R imtp 0.1.0 held this role in an earlier revision and no longer does. Its reported influence curve omits that derivative, so it can witness the point curve and cannot gate a standard error. Choosing a comparator records the full survey.

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
binary-outcome law with three incremental odds multipliers	ate_ipsi[odds x0.5 vs natural course]	difference in means under the incremental interventions “multiply the treatment odds by 0.5” against “leave the observed treatment mechanism unchanged”	cleverly	-0.000039 to 0.000378	0.9431	0.9932	pass
binary-outcome law with three incremental odds multipliers	ate_ipsi[odds x0.5 vs natural course]	difference in means under the incremental interventions “multiply the treatment odds by 0.5” against “leave the observed treatment mechanism unchanged”	R npcausal	-0.000187 to 0.000234	0.9444	0.9869	pass
binary-outcome law with three incremental odds multipliers	ate_ipsi[odds x2 vs natural course]	difference in means under the incremental interventions “multiply the treatment odds by two” against “leave the observed treatment mechanism unchanged”	cleverly	-0.000412 to 0.000034	0.9463	0.9986	pass
binary-outcome law with three incremental odds multipliers	ate_ipsi[odds x2 vs natural course]	difference in means under the incremental interventions “multiply the treatment odds by two” against “leave the observed treatment mechanism unchanged”	R npcausal	-0.000244 to 0.000209	0.9475	0.9889	pass
binary-outcome law with three incremental odds multipliers	ey_ipsi[natural course]	mean under the incremental intervention leave the observed treatment mechanism unchanged	cleverly	-0.000628 to 0.000839	0.9381	0.9824	pass
binary-outcome law with three incremental odds multipliers	ey_ipsi[natural course]	mean under the incremental intervention leave the observed treatment mechanism unchanged	R npcausal	-0.000628 to 0.000839	0.9381	0.9824	pass
binary-outcome law with three incremental odds multipliers	ey_ipsi[odds x0.5]	mean under the incremental intervention multiply the treatment odds by 0.5	cleverly	-0.000477 to 0.0010	0.9444	0.9885	pass
binary-outcome law with three incremental odds multipliers	ey_ipsi[odds x0.5]	mean under the incremental intervention multiply the treatment odds by 0.5	R npcausal	-0.000622 to 0.000881	0.9425	0.9895	pass
binary-outcome law with three incremental odds multipliers	ey_ipsi[odds x2]	mean under the incremental intervention multiply the treatment odds by two	cleverly	-0.000850 to 0.000683	0.9381	0.9762	pass
binary-outcome law with three incremental odds multipliers	ey_ipsi[odds x2]	mean under the incremental intervention multiply the treatment odds by two	R npcausal	-0.000681 to 0.000858	0.9363	0.9741	pass

Agreement with the canonical implementation

Comparator-backed: a paired comparison with npcausal 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 three incremental odds multipliers	ate_ipsi[odds x0.5 vs natural course]	difference in means under the incremental interventions “multiply the treatment odds by 0.5” against “leave the observed treatment mechanism unchanged”	0.000146	0.2993	0.9984	-0.0012	different estimator	0.0180 vs 0.0500	equivalent
binary-outcome law with three incremental odds multipliers	ate_ipsi[odds x2 vs natural course]	difference in means under the incremental interventions “multiply the treatment odds by two” against “leave the observed treatment mechanism unchanged”	-0.000172	0.3287	0.9944	-0.0012	different estimator	0.0254 vs 0.0500	equivalent
binary-outcome law with three incremental odds multipliers	ey_ipsi[natural course]	mean under the incremental intervention leave the observed treatment mechanism unchanged	6.730e-13	3.942e-10	1.0000	0	different estimator	4.226e-11 vs 0.0500	equivalent
binary-outcome law with three incremental odds multipliers	ey_ipsi[odds x0.5]	mean under the incremental intervention multiply the treatment odds by 0.5	0.000146	0.0834	1.0026	0.0019	different estimator	0.0019 vs 0.0500	equivalent
binary-outcome law with three incremental odds multipliers	ey_ipsi[odds x2]	mean under the incremental intervention multiply the treatment odds by two	-0.000172	0.0962	0.9993	0.0019	different estimator	0.0052 vs 0.0500	equivalent

Theory properties

property	cell	role	what was tested	what must hold	measured	result
interval_calibration	contrast__correctly_specified	positive	odds-x2 contrast: 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 0.9421 to 0.9598, SE ratio 0.9778 to 1.0365, empirical efficiency ratio 0.9641 to 1.0221, reported efficiency ratio 0.9987 to 0.9997	pass
interval_calibration	contrast__noise_control	control	odds-x2 contrast: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8181 to 0.8486, SE ratio 0.6895 to 0.7295, empirical efficiency ratio 1.3696 to 1.4490, reported efficiency ratio 0.9987 to 0.9997	pass
interval_calibration	contrast__shrunken_se_control	control	odds-x2 contrast: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8214 to 0.8518, SE ratio 0.6840 to 0.7256, empirical efficiency ratio 0.9642 to 1.0226, reported efficiency ratio 0.6991 to 0.6998	pass
mechanism_requirement	both_correct	positive	both the outcome regression and treatment mechanism are correctly specified	bias interval inside the equivalence margin	bias -0.000288 to 0.000233, margin 0.000875	pass
mechanism_requirement	mechanism_wrong	control	the wrong treatment mechanism is held fixed when the incremental target is evaluated	bias interval must fall entirely outside the margin	bias -0.0116 to -0.0112, margin 0.000627	pass
mechanism_requirement	outcome_wrong	positive	the treatment mechanism is correct and the outcome regression is misspecified	bias interval inside the equivalence margin	bias -0.000205 to 0.000306, margin 0.000857	pass
natural_course_identity	natural__ipsi	positive	natural course: the odds multiplier one is evaluated through the incremental axis	the paired estimate must equal the observed sample mean exactly	maximum paired difference 8.408e-11	pass
natural_course_identity	natural__mean	positive	natural course: the observed sample mean is retained as the identity control	the paired intervention estimate must equal it exactly	maximum paired difference 8.408e-11	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	contrast__n_2000	positive	odds-x2 contrast: bias, coverage and SE calibration at n = 2,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.000136, coverage 0.9416 to 0.9776, SE ratio 1.0402	pass
root_n_and_efficiency	contrast__n_500	control	odds-x2 contrast: bias, coverage and SE calibration at n = 500	coverage interval lies below nominal or clears the declared floor	bias -0.000560, coverage 0.9282 to 0.9688, SE ratio 1.0041	pass
root_n_and_efficiency	contrast__n_8000	positive	odds-x2 contrast: bias, coverage and SE calibration at n = 8,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.000050, coverage 0.9297 to 0.9698, SE ratio 0.9987	pass
root_n_rate	contrast__empirical_sd	positive	odds-x2 contrast: 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 -0.5302 to -0.4656	pass
root_n_rate	contrast__reported_se	positive	odds-x2 contrast: the same regression applied to the mean reported standard error	slope interval inside the root-n band and excluding -1/4	slope -0.5003 to -0.4987	pass
targeting_necessity	mechanism__targeted	positive	treatment-mechanism targeting: the treatment mechanism is fluctuated after outcome targeting	bias interval inside the equivalence margin	bias -0.000313 to 0.000198, margin 0.000857	pass
targeting_necessity	mechanism__untargeted	control	treatment-mechanism targeting: the identical targeted outcome regression is evaluated at the unfluctuated mechanism	bias interval must fall entirely outside the margin	bias -0.0135 to -0.0130, margin 0.000870	pass
targeting_necessity	outcome__targeted	positive	outcome-regression targeting: the outcome regression is fluctuated before the incremental target is evaluated	bias interval inside the equivalence margin	bias -0.000421 to 0.000100, margin 0.000875	pass
targeting_necessity	outcome__untargeted	control	outcome-regression targeting: the identical targeted mechanism is evaluated with the initial outcome regression	bias interval must fall entirely outside the margin	bias -0.0904 to -0.0903, margin 0.000123	pass
treatment_score_necessity	odds_x2__full_eif	positive	odds-x2 incremental mean: the incremental mean uses the complete efficient influence curve	the reported-to-empirical SE-ratio interval must stay inside the declared band	SE ratio 0.9641 to 1.0547	pass
treatment_score_necessity	odds_x2__regime_curve_control	control	odds-x2 incremental mean: the same point estimates use an influence curve with the treatment-score term removed	the SE-ratio interval must fall below the calibration band	SE ratio 0.6077 to 0.6630	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 the one-sided mechanism requirement, exact-EIF calibration, null and power controls, both targeting equations, the treatment-score term, and the natural-course identity.

Measured values and declared margins

quantity	value	source
replicates	1600	paired replications
n	2000	observations per 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	21	property cells passing
property_cells_total	21	property cells reported
max_standardized_bias	0.0546	largest primary standardized bias
min_coverage	0.9363	lowest primary coverage
max_margin_utilization	0.3287	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 in pooled SDs
margin:rmse_noninferiority	1.1000	RMSE-ratio noninferiority bound
margin:coverage_noninferiority	-0.0250	coverage-difference noninferiority bound
margin:calibration_noninferiority	0.0500	calibration-excess noninferiority 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: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

Limits

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

• Both implementations saturate that covariate. cleverly fits one-hot indicators and their interaction with treatment. npcausal fits SL.glm.interaction over the same design. The treatment mechanism is supplied exactly to cleverly and is estimated by npcausal.

• An earlier revision fitted the covariate as one numeric column. That model is misspecified, and it inflated the standard error of each contrast against the natural course by five to six percent above the exact-law efficient influence curve. Bias and coverage did not show it, because a correct mechanism keeps a targeted estimator consistent whatever the outcome model says. Consistency under a wrong outcome model is claimed by the mechanism_requirement property cells, which use exact nuisances and carry their own control.

• npcausal cross-fits its nuisances over two folds and cleverly does not. The single-fold path that ?ipsi documents selects an empty training set and is not usable.

• Only pointwise Wald inference is validated for either implementation.

• The study covers ordinary, non-cross-fitted cleverly targeting and three fixed odds multipliers.

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