DR-TMLE on a declared known treatment mechanism

This study validates complete-data DRTMLE on data that declares its treatment mechanism. The fit is Benkeser et al. (2017), Theorem 1, at the degenerate estimator \(g_n = g_0\). The branch \(g = g_0\) of the theorem’s hypothesis holds by declaration. The section a known treatment mechanism states the construction for each guard.

This study measures the complete-data known-mechanism case only. It does not measure an estimated mechanism with one wrong nuisance.

What was compared

setting	cleverly	R drtmle
datasets	1,000 samples of 2,000 rows. The four guard scenarios read the same samples	the identical rows
law	the two-arm law of the TMLE study. The three-arm scenario has the mechanism \((0.2, 0.3, 0.5)\) at \(W_2 = 0\) and \((0.4, 0.4, 0.2)\) at \(W_2 = 1\)	the same
scenarios	guard_none, guard_q, guard_g and guard_qg: each guard with the wrong outcome regression. three_arm_guard_none: the three-arm law at guard=()	the same guards. The three-arm fit passes the three levels in a_0
reported parameters	ey0, ey1 and the ATE. At three arms, each arm mean and each contrast against arm 0	the same
mechanism	the declared \(g_0\)	gn, one vector for each level, with tolg = 0.1
reduced regressions	a linear regression on the mechanism, and a logistic regression on the outcome regression	glm_Qr = "gn" and glm_gr = "Qn"
fitting	in sample, with this package’s initial outcome regression	cvFolds = 1, with the same regression as Qn
intervals	pointwise 95% Wald from the influence curve	the same

No tolg floor binds, because every declared value lies above 0.1. At guard=() the fit is the ordinary TMLE at the declared mechanism. The three-arm scenario is therefore the external comparison for the multi-arm known path. Under the "Q" guards, both sides fluctuate the mechanism along \(Q_r / g\), so the targeted \(g^*\) moves off the declaration.

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
known mechanism, outcome regression without W2, guard g	ate	average treatment effect	cleverly DR-TMLE on a declared known mechanism	-0.000825 to 0.0029	0.9520	1.0249	pass
known mechanism, outcome regression without W2, guard g	ate	average treatment effect	R drtmle with the known mechanism as gn	-0.000580 to 0.0031	0.9520	1.0256	pass
known mechanism, outcome regression without W2, guard g	ey0	counterfactual mean under no treatment	cleverly DR-TMLE on a declared known mechanism	-0.0019 to 0.000723	0.9460	1.0079	pass
known mechanism, outcome regression without W2, guard g	ey0	counterfactual mean under no treatment	R drtmle with the known mechanism as gn	-0.0020 to 0.000584	0.9420	1.0072	pass
known mechanism, outcome regression without W2, guard g	ey1	counterfactual mean under treatment	cleverly DR-TMLE on a declared known mechanism	-0.000947 to 0.0018	0.9570	1.0295	pass
known mechanism, outcome regression without W2, guard g	ey1	counterfactual mean under treatment	R drtmle with the known mechanism as gn	-0.000841 to 0.0020	0.9550	1.0305	pass
known mechanism, outcome regression without W2, no DR-TMLE guard	ate	average treatment effect	cleverly DR-TMLE on a declared known mechanism	-0.000732 to 0.0030	0.9510	1.0249	pass
known mechanism, outcome regression without W2, no DR-TMLE guard	ate	average treatment effect	R drtmle with the known mechanism as gn	-0.000732 to 0.0030	0.9590	1.0427	pass
known mechanism, outcome regression without W2, no DR-TMLE guard	ey0	counterfactual mean under no treatment	cleverly DR-TMLE on a declared known mechanism	-0.0020 to 0.000666	0.9400	1.0065	pass
known mechanism, outcome regression without W2, no DR-TMLE guard	ey0	counterfactual mean under no treatment	R drtmle with the known mechanism as gn	-0.0020 to 0.000666	0.9400	1.0114	pass
known mechanism, outcome regression without W2, no DR-TMLE guard	ey1	counterfactual mean under treatment	cleverly DR-TMLE on a declared known mechanism	-0.000903 to 0.0019	0.9540	1.0327	pass
known mechanism, outcome regression without W2, no DR-TMLE guard	ey1	counterfactual mean under treatment	R drtmle with the known mechanism as gn	-0.000903 to 0.0019	0.9650	1.0593	pass
known mechanism, outcome regression without W2, guard Q	ate	average treatment effect	cleverly DR-TMLE on a declared known mechanism	-0.000866 to 0.0029	0.9600	1.0206	pass
known mechanism, outcome regression without W2, guard Q	ate	average treatment effect	R drtmle with the known mechanism as gn	-0.000866 to 0.0029	0.9600	1.0209	pass
known mechanism, outcome regression without W2, guard Q	ey0	counterfactual mean under no treatment	cleverly DR-TMLE on a declared known mechanism	-0.0019 to 0.000733	0.9450	1.0095	pass
known mechanism, outcome regression without W2, guard Q	ey0	counterfactual mean under no treatment	R drtmle with the known mechanism as gn	-0.0019 to 0.000733	0.9430	1.0092	pass
known mechanism, outcome regression without W2, guard Q	ey1	counterfactual mean under treatment	cleverly DR-TMLE on a declared known mechanism	-0.000971 to 0.0018	0.9530	1.0258	pass
known mechanism, outcome regression without W2, guard Q	ey1	counterfactual mean under treatment	R drtmle with the known mechanism as gn	-0.000971 to 0.0018	0.9560	1.0269	pass
known mechanism, outcome regression without W2, guards Q and g	ate	average treatment effect	cleverly DR-TMLE on a declared known mechanism	-0.000874 to 0.0028	0.9580	1.0216	pass
known mechanism, outcome regression without W2, guards Q and g	ate	average treatment effect	R drtmle with the known mechanism as gn	-0.000944 to 0.0028	0.9600	1.0211	pass
known mechanism, outcome regression without W2, guards Q and g	ey0	counterfactual mean under no treatment	cleverly DR-TMLE on a declared known mechanism	-0.0019 to 0.000746	0.9460	1.0104	pass
known mechanism, outcome regression without W2, guards Q and g	ey0	counterfactual mean under no treatment	R drtmle with the known mechanism as gn	-0.0019 to 0.000764	0.9460	1.0104	pass
known mechanism, outcome regression without W2, guards Q and g	ey1	counterfactual mean under treatment	cleverly DR-TMLE on a declared known mechanism	-0.000968 to 0.0018	0.9510	1.0270	pass
known mechanism, outcome regression without W2, guards Q and g	ey1	counterfactual mean under treatment	R drtmle with the known mechanism as gn	-0.0010 to 0.0018	0.9540	1.0250	pass
three-arm law with a known mechanism, outcome regression without W2, no DR-TMLE guard	ate[1.0 vs 0.0]	difference in counterfactual means, 1.0 versus 0.0	cleverly DR-TMLE on a declared known mechanism	-0.0027 to 0.0017	0.9440	0.9741	pass
three-arm law with a known mechanism, outcome regression without W2, no DR-TMLE guard	ate[1.0 vs 0.0]	difference in counterfactual means, 1.0 versus 0.0	R drtmle with the known mechanism as gn	-0.0027 to 0.0017	0.9450	0.9771	pass
three-arm law with a known mechanism, outcome regression without W2, no DR-TMLE guard	ate[2.0 vs 0.0]	difference in counterfactual means, 2.0 versus 0.0	cleverly DR-TMLE on a declared known mechanism	-0.0015 to 0.0031	0.9440	0.9836	pass
three-arm law with a known mechanism, outcome regression without W2, no DR-TMLE guard	ate[2.0 vs 0.0]	difference in counterfactual means, 2.0 versus 0.0	R drtmle with the known mechanism as gn	-0.0015 to 0.0031	0.9440	0.9852	pass
three-arm law with a known mechanism, outcome regression without W2, no DR-TMLE guard	ey[0.0]	counterfactual mean under treatment arm ‘0.0’	cleverly DR-TMLE on a declared known mechanism	-0.0018 to 0.0016	0.9450	0.9753	pass
three-arm law with a known mechanism, outcome regression without W2, no DR-TMLE guard	ey[0.0]	counterfactual mean under treatment arm ‘0.0’	R drtmle with the known mechanism as gn	-0.0018 to 0.0016	0.9460	0.9779	pass
three-arm law with a known mechanism, outcome regression without W2, no DR-TMLE guard	ey[1.0]	counterfactual mean under treatment arm ‘1.0’	cleverly DR-TMLE on a declared known mechanism	-0.0020 to 0.000898	0.9510	1.0058	pass
three-arm law with a known mechanism, outcome regression without W2, no DR-TMLE guard	ey[1.0]	counterfactual mean under treatment arm ‘1.0’	R drtmle with the known mechanism as gn	-0.0020 to 0.000898	0.9510	1.0081	pass
three-arm law with a known mechanism, outcome regression without W2, no DR-TMLE guard	ey[2.0]	counterfactual mean under treatment arm ‘2.0’	cleverly DR-TMLE on a declared known mechanism	-0.000868 to 0.0023	0.9510	1.0164	pass
three-arm law with a known mechanism, outcome regression without W2, no DR-TMLE guard	ey[2.0]	counterfactual mean under treatment arm ‘2.0’	R drtmle with the known mechanism as gn	-0.000868 to 0.0023	0.9510	1.0171	pass

Agreement with the canonical implementation

Comparator-backed: a paired comparison with drtmle-r-known-g 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
known mechanism, outcome regression without W2, guard g	ate	average treatment effect	-0.000246	0.0719	1.0017	0	matching implementation	0.0033 vs 0.0500	equivalent
known mechanism, outcome regression without W2, guard g	ey0	counterfactual mean under no treatment	0.000140	0.0579	1.0017	0.0040	matching implementation	0.0026 vs 0.0500	equivalent
known mechanism, outcome regression without W2, guard g	ey1	counterfactual mean under treatment	-0.000106	0.0415	1.0023	0.0020	matching implementation	0.0033 vs 0.0500	equivalent
known mechanism, outcome regression without W2, no DR-TMLE guard	ate	average treatment effect	1.390e-10	4.034e-08	1.0000	-0.0080	matching implementation	0.0349 vs 0.0500	equivalent
known mechanism, outcome regression without W2, no DR-TMLE guard	ey0	counterfactual mean under no treatment	-1.384e-10	5.725e-08	1.0000	0	matching implementation	0.0097 vs 0.0500	equivalent
known mechanism, outcome regression without W2, no DR-TMLE guard	ey1	counterfactual mean under treatment	6.802e-13	2.647e-10	1.0000	-0.0110	matching implementation	0.0340 vs 0.0500	equivalent
known mechanism, outcome regression without W2, guard Q	ate	average treatment effect	4.718e-08	0.000014	1.0000	0	matching implementation	0.0011 vs 0.0500	equivalent
known mechanism, outcome regression without W2, guard Q	ey0	counterfactual mean under no treatment	-1.976e-07	0.000082	1.0000	0.0020	matching implementation	0.0011 vs 0.0500	equivalent
known mechanism, outcome regression without W2, guard Q	ey1	counterfactual mean under treatment	-1.504e-07	0.000059	1.0000	-0.0030	matching implementation	0.0027 vs 0.0500	equivalent
known mechanism, outcome regression without W2, guards Q and g	ate	average treatment effect	0.000065	0.0191	1.0004	-0.0020	matching implementation	0.0031 vs 0.0500	equivalent
known mechanism, outcome regression without W2, guards Q and g	ey0	counterfactual mean under no treatment	-0.000017	0.0070	1.0010	0	matching implementation	0.0012 vs 0.0500	equivalent
known mechanism, outcome regression without W2, guards Q and g	ey1	counterfactual mean under treatment	0.000048	0.0189	0.9996	-0.0030	matching implementation	0.0041 vs 0.0500	equivalent
three-arm law with a known mechanism, outcome regression without W2, no DR-TMLE guard	ate[1.0 vs 0.0]	difference in counterfactual means, 1.0 versus 0.0	1.256e-09	3.053e-07	1.0000	-0.0010	matching implementation	0.000135 vs 0.0500	equivalent
three-arm law with a known mechanism, outcome regression without W2, no DR-TMLE guard	ate[2.0 vs 0.0]	difference in counterfactual means, 2.0 versus 0.0	1.575e-10	3.690e-08	1.0000	0	matching implementation	0.000090 vs 0.0500	equivalent
three-arm law with a known mechanism, outcome regression without W2, no DR-TMLE guard	ey[0.0]	counterfactual mean under treatment arm ‘0.0’	-7.745e-10	2.505e-07	1.0000	-0.0010	matching implementation	0.000162 vs 0.0500	equivalent
three-arm law with a known mechanism, outcome regression without W2, no DR-TMLE guard	ey[1.0]	counterfactual mean under treatment arm ‘1.0’	4.814e-10	1.779e-07	1.0000	0	matching implementation	0.0047 vs 0.0500	equivalent
three-arm law with a known mechanism, outcome regression without W2, no DR-TMLE guard	ey[2.0]	counterfactual mean under treatment arm ‘2.0’	-6.170e-10	2.111e-07	1.0000	0	matching implementation	0.0014 vs 0.0500	equivalent

Theory properties

property	cell	role	what was tested	what must hold	measured	result
known_mechanism_accuracy	ate__drtmle_known__Q	positive	average treatment effect: DR-TMLE with guard Q on the declared mechanism, outcome regression without W2	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.0012 to 0.0027, margin 0.0060, coverage 0.9342 to 0.9694, SE ratio 0.9786	pass
known_mechanism_accuracy	ate__drtmle_known__Qg	positive	average treatment effect: DR-TMLE with guards Q and g on the declared mechanism, outcome regression without W2	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.0012 to 0.0027, margin 0.0059, coverage 0.9307 to 0.9669, SE ratio 0.9762	pass
known_mechanism_accuracy	ate__drtmle_known__g	positive	average treatment effect: DR-TMLE with guard g on the declared mechanism, outcome regression without W2	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.000977 to 0.0030, margin 0.0060, coverage 0.9295 to 0.9661, SE ratio 0.9723	pass
known_mechanism_accuracy	ate__drtmle_known__none	positive	average treatment effect: DR-TMLE with no guard on the declared mechanism, outcome regression without W2	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.0010 to 0.0029, margin 0.0060, coverage 0.9272 to 0.9644, SE ratio 0.9754	pass

Each property cell fits the ATE at one guard with the wrong outcome regression. The four cells read one draw for each replication, and these draws differ from the primary draws. At the declared mechanism the remainder is zero for any outcome regression. Each cell must therefore be unbiased, cover at the floor and report a standard error inside the SE-ratio screen.

Measured values

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	1000	primary replications per scenario
n	2000	observations per primary replication
independent_tests_total	34	implementation-estimand tests against the truth
independent_tests_passed	34	of those, passing
paired_tests_total	17	paired comparisons with R drtmle
paired_tests_passed	17	of those, passing
property_cells_total	4	repeated-sampling property cells
property_cells_passed	4	cells whose own and family verdicts pass
max_standardized_bias	0.0562	largest absolute primary bias in empirical standard deviations
min_coverage	0.9400	lowest measured primary-study coverage
min_se_ratio_ci_lower	0.9228	lowest bootstrap primary SE-ratio endpoint
max_se_ratio_ci_upper	1.1237	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. No cell of this study reads it
margin:calibration_coverage_upper	0.9800	calibration-cell coverage band, upper limit. No cell of this study reads it
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. No cell of this study reads it
margin:calibration_se_ratio_upper	1.0700	calibration-cell SE-ratio band, upper limit. No cell of this study reads it
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

Limitations

limit	what it means for use
a declared mechanism only	the study does not measure DRTMLE with an estimated mechanism. With one wrong nuisance and an estimated mechanism, the interval is not established here
complete data, in sample, with the univariate reduction	the cross-fitted, nested and bivariate paths have exact-law evidence only
no missing-outcome path	with delta=, the fit still estimates the observation mechanism. With a wrong outcome regression, the interval of that route is not established, because its remainder is linear in the observation mechanism’s error. This is a gap in the theory, not in the evidence. See a known treatment mechanism
one wrong outcome regression	a flexible outcome learner has fast-tier evidence only

Reproduction

The manifest records the seeds, the margins, the estimator configuration, the source hashes and the result hashes. Run python -m tests.canonical.known_mechanism_drtmle.regenerate to regenerate the artifacts. The replications and the property results carry every published row.