Randomized missing-outcome DR-TMLE

This study validates randomized missing-outcome DR-TMLE on a binary trial with a response mechanism that depends on treatment and baseline covariates. It supplies the known treatment probability and compares the both-correct limit with R drtmle 1.1.2 at pinned commit 538a3a2.

The comparator boundary matters. R drtmle represents treatment and response through one joint mechanism. cleverly implements separate treatment and observation mechanisms and the five reduced regressions from the randomized missing-outcome construction. The paired comparison can witness the shared both-correct limit; it cannot establish parity for that internal correction cycle. Independent property cells test the two one-correct drift directions and mutate the cycle directly.

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
binary-outcome randomized law with MAR outcomes	ate	average treatment effect	cleverly randomized missing-outcome DR-TMLE	-0.000928 to 0.0050	0.9537	1.0224	pass
binary-outcome randomized law with MAR outcomes	ate	average treatment effect	R drtmle with a joint treatment-response mechanism	-0.000917 to 0.0050	0.9550	1.0238	pass
binary-outcome randomized law with MAR outcomes	ey0	counterfactual mean under no treatment	cleverly randomized missing-outcome DR-TMLE	-0.0032 to 0.0013	0.9575	1.0294	pass
binary-outcome randomized law with MAR outcomes	ey0	counterfactual mean under no treatment	R drtmle with a joint treatment-response mechanism	-0.0032 to 0.0013	0.9613	1.0307	pass
binary-outcome randomized law with MAR outcomes	ey1	counterfactual mean under treatment	cleverly randomized missing-outcome DR-TMLE	-0.000839 to 0.0030	0.9513	1.0065	pass
binary-outcome randomized law with MAR outcomes	ey1	counterfactual mean under treatment	R drtmle with a joint treatment-response mechanism	-0.000814 to 0.0031	0.9500	1.0075	pass

Agreement with the canonical implementation

Comparator-backed: a paired comparison with drtmle-r-mar 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 randomized law with MAR outcomes	ate	average treatment effect	-0.000009	0.0018	1.0022	-0.0012	different estimator	0.0038 vs 0.0500	equivalent
binary-outcome randomized law with MAR outcomes	ey0	counterfactual mean under no treatment	-0.000015	0.0040	1.0016	-0.0037	different estimator	0.0035 vs 0.0500	equivalent
binary-outcome randomized law with MAR outcomes	ey1	counterfactual mean under treatment	-0.000023	0.0073	1.0021	0.0013	different estimator	0.0030 vs 0.0500	equivalent

Repeated-sampling properties

property	cell	role	what was tested	what must hold	measured	result
corrected_mar_inference	both_correct	positive	the outcome regression and observation mechanism are correctly specified	bias interval inside the margin, coverage clears the floor, SE ratio inside the band, SE ratio must remain between 0.1 and 10.0	bias -0.0051 to 0.0012, coverage 0.9121 to 0.9575, SE ratio 0.9701	pass
corrected_mar_inference	both_wrong	control	the outcome regression and observation mechanism are both misspecified	bias interval must fall entirely outside the margin, SE ratio must remain between 0.1 and 10.0	bias -0.1327 to -0.1268, coverage 0.0070 to 0.0316, SE ratio 0.9983	pass
corrected_mar_inference	observation_drift	positive	the outcome regression is correct and the observation mechanism is misspecified	bias interval inside the margin, coverage clears the floor, SE ratio inside the band, SE ratio must remain between 0.1 and 10.0	bias 0.000650 to 0.0063, coverage 0.9194 to 0.9627, SE ratio 1.0103	pass
corrected_mar_inference	outcome_drift	positive	the outcome regression is misspecified and the observation mechanism is correct	bias interval inside the margin, coverage clears the floor, SE ratio inside the band, SE ratio must remain between 0.1 and 10.0	bias -0.0017 to 0.0043, coverage 0.9341 to 0.9727, SE ratio 1.0282	pass
correction_necessity	five_reduction_cycle__closed_score	positive	five-reduction correction cycle: the correction scores after the complete five-reduction cycle	the upper confidence endpoint is below the declared fraction of the initial-score lower endpoint	score -3.342e-08 to 2.323e-07	pass
correction_necessity	five_reduction_cycle__initial_score_control	control	five-reduction correction cycle: the same correction scores before the cycle is run	the lower confidence endpoint clears the declared unresolved-score floor	score 0.0033 to 0.0038	pass
interval_calibration	ate__correctly_specified	positive	average treatment effect: all three required nuisance functions 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 0.9319 to 0.9563, SE ratio 0.9581 to 1.0289, empirical efficiency ratio 0.9717 to 1.0435, reported efficiency ratio 0.9985 to 1.0012	pass
interval_calibration	ate__noise_control	control	average treatment effect: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8146 to 0.8540, SE ratio 0.6781 to 0.7297, empirical efficiency ratio 1.3703 to 1.4743, reported efficiency ratio 0.9985 to 1.0013	pass
interval_calibration	ate__shrunken_se_control	control	average treatment effect: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8059 to 0.8461, SE ratio 0.6694 to 0.7213, empirical efficiency ratio 0.9701 to 1.0456, reported efficiency ratio 0.6989 to 0.7009	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 -0.000264, coverage 0.9392 to 0.9704, SE ratio 1.0032	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 0.000770, coverage 0.9248 to 0.9599, SE ratio 0.9935	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 0.000680, coverage 0.9267 to 0.9613, SE ratio 0.9861	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 -0.5237 to -0.4723	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 -0.5015 to -0.4987	pass

The property grid reports the both-correct limit, outcome drift, observation drift, and a both-wrong control. It also tests root-n contraction, exact-law efficiency, interval calibration, and the five-reduction cycle’s empirical score reduction against the same scores before correction.

Measured values and declared margins

quantity	value	source
replicates	800	paired replications
n	2000	observations per primary 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	14	property cells passing
property_cells_total	14	property cells reported
max_standardized_bias	0.0631	largest primary standardized bias
min_coverage	0.9500	lowest primary coverage
max_margin_utilization	0.0073	largest paired similarity-margin share
margin:confidence_level	0.9900	Monte Carlo confidence level
margin:alpha	0.0500	nominal 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: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: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:correction_score_ratio	0.0100	maximum closed-to-initial score-endpoint ratio
margin:uncorrected_score_floor	0.0010	unresolved initial-score floor

Limits

• The study covers binary treatment, binary outcome, one three-level baseline covariate, known 1:1 randomization, and an MAR response mechanism.

• Initial outcome and observation regressions are finite-support oracles; the reduced regressions use linear and logistic models without cross-fitting.

• The comparison uses pointwise Wald intervals. Its joint-mechanism R comparator establishes only the shared both-correct limit, not internal five-reduction parity.

• The property grid declares no type-I error, power, or targeting cell. The ordinary missing-outcome study carries those three instruments for the delta= path.

• The study excludes observational treatment assignment, weights, clusters, simultaneous bands, missing treatment, multinomial treatment, MNAR outcomes, and other DR-TMLE compositions. The default-band study measures the default band of this study’s subject fit.

• The drift cells give the consistent nuisance as its exact law table, so the conditioning terms of the theorem are zero by construction. The cells measure the targeting, not one-wrong inference with an estimated nuisance.

Reproduction

The fixture README, manifest, replications, paired decisions, and property results carry the protocol, provenance, and every published row.