Randomized multi-arm missing-outcome DR-TMLE

This study validates missing-outcome DR-TMLE at three arms. The law is L3 of tests/studies/mar_arm_indexed_laws.py: three arms named low, mid and high, a binary outcome, and a response mechanism that depends on treatment and a baseline covariate. The fit receives the W-stratified treatment table as known probabilities, keyed by level. The study compares the both-correct limit with R drtmle 1.1.2 at pinned commit 538a3a2.

The comparator boundary matters. R drtmle fits one joint treatment-response mechanism with the composite response A == a & DeltaA == 1 & DeltaY == 1. Díaz and van der Laan (2017, page 25) reject that composite construction for this problem. cleverly keeps the treatment and observation mechanisms apart, with five reductions for each arm and a separate tilt for each arm. So the paired comparison shows the shared both-correct limit only. The property cells test the armwise construction itself.

This study is the only measurement of the default simultaneous band at more than two arms for this estimator. The default-simultaneous-bands study measures the two-arm band alone.

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	ate[low vs high]	difference in counterfactual means, low versus high	cleverly randomized multi-arm missing-outcome DR-TMLE	-0.0017 to 0.0053	0.9413	0.9846	pass
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	ate[low vs high]	difference in counterfactual means, low versus high	R drtmle at three arms with a joint treatment-response mechanism	-0.0017 to 0.0053	0.9400	0.9851	pass
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	ate[mid vs high]	difference in counterfactual means, mid versus high	cleverly randomized multi-arm missing-outcome DR-TMLE	-0.0023 to 0.0043	0.9587	1.0052	pass
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	ate[mid vs high]	difference in counterfactual means, mid versus high	R drtmle at three arms with a joint treatment-response mechanism	-0.0023 to 0.0043	0.9575	1.0052	pass
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	ey[high]	counterfactual mean under treatment arm ‘high’	cleverly randomized multi-arm missing-outcome DR-TMLE	-0.0035 to 0.0014	0.9400	0.9827	pass
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	ey[high]	counterfactual mean under treatment arm ‘high’	R drtmle at three arms with a joint treatment-response mechanism	-0.0036 to 0.0014	0.9413	0.9829	pass
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	ey[low]	counterfactual mean under treatment arm ‘low’	cleverly randomized multi-arm missing-outcome DR-TMLE	-0.0019 to 0.0033	0.9300	0.9557	pass
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	ey[low]	counterfactual mean under treatment arm ‘low’	R drtmle at three arms with a joint treatment-response mechanism	-0.0019 to 0.0033	0.9300	0.9561	pass
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	ey[mid]	counterfactual mean under treatment arm ‘mid’	cleverly randomized multi-arm missing-outcome DR-TMLE	-0.0022 to 0.0021	0.9563	1.0170	pass
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	ey[mid]	counterfactual mean under treatment arm ‘mid’	R drtmle at three arms with a joint treatment-response mechanism	-0.0023 to 0.0020	0.9563	1.0172	pass
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	or[low vs high]	marginal odds ratio, low versus high, reported on the log scale	cleverly randomized multi-arm missing-outcome DR-TMLE	-0.0123 to 0.0189	0.9450	0.9832	pass
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	or[low vs high]	marginal odds ratio, low versus high, reported on the log scale	R drtmle at three arms with a joint treatment-response mechanism	-0.0122 to 0.0190	0.9437	0.9837	pass
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	or[mid vs high]	marginal odds ratio, mid versus high, reported on the log scale	cleverly randomized multi-arm missing-outcome DR-TMLE	-0.0124 to 0.0165	0.9600	1.0009	pass
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	or[mid vs high]	marginal odds ratio, mid versus high, reported on the log scale	R drtmle at three arms with a joint treatment-response mechanism	-0.0123 to 0.0165	0.9587	1.0010	pass
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	rr[low vs high]	marginal risk ratio, low versus high, reported on the log scale	cleverly randomized multi-arm missing-outcome DR-TMLE	-0.0062 to 0.0090	0.9363	0.9750	pass
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	rr[low vs high]	marginal risk ratio, low versus high, reported on the log scale	R drtmle at three arms with a joint treatment-response mechanism	-0.0061 to 0.0090	0.9363	0.9756	pass
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	rr[mid vs high]	marginal risk ratio, mid versus high, reported on the log scale	cleverly randomized multi-arm missing-outcome DR-TMLE	-0.0042 to 0.0066	0.9613	1.0102	pass
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	rr[mid vs high]	marginal risk ratio, mid versus high, reported on the log scale	R drtmle at three arms with a joint treatment-response mechanism	-0.0042 to 0.0066	0.9613	1.0102	pass

Agreement with the canonical implementation

Comparator-backed: a paired comparison with drtmle-r-multi-arm-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
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	ate[low vs high]	difference in counterfactual means, low versus high	-0.000019	0.0033	1.0021	0.0013	different estimator	0.0016 vs 0.0500	equivalent
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	ate[mid vs high]	difference in counterfactual means, mid versus high	9.748e-07	0.000181	1.0011	0.0012	different estimator	0.0015 vs 0.0500	equivalent
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	ey[high]	counterfactual mean under treatment arm ‘high’	0.000027	0.0067	1.0011	-0.0013	different estimator	0.0014 vs 0.0500	equivalent
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	ey[low]	counterfactual mean under treatment arm ‘low’	0.000008	0.0019	1.0028	0	different estimator	0.0021 vs 0.0500	equivalent
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	ey[mid]	counterfactual mean under treatment arm ‘mid’	0.000028	0.0079	1.0014	0	different estimator	0.0017 vs 0.0500	equivalent
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	or[low vs high]	marginal odds ratio, low versus high, reported on the log scale	-0.000023	0.0032	1.0024	0.0012	different estimator	0.0016 vs 0.0500	equivalent
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	or[mid vs high]	marginal odds ratio, mid versus high, reported on the log scale	-0.000009	0.000705	1.0016	0.0012	different estimator	0.0015 vs 0.0500	equivalent
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	rr[low vs high]	marginal risk ratio, low versus high, reported on the log scale	-0.000010	0.0015	1.0026	0	different estimator	0.0018 vs 0.0500	equivalent
L3: three-arm binary-outcome law with MAR outcomes and known W-stratified assignment	rr[mid vs high]	marginal risk ratio, mid versus high, reported on the log scale	0.000009	0.0012	1.0012	0	different estimator	0.0015 vs 0.0500	equivalent

Repeated-sampling properties

property	cell	role	what was tested	what must hold	measured	result
corrected_mar_inference	l3_ate_low__both_correct	positive	L3 difference, low versus high: 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 -0.0026 to 0.0043, coverage 0.9282 to 0.9688, SE ratio 1.0110	pass
corrected_mar_inference	l3_ate_low__both_wrong	control	L3 difference, low versus high: the outcome regression and observation mechanism are both misspecified	bias interval must fall entirely outside the margin, and its distance from zero must clear the declared floor, SE ratio must remain between 0.1 and 10.0	bias 0.0129 to 0.0201, coverage 0.8878 to 0.9396, SE ratio 0.9697	pass
corrected_mar_inference	l3_ate_low__observation_drift	positive	L3 difference, low versus high: 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 -0.0030 to 0.0040, coverage 0.9238 to 0.9657, SE ratio 0.9644	pass
corrected_mar_inference	l3_ate_low__outcome_drift	positive	L3 difference, low versus high: 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 -0.0043 to 0.0027, coverage 0.9150 to 0.9596, SE ratio 0.9815	pass
corrected_mar_inference	l3_ate_mid__both_correct	positive	L3 difference, mid versus high: 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 -0.0038 to 0.0026, coverage 0.9371 to 0.9747, SE ratio 1.0262	pass
corrected_mar_inference	l3_ate_mid__both_wrong	control	L3 difference, mid versus high: the outcome regression and observation mechanism are both misspecified	bias interval must fall entirely outside the margin, and its distance from zero must clear the declared floor, SE ratio must remain between 0.1 and 10.0	bias -0.0242 to -0.0168, coverage 0.8585 to 0.9167, SE ratio 0.9312	pass
corrected_mar_inference	l3_ate_mid__observation_drift	positive	L3 difference, mid versus high: 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 -0.0028 to 0.0038, coverage 0.9208 to 0.9637, SE ratio 0.9957	pass
corrected_mar_inference	l3_ate_mid__outcome_drift	positive	L3 difference, mid versus high: 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 -0.0044 to 0.0020, coverage 0.9356 to 0.9737, SE ratio 1.0466	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 7.012e-10 to 7.545e-08	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 0.0104 to 0.0114	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 0.9287 to 0.9537, SE ratio 0.9294 to 1.0000, empirical efficiency ratio 0.9994 to 1.0751, reported efficiency ratio 0.9980 to 1.0008	fail
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 0.8081 to 0.8481, SE ratio 0.6664 to 0.7185, empirical efficiency ratio 1.3910 to 1.5002, reported efficiency ratio 0.9979 to 1.0008	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 0.8003 to 0.8409, SE ratio 0.6523 to 0.7004, empirical efficiency ratio 0.9989 to 1.0725, reported efficiency ratio 0.6986 to 0.7006	pass
power	l3_ate_mid__randomized_alternative	positive	L3 difference, mid versus high: the same test applied to the randomized law’s own nonzero contrast	rejection lower bound clears the minimum power	rejection 0.9800, 0.9671 to 0.9889	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 0.0013, coverage 0.9353 to 0.9676, SE ratio 1.0027	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 -0.000473, coverage 0.9392 to 0.9704, SE ratio 1.0187	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 0.000785, coverage 0.9344 to 0.9669, SE ratio 1.0029	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 -0.5201 to -0.4675	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 -0.5008 to -0.4977	pass
simultaneous_coverage	arms__pointwise_joint_control	control	the multi-arm fit’s three arm means, two differences, two risk ratios and two odds ratios: the pointwise 95% intervals of the same fits, read jointly	joint coverage upper endpoint must fall below the nominal rate	joint coverage 0.7713 to 0.8142	pass
simultaneous_coverage	arms__simultaneous_band	positive	the multi-arm fit’s three arm means, two differences, two risk ratios and two odds ratios: 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 0.9196 to 0.9462	fail
type_i_error	l3_ate_mid__randomized_sharp_null	positive	L3 difference, mid versus high: a randomized law whose two compared arms share one outcome regression, so the contrast is exactly zero	one-sided rejection bound stays under the declared type-I ceiling, and coverage clears the floor	rejection 0.0583, 0.0423 to 0.0780	pass

The property grid reports both contrasts against high under four nuisance configurations. The drift cells use a constant outcome regression and a constant observation mechanism, as the two-arm study does. The grid also tests root-n contraction, interval calibration with its two controls, the size and power of the Wald test, the five-reduction score reduction, and the joint coverage of the band over all nine estimates.

Two positive cells are red, and both read the same 2,400 calibration fits. The SE-ratio interval of ate__correctly_specified ends at 0.9294 against a floor of 0.93. The joint-coverage interval of arms__simultaneous_band ends at 0.9196 against 0.92. The study publishes them red under the reporting policy.

The owner F4-calibration-draws in the roadmap records the diagnosis. The spread of that draw set is 1.037 times its mean standard error. Two independent cells of the same configuration read 0.997 and 0.989. Pooled over all three cells, the ratio is 1.020, with a 99% interval of 0.991 to 1.047. A Bartlett test across the cells gives p = 0.20, so the draw set differs by no more than chance. The oracle band covers as the package band does.

Measured values and declared margins

quantity	value	source
replicates	800	paired replications
n	2000	observations per primary replication
independent_tests_passed	18	truth tests passing
independent_tests_total	18	truth tests reported
paired_tests_passed	9	paired comparisons passing
paired_tests_total	9	paired comparisons reported
property_cells_passed	20	property cells passing
property_cells_total	22	property cells reported
max_standardized_bias	0.0467	largest primary standardized bias
min_coverage	0.9300	lowest primary coverage
max_margin_utilization	0.0079	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 one law with three arms, a binary outcome, one three-level baseline covariate, a known W-stratified randomization, and an MAR response mechanism.

• Every study fit passes the treatment probabilities as known. A fit with randomized=True estimates the categorical mechanism instead. Only the unit tests in tests/unit/test_drtmle_missing_multi_arm.py cover that route.

• The initial outcome and observation regressions are finite-support oracles. The reduced regressions use linear and logistic models without cross-fitting.

• The R comparator establishes the shared both-correct limit only, not parity of the armwise five-reduction construction.

• The study excludes observational assignment, weights, clusters, missing treatment, MNAR outcomes, cross-fitting, and other DR-TMLE compositions.

• 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.