Ordinary missing-outcome TMLE

This study validates ordinary, non-cross-fitted point-treatment TMLE when the outcome is missing at random. It compares cleverly with R tmle 2.1.1 on identical samples, using the same realized nuisance predictions and bounds. The observational law has a nonconstant treatment mechanism and a response mechanism that depends on treatment and baseline covariates.

The comparison isolates the observed-data targeting construction. It reports the two treatment-specific means and their average treatment-effect contrast with pointwise 95% Wald intervals.

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
binary-outcome observational law with MAR outcomes	ate	average treatment effect	cleverly missing-outcome TMLE	-0.000900 to 0.0050	0.9587	1.0324	pass
binary-outcome observational law with MAR outcomes	ate	average treatment effect	R tmle	-0.000900 to 0.0050	0.9587	1.0324	pass
binary-outcome observational law with MAR outcomes	ey0	counterfactual mean under no treatment	cleverly missing-outcome TMLE	-0.0022 to 0.0021	0.9550	1.0081	pass
binary-outcome observational law with MAR outcomes	ey0	counterfactual mean under no treatment	R tmle	-0.0022 to 0.0021	0.9550	1.0081	pass
binary-outcome observational law with MAR outcomes	ey1	counterfactual mean under treatment	cleverly missing-outcome TMLE	-0.000084 to 0.0041	0.9425	1.0097	pass
binary-outcome observational law with MAR outcomes	ey1	counterfactual mean under treatment	R tmle	-0.000084 to 0.0041	0.9425	1.0097	pass

Agreement with the canonical implementation

Comparator-backed: a paired comparison with tmle-r 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 observational law with MAR outcomes	ate	average treatment effect	1.244e-09	2.566e-07	1.0000	0	matching implementation	5.149e-09 vs 0.0500	equivalent
binary-outcome observational law with MAR outcomes	ey0	counterfactual mean under no treatment	-8.769e-10	2.483e-07	1.0000	0	matching implementation	4.212e-09 vs 0.0500	equivalent
binary-outcome observational law with MAR outcomes	ey1	counterfactual mean under treatment	3.675e-10	1.065e-07	1.0000	0	matching implementation	4.198e-09 vs 0.0500	equivalent

Theory properties

property	cell	role	what was tested	what must hold	measured	result
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 0.9334 to 0.9525, SE ratio 0.9574 to 1.0146, empirical efficiency ratio 0.9826 to 1.0413, reported efficiency ratio 0.9960 to 0.9982	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 0.8147 to 0.8455, SE ratio 0.6824 to 0.7241, empirical efficiency ratio 1.3770 to 1.4608, reported efficiency ratio 0.9960 to 0.9982	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 0.8113 to 0.8424, SE ratio 0.6700 to 0.7098, empirical efficiency ratio 0.9834 to 1.0416, reported efficiency ratio 0.6972 to 0.6988	pass
mar_robustness	both_correct	positive	the outcome regression, treatment mechanism and observation mechanism are correct	bias interval inside the equivalence margin, SE ratio must remain between 0.1 and 10.0	bias -0.0026 to 0.0024, margin 0.0083, SE ratio 1.0011	pass
mar_robustness	mechanisms_correct	positive	the treatment and observation mechanisms are correct and the outcome regression is not	bias interval inside the equivalence margin, SE ratio must remain between 0.1 and 10.0	bias -0.0039 to 0.0023, margin 0.0103, SE ratio 0.9785	pass
mar_robustness	observation_wrong	control	only the treatment mechanism is correct	bias interval must fall entirely outside the margin, SE ratio must remain between 0.1 and 10.0	bias -0.4824 to -0.4783, margin 0.0069, SE ratio 1.6716	pass
mar_robustness	outcome_correct	positive	only the outcome regression is correct	bias interval inside the equivalence margin, SE ratio must remain between 0.1 and 10.0	bias -0.000857 to 0.0023, margin 0.0053, SE ratio 2.6184	pass
mar_robustness	treatment_wrong	control	only the observation mechanism is correct	bias interval must fall entirely outside the margin, SE ratio must remain between 0.1 and 10.0	bias -0.2110 to -0.2059, margin 0.0084, SE ratio 1.5397	pass
missingness_necessity	ate__complete_case_control	control	average treatment effect: the identical estimator silently discards unobserved outcomes and ignores selection	bias interval must fall entirely outside the margin	bias -0.1229 to -0.1172, margin 0.0096	pass
missingness_necessity	ate__declared	positive	average treatment effect: the observation indicator is declared, so correct mechanisms carry a wrong outcome model	bias interval inside the equivalence margin	bias -0.0055 to 0.000397, margin 0.0099	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 0.000585, coverage 0.9252 to 0.9667, SE ratio 0.9909	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 0.0012, coverage 0.9165 to 0.9606, SE ratio 0.9635	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 0.000099, coverage 0.9165 to 0.9606, SE ratio 1.0001	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 -0.5441 to -0.4770	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 -0.4993 to -0.4954	pass
targeting_necessity	ate__targeted	positive	average treatment effect: the estimator fluctuates a misspecified outcome model, so targeting does all the adjusting	bias interval inside the equivalence margin	bias -0.0043 to 0.0018, margin 0.0101	pass
targeting_necessity	ate__untargeted	control	average treatment effect: the identical fit with every fluctuation step removed	bias interval must fall entirely outside the margin	bias -0.4208 to -0.4198, margin 0.0017	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 0.0450, 0.0283 to 0.0674	pass

The property grid separates the three-nuisance robustness contract. A correct outcome regression must rescue wrong treatment and observation mechanisms; when the outcome regression is wrong, both mechanisms must be correct. Separate controls leave only one mechanism correct. Further cells test root-n contraction, exact-law efficiency, interval calibration, type-I error and power, targeting, and the consequence of silently analyzing complete cases.

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	19	property cells passing
property_cells_total	19	property cells reported
max_standardized_bias	0.0877	largest primary standardized bias
min_coverage	0.9425	lowest primary coverage
max_margin_utilization	2.566e-07	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:targeting_displacement	0.2500	minimum targeting displacement
margin:missingness_displacement	0.2500	minimum complete-case displacement

Limits

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

• The nuisance functions are supplied as finite-support oracles. This isolates targeting and inference; it does not validate flexible learner wrappers or cross-fitting.

• The comparison uses pointwise Wald intervals and does not cover simultaneous bands. The default-band study measures the default band of this study’s subject fit.

• The study reports the two arm means and their difference. The att, atc, rr and or estimands keep only their exact-law Gateaux and remainder evidence under delta=.

• The study excludes weights, clusters, missing treatment, multinomial treatment, MNAR outcomes, longitudinal data, and DR-TMLE correction cycles.

Reproduction

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