Learned weighted point-treatment TMLE

This study validates ordinary point-treatment TMLE when fixed observation weights reach the nuisance learners. The fit sends the weights to both learner classes. The repeated-sampling witness resolves the outcome regression. The study draws 2,000 observations from a selected population. The weighted fit targets a uniform population and learns both nuisance regressions from each sample.

The selected density of W1 is (1 + 0.75 * W1) / 2 on [-1, 1]. W2 is uniform on the same interval. The fixed weight 1 / (1 + 0.75 * W1) recovers the uniform target law. Treatment is randomized with probability 0.5. The continuous outcome has unit-variance Gaussian noise and conditional mean 0.5 + 0.5 * W1 + 0.25 * W2 + A * (1 + 2 * W1).

The target arm means are 0.5 and 1.5, so the target ATE is 1.0. The selected ATE is 1.5. That difference supplies a nonvacuous witness that learner weights change the fitted outcome regression.

What was compared

setting	cleverly	R tmle
estimator	ordinary TMLE, cross-fitting disabled	ordinary tmle
samples	exact-size draws from the selected continuous law	the identical rows
outcome nuisance	weighted main-effects linear regression	weighted Gaussian main-effects GLM
treatment nuisance	weighted unpenalized logistic regression	weighted binomial main-effects GLM
probability weights	exact target-to-selected density ratio	the same values through obsWeights=
fluctuation	linear	linear
estimands	both arm means and ATE	the same three parameters
intervals	pointwise 95% Wald	pointwise 95% Wald

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
continuous-outcome law selected by a covariate-dependent density	ate	average treatment effect	cleverly weighted point-treatment TMLE with learned nuisances	-0.0036 to 0.0075	0.9413	0.9894	pass
continuous-outcome law selected by a covariate-dependent density	ate	average treatment effect	R tmle with learned weighted nuisances	-0.0036 to 0.0075	0.9413	0.9894	pass
continuous-outcome law selected by a covariate-dependent density	ey0	counterfactual mean under no treatment	cleverly weighted point-treatment TMLE with learned nuisances	-0.0029 to 0.0037	0.9712	1.1152	pass
continuous-outcome law selected by a covariate-dependent density	ey0	counterfactual mean under no treatment	R tmle with learned weighted nuisances	-0.0029 to 0.0037	0.9712	1.1153	pass
continuous-outcome law selected by a covariate-dependent density	ey1	counterfactual mean under treatment	cleverly weighted point-treatment TMLE with learned nuisances	-0.0026 to 0.0072	0.9563	1.0606	pass
continuous-outcome law selected by a covariate-dependent density	ey1	counterfactual mean under treatment	R tmle with learned weighted nuisances	-0.0026 to 0.0072	0.9563	1.0607	pass

The ey0 standard error is conservative. It sits about 11.5% above the sampling spread, and the coverage is 0.9712. The two implementations agree to four decimal places, at 1.1152 and 1.1153. The ratio is therefore a property of the estimator under the misspecified outcome regression. It stays inside the 0.8 to 1.2 sanity screen.

Agreement with the canonical implementation

Comparator-backed: a paired comparison with tmle-r-learned-weighted 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
continuous-outcome law selected by a covariate-dependent density	ate	average treatment effect	2.830e-07	0.000031	1.0000	0	matching implementation	0.000008 vs 0.0500	equivalent
continuous-outcome law selected by a covariate-dependent density	ey0	counterfactual mean under no treatment	0.000002	0.000325	1.0002	0	matching implementation	0.000146 vs 0.0500	equivalent
continuous-outcome law selected by a covariate-dependent density	ey1	counterfactual mean under treatment	0.000002	0.000257	1.0002	0	matching implementation	0.000111 vs 0.0500	equivalent

The comparator-backed comparison shows that the implementations agree on all three parameters. This comparison is independent because R tmle fits its own weighted Gaussian and binomial nuisance regressions on the shared samples.

Theory properties

property	cell	role	what was tested	what must hold	measured	result
interval_calibration	ate__noise_control	control	average treatment effect: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8238 to 0.8623, SE ratio 0.6964 to 0.7493	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 0.8203 to 0.8592, SE ratio 0.6804 to 0.7328	pass
interval_calibration	ate__treatment_correct	positive	average treatment effect: the randomized treatment mechanism is correct while the outcome regression omits effect modification	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9420 to 0.9645, SE ratio 0.9715 to 1.0471	pass
learner_weight_necessity	ate__unweighted_plugin_control	control	average treatment effect: the untargeted plug-in from nuisance fits that omit weights	target bias outside its margin, selected-population bias inside its margin, and paired displacement above its threshold	bias 0.4962 to 0.5038, margin 0.0128, selected-target bias -0.0038 to 0.0038, margin 0.0128	pass
learner_weight_necessity	ate__unweighted_targeted	positive	average treatment effect: nuisance fits omit weights, while targeting and averaging retain them	target-population bias interval inside the equivalence margin	bias -0.0047 to 0.0047, margin 0.0157	pass
learner_weight_necessity	ate__weighted_plugin	positive	average treatment effect: the untargeted plug-in from both weighted nuisance fits	target-population bias interval inside the equivalence margin	bias -0.0038 to 0.0056, margin 0.0157	pass
learner_weight_necessity	ate__weighted_targeted	positive	average treatment effect: the weighted nuisance fits followed by weighted targeting and averaging	target-population bias interval inside the equivalence margin	bias -0.0045 to 0.0048, margin 0.0157	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 0.0032, coverage 0.9356 to 0.9737, SE ratio 0.9941	pass
root_n_and_efficiency	n_500	positive	bias, coverage and SE calibration at n = 500	bias inside the margin, coverage clears the floor, SE ratio inside the band	bias -0.0031, coverage 0.9282 to 0.9688, SE ratio 0.9982	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 0.000978, coverage 0.9179 to 0.9616, SE ratio 0.9805	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 -0.5278 to -0.4580	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 -0.5009 to -0.4969	pass
type_i_error	target_null	positive	a selected law whose weighted target contrast is exactly zero	one-sided rejection bound stays under the declared type-I ceiling	rejection 0.0400, 0.0243 to 0.0614	pass

The learner-weight control changes only nuisance fitting. Both arms retain weights in targeting, plug-in averaging, covariance, and inference. The weighted untargeted plug-in recovers the target ATE, while the unweighted plug-in recovers the selected ATE. Their paired separation exceeds the declared threshold.

Both targeted arms recover the target ATE. This is expected because the correctly specified randomized treatment mechanism repairs the unweighted outcome learner through double robustness. The two targeted passes therefore do not replace the untargeted learner-weight witness.

Measured values and declared margins

quantity	value	source
replicates	800	paired primary 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	0.0432	largest primary standardized bias
min_coverage	0.9413	lowest primary coverage
max_margin_utilization	0.000325	largest paired similarity-margin share
properties[learner_weight_necessity/ate__unweighted_plugin_control]:necessity_displacement	7.9655	paired learner-weight displacement
properties[learner_weight_necessity/ate__unweighted_plugin_control]:alternative_truth	1.5000	exact selected-population ATE
properties[learner_weight_necessity/ate__unweighted_plugin_control]:alternative_bias_ci_lower	-0.0038	lower bias endpoint against the selected target
properties[learner_weight_necessity/ate__unweighted_plugin_control]:alternative_bias_ci_upper	0.0038	upper bias endpoint against the selected target
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: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:shrunken_se_factor	0.7000	negative-control SE multiplier
margin:learner_weight_displacement	0.5000	minimum learner-weight displacement

Limits

• The law has binary treatment, a continuous outcome, two continuous baseline covariates, and known fixed weights.

• The nuisance learners are ordinary main-effects regressions, and the treatment assignment is randomized. Flexible learners and observational treatment remain outside this row.

• The weights are exact and moderately variable. Estimated, trimmed, calibrated, normalized, and replicate weights remain outside this row.

• The study covers ordinary pointwise inference for two arm means and their difference. It excludes ratios, ATT, ATC, missing outcomes, clusters, strata, and simultaneous bands. The default-band study measures the default band of this study’s subject fit.

• Cross-fitted weighted point-treatment nuisances remain outside this row. Weighted longitudinal targeting has separate ordinary and cross-fitted reporting studies.

• The learner-weight witness resolves the outcome regression only. Treatment is randomized with probability 0.5, and A ~ W1 + W2 contains the truth. A weighted treatment fit and an unweighted treatment fit therefore converge to the same regression. Only tests/unit/test_learned_weighted_tmle_method_study.py::test_both_nuisance_learners_consume_the_declared_weights checks the weights the treatment learner receives.

• The study declares two calibration controls and judges both with one detector. The shrunken-standard-error arm and the noise arm answer to the SE-ratio rule, because the study publishes no exact efficiency ratio. A study that publishes one judges its noise arm on the empirical efficiency ratio instead.

Reproduction

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