Outcome-adaptive per-arm C-TMLE

This study measures CTMLE(strategy="oat") with the default oat_design="per_arm". For each arm \(a\), the fit regresses \(1\{A = a\}\) on the one column \(\bar Q_n(a, W)\). The section the per-arm outcome-adaptive design states the estimator and its curve. The base result is Theorem 1 of Benkeser, Cai and van der Laan (2020) for one treatment-specific mean, applied to each arm.

The study pairs the fit with R drtmle 1.1.2 at commit 538a3a2, with adapt_g = TRUE. The runner calls drtmle once per arm on the binary recode \(1\{A = a\}\). Both sides read this package’s initial outcome regression, so the pairs test the design, the mechanism and the targeting step.

The study publishes under the reporting policy. Its declared revert cells are red, so every per-arm fit takes the generated_design_plugin status and reports no interval. A shared-design fit already withheld its interval before this study. The coverage and standard-error columns below measure plugin_interval and plugin_std_error as diagnostics. Their pass labels do not authorize ci, pvalue or std_error. The applied revert gives the diagnosis, and F19 owns every red row.

What was compared

setting	cleverly	R drtmle
datasets	800 samples of 1,500 rows for each law	the identical rows
law	\(W = (W_1, W_2, W_3)\) is independent standard normal. The outcome is binary with \(\operatorname{logit} \bar Q_0(a, W) = c_a + u_{a1} W_1 + u_{a2} W_2\), so each arm reads its own index. The treatment law is a softmax in \(W_1\) and \(W_3\). \(W_3\) enters only the treatment law, so it is an instrument	the same
scenarios	binary_active: two arms. three_arm_active: three arms, with arm 0 as the contrast arm	the same
reported parameters	at two arms, the arm means, the ATE, RR, OR, PAR and PAF. At three arms, the three arm means and the ATE, RR and OR of arms 1 and 2 against arm 0	the same
outcome regression	an unpenalized logistic regression on the arm indicators, \(W\) and \(A \times W\). It is correctly specified	this package’s initial fit, passed as Qn
mechanism	for each arm, an unpenalized logistic regression of \(1\{A = a\}\) on \((l, l^2, l^3)\), \(l = \operatorname{logit} \bar Q_n(a, W)\), with \(\bar Q_n\) clipped to \([10^{-6}, 1 - 10^{-6}]\). g_bounds=(0.01, 0.99)	glm_g with the same cubic, tolg = 0.01
targeting	one logistic fluctuation for each arm on that arm’s clever covariate	the TMLE stage of drtmle
intervals	pointwise 95% Wald from the per-arm curve, with RR and OR on the log scale. They are diagnostics	the same, from tmle$cov

The drtmle result carries no TMLE curve. The runner rebuilds each arm’s curve on its one-dimensional logistic submodel at the reported estimate. It refuses a fit unless the rebuilt standard error equals the square root of tmle$cov[1, 1] to a relative 1e-8. The contrasts use the rebuilt curves. A call with a_0 equal to every level does not fit this design, because drtmle then builds a sequential chain of conditional binary regressions.

The table gives the law-level numbers. A \(10^6\)-draw Monte Carlo with Gauss-Hermite projections computes them, in tests/studies/oat_per_arm_laws.py:design_numbers. The efficiency ratio is the variance of Theorem 1’s curve over the variance of the efficient influence function.

quantity	binary_active	three_arm_active
arm means	0.4406, 0.5593	0.4178, 0.4999, 0.5834
efficiency ratio, arm means	0.428, 0.428	0.792, 0.178, 0.178
\(L_2\) gap between the per-arm and the shared design limits	0.059, 0.059	0.037, 0.114, 0.059
share of rows with the true \(g_0\) outside \((0.025, 0.975)\)	0.029	0.004, 0.072, 0.065
\(L_2\) distance from the learner limit to the projection	below \(10^{-6}\)	below \(2 \times 10^{-4}\)

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
two arms, an arm-specific outcome index, an instrument, correct outcome regression	ate	average treatment effect	cleverly per-arm outcome-adaptive C-TMLE	-0.0021 to 0.0031	0.9087	0.8500	fail
two arms, an arm-specific outcome index, an instrument, correct outcome regression	ate	average treatment effect	R drtmle with adapt_g, by a binary recode per arm	-0.0021 to 0.0031	0.9087	0.8500	fail
two arms, an arm-specific outcome index, an instrument, correct outcome regression	ey0	counterfactual mean under no treatment	cleverly per-arm outcome-adaptive C-TMLE	-0.0024 to 0.0013	0.9025	0.8526	fail
two arms, an arm-specific outcome index, an instrument, correct outcome regression	ey0	counterfactual mean under no treatment	R drtmle with adapt_g, by a binary recode per arm	-0.0024 to 0.0013	0.9025	0.8526	fail
two arms, an arm-specific outcome index, an instrument, correct outcome regression	ey1	counterfactual mean under treatment	cleverly per-arm outcome-adaptive C-TMLE	-0.0019 to 0.0018	0.9062	0.8463	fail
two arms, an arm-specific outcome index, an instrument, correct outcome regression	ey1	counterfactual mean under treatment	R drtmle with adapt_g, by a binary recode per arm	-0.0019 to 0.0018	0.9062	0.8463	fail
two arms, an arm-specific outcome index, an instrument, correct outcome regression	or	marginal odds ratio, reported on the log scale	cleverly per-arm outcome-adaptive C-TMLE	-0.0078 to 0.0136	0.9087	0.8493	fail
two arms, an arm-specific outcome index, an instrument, correct outcome regression	or	marginal odds ratio, reported on the log scale	R drtmle with adapt_g, by a binary recode per arm	-0.0078 to 0.0136	0.9087	0.8493	fail
two arms, an arm-specific outcome index, an instrument, correct outcome regression	paf	population attributable fraction	cleverly per-arm outcome-adaptive C-TMLE	-0.0024 to 0.0036	0.8875	0.7835	fail
two arms, an arm-specific outcome index, an instrument, correct outcome regression	paf	population attributable fraction	R drtmle with adapt_g, by a binary recode per arm	-0.0024 to 0.0036	0.8875	0.7835	fail
two arms, an arm-specific outcome index, an instrument, correct outcome regression	par	population attributable risk	cleverly per-arm outcome-adaptive C-TMLE	-0.0013 to 0.0018	0.8875	0.7859	fail
two arms, an arm-specific outcome index, an instrument, correct outcome regression	par	population attributable risk	R drtmle with adapt_g, by a binary recode per arm	-0.0013 to 0.0018	0.8875	0.7859	fail
two arms, an arm-specific outcome index, an instrument, correct outcome regression	rr	marginal risk ratio, reported on the log scale	cleverly per-arm outcome-adaptive C-TMLE	-0.0038 to 0.0070	0.9163	0.8517	fail
two arms, an arm-specific outcome index, an instrument, correct outcome regression	rr	marginal risk ratio, reported on the log scale	R drtmle with adapt_g, by a binary recode per arm	-0.0038 to 0.0070	0.9163	0.8517	fail
three arms, an arm-specific outcome index, an instrument, correct outcome regression	ate[1 vs 0]	difference in counterfactual means, 1 versus 0	cleverly per-arm outcome-adaptive C-TMLE	-0.0039 to 0.0030	0.8888	0.8311	fail
three arms, an arm-specific outcome index, an instrument, correct outcome regression	ate[1 vs 0]	difference in counterfactual means, 1 versus 0	R drtmle with adapt_g, by a binary recode per arm	-0.0039 to 0.0030	0.8888	0.8311	fail
three arms, an arm-specific outcome index, an instrument, correct outcome regression	ate[2 vs 0]	difference in counterfactual means, 2 versus 0	cleverly per-arm outcome-adaptive C-TMLE	-0.0039 to 0.0028	0.9087	0.8631	fail
three arms, an arm-specific outcome index, an instrument, correct outcome regression	ate[2 vs 0]	difference in counterfactual means, 2 versus 0	R drtmle with adapt_g, by a binary recode per arm	-0.0039 to 0.0028	0.9087	0.8631	fail
three arms, an arm-specific outcome index, an instrument, correct outcome regression	ey[0]	counterfactual mean under treatment arm ‘0’	cleverly per-arm outcome-adaptive C-TMLE	-0.0021 to 0.0027	0.9363	0.9212	pass
three arms, an arm-specific outcome index, an instrument, correct outcome regression	ey[0]	counterfactual mean under treatment arm ‘0’	R drtmle with adapt_g, by a binary recode per arm	-0.0021 to 0.0027	0.9363	0.9212	pass
three arms, an arm-specific outcome index, an instrument, correct outcome regression	ey[1]	counterfactual mean under treatment arm ‘1’	cleverly per-arm outcome-adaptive C-TMLE	-0.0026 to 0.0023	0.8588	0.7608	fail
three arms, an arm-specific outcome index, an instrument, correct outcome regression	ey[1]	counterfactual mean under treatment arm ‘1’	R drtmle with adapt_g, by a binary recode per arm	-0.0026 to 0.0023	0.8588	0.7608	fail
three arms, an arm-specific outcome index, an instrument, correct outcome regression	ey[2]	counterfactual mean under treatment arm ‘2’	cleverly per-arm outcome-adaptive C-TMLE	-0.0027 to 0.0022	0.8738	0.7740	fail
three arms, an arm-specific outcome index, an instrument, correct outcome regression	ey[2]	counterfactual mean under treatment arm ‘2’	R drtmle with adapt_g, by a binary recode per arm	-0.0027 to 0.0022	0.8738	0.7740	fail
three arms, an arm-specific outcome index, an instrument, correct outcome regression	or[1 vs 0]	marginal odds ratio, 1 versus 0, reported on the log scale	cleverly per-arm outcome-adaptive C-TMLE	-0.0148 to 0.0133	0.8900	0.8329	fail
three arms, an arm-specific outcome index, an instrument, correct outcome regression	or[1 vs 0]	marginal odds ratio, 1 versus 0, reported on the log scale	R drtmle with adapt_g, by a binary recode per arm	-0.0148 to 0.0133	0.8900	0.8329	fail
three arms, an arm-specific outcome index, an instrument, correct outcome regression	or[2 vs 0]	marginal odds ratio, 2 versus 0, reported on the log scale	cleverly per-arm outcome-adaptive C-TMLE	-0.0142 to 0.0137	0.9113	0.8627	fail
three arms, an arm-specific outcome index, an instrument, correct outcome regression	or[2 vs 0]	marginal odds ratio, 2 versus 0, reported on the log scale	R drtmle with adapt_g, by a binary recode per arm	-0.0142 to 0.0137	0.9113	0.8627	fail
three arms, an arm-specific outcome index, an instrument, correct outcome regression	rr[1 vs 0]	marginal risk ratio, 1 versus 0, reported on the log scale	cleverly per-arm outcome-adaptive C-TMLE	-0.0081 to 0.0072	0.9000	0.8434	fail
three arms, an arm-specific outcome index, an instrument, correct outcome regression	rr[1 vs 0]	marginal risk ratio, 1 versus 0, reported on the log scale	R drtmle with adapt_g, by a binary recode per arm	-0.0081 to 0.0072	0.9000	0.8434	fail
three arms, an arm-specific outcome index, an instrument, correct outcome regression	rr[2 vs 0]	marginal risk ratio, 2 versus 0, reported on the log scale	cleverly per-arm outcome-adaptive C-TMLE	-0.0073 to 0.0068	0.9175	0.8817	fail
three arms, an arm-specific outcome index, an instrument, correct outcome regression	rr[2 vs 0]	marginal risk ratio, 2 versus 0, reported on the log scale	R drtmle with adapt_g, by a binary recode per arm	-0.0073 to 0.0068	0.9175	0.8817	fail

Agreement with the canonical implementation

Comparator-backed: a paired comparison with drtmle-r-oat-per-arm 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
two arms, an arm-specific outcome index, an instrument, correct outcome regression	ate	average treatment effect	1.809e-10	4.183e-08	1.0000	0	matching implementation	6.837e-09 vs 0.0500	equivalent
two arms, an arm-specific outcome index, an instrument, correct outcome regression	ey0	counterfactual mean under no treatment	-6.747e-11	2.215e-08	1.0000	0	matching implementation	5.179e-09 vs 0.0500	equivalent
two arms, an arm-specific outcome index, an instrument, correct outcome regression	ey1	counterfactual mean under treatment	1.134e-10	3.691e-08	1.0000	0	matching implementation	3.063e-09 vs 0.0500	equivalent
two arms, an arm-specific outcome index, an instrument, correct outcome regression	or	marginal odds ratio, reported on the log scale	1.175e-09	4.064e-08	1.0000	0	matching implementation	6.885e-09 vs 0.0500	equivalent
two arms, an arm-specific outcome index, an instrument, correct outcome regression	paf	population attributable fraction	1.375e-10	2.758e-08	1.0000	0	related target	3.761e-09 vs 0.0500	equivalent
two arms, an arm-specific outcome index, an instrument, correct outcome regression	par	population attributable risk	6.747e-11	2.702e-08	1.0000	0	related target	3.051e-09 vs 0.0500	equivalent
two arms, an arm-specific outcome index, an instrument, correct outcome regression	rr	marginal risk ratio, reported on the log scale	4.586e-10	4.061e-08	1.0000	0	matching implementation	7.033e-09 vs 0.0500	equivalent
three arms, an arm-specific outcome index, an instrument, correct outcome regression	ate[1 vs 0]	difference in counterfactual means, 1 versus 0	2.357e-09	4.145e-07	1.0000	0	matching implementation	1.347e-07 vs 0.0500	equivalent
three arms, an arm-specific outcome index, an instrument, correct outcome regression	ate[2 vs 0]	difference in counterfactual means, 2 versus 0	-4.589e-11	8.271e-09	1.0000	0	matching implementation	5.229e-09 vs 0.0500	equivalent
three arms, an arm-specific outcome index, an instrument, correct outcome regression	ey[0]	counterfactual mean under treatment arm ‘0’	1.237e-11	3.177e-09	1.0000	0	matching implementation	2.161e-09 vs 0.0500	equivalent
three arms, an arm-specific outcome index, an instrument, correct outcome regression	ey[1]	counterfactual mean under treatment arm ‘1’	2.369e-09	5.857e-07	1.0000	0	matching implementation	1.805e-07 vs 0.0500	equivalent
three arms, an arm-specific outcome index, an instrument, correct outcome regression	ey[2]	counterfactual mean under treatment arm ‘2’	-3.352e-11	8.294e-09	1.0000	0	matching implementation	8.831e-09 vs 0.0500	equivalent
three arms, an arm-specific outcome index, an instrument, correct outcome regression	or[1 vs 0]	marginal odds ratio, 1 versus 0, reported on the log scale	9.058e-09	2.777e-07	1.0000	0	matching implementation	1.341e-07 vs 0.0500	equivalent
three arms, an arm-specific outcome index, an instrument, correct outcome regression	or[2 vs 0]	marginal odds ratio, 2 versus 0, reported on the log scale	-1.735e-10	3.794e-09	1.0000	0	matching implementation	5.147e-09 vs 0.0500	equivalent
three arms, an arm-specific outcome index, an instrument, correct outcome regression	rr[1 vs 0]	marginal risk ratio, 1 versus 0, reported on the log scale	5.324e-09	3.524e-07	1.0000	0	matching implementation	1.413e-07 vs 0.0500	equivalent
three arms, an arm-specific outcome index, an instrument, correct outcome regression	rr[2 vs 0]	marginal risk ratio, 2 versus 0, reported on the log scale	-9.641e-11	5.893e-09	1.0000	0	matching implementation	4.436e-09 vs 0.0500	equivalent

Theory properties

property	cell	role	what was tested	what must hold	measured	result
crossfit_overfitting	cross_fitted_oat	positive	outcome-adaptive C-TMLE with cross-fitted nuisances and a flexible learner	SE ratio clears the overfitting floor and stays inside the sanity band	SE ratio 0.8842 to 1.0686	pass
crossfit_overfitting	in_sample_control	control	the same flexible learner fitted in sample, with no cross-fitting	SE ratio must fall below the overfitting ceiling	SE ratio 0.3638 to 0.4341	pass
generated_design	binary_active__estimated	positive	two-arm law with an arm-specific outcome index and an instrument: the same design is estimated from the data, as a real fit does	SE ratio and coverage intervals both inside their calibration bands; the paired deficit is reported only	coverage 0.8892 to 0.9116, SE ratio 0.8217 to 0.8671	fail
generated_design	binary_active__oracle_design	positive	two-arm law with an arm-specific outcome index and an instrument: the outcome-adaptive design is supplied rather than estimated	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9375 to 0.9545, SE ratio 0.9715 to 1.0241	pass
generated_design	three_arm_active__estimated	positive	three-arm law with an arm-specific outcome index and an instrument: the same design is estimated from the data, as a real fit does	SE ratio and coverage intervals both inside their calibration bands; the paired deficit is reported only	coverage 0.9085 to 0.9290, SE ratio 0.8674 to 0.9131	fail
generated_design	three_arm_active__oracle_design	positive	three-arm law with an arm-specific outcome index and an instrument: the outcome-adaptive design is supplied rather than estimated	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9413 to 0.9578, SE ratio 0.9836 to 1.0360	pass
interval_calibration	binary_active__correctly_specified	positive	two-arm law with an arm-specific outcome index and an instrument: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.8820 to 0.9170, SE ratio 0.8202 to 0.8874	fail
interval_calibration	binary_active__noise_control	control	two-arm law with an arm-specific outcome index and an instrument: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.7776 to 0.8240, SE ratio 0.6185 to 0.6726	pass
interval_calibration	binary_active__shrunken_se_control	control	two-arm law with an arm-specific outcome index and an instrument: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.7309 to 0.7808, SE ratio 0.5742 to 0.6214	pass
interval_calibration	binary_repeats__correctly_specified	positive	the two-arm law, the median estimate over three cross-fitting splits: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.8842 to 0.9189, SE ratio 0.8081 to 0.8753	fail
interval_calibration	binary_weighted__correctly_specified	positive	the two-arm law under the declared known weight 0.5 + 1{W1 > 0}: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.8863 to 0.9207, SE ratio 0.8060 to 0.8714	fail
interval_calibration	three_arm_active__correctly_specified	positive	three-arm law with an arm-specific outcome index and an instrument: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.8928 to 0.9262, SE ratio 0.8509 to 0.9240	fail
interval_calibration	three_arm_active__noise_control	control	three-arm law with an arm-specific outcome index and an instrument: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.7864 to 0.8321, SE ratio 0.6425 to 0.6982	pass
interval_calibration	three_arm_active__shrunken_se_control	control	three-arm law with an arm-specific outcome index and an instrument: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.7433 to 0.7924, SE ratio 0.5952 to 0.6465	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
robustness_contract	outcome_correct	positive	the outcome regression is correct and the mechanism is not	bias interval inside the equivalence margin, SE ratio must remain between 0.1 and 10.0	bias -0.000703 to 0.0033, margin 0.0061, SE ratio 0.8757	pass
robustness_contract	outcome_wrong	control	the outcome regression is misspecified	bias interval must fall entirely outside the margin, SE ratio must remain between 0.1 and 10.0	bias 0.0634 to 0.0681, margin 0.0073, SE ratio 0.9857	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 -0.000434, coverage 0.8786 to 0.9357, SE ratio 0.8636	fail
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 0.0044, coverage 0.8721 to 0.9308, SE ratio 0.8275	fail
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 -0.000358, coverage 0.8850 to 0.9406, SE ratio 0.8598	fail
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 -0.5579 to -0.4843	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 -0.5090 to -0.5059	pass
simultaneous_coverage	three_arm_active__pointwise_joint_control	control	three-arm law with an arm-specific outcome index and an instrument: the pointwise 95% intervals of the same fits, read jointly	joint coverage upper endpoint must fall below the nominal rate	joint coverage 0.6753 to 0.7239	pass
simultaneous_coverage	three_arm_active__simultaneous_band	positive	three-arm law with an arm-specific outcome index and an instrument: 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 0.8527 to 0.8883	fail
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 0.0963, 0.0712 to 0.1262	fail

Before its gated run, the study declared its revert cells.

revert cells	count
primary coverage, one cell for each estimand	16
interval_calibration positive cells	4
generated_design cells	4
the simultaneous band	1

One red revert cell with no defect withholds the per-arm interval.

The two oracle_design cells pin \(\bar Q\) to the truth. They still estimate the mechanism. Each passes its own rule, which the table shows. The red-cell ledger counts each one red by the family’s joint clause, because its estimated twin fails. That contrast locates the deficit in the estimated design.

The simultaneous band cell calls the public simultaneous_bands function on the plug-in curves of its own fit. It measures the band that the per-arm design would build from those curves. The shipped fit withholds the band. A replay of 20 joint replications gave the same bands, critical values and coverage rows as the earlier cell, bit for bit. The earlier cell enabled inference for its own fit through a global flag that the package no longer has.

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	800	primary replications per law
n	1500	observations per primary replication
independent_tests_total	32	implementation-estimand tests against the truth
independent_tests_passed	2	of those, passing
paired_tests_total	16	paired comparisons with R drtmle
paired_tests_passed	16	of those, passing
property_cells_total	25	repeated-sampling property cells
property_cells_passed	12	cells whose own and family verdicts pass
max_standardized_bias	0.0269	largest absolute primary bias in empirical standard deviations
min_coverage	0.8588	lowest primary coverage of the diagnostic interval
min_se_ratio_ci_lower	0.7142	lowest bootstrap primary SE-ratio endpoint
max_se_ratio_ci_upper	0.9919	highest bootstrap primary SE-ratio endpoint
max_margin_utilization	5.857e-07	largest share of the paired margin used
mechanism_fits_with_rows_at_bound	157	primary fits with one or more rows of the per-arm mechanism at a declared bound
mechanism_max_rows_at_bound	21	most rows at a bound in one fit
mechanism_max_gap_off_bound	4.658e-07	largest gap between the two mechanisms in a fit with no row at a bound
mechanism_max_gap	0.0080	largest gap between the two mechanisms in any fit
design_gap[binary_active/ate]:mean_design_gap	0.0458	mean absolute shift from the shared design, in this fit’s standard errors, two-arm ate
design_gap[three_arm_active/ey[1]]:mean_design_gap	1.0825	the same for the three-arm ey[1]
properties[crossfit_overfitting/cross_fitted_oat]:standardized_bias	0.9461	bias of the cross-fitted fit with a fully grown tree, in empirical standard deviations
properties[crossfit_overfitting/cross_fitted_oat]:coverage	0.8275	coverage of that fit’s diagnostic interval
properties[root_n_and_efficiency/n_2000]:empirical_se	0.0247	empirical spread of the two-arm ate at n = 2,000
properties[root_n_and_efficiency/n_2000]:mean_std_error	0.0213	mean reported diagnostic standard error at n = 2,000
margin:alpha	0.0500	nominal size of the reported intervals
margin:bootstrap_replicates	10000	resamples behind every bootstrap interval
margin:calibration_coverage_lower	0.9200	calibration-cell coverage band, lower limit
margin:calibration_coverage_upper	0.9800	calibration-cell coverage band, upper limit
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
margin:calibration_se_ratio_upper	1.0700	calibration-cell SE-ratio band, upper limit
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
margin:minimum_power	0.8000	rejection lower bound the power cell must clear
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:overfit_control_ceiling	0.7500	coverage the in-sample overfitting control must stay under
margin:overfit_coverage_gain	0.1500	coverage gain cross-fitting must show over that control
margin:overfit_se_floor	0.8500	SE-ratio floor of the cross-fitted overfitting cell
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
margin:root_n_slope_lower	-0.6250	accepted root-n slope band, lower limit
margin:root_n_slope_upper	-0.3750	accepted root-n slope band, upper limit
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:shrunken_se_factor	0.7000	negative-control SE multiplier
margin:standardized_bias	0.2500	bias equivalence margin, in empirical standard deviations
margin:type_i_ceiling	0.1000	the rate the null cell must bound
margin:union_model_se_lower	0.1000	robustness-contract SE-ratio screen, lower limit
margin:union_model_se_upper	10	robustness-contract SE-ratio screen, upper limit

The reported standard error is too small, and the estimator is still more precise than the efficient bound. At n = 2,000 the empirical spread times \(\sqrt{n}\) is about 1.10. The per-arm curve standard deviation of the law is 0.949, and the efficient influence function’s is 1.451 (POWER_CURVE_SD and EIF_CURVE_SD in tests/studies/ctmle_oat_per_arm_properties.py). The study fits no ordinary TMLE on the same draws, so this comparison is reported and not gated.

Limitations

limitation	what it means for use
No cell measures published inference	Every oat fit withholds ci, pvalue and std_error. The std_error, ci_lower, ci_upper and covered columns read plugin_std_error and plugin_interval. The coverage, calibration, null-size, band and power cells describe that diagnostic
The diagnostic standard error is too small on both laws	The primary SE ratios run from min_se_ratio_ci_lower to max_se_ratio_ci_upper, and the ladder does not close the gap as n grows. Condition (v) of Theorem 1 fails for an estimated outcome regression, so the remainder \(R_{24}\) is first order. R drtmle reports the same standard errors, so the pairing cannot detect the gap
The points are paired, not the intervals	All 16 paired rows are equivalent, with max_margin_utilization of the margin used. The truth rows of both implementations are red alike. The per-arm design moves the three-arm points away from the shared design by design_gap[three_arm_active/ey[1]]:mean_design_gap standard errors on ey[1]
The mechanisms agree only off the bounds	Where no row of the per-arm mechanism sits at a bound, the two mechanisms agree to mechanism_max_gap_off_bound. In mechanism_fits_with_rows_at_bound fits some rows sit at a bound, and the gap reaches mechanism_max_gap. The rows at a bound are reported, not gated
A flexible outcome learner biases the point estimate	The crossfit_overfitting pair fits a fully grown regression tree. Cross-fitting restores the SE ratio, which is all that pair gates. The cross-fitted estimate still has a bias of properties[crossfit_overfitting/cross_fitted_oat]:standardized_bias empirical standard deviations, and its diagnostic interval covers properties[crossfit_overfitting/cross_fitted_oat]:coverage. The point-estimate evidence holds for the correctly specified GLM
Two parametric laws with a binary outcome	Each law has a normal index for each arm and one instrument. A continuous outcome and more than three arms have fast-tier evidence only
Strata and missing data have a study of their own	The laws above have no strata and no missing data. The strata and missing-data study measures those fits. Clusters stay refused
One unstratified split does the cross-fitting	The property cells declare stratify_folds="none". The primary fit is in sample, so the pairing says nothing about cross-fitted parity

Reproduction

The manifest records the seeds, the margins, the declared revert cells, the expected red rows, the estimator configuration and the hashes. Run python -m tests.canonical.drtmle_oat_per_arm.regenerate to regenerate the artifacts. The replications, paired decisions, design gap and property results carry every published row. tests/diagnostics/x17_generated_design/ reproduces the diagnosis.

The committed R phase ran under the harness before RM39. RM39’s revision row for tests/canonical/study_harness.R covers that hash. tests/canonical/harness_fixture/determinism.csv has no row for run_drtmle_oat_per_arm.R yet. The runner itself calls no random number generator. It fits with cvFolds = 1, glm_g and uniroot.