Outcome-adaptive per-arm C-TMLE (parametric)

This study measures the interval of CTMLE(strategy="oat", oat_design="per_arm", cross_fit=False) with ParametricOutcome and ParametricMechanism. The fit is a stacked Z-estimator. Its outcome GLM, each arm’s mechanism, each arm’s fluctuation and each arm’s mean solve one system. The section Parametric per-arm intervals states the construction, its conditions and what the interval covers.

Python property evidence, no comparator. No canonical implementation is compared. R drtmle with adapt_g = TRUE reports the shipped curve of Theorem 1 of Benkeser, Cai and van der Laan (2020). The stack corrects that curve, so no R program measures the corrected uncertainty. The point estimate agrees with the pipelines of the registered study ctmle-oat-per-arm to 4e-14 (tests/unit/test_parametric_learners.py), and that study pairs those pipelines with drtmle. So the empty equivalence artifact records the absence of a comparator. The evidence is the exact truth of each law, the declared calibration bands, and two negative controls that must read red.

The study publishes under the gated policy. Every gated row passed, and both negative controls read red as declared. The red-cell ledger lists no cell of this study.

What was tested

setting	declaration
subject	CTMLE(strategy="oat", oat_design="per_arm", cross_fit=False, simultaneous=False) with ParametricOutcome() and ParametricMechanism(degree=3). The default alpha and max_iter=100, tol=1e-10
primary phase	binary_active at \(n = 1500\), 800 replications, with ey0, ey1, ate, rr and or. Each replication must report the status "influence_curve" for every estimand
property phase	13 fit groups. Each group fits one shared sample per replicate, and its cells read the same fits
truth	a quadrature of each law (tests/studies/oat_parametric_quadrature.py), so no truth comes from a simulation. A cell of the pseudo-true law reads the limit \(\psi^*\)
verdicts	each calibration cell needs its SE-ratio and coverage intervals inside the calibration bands. The type-I cell needs its rejection bound under the ceiling. The power cell needs its rejection lower bound above the minimum power
failed replicates	a fit that raises, or a fit whose stack status is not "influence_curve", is a failed replicate. The run records it and draws no replacement. A cell may lose floor(0.01 R) replicates, and one more reads bound exceeded. No replicate was lost

The seven laws share \(W = (W_1, W_2, W_3)\), three independent standard normals.

law	role	construction
binary_active	primary and calibration	a binary outcome with an arm-specific index in \(W_1\) and \(W_2\). The treatment law is a softmax in \(W_1\) and \(W_3\), so \(W_3\) is an instrument. The outcome model is correct
binary_null	type-I error	the treatment law of binary_active, with arm 1 reading the index of arm 0, so the ATE is zero
shared_prognosis	covariance, rr and or	the treatment law of binary_active and \(\operatorname{logit} \bar Q_0(a, W) = \pm 0.3 + 3.5 W_1 + 0.8 W_2\). The two arm estimates correlate
u_shaped	\(\kappa_a \ne 1\)	a mechanism U-shaped in \(W_1\), which the cubic basis misses. The study also fits the linear basis
steep	binding bound	a steep mechanism with g_bounds=(0.05, 0.95), which clips a share of the rows at the limit
bounded	outcome scale	\(Y = 2 + 8B\) with \(B \sim \operatorname{Beta}(4 \bar Q_0, 4(1 - \bar Q_0))\) and the \(\bar Q_0\) of binary_active, fitted with q_bounds=(2, 10)
pseudo_true	wrong outcome model	binary_active with \(W_1\) dropped from the outcome features. The truth of the cell is the limit \(\psi^*\)

The budgets in the table are the declared replications of each fit group.

group	law	\(n\)	replications
binary_active	binary_active	1,500	17,200
ladder_500, ladder_2000, ladder_8000	binary_active	500, 2,000, 8,000	63,600, 8,400, 9,200
shared_prognosis	shared_prognosis, with rr and or	1,500	10,000
kappa_law	u_shaped, cubic and linear basis	1,500	19,600
bounds_active	steep, g_bounds=(0.05, 0.95)	1,500	12,400
auto_bounds	binary_active, g_bounds="auto"	1,500	9,200
bounded_outcome	bounded, q_bounds=(2, 10)	1,500	16,400
pseudo_true	pseudo_true	1,500	7,600
generated_design	binary_active, the estimated outcome and a pinned true outcome	1,500	9,200
sharp_null	binary_null	1,000	800
power	binary_active	800	1,200

The study sizes each calibration group so that the whole study passes with probability at least 0.95 when nothing is wrong. Each group needs a pass probability of \(0.95^{1/12}\). The declaration counted 12 groups. The study has 11 gated calibration groups, so the study-wise pass probability is at least \(0.95^{11/12}\).

A predictive model sets the size. It centres each row on its own pilot SE ratio, with variance \(s^2 + \tau_{80}^2\). The value \(\tau_{80} = 0.0080\) is pooled over the gated rows of the study.

The pilots raised no separation refusal and no stack failure. They drew 5,000 replications for each cell, then ran each group at its declared budget on pilot-only seeds. The power cell has an exact-law power of 0.8475 at \(n = 800\). At 1,200 replications, the chance that its lower rejection endpoint clears 0.80 is 0.957.

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 with a parametric outcome and mechanism	-0.0025 to 0.0028	0.9450	1.0004	pass
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 with a parametric outcome and mechanism	-0.0019 to 0.0018	0.9450	0.9924	pass
two arms, an arm-specific outcome index, an instrument, correct outcome regression	ey1	counterfactual mean under treatment	cleverly per-arm outcome-adaptive C-TMLE with a parametric outcome and mechanism	-0.0018 to 0.0020	0.9437	0.9705	pass
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 with a parametric outcome and mechanism	-0.0094 to 0.0121	0.9450	1.0006	pass
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 with a parametric outcome and mechanism	-0.0048 to 0.0060	0.9487	1.0008	pass

Theory properties

property	cell	role	what was tested	what must hold	measured	result
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 0.9417 to 0.9537, SE ratio 0.9779 to 1.0160	pass
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 0.9414 to 0.9535, SE ratio 0.9742 to 1.0111	pass
interval_calibration	auto_bounds__ate__corrected	positive	the stacked curve with the default g_bounds of “auto”: ATE	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9445 to 0.9563, SE ratio 0.9830 to 1.0211	pass
interval_calibration	auto_bounds__ey0__corrected	positive	the stacked curve with the default g_bounds of “auto”: arm-0 mean	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9482 to 0.9596, SE ratio 0.9846 to 1.0226	pass
interval_calibration	auto_bounds__ey1__corrected	positive	the stacked curve with the default g_bounds of “auto”: arm-1 mean	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9423 to 0.9542, SE ratio 0.9786 to 1.0163	pass
interval_calibration	binary_active__ate__corrected	positive	two-arm law with an arm-specific outcome index and an instrument: the stacked curve at n = 1,500: ATE	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9462 to 0.9548, SE ratio 0.9883 to 1.0164	pass
interval_calibration	binary_active__ate__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 ATE estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8253 to 0.8401, SE ratio 0.6975 to 0.7171	pass
interval_calibration	binary_active__ate__shipped_curve__shrunken_se_control	control	two-arm law with an arm-specific outcome index and an instrument: the ATE standard errors come from the shipped adaptive-propensity curve of the same fits, which omits the generated-design term	the SE-ratio interval must fall below the calibration band	coverage 0.8944 to 0.9063, SE ratio 0.8358 to 0.8595	pass
interval_calibration	binary_active__ate__shrunken_se_control	control	two-arm law with an arm-specific outcome index and an instrument: the ATE standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8206 to 0.8355, SE ratio 0.6920 to 0.7115	pass
interval_calibration	binary_active__ey0__corrected	positive	two-arm law with an arm-specific outcome index and an instrument: the stacked curve at n = 1,500: arm-0 mean	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9447 to 0.9534, SE ratio 0.9880 to 1.0158	pass
interval_calibration	binary_active__ey1__corrected	positive	two-arm law with an arm-specific outcome index and an instrument: the stacked curve at n = 1,500: arm-1 mean	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9425 to 0.9513, SE ratio 0.9833 to 1.0112	pass
interval_calibration	bounded_outcome__ate__corrected	positive	the stacked curve for an outcome bounded in 2 to 10, fitted with q_bounds on the scaled outcome: ATE	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9442 to 0.9531, SE ratio 0.9800 to 1.0082	pass
interval_calibration	bounded_outcome__ey0__corrected	positive	the stacked curve for an outcome bounded in 2 to 10, fitted with q_bounds on the scaled outcome: arm-0 mean	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9428 to 0.9518, SE ratio 0.9759 to 1.0040	pass
interval_calibration	bounded_outcome__ey1__corrected	positive	the stacked curve for an outcome bounded in 2 to 10, fitted with q_bounds on the scaled outcome: arm-1 mean	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9455 to 0.9543, SE ratio 0.9835 to 1.0116	pass
interval_calibration	bounds_active__ate__corrected	positive	the stacked curve with g_bounds of (0.05, 0.95), which clip a share of the rows at the limit: ATE	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9394 to 0.9501, SE ratio 0.9722 to 1.0043	pass
interval_calibration	bounds_active__ey0__corrected	positive	the stacked curve with g_bounds of (0.05, 0.95), which clip a share of the rows at the limit: arm-0 mean	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9414 to 0.9518, SE ratio 0.9797 to 1.0115	pass
interval_calibration	bounds_active__ey1__corrected	positive	the stacked curve with g_bounds of (0.05, 0.95), which clip a share of the rows at the limit: arm-1 mean	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9392 to 0.9498, SE ratio 0.9688 to 1.0012	pass
interval_calibration	kappa_cubic__ate__corrected	positive	the stacked curve on a U-shaped mechanism, with the cubic basis in logit q: ATE	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9430 to 0.9513, SE ratio 0.9726 to 0.9979	pass
interval_calibration	kappa_cubic__ey0__corrected	positive	the stacked curve on a U-shaped mechanism, with the cubic basis in logit q: arm-0 mean	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9381 to 0.9467, SE ratio 0.9519 to 0.9771	pass
interval_calibration	kappa_cubic__ey1__corrected	positive	the stacked curve on a U-shaped mechanism, with the cubic basis in logit q: arm-1 mean	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9432 to 0.9515, SE ratio 0.9872 to 1.0129	pass
interval_calibration	kappa_linear__ate__corrected	positive	the stacked curve on a U-shaped mechanism, with the linear basis in logit q: ATE	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9436 to 0.9518, SE ratio 0.9824 to 1.0083	pass
interval_calibration	kappa_linear__ey0__corrected	positive	the stacked curve on a U-shaped mechanism, with the linear basis in logit q: arm-0 mean	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9439 to 0.9521, SE ratio 0.9805 to 1.0065	pass
interval_calibration	kappa_linear__ey1__corrected	positive	the stacked curve on a U-shaped mechanism, with the linear basis in logit q: arm-1 mean	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9447 to 0.9528, SE ratio 0.9814 to 1.0068	pass
interval_calibration	n_2000__ate__corrected	positive	the stacked curve at n = 2,000: ATE	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9457 to 0.9578, SE ratio 0.9938 to 1.0338	pass
interval_calibration	n_2000__ey0__corrected	positive	the stacked curve at n = 2,000: arm-0 mean	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9419 to 0.9545, SE ratio 0.9763 to 1.0161	pass
interval_calibration	n_2000__ey1__corrected	positive	the stacked curve at n = 2,000: arm-1 mean	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9453 to 0.9575, SE ratio 0.9946 to 1.0359	pass
interval_calibration	n_500__ate__corrected	positive	the stacked curve at n = 500: ATE	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9443 to 0.9489, SE ratio 0.9895 to 1.0039	pass
interval_calibration	n_500__ey0__corrected	positive	the stacked curve at n = 500: arm-0 mean	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9421 to 0.9468, SE ratio 0.9827 to 0.9970	pass
interval_calibration	n_500__ey1__corrected	positive	the stacked curve at n = 500: arm-1 mean	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9419 to 0.9466, SE ratio 0.9870 to 1.0014	pass
interval_calibration	n_8000__ate__corrected	positive	the stacked curve at n = 8,000: ATE	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9478 to 0.9592, SE ratio 0.9934 to 1.0311	pass
interval_calibration	n_8000__ey0__corrected	positive	the stacked curve at n = 8,000: arm-0 mean	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9455 to 0.9571, SE ratio 0.9887 to 1.0274	pass
interval_calibration	n_8000__ey1__corrected	positive	the stacked curve at n = 8,000: arm-1 mean	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9473 to 0.9587, SE ratio 0.9926 to 1.0312	pass
interval_calibration	pseudo_true__ate__corrected	positive	the stacked curve with W1 dropped from the outcome features, so the interval covers the limit psi*: ATE	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9411 to 0.9544, SE ratio 0.9767 to 1.0184	pass
interval_calibration	pseudo_true__ey0__corrected	positive	the stacked curve with W1 dropped from the outcome features, so the interval covers the limit psi*: arm-0 mean	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9414 to 0.9546, SE ratio 0.9773 to 1.0197	pass
interval_calibration	pseudo_true__ey1__corrected	positive	the stacked curve with W1 dropped from the outcome features, so the interval covers the limit psi*: arm-1 mean	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9429 to 0.9560, SE ratio 0.9737 to 1.0156	pass
interval_calibration	shared_prognosis__ate__corrected	positive	the stacked curve on a law whose two arm means share W1 and W2 in their outcome index: ATE	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9429 to 0.9543, SE ratio 0.9729 to 1.0085	pass
interval_calibration	shared_prognosis__ate__cross_term_dropped__inflated_se_control	control	the ATE standard errors are built from the two arm variances without the cross-arm covariance, on a law whose arm estimates correlate	the SE-ratio interval must fall above the calibration band	coverage 0.9811 to 0.9876, SE ratio 1.1958 to 1.2396	pass
interval_calibration	shared_prognosis__log_or__corrected	positive	the stacked curve on a law whose two arm means share W1 and W2 in their outcome index: log odds ratio	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9429 to 0.9543, SE ratio 0.9739 to 1.0095	pass
interval_calibration	shared_prognosis__log_rr__corrected	positive	the stacked curve on a law whose two arm means share W1 and W2 in their outcome index: log risk ratio	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9425 to 0.9540, SE ratio 0.9725 to 1.0088	pass
power	alternative	positive	the same test applied to a law with a real effect	rejection lower bound clears the minimum power	rejection 0.8442, 0.8154 to 0.8701	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 0.0550, 0.0363 to 0.0792	pass

Read the two generated_design cells with this study’s setting. The shared descriptions come from the parent study. In this study the oracle_design cell pins the true outcome regression and fits none, so the shipped curve is its full curve. The estimated cell fits the outcome model and reads the stacked curve. The study gates both cells. The phrase “the paired deficit is reported only” does not apply here.

Declared reading

The study fixed its reading before the first run, in the module tests/studies/canonical_ctmle_oat_per_arm_parametric.py.

rule	text
red calibration row	a finite-sample result only when its 99% SE-ratio interval contains 1, its coverage agrees with 0.95, and the diagnosis finds no defect. Any other red row is a defect suspect, diagnosed before publication
negative control	each control must read red. A control row publishes passed = True exactly when its SE-ratio interval leaves the calibration band on the declared side. A control that stays inside the band, or straddles it, cannot see its term. The study is then invalid and the design changes before any reading
reported only	the quantities of the next section gate nothing
failures	a failed replicate is recorded and not redrawn. Dropping it would select on success. Substituting the shipped standard error would mix two estimators

Negative controls

Both controls read red. The table gives each control, the term it removes, and the measured SE ratio.

control	cell	term removed	expected SE ratio	side	measured
shipped curve	binary_active__ate__shipped_curve__shrunken_se_control	the generated-design term. The standard error comes from the shipped adaptive-propensity curve of the same fits	0.845	below the band	properties[interval_calibration/binary_active__ate__shipped_curve__shrunken_se_control]:se_ratio, with an upper endpoint of properties[interval_calibration/binary_active__ate__shipped_curve__shrunken_se_control]:se_ratio_ci_upper
cross term dropped	shared_prognosis__ate__cross_term_dropped__inflated_se_control	the cross-arm covariance. The ATE standard error is \(\sqrt{se_0^2 + se_1^2}\)	1.228	above the band	properties[interval_calibration/shared_prognosis__ate__cross_term_dropped__inflated_se_control]:se_ratio, with a lower endpoint of properties[interval_calibration/shared_prognosis__ate__cross_term_dropped__inflated_se_control]:se_ratio_ci_lower

The first control uses the same fits as binary_active__ate__corrected. Its coverage is properties[interval_calibration/binary_active__ate__shipped_curve__shrunken_se_control]:coverage, so the shipped curve under-covers where the stacked curve calibrates. The second control uses the fits of shared_prognosis__ate__corrected. The two arm estimates correlate on this law, so the dropped term is material. The harness also derives a shrunken_se_control and a noise_control from the primary ATE row. Both read below the band.

Quantities reported and not gated

These quantities describe the fit, and no verdict reads them.

quantity	cell	what it shows	measured
coverage of \(\psi_0\)	pseudo_true__ey0__corrected, pseudo_true__ey1__corrected, pseudo_true__ate__corrected	the interval covers \(\psi^*\) and not \(\psi_0\) when the outcome model is wrong. The gated coverage of \(\psi^*\) in the same cells passes	properties[interval_calibration/pseudo_true__ey0__corrected]:psi0_coverage, properties[interval_calibration/pseudo_true__ey1__corrected]:psi0_coverage and properties[interval_calibration/pseudo_true__ate__corrected]:psi0_coverage
covariance of the arm means	shared_prognosis__ate__corrected	the Monte Carlo covariance of the two arm estimates against the mean reported covariance	a ratio of properties[interval_calibration/shared_prognosis__ate__corrected]:covariance_ratio, with a 99% interval from properties[interval_calibration/shared_prognosis__ate__corrected]:covariance_ratio_ci_lower to properties[interval_calibration/shared_prognosis__ate__corrected]:covariance_ratio_ci_upper
\(\sqrt{n}\) SD of the ladder	n_500__ate__corrected, n_2000__ate__corrected, n_8000__ate__corrected	the empirical and reported \(\sqrt{n}\) SD of the ATE against the quadrature SD	the table of the ladder
clipped share	bounds_active__ate__corrected	the mean share of rows at a g_bounds limit, over arms and replications	properties[interval_calibration/bounds_active__ate__corrected]:clipped_share

The ladder

The quadrature gives the asymptotic \(\sqrt{n}\) SD of the ATE curve as properties[interval_calibration/n_500__ate__corrected]:quadrature_sd. The table gives the empirical SD and the mean reported standard error, each times \(\sqrt{n}\).

\(n\)	empirical \(\sqrt{n}\) SD	reported \(\sqrt{n}\) SE
500	properties[interval_calibration/n_500__ate__corrected]:sqrt_n_empirical_sd	properties[interval_calibration/n_500__ate__corrected]:sqrt_n_reported_se
2,000	properties[interval_calibration/n_2000__ate__corrected]:sqrt_n_empirical_sd	properties[interval_calibration/n_2000__ate__corrected]:sqrt_n_reported_se
8,000	properties[interval_calibration/n_8000__ate__corrected]:sqrt_n_empirical_sd	properties[interval_calibration/n_8000__ate__corrected]:sqrt_n_reported_se

The covariance

The study reports the Monte Carlo covariance and does not gate it. At 10,000 replications the 99% interval of the ratio runs from properties[interval_calibration/shared_prognosis__ate__corrected]:covariance_ratio_ci_lower to properties[interval_calibration/shared_prognosis__ate__corrected]:covariance_ratio_ci_upper. That interval is wider than the SE-ratio band of margin:calibration_se_ratio_lower to margin:calibration_se_ratio_upper, so the SE-ratio band does not transfer to the covariance. The study declared no covariance band before the run. The ATE row and the cross-term control carry the covariance instead. The ATE variance contains \(-2 \operatorname{Cov}\), and the control moves the ATE SE ratio out of the band when that term is missing.

Deviations from the plan

The plan is the X58 plan, section 5. The table lists each deviation and its reason.

plan	study	reason
each cell sized to a family-wise pass probability of 0.95 (section 11)	each of the 11 calibration groups sized to \(0.95^{1/12}\), so the whole study passes with probability at least 0.95	per-cell sizing gives a study-wise pass probability near 0.63, as the pre-queue design review found
the Monte Carlo covariance of the arm means gated, with a bootstrap interval	the covariance reported and not gated	at 10,000 replications the 99% interval of the ratio is wider than the SE-ratio band, and the study declared no covariance band, as the covariance shows. The ATE row and the cross-term control carry the covariance
the ladder in the root_n_and_efficiency family and the joint cells in a joint family	the ladder and the joint cells are interval_calibration cells named <label>__<estimand>__corrected	the shared calibration verdict then applies to each estimand. The sqrt(n) SD of the ladder stays a reported column
the generated_design pair a reporting diagnostic, as in the parent study	the pair is a gated family. Both cells pass	the pinned cell fits no outcome model, so the shipped curve is its full curve. The estimated cell reads the stacked curve

Result

Every one of the property_cells_total cells passes, and so does each of the independent_tests_total primary tests. The lowest primary coverage is min_coverage, with a lower endpoint of min_coverage_ci_lower. The primary SE-ratio endpoints run from min_se_ratio_ci_lower to max_se_ratio_ci_upper. The largest standardized bias is max_standardized_bias.

The stacked curve calibrates on every law the study fits. The sample sizes run from 500 to 8,000. The mechanism misses the projection on u_shaped, a g_bounds limit binds on steep, the default "auto" bound resolves on binary_active, and a bounded outcome fits on its own scale. The shipped curve fails the same band on binary_active, so the correction is what the study measures.

The coverage of \(\psi^*\) in the pseudo-true cells is calibrated, and the coverage of \(\psi_0\) is not. A user who reads the interval as an interval for \(\psi_0\) under a wrong outcome model gets the second number. Parametric per-arm intervals states the claim.

The sharp-null cell rejects at properties[type_i_error/sharp_null]:rejection_rate, with an upper endpoint of properties[type_i_error/sharp_null]:rejection_ci_upper against the ceiling margin:type_i_ceiling. The power cell rejects at properties[power/alternative]:rejection_rate, with a lower endpoint of properties[power/alternative]:rejection_ci_lower against margin:minimum_power.

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
n	1500	observations per primary replication
independent_tests_total	5	primary tests against truth
independent_tests_passed	5	of those, passing
paired_tests_total	0	external comparisons declared
paired_tests_passed	0	external comparisons passing
property_cells_total	41	repeated-sampling property cells
property_cells_passed	41	cells whose own and family verdicts pass
max_standardized_bias	0.0116	largest absolute primary bias in empirical standard deviations
min_coverage	0.9437	lowest measured primary-study coverage
min_coverage_ci_lower	0.9194	lowest exact 99% primary coverage endpoint
min_se_ratio_ci_lower	0.9157	lowest bootstrap primary SE-ratio endpoint
max_se_ratio_ci_upper	1.0709	highest bootstrap primary SE-ratio endpoint
subject_failures	0	primary fits lost
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. A control may rise above it
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. No cell of this study reads it
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:paired_difference	0.1500	paired similarity margin. No cell of this study reads it
margin:rmse_noninferiority	1.1000	largest external-comparison RMSE ratio bound
margin:root_n_slope	-0.5000	contraction rate root-n asymptotics predict. No cell of this study reads it
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:subject_failure_bound	0.0100	share of replicates a cell may lose
margin:subject_failures_allowed	8	replicates one primary cell may lose
margin:type_i_ceiling	0.1000	the rate the null cell must bound
properties[interval_calibration/binary_active__ate__shipped_curve__shrunken_se_control]:se_ratio	0.8476	SE ratio, shipped-curve control
properties[interval_calibration/binary_active__ate__shipped_curve__shrunken_se_control]:se_ratio_ci_upper	0.8595	its 99% upper endpoint
properties[interval_calibration/binary_active__ate__shipped_curve__shrunken_se_control]:coverage	0.9005	coverage, shipped-curve control
properties[interval_calibration/shared_prognosis__ate__cross_term_dropped__inflated_se_control]:se_ratio	1.2173	SE ratio, cross-term control
properties[interval_calibration/shared_prognosis__ate__cross_term_dropped__inflated_se_control]:se_ratio_ci_lower	1.1958	its 99% lower endpoint
properties[interval_calibration/pseudo_true__ey0__corrected]:psi0_coverage	0.5691	coverage of \(\psi_0\), arm-0 mean, wrong outcome model
properties[interval_calibration/pseudo_true__ey1__corrected]:psi0_coverage	0.5625	coverage of \(\psi_0\), arm-1 mean, wrong outcome model
properties[interval_calibration/pseudo_true__ate__corrected]:psi0_coverage	0.3053	coverage of \(\psi_0\), ATE, wrong outcome model
properties[interval_calibration/shared_prognosis__ate__corrected]:covariance_ratio	1.0055	Monte Carlo covariance over mean reported covariance
properties[interval_calibration/shared_prognosis__ate__corrected]:covariance_ratio_ci_lower	0.9269	its 99% lower endpoint
properties[interval_calibration/shared_prognosis__ate__corrected]:covariance_ratio_ci_upper	1.0956	its 99% upper endpoint
properties[interval_calibration/n_500__ate__corrected]:quadrature_sd	1.1234	asymptotic \(\sqrt{n}\) SD of the ATE curve
properties[interval_calibration/n_500__ate__corrected]:sqrt_n_empirical_sd	1.1240	empirical \(\sqrt{n}\) SD, \(n = 500\)
properties[interval_calibration/n_500__ate__corrected]:sqrt_n_reported_se	1.1204	reported \(\sqrt{n}\) SE, \(n = 500\)
properties[interval_calibration/n_2000__ate__corrected]:sqrt_n_empirical_sd	1.1076	empirical \(\sqrt{n}\) SD, \(n = 2000\)
properties[interval_calibration/n_2000__ate__corrected]:sqrt_n_reported_se	1.1226	reported \(\sqrt{n}\) SE, \(n = 2000\)
properties[interval_calibration/n_8000__ate__corrected]:sqrt_n_empirical_sd	1.1097	empirical \(\sqrt{n}\) SD, \(n = 8000\)
properties[interval_calibration/n_8000__ate__corrected]:sqrt_n_reported_se	1.1231	reported \(\sqrt{n}\) SE, \(n = 8000\)
properties[interval_calibration/bounds_active__ate__corrected]:clipped_share	0.1440	share of rows at a g_bounds limit, steep law
properties[type_i_error/sharp_null]:rejection_rate	0.0550	rejection rate at the sharp null
properties[type_i_error/sharp_null]:rejection_ci_upper	0.0792	its 99% upper endpoint
properties[power/alternative]:rejection_rate	0.8442	rejection rate at the alternative
properties[power/alternative]:rejection_ci_lower	0.8154	its 99% lower endpoint

Limits

limit	what it means
In sample, one split, one draw	The fit is not cross-fitted, and repeats is 1. This study measures no cross-fitted or repeated fit. The cross-fitted study of X58 part (c) measures them
Two arms	The treatment is binary. The extensions study of X58 part (b) measures three arms, ey_obs, par, paf and the simultaneous band
Complete, unweighted, unstratified, unclustered data	The study measures no weight, stratum, missing outcome, missing treatment or cluster. The extensions study of X58 part (d) measures a known fixed weight, and X52 holds the rest
Two parametric learners only	Only ParametricOutcome and ParametricMechanism are admitted, as exact classes. A scikit-learn learner keeps the "generated_design_plugin" status, and F19 owns its interval
The interval covers \(\psi^*\)	Under a wrong outcome model the interval covers the limit \(\psi^*\) and not \(\psi_0\). The pseudo-true cells report the coverage of \(\psi_0\) and gate nothing on it. The claim is pointwise, not locally uniform over the nonparametric model
Default alpha only	Every cell runs at the default alpha. No cell sets alpha below it. The clip caveat for a user alpha such as 0.9 has no study cell
Binary outcomes and one bounded outcome	Seven laws on three normal covariates. The bounded outcome is a scaled binary draw. A continuous outcome with a Gaussian family has fast-tier evidence only
The covariance is reported, not gated	The covariance explains why. Only the ATE row and the cross-term control test it
Intervals only	The bootstrap of an oat fit stays a diagnostic. This study does not measure it

Reproduction

Run python -m tests.canonical.ctmle_oat_per_arm_parametric.regenerate to regenerate the artifacts. The directory tests/canonical/ctmle_oat_per_arm_parametric holds manifest.json, run.log and the committed results. The replications and property results carry every published row. The run used 4,788 wall seconds on four cores.