Stratified DR-TMLE#

This study validates DRTMLE(guard=("Q", "g")) with baseline strata, strata=["V"]. The fit solves every equation of the doubly robust alternation with one block per stratum. It fits every reduced regression inside each stratum. Each stratum’s estimate is then the DR-TMLE of the law given \(V = s\). DR-TMLE estimands states the construction.

What was tested#

setting	declaration
law	the complete-data binary law of Benkeser et al. (2017), Section 5.1, with a stratum \(V \in \{0, 1, 2\}\) (probabilities 0.5, 0.3 and 0.2) that shifts both intercepts and the treatment effect. `tests/studies/stratified_alternating_law.py` computes each truth by one-dimensional quadrature
nuisances	a logistic outcome GLM on \((A, W_1, W_2, W_1 W_2, V, A V)\) and a logistic treatment GLM on \((W_1, W_2, W_1 W_2, V)\), both correct. A wrong GLM omits \(W_1 W_2\)
fit	cross-fitted on ten folds, `reduction="univariate"`, Gaussian and binomial reduced GLMs, `g_bounds=(0.01, 0.99)`
primary scenario	`stratified_both_correct`: the two arm means and the ATE in each stratum, \(n = 2000\), 800 replications
Monte Carlo inference	exact 99% intervals around every rate, and 99% intervals around every primary endpoint

The comparator is R drtmle 1.1.2 at commit 538a3a2. drtmle has no stratified form, so the runner calls it once per stratum subset. Both sides read the same rows, the same ten-fold assignment, and the subject’s initial nuisance arrays. Inside a stratum, drtmle fits its reduced regressions on the stratum’s rows, which is the construction the subject uses.

difference	effect
the subject’s outer loop stops when every stratum’s equations meet the stop rule. R `drtmle` stops on each subset alone	a stratum can take more rounds in the subject than in R
the marginal estimate of a stratified fit reduces on \((g_n, V)\). A marginal `drtmle` call reduces on \(g_n\) alone	the three marginal names are not paired. The property cells measure the marginal ATE against its exact truth

Accuracy against known truth#

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ate[V=0]`	average treatment effect, in stratum V=0	`cleverly` DR-TMLE with baseline strata	-0.0012 to 0.0046	0.9587	1.0690	pass
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ate[V=0]`	average treatment effect, in stratum V=0	R `drtmle`, once per stratum subset	-0.0015 to 0.0043	0.9525	1.0630	pass
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ate[V=1]`	average treatment effect, in stratum V=1	`cleverly` DR-TMLE with baseline strata	-0.000489 to 0.0072	0.9450	1.0096	pass
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ate[V=1]`	average treatment effect, in stratum V=1	R `drtmle`, once per stratum subset	-0.0012 to 0.0066	0.9437	1.0057	pass
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ate[V=2]`	average treatment effect, in stratum V=2	`cleverly` DR-TMLE with baseline strata	0.0022 to 0.0123	0.9287	0.9848	pass
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ate[V=2]`	average treatment effect, in stratum V=2	R `drtmle`, once per stratum subset	0.0021 to 0.0122	0.9387	0.9727	pass
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ey[0][V=0]`	counterfactual mean under treatment arm ‘0’, in stratum V=0	`cleverly` DR-TMLE with baseline strata	-0.0038 to 0.0012	0.9537	1.0364	pass
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ey[0][V=0]`	counterfactual mean under treatment arm ‘0’, in stratum V=0	R `drtmle`, once per stratum subset	-0.0035 to 0.0014	0.9450	1.0302	pass
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ey[0][V=1]`	counterfactual mean under treatment arm ‘0’, in stratum V=1	`cleverly` DR-TMLE with baseline strata	-0.0054 to 0.000269	0.9500	0.9912	pass
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ey[0][V=1]`	counterfactual mean under treatment arm ‘0’, in stratum V=1	R `drtmle`, once per stratum subset	-0.0053 to 0.000332	0.9513	0.9952	pass
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ey[0][V=2]`	counterfactual mean under treatment arm ‘0’, in stratum V=2	`cleverly` DR-TMLE with baseline strata	-0.0033 to 0.0022	0.9587	1.0685	pass
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ey[0][V=2]`	counterfactual mean under treatment arm ‘0’, in stratum V=2	R `drtmle`, once per stratum subset	-0.0032 to 0.0023	0.9575	1.0714	pass
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ey[1][V=0]`	counterfactual mean under treatment arm ‘1’, in stratum V=0	`cleverly` DR-TMLE with baseline strata	-0.0015 to 0.0023	0.9575	1.0317	pass
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ey[1][V=0]`	counterfactual mean under treatment arm ‘1’, in stratum V=0	R `drtmle`, once per stratum subset	-0.0016 to 0.0022	0.9587	1.0330	pass
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ey[1][V=1]`	counterfactual mean under treatment arm ‘1’, in stratum V=1	`cleverly` DR-TMLE with baseline strata	-0.0021 to 0.0037	0.9587	1.0100	pass
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ey[1][V=1]`	counterfactual mean under treatment arm ‘1’, in stratum V=1	R `drtmle`, once per stratum subset	-0.0027 to 0.0032	0.9537	1.0052	pass
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ey[1][V=2]`	counterfactual mean under treatment arm ‘1’, in stratum V=2	`cleverly` DR-TMLE with baseline strata	0.0023 to 0.0112	0.9000	0.9388	fail
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ey[1][V=2]`	counterfactual mean under treatment arm ‘1’, in stratum V=2	R `drtmle`, once per stratum subset	0.0022 to 0.0111	0.8925	0.9205	fail

Agreement with the canonical implementation#

Comparator-backed: a paired comparison with drtmle-r-stratified 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
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ate[V=0]`	average treatment effect, in stratum V=0	0.000339	0.0715	1.0052	0.0062	related target	0.0121 vs 0.0500	superior
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ate[V=1]`	average treatment effect, in stratum V=1	0.000652	0.1025	1.0093	0.0012	related target	0.0118 vs 0.0500	equivalent
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ate[V=2]`	average treatment effect, in stratum V=2	0.000169	0.0203	1.0251	-0.0100	related target	0.0404 vs 0.0500	equivalent
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ey[0][V=0]`	counterfactual mean under treatment arm ‘0’, in stratum V=0	-0.000248	0.0608	1.0073	0.0088	related target	0.0135 vs 0.0500	equivalent
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ey[0][V=1]`	counterfactual mean under treatment arm ‘0’, in stratum V=1	-0.000066	0.0141	1.0146	-0.0013	related target	0.0112 vs 0.0500	equivalent
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ey[0][V=2]`	counterfactual mean under treatment arm ‘0’, in stratum V=2	-0.000126	0.0276	1.0098	0.0012	related target	0.0105 vs 0.0500	equivalent
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ey[1][V=0]`	counterfactual mean under treatment arm ‘1’, in stratum V=0	0.000091	0.0292	1.0050	-0.0012	related target	0.0047 vs 0.0500	equivalent
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ey[1][V=1]`	counterfactual mean under treatment arm ‘1’, in stratum V=1	0.000587	0.1212	1.0089	0.0050	related target	0.0123 vs 0.0500	equivalent
Benkeser et al. (2017) binary law with three baseline strata, correct GLM nuisances	`ey[1][V=2]`	counterfactual mean under treatment arm ‘1’, in stratum V=2	0.000043	0.0058	1.0238	0.0075	related target	0.0277 vs 0.0500	equivalent

Repeated-sampling properties#

property	cell	role	what was tested	what must hold	measured	result
`double_robustness`	`marginal_ate__both_correct`	positive	average treatment effect over every stratum, from the stratified fit: both the outcome regression and the treatment mechanism are correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias 0.0013 to 0.0042, margin 0.0062, SE ratio 0.9629	pass
`double_robustness`	`marginal_ate__both_wrong`	control	average treatment effect over every stratum, from the stratified fit: both nuisances are misspecified	bias interval must fall entirely outside the margin, with the reported standard error still on the scale of the empirical spread	bias 0.2197 to 0.2237, margin 0.0055, SE ratio 2.4268	pass
`double_robustness`	`marginal_ate__outcome_correct`	positive	average treatment effect over every stratum, from the stratified fit: only the outcome regression is correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias 0.000440 to 0.0048, margin 0.0059, SE ratio 1.1031	pass
`double_robustness`	`marginal_ate__treatment_correct`	positive	average treatment effect over every stratum, from the stratified fit: only the treatment mechanism is correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias 0.0136 to 0.0185, margin 0.0067, SE ratio 0.9902	fail
`double_robustness`	`v0_ate__both_correct`	positive	average treatment effect in stratum V = 0: both the outcome regression and the treatment mechanism are correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.0011 to 0.0029, margin 0.0087, SE ratio 0.9659	pass
`double_robustness`	`v0_ate__both_wrong`	control	average treatment effect in stratum V = 0: both nuisances are misspecified	bias interval must fall entirely outside the margin, with the reported standard error still on the scale of the empirical spread	bias 0.2267 to 0.2323, margin 0.0077, SE ratio 2.0577	pass
`double_robustness`	`v0_ate__outcome_correct`	positive	average treatment effect in stratum V = 0: only the outcome regression is correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.000165 to 0.0060, margin 0.0084, SE ratio 1.0270	pass
`double_robustness`	`v0_ate__treatment_correct`	positive	average treatment effect in stratum V = 0: only the treatment mechanism is correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias 0.0094 to 0.0162, margin 0.0093, SE ratio 0.9505	fail
`double_robustness`	`v1_ate__both_correct`	positive	average treatment effect in stratum V = 1: both the outcome regression and the treatment mechanism are correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias 0.000831 to 0.0059, margin 0.0109, SE ratio 0.9792	pass
`double_robustness`	`v1_ate__both_wrong`	control	average treatment effect in stratum V = 1: both nuisances are misspecified	bias interval must fall entirely outside the margin, with the reported standard error still on the scale of the empirical spread	bias 0.2218 to 0.2289, margin 0.0097, SE ratio 1.8668	pass
`double_robustness`	`v1_ate__outcome_correct`	positive	average treatment effect in stratum V = 1: only the outcome regression is correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.0015 to 0.0061, margin 0.0104, SE ratio 1.0494	pass
`double_robustness`	`v1_ate__treatment_correct`	positive	average treatment effect in stratum V = 1: only the treatment mechanism is correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias 0.0123 to 0.0214, margin 0.0124, SE ratio 0.9364	fail
`double_robustness`	`v2_ate__both_correct`	positive	average treatment effect in stratum V = 2: both the outcome regression and the treatment mechanism are correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias 0.0031 to 0.0097, margin 0.0143, SE ratio 0.9544	pass
`double_robustness`	`v2_ate__both_wrong`	control	average treatment effect in stratum V = 2: both nuisances are misspecified	bias interval must fall entirely outside the margin, with the reported standard error still on the scale of the empirical spread	bias 0.1924 to 0.2010, margin 0.0118, SE ratio 1.7752	pass
`double_robustness`	`v2_ate__outcome_correct`	positive	average treatment effect in stratum V = 2: only the outcome regression is correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.0027 to 0.0071, margin 0.0134, SE ratio 1.2492	pass
`double_robustness`	`v2_ate__treatment_correct`	positive	average treatment effect in stratum V = 2: only the treatment mechanism is correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias 0.0179 to 0.0286, margin 0.0147, SE ratio 1.0627	fail
`interval_calibration`	`marginal_ate__correctly_specified`	positive	average treatment effect over every stratum, from the stratified fit: both nuisances 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 0.9355 to 0.9613, SE ratio 0.9735 to 1.0565, empirical efficiency ratio 0.9755 to 1.0587, reported efficiency ratio 1.0259 to 1.0383	pass
`interval_calibration`	`marginal_ate__noise_control`	control	average treatment effect over every stratum, from the stratified fit: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8157 to 0.8586, SE ratio 0.6843 to 0.7457, empirical efficiency ratio 1.3832 to 1.5074, reported efficiency ratio 1.0258 to 1.0387	pass
`interval_calibration`	`marginal_ate__shrunken_se_control`	control	average treatment effect over every stratum, from the stratified fit: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8141 to 0.8572, SE ratio 0.6818 to 0.7407, empirical efficiency ratio 0.9751 to 1.0585, reported efficiency ratio 0.7181 to 0.7269	pass
`interval_calibration`	`v0_ate__correctly_specified`	positive	average treatment effect in stratum V = 0: both nuisances 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 0.9305 to 0.9573, SE ratio 0.9682 to 1.0534, empirical efficiency ratio 0.9672 to 1.0520, reported efficiency ratio 1.0144 to 1.0226	pass
`interval_calibration`	`v0_ate__noise_control`	control	average treatment effect in stratum V = 0: 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 0.8577, SE ratio 0.6875 to 0.7483, empirical efficiency ratio 1.3617 to 1.4803, reported efficiency ratio 1.0145 to 1.0227	pass
`interval_calibration`	`v0_ate__shrunken_se_control`	control	average treatment effect in stratum V = 0: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8115 to 0.8549, SE ratio 0.6766 to 0.7378, empirical efficiency ratio 0.9663 to 1.0532, reported efficiency ratio 0.7103 to 0.7160	pass
`interval_calibration`	`v1_ate__correctly_specified`	positive	average treatment effect in stratum V = 1: both nuisances 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 0.9266 to 0.9542, SE ratio 0.9471 to 1.0298, empirical efficiency ratio 0.9951 to 1.0817, reported efficiency ratio 1.0201 to 1.0291	pass
`interval_calibration`	`v1_ate__noise_control`	control	average treatment effect in stratum V = 1: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8131 to 0.8563, SE ratio 0.6884 to 0.7467, empirical efficiency ratio 1.3734 to 1.4887, reported efficiency ratio 1.0202 to 1.0291	pass
`interval_calibration`	`v1_ate__shrunken_se_control`	control	average treatment effect in stratum V = 1: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8005 to 0.8449, SE ratio 0.6633 to 0.7202, empirical efficiency ratio 0.9966 to 1.0809, reported efficiency ratio 0.7141 to 0.7204	pass
`interval_calibration`	`v2_ate__correctly_specified`	positive	average treatment effect in stratum V = 2: both nuisances 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 0.9069 to 0.9380, SE ratio 0.9294 to 1.0112, empirical efficiency ratio 1.0348 to 1.1237, reported efficiency ratio 1.0296 to 1.0653	fail
`interval_calibration`	`v2_ate__noise_control`	control	average treatment effect in stratum V = 2: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.7984 to 0.8430, SE ratio 0.6702 to 0.7304, empirical efficiency ratio 1.4368 to 1.5556, reported efficiency ratio 1.0294 to 1.0659	pass
`interval_calibration`	`v2_ate__shrunken_se_control`	control	average treatment effect in stratum V = 2: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.7843 to 0.8302, SE ratio 0.6501 to 0.7072, empirical efficiency ratio 1.0361 to 1.1246, reported efficiency ratio 0.7207 to 0.7458	pass

The property grid calibrates the marginal ATE and each stratum ATE of the both-correct fit, each with a shrunken-SE control and a noise control. The robustness family fits the same targets under four nuisance configurations. The marginal cells read the same fits as the stratum cells.

Seven results are red. The four treatment_correct robustness cells carry a positive bias of 0.34 to 0.60 of their spread. The owner X8-drtmle-one-sided-bias in the roadmap records a declared diagnostic.

Inside each stratum, the shipped unstratified DRTMLE and R drtmle, handed the same initial arrays, carry a positive bias too. The direction is shared in every stratum, and the size mostly is. In stratum 2 the package’s bias, 0.0237, exceeds R’s, 0.0157, on the same arrays: the paired difference is 0.0080 with a standard error of 0.0018. The shipped unstratified fit on the stratum’s rows has the package’s bias there to 2e-5, so the excess belongs to DRTMLE at about 400 rows and not to the strata. This is the one-sided bias that RM18-one-sided-bias records for the unstratified study on the same law.

The marginal estimate is the mixture of the stratum estimates, so it inherits their bias. Its bias is 0.56 of its spread at n = 2,000 (99% interval 0.43 to 0.69), and 0.43 at n = 8,000 (0.25 to 0.61). The two intervals overlap, so the diagnostic does not establish that the bias contracts faster than the spread.

The other three reds sit in stratum 2, which holds about 400 of the 2,000 rows. The two truth rows of ey[1][V=2] cover 0.900 and 0.8925 in the two implementations on the same draws. The v2_ate__correctly_specified cell has an SE-ratio interval that ends at 0.9294 against a floor of 0.93. The owner X8-drtmle-small-stratum records the reading. The study was declared under reporting.

Measured values and declared margins#

quantity	value	source
`replicates`	800	paired replications
`n`	2000	observations per primary replication
`independent_tests_passed`	16	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`	23	property cells passing
`property_cells_total`	28	property cells reported
`max_standardized_bias`	0.1375	largest primary standardized bias
`min_coverage`	0.8925	lowest primary coverage
`max_margin_utilization`	0.1212	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
`bound:marginal_ate_standard_error`	0.0233	efficiency-bound standard error by quadrature at primary n
`bound:v0_ate_standard_error`	0.0330	efficiency-bound standard error by quadrature at primary n
`bound:v1_ate_standard_error`	0.0417	efficiency-bound standard error by quadrature at primary n
`bound:v2_ate_standard_error`	0.0526	efficiency-bound standard error by quadrature at primary n

Limits#

The study covers one law with one three-level stratum and a complete binary outcome.
The study fits only the univariate reduction at guard=("Q", "g"). The exact-law tests in tests/unit/test_stratified_drtmle_exact.py cover the other guards, the bivariate reduction, three arms, and the missing-outcome and composite routes.
The marginal estimate has no paired comparator.
The law keeps the symmetry of the paper law in each stratum, so it is H1 in both branches. The one-correct cells therefore measure second-order terms only. tests/unit/test_drtmle_h1_symmetry.py pins it for the unstratified law, and the argument is the same per stratum, and the theorem states why it matters.

Reproduction#

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