Deterministic point-treatment regimes

This study validates ordinary static and covariate-dependent deterministic regimes. It compares cleverly with pinned R lmtp 1.5.4 on identical binary-outcome samples and exact treatment probabilities. The dynamic rule assigns both arms and differs from either static endpoint.

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
binary-outcome law with a covariate-dependent deterministic rule	ate_regime[rule vs never]	difference in means under the regimes “follow the covariate-dependent rule” against “assign no treatment”	cleverly	-0.0027 to 0.000738	0.9525	0.9990	pass
binary-outcome law with a covariate-dependent deterministic rule	ate_regime[rule vs never]	difference in means under the regimes “follow the covariate-dependent rule” against “assign no treatment”	R lmtp	-0.0026 to 0.000768	0.9513	1.0014	pass
binary-outcome law with a covariate-dependent deterministic rule	ey_regime[never]	mean under the regime assign no treatment	cleverly	-0.000873 to 0.0018	0.9500	1.0070	pass
binary-outcome law with a covariate-dependent deterministic rule	ey_regime[never]	mean under the regime assign no treatment	R lmtp	-0.000902 to 0.0017	0.9500	1.0071	pass
binary-outcome law with a covariate-dependent deterministic rule	ey_regime[rule]	mean under the regime follow the covariate-dependent rule	cleverly	-0.0020 to 0.0010	0.9563	1.0322	pass
binary-outcome law with a covariate-dependent deterministic rule	ey_regime[rule]	mean under the regime follow the covariate-dependent rule	R lmtp	-0.0021 to 0.0010	0.9550	1.0335	pass

Agreement with the canonical implementation

Comparator-backed: a paired comparison with lmtp on the same samples. Each row reports estimate equivalence within the declared margin, the RMSE ratio and the coverage difference, and names its comparison kind.

law	estimand	what was compared	paired difference	share of margin used	RMSE ratio bound	coverage difference	comparison kind	calibration resolution	result
binary-outcome law with a covariate-dependent deterministic rule	ate_regime[rule vs never]	difference in means under the regimes “follow the covariate-dependent rule” against “assign no treatment”	-0.000025	0.0091	1.0009	0.0012	different estimator	0.0054 vs 0.0500	equivalent
binary-outcome law with a covariate-dependent deterministic rule	ey_regime[never]	mean under the regime assign no treatment	0.000029	0.0132	1.0011	0	different estimator	0.000900 vs 0.0500	equivalent
binary-outcome law with a covariate-dependent deterministic rule	ey_regime[rule]	mean under the regime follow the covariate-dependent rule	0.000003	0.0013	0.9957	0.0013	different estimator	0.0044 vs 0.0500	equivalent

Theory properties

property	cell	role	what was tested	what must hold	measured	result
double_robustness	both_correct	positive	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 0.0014, margin 0.0046, SE ratio 0.9911	pass
double_robustness	both_wrong	control	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 -0.2079 to -0.2047, margin 0.0054, SE ratio 1.5052	pass
double_robustness	outcome_correct	positive	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 -0.0020 to 0.000562, margin 0.0043, SE ratio 1.7493	pass
double_robustness	treatment_correct	positive	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 -0.0020 to 0.0010, margin 0.0050, SE ratio 0.9999	pass
interval_calibration	rule__correctly_specified	positive	covariate-dependent rule: 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 0.9451 to 0.9623, SE ratio 0.9829 to 1.0392, empirical efficiency ratio 0.9615 to 1.0169, reported efficiency ratio 0.9986 to 1.0001	pass
interval_calibration	rule__noise_control	control	covariate-dependent rule: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8302 to 0.8599, SE ratio 0.6993 to 0.7411, empirical efficiency ratio 1.3484 to 1.4288, reported efficiency ratio 0.9985 to 1.0001	pass
interval_calibration	rule__shrunken_se_control	control	covariate-dependent rule: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8183 to 0.8489, SE ratio 0.6875 to 0.7277, empirical efficiency ratio 0.9612 to 1.0177, reported efficiency ratio 0.6990 to 0.7001	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 0.000937, coverage 0.9447 to 0.9796, SE ratio 1.0152	pass
root_n_and_efficiency	n_500	control	bias, coverage and SE calibration at n = 500	coverage interval lies below nominal or clears the declared floor	bias 0.000019, coverage 0.9267 to 0.9677, SE ratio 0.9942	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 0.000003, coverage 0.9223 to 0.9647, SE ratio 0.9797	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 -0.5263 to -0.4613	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 -0.4983	pass
rule_necessity	rule__declared	positive	covariate-dependent rule: the declared covariate-dependent rule assigns both treatment arms	bias interval inside the equivalence margin	bias -0.0015 to 0.0012, margin 0.0046	pass
rule_necessity	rule__static_control	control	covariate-dependent rule: the same fit replaces the rule with an always-treated static plan	bias interval must fall entirely outside the margin	bias 0.0885 to 0.0916, margin 0.0052	pass
static_reduction	never__arm	positive	never-treated plan: the same target is requested as the ordinary untreated-arm mean	the paired estimate must equal the regime estimate exactly	maximum paired difference 0	pass
static_reduction	never__regime	positive	never-treated plan: the never-treated target is requested through the regime axis	the paired estimate must equal the treatment-arm estimate exactly	maximum paired difference 0	pass
targeting_necessity	rule__targeted	positive	covariate-dependent rule: the estimator fluctuates a misspecified outcome model, so targeting does all the adjusting	bias interval inside the equivalence margin	bias -0.000676 to 0.0022, margin 0.0048	pass
targeting_necessity	rule__untargeted	control	covariate-dependent rule: the identical fit with every fluctuation step removed	bias interval must fall entirely outside the margin	bias -0.3803 to -0.3798, margin 0.000718	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 0.0375, 0.0224 to 0.0584	pass

The property study separately tests both sides of double robustness, exact-EIF calibration, a confounded sharp null, targeting, the declared rule, and exact static reduction.

Measured values and declared margins

quantity	value	source
replicates	800	paired replications
n	2000	observations per 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	20	property cells passing
property_cells_total	20	property cells reported
max_standardized_bias	0.0517	largest primary standardized bias
min_coverage	0.9500	lowest primary coverage
max_margin_utilization	0.0132	largest paired similarity-margin share
margin:confidence_level	0.9900	Monte Carlo confidence level
margin:alpha	0.0500	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: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:efficiency_ratio_lower	0.9000	efficiency-ratio lower bound
margin:efficiency_ratio_upper	1.1000	efficiency-ratio upper bound
margin:shrunken_se_factor	0.7000	negative-control SE multiplier
margin:targeting_displacement	0.2500	minimum targeting displacement
margin:necessity_displacement	0.2500	minimum rule-control displacement

Limits

• The laws use binary treatment, binary outcome, and one three-level baseline covariate.

• The primary working outcome model is logistic and the treatment mechanism is supplied exactly.

• The study covers ordinary, non-cross-fitted targeting and pointwise Wald intervals.

• It excludes missing outcomes, observation weights, clusters, simultaneous bands, and flexible learners. The default-band study measures the default band of this study’s subject fit.

• It does not validate stochastic, continuous-dose, incremental, or longitudinal interventions.