MTP tuning

This study validates a modified treatment policy whose cap is tuned on the data. The policy is Shift(1.0, cap=c(h)) at both nodes of one law. The cap is a dose quantile (DoseQuantile) or a linear quantile regression of the dose (SupportBoundary(LinearQuantileRegression(...))). The modified treatment policies at a node section states the estimator and its conditions.

Status of this page. The study is registered, and its Python phases have run. No other implementation fits the tuning term of a tuned cap, so the study has no comparator and no pending reference. The publication policy is reporting. A red cell stays red on this page and in the red-cell ledger, with the owner row that names it.

part	what it is	status
Python property evidence	exact law by quadrature, interval calibration with three controls for each positive	the tables below give the verdicts. It decides whether the method ships
comparator runs	none	no implementation fits a tuned cap, so the row rests on the other two columns

What was compared

The primary phase draws 1,000 samples of 2,000 rows for each of two scenarios.

scenario	subject
population_quantile_ltmle	LTMLE at one fold, population target, DoseQuantile caps, the oracle density on the density route
population_quantile_sdr	the same draw, SDR

Law Q has two nodes. The baseline \(W\) is a truncated normal, the first dose \(A_1\) is normal given \(W\), the covariate \(L_2\) is Bernoulli given \(A_1\) and \(W\), and the second dose \(A_2\) is normal given the history. A Beta law draws the outcome. The policy adds 1 to the dose, and holds the dose when the sum passes the cap. The truth and the influence curve come from quadrature (tests/law_q_tuned.py), and the implementation never enters that module.

target	definition	what the curve adds
population	\(\Psi_0 = \Phi(\theta_0)\) at the population value of the tuner	the tuning term \(\sum_t \partial_t\Phi' \mathrm{IF}_t\)
fold average	the mean of \(\Phi(\theta^{(-k)})\) over the fold maps	none. The truth varies by replication

The first declaration used 200 primary replications. They could not pass the bias-equivalence and coverage-floor gates at any plausible bias, and both rows read coverage 0.935 and an SE ratio of 0.92. The second declaration sizes the primary at 1,000. The claims are the property cells.

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
two-node law Q with a dose-quantile cap, LTMLE	ey_regimen[tuned_cap]	mean outcome under the plan add 1 to the dose at both times where the result stays below the tuned cap	cleverly LTMLE and SDR under a shift with a tuned cap	-0.0012 to 0.000719	0.9470	1.0024	pass
two-node law Q with a dose-quantile cap, SDR	ey_regimen[tuned_cap]	mean outcome under the plan add 1 to the dose at both times where the result stays below the tuned cap	cleverly LTMLE and SDR under a shift with a tuned cap	-0.0012 to 0.000718	0.9470	1.0024	pass

Agreement with the canonical implementation

None claimed. No canonical implementation is compared, because no other implementation fits the tuning term of a tuned cap. [0/0 external comparisons; the row rests on the other two columns]

Theory properties

property	cell	role	what was tested	what must hold	measured	result
interval_calibration	fold_average_dose_quantile_tau2__correctly_specified	positive	the fold-average target of the dose-quantile cap: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9398 to 0.9535, SE ratio 0.9726 to 1.0160	pass
interval_calibration	fold_average_dose_quantile_tau2__noise_control	control	the fold-average target of the dose-quantile cap: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8264 to 0.8489, SE ratio 0.6891 to 0.7193	pass
interval_calibration	fold_average_dose_quantile_tau2__population_truth_control	control	the fold-average target of the dose-quantile cap: the fold-average rows are scored on the population target	the SE-ratio interval must fall below the calibration band	coverage 0.7628 to 0.7882, SE ratio 0.6024 to 0.6285	pass
interval_calibration	fold_average_dose_quantile_tau2__shrunken_se_control	control	the fold-average target of the dose-quantile cap: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8187 to 0.8417, SE ratio 0.6812 to 0.7116	pass
interval_calibration	fold_average_quantile_regression_tau2__correctly_specified	positive	the fold-average target of the quantile-regression cap: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9390 to 0.9529, SE ratio 0.9615 to 1.0043	pass
interval_calibration	fold_average_quantile_regression_tau2__noise_control	control	the fold-average target of the quantile-regression cap: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8147 to 0.8379, SE ratio 0.6815 to 0.7122	pass
interval_calibration	fold_average_quantile_regression_tau2__population_truth_control	control	the fold-average target of the quantile-regression cap: the fold-average rows are scored on the population target	the SE-ratio interval must fall below the calibration band	coverage 0.8771 to 0.8965, SE ratio 0.7968 to 0.8325	pass
interval_calibration	fold_average_quantile_regression_tau2__shrunken_se_control	control	the fold-average target of the quantile-regression cap: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8093 to 0.8327, SE ratio 0.6731 to 0.7030	pass
interval_calibration	population_dose_quantile_crossfit_classifier_tau2__correctly_specified	positive	the population target of the dose-quantile cap, five folds, classifier route: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9440 to 0.9573, SE ratio 0.9887 to 1.0338	pass
interval_calibration	population_dose_quantile_crossfit_classifier_tau2__noise_control	control	the population target of the dose-quantile cap, five folds, classifier route: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8213 to 0.8441, SE ratio 0.6951 to 0.7252	pass
interval_calibration	population_dose_quantile_crossfit_classifier_tau2__shrunken_se_control	control	the population target of the dose-quantile cap, five folds, classifier route: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8236 to 0.8462, SE ratio 0.6927 to 0.7237	pass
interval_calibration	population_dose_quantile_crossfit_classifier_tau2__term_dropped_control	control	the population target of the dose-quantile cap, five folds, classifier route: the reported standard error omits the tuning term of the curve	the SE-ratio interval must fall below the calibration band	coverage 0.7622 to 0.7877, SE ratio 0.6085 to 0.6356	pass
interval_calibration	population_dose_quantile_crossfit_tau2__correctly_specified	positive	the population target of the dose-quantile cap, five folds, density route: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9451 to 0.9582, SE ratio 1.0017 to 1.0446	pass
interval_calibration	population_dose_quantile_crossfit_tau2__noise_control	control	the population target of the dose-quantile cap, five folds, density route: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8293 to 0.8516, SE ratio 0.7036 to 0.7348	pass
interval_calibration	population_dose_quantile_crossfit_tau2__shrunken_se_control	control	the population target of the dose-quantile cap, five folds, density route: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8269 to 0.8494, SE ratio 0.7012 to 0.7323	pass
interval_calibration	population_dose_quantile_crossfit_tau2__term_dropped_control	control	the population target of the dose-quantile cap, five folds, density route: the reported standard error omits the tuning term of the curve	the SE-ratio interval must fall below the calibration band	coverage 0.7675 to 0.7927, SE ratio 0.6163 to 0.6432	pass
interval_calibration	population_dose_quantile_tau2__correctly_specified	positive	the population target of the dose-quantile cap, one fold: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9430 to 0.9564, SE ratio 0.9954 to 1.0382	pass
interval_calibration	population_dose_quantile_tau2__noise_control	control	the population target of the dose-quantile cap, one fold: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8259 to 0.8484, SE ratio 0.6981 to 0.7286	pass
interval_calibration	population_dose_quantile_tau2__shrunken_se_control	control	the population target of the dose-quantile cap, one fold: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8194 to 0.8423, SE ratio 0.6964 to 0.7263	pass
interval_calibration	population_dose_quantile_tau2__term_dropped_control	control	the population target of the dose-quantile cap, one fold: the reported standard error omits the tuning term of the curve	the SE-ratio interval must fall below the calibration band	coverage 0.7625 to 0.7880, SE ratio 0.6123 to 0.6387	pass
interval_calibration	population_quantile_regression_tau2__correctly_specified	positive	the population target of the quantile-regression cap, one fold: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9399 to 0.9537, SE ratio 0.9749 to 1.0183	pass
interval_calibration	population_quantile_regression_tau2__noise_control	control	the population target of the quantile-regression cap, one fold: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8243 to 0.8469, SE ratio 0.6951 to 0.7257	pass
interval_calibration	population_quantile_regression_tau2__shrunken_se_control	control	the population target of the quantile-regression cap, one fold: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8206 to 0.8434, SE ratio 0.6823 to 0.7130	pass
interval_calibration	population_quantile_regression_tau2__term_dropped_control	control	the population target of the quantile-regression cap, one fold: the reported standard error omits the tuning term of the curve	the SE-ratio interval must fall below the calibration band	coverage 0.8865 to 0.9052, SE ratio 0.8098 to 0.8457	pass

calibration positive	construction and truth
population_dose_quantile_tau2	one fold, DoseQuantile, oracle density. \(\Psi_0\)
population_quantile_regression_tau2	one fold, LinearQuantileRegression. \(\Psi_0\)
population_dose_quantile_crossfit_tau2	five folds, oracle density, ratio at \(\theta_n\). \(\Psi_0\)
population_dose_quantile_crossfit_classifier_tau2	five folds, the declared classifier at \(\theta^{(-k)}\). \(\Psi_0\)
fold_average_dose_quantile_tau2	five folds, fold-average target
fold_average_quantile_regression_tau2	the same with LinearQuantileRegression

Each positive has a shrunken-SE control and a noise control. It also has one derived control that removes a term which can vanish at the truth.

control	cells	what it removes	expected SE ratio
term_dropped_control	population cells	the standard error of the curve without the tuning term	0.612 (dose quantile) and 0.832 (regression)
population_truth_control	fold-average cells	the target: the same rows scored on \(\Psi_0\)	the same reading

Both controls pass when the SE ratio leaves the band below. The sizes and the pilot are those of mtp-pricing, because the six arms enter its 18-arm pooled rule.

Measured values and declared margins

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	1000	paired replications
n	2000	observations per paired replication
independent_tests_passed	2	truth tests passing
independent_tests_total	2	truth tests reported
paired_tests_passed	0	paired comparisons passing
paired_tests_total	0	paired comparisons reported
property_cells_passed	24	property cells passing
property_cells_total	24	property cells reported
max_standardized_bias	0.0208	largest primary standardized bias
min_coverage	0.9470	lowest primary coverage
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	power lower bound
margin:root_n_slope	-0.5000	expected root-n slope
margin:root_n_slope_lower	-0.6250	root-n slope lower bound
margin:root_n_slope_upper	-0.3750	root-n slope upper bound
margin:excluded_slope	-0.2500	slower rate the interval must exclude
margin:shrunken_se_factor	0.7000	negative-control SE multiplier
margin:subject_failure_bound	0.0100	declared share of replications a cell may lose to a recordable fitting failure
margin:subject_failures_allowed	10	the same bound, in primary replications

Limits

limit	what it means for use
one law with two nodes and a Beta outcome	the evidence covers one tuned-cap law. The study has no size ladder, so it claims no rate
the density is an oracle on the one-fold cells	the density route reads the generating density. The cross-fitted classifier cell reads a learned ratio
the caps are two tuners	a dose quantile and a linear quantile regression. Other tuners are outside this evidence
the kernel bandwidth follows a declared rule	undersmoothed for the dose quantile and Silverman for the regression, set by the relative bias of the kernel terms at n = 2,000
there is no external comparator	no other implementation fits a tuned cap