Point modified treatment policies

This study validates point-treatment TMLE under modified treatment policies beyond the additive shift. The law has one dose, a normal truncated to \([0, 6]\) given two covariates, and a binary outcome whose mean is quadratic in the dose. The modified treatment policies section states the estimator, its conditions and its exact-law evidence.

The study pairs each fit with pinned lmtp 1.5.4. Both sides read the same density ratio, the one this package computed, and the same policy dose. The adapter tests/canonical/lmtp_mtp_adapter.R writes both into an lmtp task, so lmtp fits no ratio.

What was compared

setting	cleverly	R lmtp
datasets	1,000 samples of 2,000 rows	the identical rows
policies	the natural course; Scale(1.25, cap=5.5); a Piecewise policy that lowers doses above 3 by 0.5; the declared ModifiedPolicy “halve below 3”, which moves a dose at or below 3 halfway toward 3	the same policy doses, read from the replicate data
density ratio	Equation (3) of Díaz, Williams, Hoffman and Schenck (2023) on an oracle binned density with ceil(320 (n / 2000)^(2/3)) bins (320 at n = 2,000)	the same ratio array, read from the replicate data
outcome regression	a logistic regression on the dose, its square and both covariates	SL.glm on the same design, through SL.glm.quadratic
targeting	the covariate submodel over the observed and policy doses	the intercept fluctuation weighted by the ratio
intervals	pointwise 95% Wald from the influence curve	the same, from lmtp’s influence curve

The untargeted estimates agree to rounding. The two fluctuations differ by construction, so the paired rows are read under the default margins and are not an exactness check.

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
one truncated-normal dose on [0, 6] with an outcome quadratic in the dose	ate_policy[halve below 3 vs natural course]	difference in means under the modified treatment policies “move a dose at or below 3 halfway toward 3” against “leave the observed treatment mechanism unchanged”	cleverly point-treatment TMLE under modified treatment policies	-0.000050 to 0.000822	0.9520	0.9931	pass
one truncated-normal dose on [0, 6] with an outcome quadratic in the dose	ate_policy[halve below 3 vs natural course]	difference in means under the modified treatment policies “move a dose at or below 3 halfway toward 3” against “leave the observed treatment mechanism unchanged”	R lmtp	-0.000058 to 0.000813	0.9510	0.9945	pass
one truncated-normal dose on [0, 6] with an outcome quadratic in the dose	ate_policy[piecewise vs natural course]	difference in means under the modified treatment policies “lower the dose by 0.5 above 3, and leave it unchanged at or below 3” against “leave the observed treatment mechanism unchanged”	cleverly point-treatment TMLE under modified treatment policies	-0.000197 to 0.000675	0.9550	1.0481	pass
one truncated-normal dose on [0, 6] with an outcome quadratic in the dose	ate_policy[piecewise vs natural course]	difference in means under the modified treatment policies “lower the dose by 0.5 above 3, and leave it unchanged at or below 3” against “leave the observed treatment mechanism unchanged”	R lmtp	-0.000199 to 0.000677	0.9540	1.0429	pass
one truncated-normal dose on [0, 6] with an outcome quadratic in the dose	ate_policy[x1.25 vs natural course]	difference in means under the modified treatment policies “multiply the dose by 1.25, holding doses whose product exceeds 5.5” against “leave the observed treatment mechanism unchanged”	cleverly point-treatment TMLE under modified treatment policies	-0.000575 to 0.000992	0.9510	1.0150	pass
one truncated-normal dose on [0, 6] with an outcome quadratic in the dose	ate_policy[x1.25 vs natural course]	difference in means under the modified treatment policies “multiply the dose by 1.25, holding doses whose product exceeds 5.5” against “leave the observed treatment mechanism unchanged”	R lmtp	-0.000589 to 0.000964	0.9560	1.0275	pass
one truncated-normal dose on [0, 6] with an outcome quadratic in the dose	ey_policy[halve below 3]	mean under the modified treatment policy move a dose at or below 3 halfway toward 3	cleverly point-treatment TMLE under modified treatment policies	-0.000142 to 0.0018	0.9550	1.0256	pass
one truncated-normal dose on [0, 6] with an outcome quadratic in the dose	ey_policy[halve below 3]	mean under the modified treatment policy move a dose at or below 3 halfway toward 3	R lmtp	-0.000150 to 0.0018	0.9550	1.0264	pass
one truncated-normal dose on [0, 6] with an outcome quadratic in the dose	ey_policy[natural course]	mean under the modified treatment policy leave the observed treatment mechanism unchanged	cleverly point-treatment TMLE under modified treatment policies	-0.000408 to 0.0013	0.9630	1.0373	pass
one truncated-normal dose on [0, 6] with an outcome quadratic in the dose	ey_policy[natural course]	mean under the modified treatment policy leave the observed treatment mechanism unchanged	R lmtp	-0.000408 to 0.0013	0.9630	1.0373	pass
one truncated-normal dose on [0, 6] with an outcome quadratic in the dose	ey_policy[piecewise]	mean under the modified treatment policy lower the dose by 0.5 above 3, and leave it unchanged at or below 3	cleverly point-treatment TMLE under modified treatment policies	-0.000280 to 0.0017	0.9560	1.0267	pass
one truncated-normal dose on [0, 6] with an outcome quadratic in the dose	ey_policy[piecewise]	mean under the modified treatment policy lower the dose by 0.5 above 3, and leave it unchanged at or below 3	R lmtp	-0.000281 to 0.0017	0.9590	1.0258	pass
one truncated-normal dose on [0, 6] with an outcome quadratic in the dose	ey_policy[x1.25]	mean under the modified treatment policy multiply the dose by 1.25, holding doses whose product exceeds 5.5	cleverly point-treatment TMLE under modified treatment policies	-0.000507 to 0.0018	0.9490	1.0332	pass
one truncated-normal dose on [0, 6] with an outcome quadratic in the dose	ey_policy[x1.25]	mean under the modified treatment policy multiply the dose by 1.25, holding doses whose product exceeds 5.5	R lmtp	-0.000521 to 0.0018	0.9540	1.0418	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
one truncated-normal dose on [0, 6] with an outcome quadratic in the dose	ate_policy[halve below 3 vs natural course]	difference in means under the modified treatment policies “move a dose at or below 3 halfway toward 3” against “leave the observed treatment mechanism unchanged”	0.000008	0.0104	1.0065	0.0010	different estimator	0.0039 vs 0.0500	equivalent
one truncated-normal dose on [0, 6] with an outcome quadratic in the dose	ate_policy[piecewise vs natural course]	difference in means under the modified treatment policies “lower the dose by 0.5 above 3, and leave it unchanged at or below 3” against “leave the observed treatment mechanism unchanged”	1.898e-07	0.000236	1.0014	0.0010	different estimator	0.0061 vs 0.0500	equivalent
one truncated-normal dose on [0, 6] with an outcome quadratic in the dose	ate_policy[x1.25 vs natural course]	difference in means under the modified treatment policies “multiply the dose by 1.25, holding doses whose product exceeds 5.5” against “leave the observed treatment mechanism unchanged”	0.000021	0.0147	1.0183	-0.0050	different estimator	0.0293 vs 0.0500	equivalent
one truncated-normal dose on [0, 6] with an outcome quadratic in the dose	ey_policy[halve below 3]	mean under the modified treatment policy move a dose at or below 3 halfway toward 3	0.000008	0.0046	1.0021	0	different estimator	0.0023 vs 0.0500	equivalent
one truncated-normal dose on [0, 6] with an outcome quadratic in the dose	ey_policy[natural course]	mean under the modified treatment policy leave the observed treatment mechanism unchanged	-1.977e-10	1.249e-07	1.0000	0	different estimator	1.439e-08 vs 0.0500	equivalent
one truncated-normal dose on [0, 6] with an outcome quadratic in the dose	ey_policy[piecewise]	mean under the modified treatment policy lower the dose by 0.5 above 3, and leave it unchanged at or below 3	1.896e-07	0.000106	1.0007	-0.0030	different estimator	0.0020 vs 0.0500	equivalent
one truncated-normal dose on [0, 6] with an outcome quadratic in the dose	ey_policy[x1.25]	mean under the modified treatment policy multiply the dose by 1.25, holding doses whose product exceeds 5.5	0.000021	0.0099	1.0117	-0.0050	different estimator	0.0174 vs 0.0500	equivalent

Theory properties

property	cell	role	what was tested	what must hold	measured	result
double_robustness	x1_25__both_correct	positive	the contrast of the dose times 1.25, capped at 5.5, against the natural course: 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 -0.000902 to 0.000752, margin 0.0025, SE ratio 0.9623	pass
double_robustness	x1_25__both_wrong	control	the contrast of the dose times 1.25, capped at 5.5, against the natural course: 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 0.0293 to 0.0308, margin 0.0023, SE ratio 1.3549	pass
double_robustness	x1_25__density_correct	positive	the contrast of the dose times 1.25, capped at 5.5, against the natural course: only the continuous-dose density ratio is correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.000541 to 0.0012, margin 0.0027, SE ratio 0.9778	pass
double_robustness	x1_25__outcome_correct	positive	the contrast of the dose times 1.25, capped at 5.5, against the natural course: 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 -0.000585 to 0.000905, margin 0.0023, SE ratio 1.2833	pass
interval_calibration	classifier_route__correctly_specified	positive	the primary policy contrast, its ratio estimated by a classifier on the true log ratio: 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 0.9255 to 0.9533, SE ratio 0.9431 to 1.0244, empirical efficiency ratio 0.9598 to 1.0421, reported efficiency ratio 0.9793 to 0.9860	pass
interval_calibration	classifier_route__noise_control	control	the primary policy contrast, its ratio estimated by a classifier on the true log ratio: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.7916 to 0.8368, SE ratio 0.6667 to 0.7239, empirical efficiency ratio 1.3578 to 1.4732, reported efficiency ratio 0.9793 to 0.9861	pass
interval_calibration	classifier_route__shrunken_se_control	control	the primary policy contrast, its ratio estimated by a classifier on the true log ratio: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8010 to 0.8454, SE ratio 0.6605 to 0.7178, empirical efficiency ratio 0.9587 to 1.0408, reported efficiency ratio 0.6855 to 0.6902	pass
interval_calibration	halve__correctly_specified	positive	the contrast of the declared policy halve below 3 against the natural course: 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 0.9479 to 0.9709, SE ratio 1.0011 to 1.0881, empirical efficiency ratio 0.9208 to 1.0003, reported efficiency ratio 1.0003 to 1.0034	fail
interval_calibration	halve__noise_control	control	the contrast of the declared policy halve below 3 against the natural course: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8268 to 0.8685, SE ratio 0.6982 to 0.7588, empirical efficiency ratio 1.3202 to 1.4346, reported efficiency ratio 1.0004 to 1.0033	pass
interval_calibration	halve__shrunken_se_control	control	the contrast of the declared policy halve below 3 against the natural course: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8315 to 0.8728, SE ratio 0.7014 to 0.7624, empirical efficiency ratio 0.9196 to 0.9996, reported efficiency ratio 0.7002 to 0.7023	pass
interval_calibration	piecewise__correctly_specified	positive	the contrast of the dose lowered by 0.5 above 3 against the natural course: 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 0.9422 to 0.9665, SE ratio 0.9660 to 1.0414, empirical efficiency ratio 0.9608 to 1.0360, reported efficiency ratio 0.9993 to 1.0018	pass
interval_calibration	piecewise__noise_control	control	the contrast of the dose lowered by 0.5 above 3 against the natural course: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8068 to 0.8506, SE ratio 0.6751 to 0.7295, empirical efficiency ratio 1.3718 to 1.4819, reported efficiency ratio 0.9994 to 1.0018	pass
interval_calibration	piecewise__shrunken_se_control	control	the contrast of the dose lowered by 0.5 above 3 against the natural course: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8042 to 0.8482, SE ratio 0.6751 to 0.7302, empirical efficiency ratio 0.9591 to 1.0374, reported efficiency ratio 0.6996 to 0.7013	pass
interval_calibration	x1_25__correctly_specified	positive	the contrast of the dose times 1.25, capped at 5.5, against the natural course: 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 0.9250 to 0.9529, SE ratio 0.9533 to 1.0320, empirical efficiency ratio 0.9548 to 1.0312, reported efficiency ratio 0.9804 to 0.9877	pass
interval_calibration	x1_25__noise_control	control	the contrast of the dose times 1.25, capped at 5.5, against the natural course: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8210 to 0.8634, SE ratio 0.6915 to 0.7495, empirical efficiency ratio 1.3139 to 1.4224, reported efficiency ratio 0.9805 to 0.9876	pass
interval_calibration	x1_25__shrunken_se_control	control	the contrast of the dose times 1.25, capped at 5.5, against the natural course: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.7995 to 0.8440, SE ratio 0.6677 to 0.7225, empirical efficiency ratio 0.9542 to 1.0313, reported efficiency ratio 0.6863 to 0.6914	pass
inverse_necessity	halve__declared_inverse	positive	the contrast of the declared policy halve below 3 against the natural course: the density ratio reads each piece at its declared inverse	bias interval inside the equivalence margin	bias -0.000694 to 0.000252, margin 0.0014	pass
inverse_necessity	halve__inverse_dropped_control	control	the contrast of the declared policy halve below 3 against the natural course: the same fit reads each moving piece at the dose itself	bias interval must fall entirely outside the margin	bias -0.0149 to -0.0149, margin 0	pass
power	x1_25__alternative	positive	the contrast of the dose times 1.25, capped at 5.5, against the natural course: the same test applied to a law with a real effect	rejection lower bound clears the minimum power	rejection 0.9617, 0.9367 to 0.9790	pass
root_n_and_efficiency	x1_25__n_2000	positive	the contrast of the dose times 1.25, capped at 5.5, against the natural course: bias, coverage and SE calibration at n = 2,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias 0.000496, coverage 0.9204 to 0.9688, SE ratio 0.9897	pass
root_n_and_efficiency	x1_25__n_500	control	the contrast of the dose times 1.25, capped at 5.5, against the natural course: bias, coverage and SE calibration at n = 500	coverage interval lies below nominal or clears the declared floor	bias 0.000758, coverage 0.8833 to 0.9432, SE ratio 0.9515	pass
root_n_and_efficiency	x1_25__n_8000	positive	the contrast of the dose times 1.25, capped at 5.5, against the natural course: bias, coverage and SE calibration at n = 8,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias 0.000072, coverage 0.9125 to 0.9636, SE ratio 1.0197	pass
root_n_rate	x1_25__empirical_sd	positive	the contrast of the dose times 1.25, capped at 5.5, against the natural course: 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 -0.5503 to -0.4760	pass
root_n_rate	x1_25__reported_se	positive	the contrast of the dose times 1.25, capped at 5.5, against the natural course: the same regression applied to the mean reported standard error	slope interval inside the root-n band and excluding -1/4	slope -0.4922 to -0.4826	pass
targeting_necessity	x1_25__targeted	positive	the contrast of the dose times 1.25, capped at 5.5, against the natural course: the estimator fluctuates a misspecified outcome model, so targeting does all the adjusting	bias interval inside the equivalence margin	bias -0.000143 to 0.0015, margin 0.0026	pass
targeting_necessity	x1_25__untargeted	control	the contrast of the dose times 1.25, capped at 5.5, against the natural course: the identical fit with every fluctuation step removed	bias interval must fall entirely outside the margin	bias -0.0247 to -0.0247, margin 0	pass
type_i_error	x1_25__sharp_null	positive	the contrast of the dose times 1.25, capped at 5.5, against the natural course: a confounded law whose true contrast is exactly zero	one-sided rejection bound stays under the declared type-I ceiling	rejection 0.0667, 0.0431 to 0.0973	pass

Every family reads the law with the declared policies. The truths come from Gauss-Legendre quadrature over the dose and Gauss-Hermite quadrature over the continuous covariate, checked at twice the order. The efficiency bounds are the standard deviations of the closed-form efficient influence functions, over 400,000 draws.

classifier_route estimates the ratio by stacked classification. Its classifier is a logistic regression on the true log ratio, so it is correctly specified. inverse_necessity fits “halve below 3” twice on one sample. The positive arm reads Equation (3) at the declared inverse, and the control reads it at the dose itself.

The oracle’s bin count grows as \(n^{2/3}\). The first run read a fixed 160 bins at every size. After longitudinal-mtp found that a fixed count leaves the binned ratio inconsistent, this study re-ran under a fresh declaration.

The halve__correctly_specified calibration cell is red on its SE-ratio band, in both runs. The SE ratio is 1.0429, and its 99% interval is 1.0011 to 1.0881 against an upper bound of 1.07. It was 1.0430 at 160 bins, so the bin count does not move it. Its coverage is 0.9605, and the reported standard error matches the efficiency bound (ratio 1.0018).

The influence curve with the binned ratio predicts a ratio of 0.9998. A diagnostic refitted the same configuration on 2,000 fresh draws, the cell’s own budget. It read an SE ratio of 0.999, with a bootstrap 99% interval of 0.959 to 1.043. Fewer than 1% of its bootstrap ratios reach 1.0429 (tests/diagnostics/mtp_point_halve_excursion/). So the registered draw set is a Monte Carlo excursion. The diagnostic cannot change the verdict.

The diagnosis finds no defect. By the red-cell rule the study moved to reporting, with the owner row mtp-point-calibration, and the run repeated with no other change. The repeat reproduced every artifact.

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	14	truth tests passing
independent_tests_total	14	truth tests reported
paired_tests_passed	7	paired comparisons passing
paired_tests_total	7	paired comparisons reported
property_cells_passed	26	property cells passing
property_cells_total	27	property cells reported
max_standardized_bias	0.0722	largest primary standardized bias
min_coverage	0.9490	lowest primary coverage
max_margin_utilization	0.0147	largest paired similarity-margin share
margin:confidence_level	0.9900	Monte Carlo confidence level
margin:standardized_bias	0.2500	standardized-bias margin
margin:coverage_floor	0.9000	primary coverage floor
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:paired_difference	0.1500	paired similarity margin in pooled SDs
margin:alpha	0.0500	test size
margin:bootstrap_replicates	10000	bootstrap replications
margin:nominal_coverage	0.9500	nominal interval coverage
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:coverage_noninferiority	-0.0250	coverage-difference noninferiority bound
margin:rmse_noninferiority	1.1000	RMSE-ratio noninferiority bound
margin:calibration_noninferiority	0.0500	calibration-excess noninferiority 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	exact-EIF ratio lower bound
margin:efficiency_ratio_upper	1.1000	exact-EIF ratio upper bound
margin:shrunken_se_factor	0.7000	negative-control SE multiplier
margin:targeting_displacement	0.1000	minimum targeting displacement
margin:inverse_displacement	0.1000	minimum inverse displacement
margin:type_i_ceiling	0.1000	type-I upper bound
margin:minimum_power	0.8000	power lower bound

Limits

limit	what it means for use
one law, one dose and two covariates	the evidence covers one bounded dose law with policies fixed before the run
the density is an oracle	the ratio reads the generating density through a bin count that grows with \(n\). The bin count paragraph states why a fixed count is not consistent
one calibration cell is red under reporting	the halve cell’s draw set sits in the upper tail of a calibrated cell’s spread. On fresh draws the same fit reads an SE ratio of 0.999. The mtp-point-calibration owner holds it
the paired rows are not an exactness check	the two fluctuations differ by construction, so the pairs read under the default margins
a policy must be known and fixed	a policy learned from the sample, or one that depends on the law, is outside this evidence
no cell for a categorical treatment, a risk-ratio tilt or a randomized policy	these compositions have the exact-law and end-to-end evidence of the point-treatment section only