LTMLE on declared known node mechanisms

This study validates longitudinal TMLE on data that declares its treatment and censoring factors at each node. The fit reads the declared factors and runs no mechanism learner. With every factor known, the remainder of the sequential TMLE is zero. The curve is then \(D^*(\bar Q_\infty, g_0)\) (van der Laan and Gruber 2012, Theorem 2 and Section 4). The section known node mechanisms states the estimator and its exact-law evidence.

What was compared

setting	cleverly	R ltmle
datasets	1,000 samples of 2,000 rows. The two scenarios read the same samples	the identical rows
law	a two-node SMART. \(A_1 \sim \operatorname{Bernoulli}(0.5)\). Retention after node 1 is 0.9 at \(L_0 = 0\) and 0.8 at \(L_0 = 1\). \(A_2\) is \(\operatorname{Bernoulli}(0.3)\) at \(L_1 = 0\) and \(\operatorname{Bernoulli}(0.6)\) at \(L_1 = 1\). The outcome \(Y\) is binary	the same
scenarios	smart_q_correct: \(Y\) on \(L_0 + L_1\) and \(L_1\) on \(L_0\), saturated within each plan. smart_q_wrong: \(Y\) on \(L_0\) and \(L_1\) on an intercept	the same Qform
reported parameters	the means of the plans always and never, and their contrast	the same
mechanism	the declared treatment and censoring factors, read from probability columns	a numeric gform with the same factors, and gbounds = c(1e-8, 1)
sequential regressions	follower-stratified quasibinomial regressions	the same
intervals	pointwise 95% Wald from the influence curve	variance.method = "ic"

The R runner refuses a fit where a cumulative bound binds. Each truth is an exact enumeration of the law.

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
two-node SMART with known factors, correct outcome regressions	ate_regimen[always vs never]	difference in mean outcome between the plans “treat at both times” against “treat at neither time”	cleverly LTMLE on declared known node mechanisms	-0.0032 to 0.0018	0.9480	0.9804	pass
two-node SMART with known factors, correct outcome regressions	ate_regimen[always vs never]	difference in mean outcome between the plans “treat at both times” against “treat at neither time”	R ltmle with the known factors as a numeric gform	-0.0032 to 0.0018	0.9480	0.9804	pass
two-node SMART with known factors, correct outcome regressions	ey_regimen[always]	mean outcome under the plan treat at both times	cleverly LTMLE on declared known node mechanisms	-0.0021 to 0.0012	0.9350	0.9589	pass
two-node SMART with known factors, correct outcome regressions	ey_regimen[always]	mean outcome under the plan treat at both times	R ltmle with the known factors as a numeric gform	-0.0021 to 0.0012	0.9350	0.9589	pass
two-node SMART with known factors, correct outcome regressions	ey_regimen[never]	mean outcome under the plan treat at neither time	cleverly LTMLE on declared known node mechanisms	-0.0017 to 0.0022	0.9550	0.9896	pass
two-node SMART with known factors, correct outcome regressions	ey_regimen[never]	mean outcome under the plan treat at neither time	R ltmle with the known factors as a numeric gform	-0.0017 to 0.0022	0.9550	0.9896	pass
two-node SMART with known factors, outcome regressions without L1	ate_regimen[always vs never]	difference in mean outcome between the plans “treat at both times” against “treat at neither time”	cleverly LTMLE on declared known node mechanisms	-0.0033 to 0.0018	0.9530	0.9801	pass
two-node SMART with known factors, outcome regressions without L1	ate_regimen[always vs never]	difference in mean outcome between the plans “treat at both times” against “treat at neither time”	R ltmle with the known factors as a numeric gform	-0.0033 to 0.0018	0.9530	0.9801	pass
two-node SMART with known factors, outcome regressions without L1	ey_regimen[always]	mean outcome under the plan treat at both times	cleverly LTMLE on declared known node mechanisms	-0.0021 to 0.0011	0.9400	0.9617	pass
two-node SMART with known factors, outcome regressions without L1	ey_regimen[always]	mean outcome under the plan treat at both times	R ltmle with the known factors as a numeric gform	-0.0021 to 0.0011	0.9400	0.9617	pass
two-node SMART with known factors, outcome regressions without L1	ey_regimen[never]	mean outcome under the plan treat at neither time	cleverly LTMLE on declared known node mechanisms	-0.0017 to 0.0022	0.9450	0.9866	pass
two-node SMART with known factors, outcome regressions without L1	ey_regimen[never]	mean outcome under the plan treat at neither time	R ltmle with the known factors as a numeric gform	-0.0017 to 0.0022	0.9450	0.9866	pass

Agreement with the canonical implementation

Comparator-backed: a paired comparison with ltmle-known-gform 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
two-node SMART with known factors, correct outcome regressions	ate_regimen[always vs never]	difference in mean outcome between the plans “treat at both times” against “treat at neither time”	1.932e-10	4.172e-08	1.0000	0	matching implementation	1.890e-09 vs 0.0500	equivalent
two-node SMART with known factors, correct outcome regressions	ey_regimen[always]	mean outcome under the plan treat at both times	1.899e-10	6.390e-08	1.0000	0	matching implementation	8.035e-09 vs 0.0500	equivalent
two-node SMART with known factors, correct outcome regressions	ey_regimen[never]	mean outcome under the plan treat at neither time	-3.305e-12	9.239e-10	1.0000	0	matching implementation	2.146e-10 vs 0.0500	equivalent
two-node SMART with known factors, outcome regressions without L1	ate_regimen[always vs never]	difference in mean outcome between the plans “treat at both times” against “treat at neither time”	2.441e-10	5.203e-08	1.0000	0	matching implementation	9.728e-10 vs 0.0500	equivalent
two-node SMART with known factors, outcome regressions without L1	ey_regimen[always]	mean outcome under the plan treat at both times	1.859e-10	6.253e-08	1.0000	0	matching implementation	5.748e-09 vs 0.0500	equivalent
two-node SMART with known factors, outcome regressions without L1	ey_regimen[never]	mean outcome under the plan treat at neither time	-5.819e-11	1.596e-08	1.0000	0	matching implementation	4.328e-10 vs 0.0500	equivalent

Theory properties

property	cell	role	what was tested	what must hold	measured	result
known_mechanism_accuracy	ate_regimen__ltmle_known__q_wrong	positive	contrast of always against never: LTMLE on a two-node SMART with every factor declared, outcome regressions without L1	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.000880 to 0.0041, margin 0.0077, coverage 0.9249 to 0.9627, SE ratio 0.9988	pass
known_mechanism_accuracy	ate_regimen__ltmle_known_treatment__censoring_estimated	positive	contrast of always against never: LTMLE on a two-node SMART with the treatment declared and the censoring estimated, outcome regressions without L1	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.000880 to 0.0041, margin 0.0077, coverage 0.9272 to 0.9644, SE ratio 0.9974	pass

Both property cells fit the contrast with the wrong outcome regressions. They read one draw for each replication, and these draws differ from the primary draws.

The first cell declares every factor. The second declares the treatment and estimates the censoring factor with a logistic regression on \((L_0, A_1)\), which is correctly specified for this law. That is the usual SMART analysis. With a wrong outcome regression, its interval is conservative for a correct parametric censoring model, and it is not established for a flexible learner. Each cell must be unbiased, cover at the floor and report a standard error inside the SE-ratio screen.

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	1000	primary replications per scenario
n	2000	observations per primary replication
independent_tests_total	12	implementation-estimand tests against the truth
independent_tests_passed	12	of those, passing
paired_tests_total	6	paired comparisons with R ltmle
paired_tests_passed	6	of those, passing
property_cells_total	2	repeated-sampling property cells
property_cells_passed	2	cells whose own and family verdicts pass
max_standardized_bias	0.0242	largest absolute primary bias in empirical standard deviations
min_coverage	0.9350	lowest measured primary-study coverage
min_se_ratio_ci_lower	0.9063	lowest bootstrap primary SE-ratio endpoint
max_se_ratio_ci_upper	1.0475	highest bootstrap primary SE-ratio endpoint
margin:alpha	0.0500	nominal size of the reported intervals, and the family level of the rule
margin:bootstrap_replicates	10000	resamples behind every bootstrap interval
margin:calibration_coverage_lower	0.9200	calibration-cell coverage band, lower limit. No cell of this study reads it
margin:calibration_coverage_upper	0.9800	calibration-cell coverage band, upper limit. No cell of this study reads it
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. No cell of this study reads it
margin:calibration_se_ratio_upper	1.0700	calibration-cell SE-ratio band, upper limit. No cell of this study reads 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 a power cell must clear. No cell of this study reads it
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, in pooled empirical standard deviations
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. No cell of this study reads it
margin:root_n_slope_upper	-0.3750	accepted root-n slope band, upper limit. No cell of this study reads it
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:standardized_bias	0.2500	bias equivalence margin, in empirical standard deviations
margin:type_i_ceiling	0.1000	the rate a positive cell must bound and a control must exceed. No cell of this study reads it

Limitations

limit	what it means for use
one two-node SMART with static plans	a modified treatment policy, a held survival design, competing risks and time-to-event input on declared factors have exact-law evidence only
in sample, with parametric regressions	cross-fitting and flexible learners on declared factors have fast-tier evidence only
no estimated-mechanism control	the study does not show a bias when the declaration is removed. The exact-law tests show that a perturbed declaration moves the estimate

Reproduction

The manifest records the seeds, the margins, the estimator configuration, the source hashes and the result hashes. Run python -m tests.canonical.known_node_mechanisms.regenerate to regenerate the artifacts. The replications and the property results carry every published row.