Cross-fitted longitudinal MSM projection

This study validates the cross-fitted identity-link projection over four censored two-time-point treatment plans. The subject is the ordinary longitudinal MSM projection subject at five outer folds. Each fold runs the untargeted recursion of every plan on its training rows, and one stacked fluctuation per node targets every follower of every plan. The longitudinal projection states the construction and its theory.

No pinned comparator cross-fits a longitudinal MSM. The R comparison fits each plan with pinned lmtp 1.5.4 on the same realized folds, then applies the declared projection to the four estimates and their joint influence curves. The design has no effect modifier, so that projection is the same estimand.

What was compared

setting	cleverly	projected R lmtp fits
datasets	800 censored two-time-point samples of 2,500 rows	the identical rows
folds	five outer folds, from the row count and random_state=0	the realized fold column
plans	never, always, early-only, and a dynamic second-node rule	the same four plans
working model	identity-link intercept and treatment-duration terms	the same fixed projection after four cross-fitted fits
projection weights	0.1, 10, 0.1, and 10 in declared plan order	the same weights
mechanisms	the generating treatment and censoring probabilities	the same, as exact per-node density ratios
targeting	one stacked fluctuation per node over every follower of every plan	one scalar fluctuation per plan, on each training fold
intervals	pointwise 95% Wald from the joint coefficient curve	the projected joint regimen curves

The two constructions differ twice. lmtp fluctuates on each training fold, and it targets each plan alone. Upstream commit 9996b04 classifies the 1.5.4 training-fold update as a bug, so the paired verdicts compare two constructions and validate neither fold-local update.

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
two-time-point law with monotone censoring and four projected treatment plans, fitted with five outer folds	msm_regimen[(intercept)]	longitudinal regimen MSM projection intercept coefficient	cleverly cross-fitted longitudinal MSM projection	-0.000671 to 0.0047	0.9375	0.9761	pass
two-time-point law with monotone censoring and four projected treatment plans, fitted with five outer folds	msm_regimen[(intercept)]	longitudinal regimen MSM projection intercept coefficient	projected R lmtp cross-fitted regimen fits	-0.000757 to 0.0045	0.9387	0.9822	pass
two-time-point law with monotone censoring and four projected treatment plans, fitted with five outer folds	msm_regimen[duration]	longitudinal regimen MSM projection treatment-duration coefficient	cleverly cross-fitted longitudinal MSM projection	-0.0025 to 0.000337	0.9563	1.0158	pass
two-time-point law with monotone censoring and four projected treatment plans, fitted with five outer folds	msm_regimen[duration]	longitudinal regimen MSM projection treatment-duration coefficient	projected R lmtp cross-fitted regimen fits	-0.0024 to 0.000370	0.9463	1.0252	pass

Agreement with the canonical implementation

Comparator-backed: a paired comparison with lmtp projected regimen fits 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-time-point law with monotone censoring and four projected treatment plans, fitted with five outer folds	msm_regimen[(intercept)]	longitudinal regimen MSM projection intercept coefficient	0.000138	0.0314	1.0315	-0.0012	related target	0.0105 vs 0.0500	equivalent
two-time-point law with monotone censoring and four projected treatment plans, fitted with five outer folds	msm_regimen[duration]	longitudinal regimen MSM projection treatment-duration coefficient	-0.000064	0.0277	1.0345	0.0100	related target	0.0247 vs 0.0500	equivalent

Theory properties

property	cell	role	what was tested	what must hold	measured	result
crossfit_overfitting	cross_fitted_msm	positive	five-fold longitudinal MSM projection with fully grown outcome trees	SE ratio clears the overfitting floor and stays inside the sanity band	SE ratio 0.9840 to 1.0256	pass
crossfit_overfitting	in_sample_control	control	the same flexible learner fitted in sample, with no cross-fitting	SE ratio must fall below the overfitting ceiling	SE ratio 0.3802 to 0.3963	pass
double_robustness	(intercept)__both_correct	positive	intercept coefficient: 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 -0.0044 to 0.0071, margin 0.0176, SE ratio 1.0224	pass
double_robustness	(intercept)__both_wrong	control	intercept coefficient: 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 0.0459 to 0.0574, margin 0.0176, SE ratio 0.9170	pass
double_robustness	(intercept)__mechanism_correct	positive	intercept coefficient: only the treatment and censoring mechanisms are correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.000076 to 0.0118, margin 0.0182, SE ratio 1.0094	pass
double_robustness	(intercept)__outcome_correct	positive	intercept coefficient: 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 -0.0056 to 0.0063, margin 0.0181, SE ratio 0.8592	pass
double_robustness	duration__both_correct	positive	treatment-duration coefficient: 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 -0.0051 to 0.0039, margin 0.0137, SE ratio 1.0063	pass
double_robustness	duration__both_wrong	control	treatment-duration coefficient: 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 -0.0431 to -0.0342, margin 0.0136, SE ratio 0.7182	pass
double_robustness	duration__mechanism_correct	positive	treatment-duration coefficient: only the treatment and censoring mechanisms are correctly specified	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.0080 to 0.000908, margin 0.0137, SE ratio 1.0261	pass
double_robustness	duration__outcome_correct	positive	treatment-duration coefficient: 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 -0.0056 to 0.0035, margin 0.0139, SE ratio 0.6871	pass
in_sample_agreement	(intercept)__in_sample_agreement	diagnostic	intercept coefficient: the absolute difference from the in-sample coefficient on the same draw, over the cross-fitted standard error; the summary publishes its mean	none; the row reports the statistic	mean absolute difference / SE 0.0287, share of draws with the in-sample estimate inside the cross-fitted interval 1.0000	reported
in_sample_agreement	duration__in_sample_agreement	diagnostic	treatment-duration coefficient: the absolute difference from the in-sample coefficient on the same draw, over the cross-fitted standard error; the summary publishes its mean	none; the row reports the statistic	mean absolute difference / SE 0.0335, share of draws with the in-sample estimate inside the cross-fitted interval 1.0000	reported
interval_calibration	(intercept)__correctly_specified	positive	intercept coefficient: 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 0.9297 to 0.9493, SE ratio 0.9548 to 1.0099, empirical efficiency ratio 1.0054 to 1.0629, reported efficiency ratio 1.0134 to 1.0164	pass
interval_calibration	(intercept)__noise_control	control	intercept coefficient: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8199 to 0.8503, SE ratio 0.6894 to 0.7297, empirical efficiency ratio 1.3910 to 1.4721, reported efficiency ratio 1.0134 to 1.0165	pass
interval_calibration	(intercept)__shrunken_se_control	control	intercept coefficient: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8020 to 0.8337, SE ratio 0.6689 to 0.7081, empirical efficiency ratio 1.0037 to 1.0619, reported efficiency ratio 0.7094 to 0.7115	pass
interval_calibration	duration__correctly_specified	positive	treatment-duration coefficient: 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 0.9275 to 0.9474, SE ratio 0.9545 to 1.0113, empirical efficiency ratio 1.0103 to 1.0689, reported efficiency ratio 1.0185 to 1.0232	pass
interval_calibration	duration__noise_control	control	treatment-duration coefficient: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8181 to 0.8486, SE ratio 0.6765 to 0.7183, empirical efficiency ratio 1.4214 to 1.5082, reported efficiency ratio 1.0185 to 1.0231	pass
interval_calibration	duration__shrunken_se_control	control	treatment-duration coefficient: 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 0.8327, SE ratio 0.6686 to 0.7075, empirical efficiency ratio 1.0104 to 1.0682, reported efficiency ratio 0.7129 to 0.7163	pass
interval_calibration	duration_logit__correctly_specified	positive	the treatment-duration coefficient of a logit working model: 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 0.9404 to 0.9585, SE ratio 0.9732 to 1.0306, empirical efficiency ratio 1.0000 to 1.0596, reported efficiency ratio 1.0286 to 1.0332	pass
interval_calibration	duration_logit__noise_control	control	the treatment-duration coefficient of a logit working model: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8323 to 0.8618, SE ratio 0.7086 to 0.7505, empirical efficiency ratio 1.3733 to 1.4549, reported efficiency ratio 1.0286 to 1.0330	pass
interval_calibration	duration_logit__shrunken_se_control	control	the treatment-duration coefficient of a logit working model: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8165 to 0.8472, SE ratio 0.6816 to 0.7211, empirical efficiency ratio 1.0007 to 1.0590, reported efficiency ratio 0.7200 to 0.7231	pass
power	duration__alternative	positive	treatment-duration coefficient: the same test applied to a law with a real effect	rejection lower bound clears the minimum power	rejection 0.9938, 0.9824 to 0.9987	pass
projection_necessity	duration__declared_weights	positive	treatment-duration coefficient: the working model uses its declared nonuniform projection weights	bias interval inside the equivalence margin	bias -0.0064 to 0.0025, margin 0.0137	pass
projection_necessity	duration__uniform_weights	control	treatment-duration coefficient: the identical working model is projected under uniform weights	bias interval must fall entirely outside the margin	bias -0.1051 to -0.1002, margin 0.0075	pass
root_n_and_efficiency	(intercept)__n_1000	control	intercept coefficient: bias, coverage and SE calibration at n = 1,000	coverage interval lies below nominal or clears the declared floor	bias -0.0029, coverage 0.9265 to 0.9699, SE ratio 1.0428	pass
root_n_and_efficiency	(intercept)__n_2000	positive	intercept coefficient: bias, coverage and SE calibration at n = 2,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias -0.0023, coverage 0.9147 to 0.9619, SE ratio 0.9773	pass
root_n_and_efficiency	(intercept)__n_8000	positive	intercept coefficient: bias, coverage and SE calibration at n = 8,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias 0.000142, coverage 0.9351 to 0.9755, SE ratio 1.0385	pass
root_n_and_efficiency	duration__n_1000	control	treatment-duration coefficient: bias, coverage and SE calibration at n = 1,000	coverage interval lies below nominal or clears the declared floor	bias 0.0015, coverage 0.9214 to 0.9665, SE ratio 1.0182	pass
root_n_and_efficiency	duration__n_2000	positive	treatment-duration coefficient: bias, coverage and SE calibration at n = 2,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias 0.000688, coverage 0.9231 to 0.9677, SE ratio 0.9991	pass
root_n_and_efficiency	duration__n_8000	positive	treatment-duration coefficient: bias, coverage and SE calibration at n = 8,000	bias inside the margin, coverage clears the floor, SE ratio inside the sanity band	bias 0.000276, coverage 0.9369 to 0.9766, SE ratio 1.0192	pass
root_n_rate	(intercept)__empirical_sd	positive	intercept coefficient: 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 -0.5670 to -0.4661	pass
root_n_rate	(intercept)__reported_se	positive	intercept coefficient: the same regression applied to the mean reported standard error	slope interval inside the root-n band and excluding -1/4	slope -0.5143 to -0.5097	pass
root_n_rate	duration__empirical_sd	positive	treatment-duration coefficient: 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 -0.5697 to -0.4687	pass
root_n_rate	duration__reported_se	positive	treatment-duration coefficient: the same regression applied to the mean reported standard error	slope interval inside the root-n band and excluding -1/4	slope -0.5206 to -0.5133	pass
simultaneous_coverage	cross_fitted_msm__pointwise_joint_control	control	the two terms of the cross-fitted longitudinal MSM projection: the pointwise 95% intervals of the same fits, read jointly	joint coverage upper endpoint must fall below the nominal rate	joint coverage 0.9120 to 0.9339	pass
simultaneous_coverage	cross_fitted_msm__simultaneous_band	positive	the two terms of the cross-fitted longitudinal MSM projection: the max-t multiplier band over every estimand the law reports, from the same-row centered influence curves	joint coverage interval inside the calibration coverage band	joint coverage 0.9391 to 0.9573	pass
targeting_necessity	duration__targeted	positive	treatment-duration coefficient: the estimator fluctuates a misspecified outcome model, so targeting does all the adjusting	bias interval inside the equivalence margin	bias -0.0036 to 0.0056, margin 0.0140	pass
targeting_necessity	duration__untargeted	control	treatment-duration coefficient: the identical fit with every fluctuation step removed	bias interval must fall entirely outside the margin	bias -0.0486 to -0.0394, margin 0.0141	pass
type_i_error	duration__sharp_null	positive	treatment-duration coefficient: a confounded law whose true contrast is exactly zero	one-sided rejection bound stays under the declared type-I ceiling	rejection 0.0700, 0.0488 to 0.0965	pass

Every property family but two uses the exact censored binary support law of the ordinary study. Its regimen truths and influence curves are derived independently before the fixed projection is applied. The logit truth is a longhand Newton solve of the weighted logistic projection. The overfitting pair and the band cell use the primary law. The untargeted arm is the projection of the same fit’s stitched initial fit.

in_sample_agreement is reported, not gated. Each row is \(\lvert\hat\beta_{\mathrm{cf}} - \hat\beta_{\mathrm{in}}\rvert / \widehat{SE}_{\mathrm{cf}}\) on one draw of the n = 8,000 rung, and the summary publishes its mean.

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	800	paired replications
n	2500	observations per paired replication
independent_tests_passed	4	truth tests passing
independent_tests_total	4	truth tests reported
paired_tests_passed	2	paired comparisons passing
paired_tests_total	2	paired comparisons reported
property_cells_passed	37	property cells passing
property_cells_total	37	property cells reported
max_standardized_bias	0.0696	largest primary standardized bias
min_coverage	0.9375	lowest primary coverage
max_margin_utilization	0.0314	largest paired similarity-margin share
properties[in_sample_agreement/(intercept)__in_sample_agreement]:mean_abs_difference_over_se	0.0287	mean in-sample distance, intercept, in cross-fitted SEs
properties[in_sample_agreement/duration__in_sample_agreement]:mean_abs_difference_over_se	0.0335	mean in-sample distance, duration, in cross-fitted SEs
properties[crossfit_overfitting/cross_fitted_msm]:coverage	0.9504	cross-fitted tree coverage
properties[crossfit_overfitting/in_sample_control]:coverage	0.5490	in-sample tree coverage
properties[crossfit_overfitting/cross_fitted_msm]:replicates	8000	paired overfitting replications
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:overfit_control_ceiling	0.7500	in-sample tree SE-ratio upper bound
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	0.2500	minimum targeting displacement
margin:projection_displacement	0.2500	minimum projection displacement
margin:type_i_ceiling	0.1000	type-I upper bound
margin:minimum_power	0.8000	power lower bound
margin:overfit_se_floor	0.8500	cross-fitted tree SE-ratio lower bound
margin:overfit_coverage_gain	0.1500	minimum paired coverage gain

Limits

limit	what it means for use
one law, four plans and two time points	the evidence covers one censored binary law, with the projection weights, plan durations and terms fixed before fitting
no repeated-sampling cell for survival, competing-risk, weighted or clustered projections	these compositions have the fast-tier exact identities of tests/unit/test_cross_fitted_longitudinal_msm.py only
the n = 500 rung was dropped before the run	its failure-only probe found 5 failures in 2,000 draws: a training fold can hold only events among the followers of always. The n = 1,000 rung replaced it as the ladder’s control rung
the comparator runs a different construction	lmtp 1.5.4 fluctuates on each training fold and targets each plan alone, so each paired verdict compares two constructions
the mechanism is supplied	both implementations receive the generating probabilities, so the paired row says nothing about mechanism estimation. The double-robustness cells cover misspecified mechanisms
one fixed five-fold assignment	the row does not validate repeated folds or time-respecting splits
learner caution	the tree pair validates held-out prediction with flexible learners, not learner-library selection. With id=, Wang et al. (2024), Remark 3, caution against complex learners at few clusters