Ordinary longitudinal MSM projection#

This study validates the ordinary identity-link projection over four censored two-time-point treatment plans. The working model has an intercept and treatment-duration term, so four regimen means are reduced to two coefficients under a fixed nonuniform measure. The R comparison fits each plan with pinned ltmle, then applies the declared projection to the estimates and their joint influence curves. Raw ltmleMSM coefficients are not compared because they use a quasibinomial projection rather than this outcome-scale weighted least-squares parameter.

What was compared#

setting	`cleverly`	projected R `ltmle` fits
datasets	800 censored two-time-point samples	the identical rows
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 ordinary 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
intervals	pointwise 95% Wald from the joint coefficient curve	the projected joint regimen curves

Projecting the joint curves is material. Projecting four marginal standard errors would discard the covariance created because every regimen fit uses the same sample.

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	`msm_regimen[(intercept)]`	longitudinal regimen MSM projection intercept coefficient	`cleverly`	-0.0037 to 0.0015	0.9413	0.9944	pass
two-time-point law with monotone censoring and four projected treatment plans	`msm_regimen[(intercept)]`	longitudinal regimen MSM projection intercept coefficient	projected R `ltmle` regimen fits	-0.0037 to 0.0015	0.9413	0.9945	pass
two-time-point law with monotone censoring and four projected treatment plans	`msm_regimen[duration]`	longitudinal regimen MSM projection treatment-duration coefficient	`cleverly`	-0.000999 to 0.0019	0.9337	0.9805	pass
two-time-point law with monotone censoring and four projected treatment plans	`msm_regimen[duration]`	longitudinal regimen MSM projection treatment-duration coefficient	projected R `ltmle` regimen fits	-0.000998 to 0.0019	0.9337	0.9807	pass

Agreement with the canonical implementation#

Comparator-backed: a paired comparison with ltmle 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	`msm_regimen[(intercept)]`	longitudinal regimen MSM projection intercept coefficient	0.000002	0.000370	1.0003	0	related target	0.000171 vs 0.0500	equivalent
two-time-point law with monotone censoring and four projected treatment plans	`msm_regimen[duration]`	longitudinal regimen MSM projection treatment-duration coefficient	-8.634e-07	0.000362	1.0004	0	related target	0.000181 vs 0.0500	equivalent

Theory properties#

property	cell	role	what was tested	what must hold	measured	result
`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 -0.0054 to 0.0067, margin 0.0185, SE ratio 0.9594	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 0.0438 to 0.0555, margin 0.0180, SE ratio 0.8918	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 -0.0058 to 0.0058, margin 0.0178, SE ratio 1.0374	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 -0.0110 to 0.000998, margin 0.0184, SE ratio 0.8304	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 -0.0057 to 0.0035, margin 0.0141, SE ratio 0.9628	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 -0.0426 to -0.0334, margin 0.0141, SE ratio 0.6868	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 -0.0054 to 0.0034, margin 0.0136, SE ratio 1.0418	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 -0.000797 to 0.0085, margin 0.0143, SE ratio 0.6525	pass
`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 0.9383 to 0.9566, SE ratio 0.9569 to 1.0136, empirical efficiency ratio 0.9850 to 1.0434, reported efficiency ratio 0.9969 to 1.0000	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 0.8175 to 0.8482, SE ratio 0.6816 to 0.7222, empirical efficiency ratio 1.3825 to 1.4644, reported efficiency ratio 0.9969 to 1.0000	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 0.8023 to 0.8339, SE ratio 0.6701 to 0.7091, empirical efficiency ratio 0.9856 to 1.0432, reported efficiency ratio 0.6978 to 0.7000	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 0.9364 to 0.9550, SE ratio 0.9469 to 1.0023, empirical efficiency ratio 0.9944 to 1.0518, reported efficiency ratio 0.9939 to 0.9987	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 0.8049 to 0.8363, SE ratio 0.6747 to 0.7152, empirical efficiency ratio 1.3933 to 1.4765, reported efficiency ratio 0.9940 to 0.9988	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 0.7953 to 0.8274, SE ratio 0.6630 to 0.7013, empirical efficiency ratio 0.9945 to 1.0521, reported efficiency ratio 0.6958 to 0.6991	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 0.9975, 0.9885 to 0.9999	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 -0.0051 to 0.0039, 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 -0.1054 to -0.1006, margin 0.0073	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 0.0019, coverage 0.9047 to 0.9549, SE ratio 0.9893	pass
`root_n_and_efficiency`	`(intercept)__n_500`	control	intercept coefficient: bias, coverage and SE calibration at n = 500	coverage interval lies below nominal or clears the declared floor	bias 0.0032, coverage 0.8769 to 0.9345, SE ratio 0.9130	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 0.0011, coverage 0.9164 to 0.9631, SE ratio 0.9871	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 -0.0028, coverage 0.9147 to 0.9619, SE ratio 0.9725	pass
`root_n_and_efficiency`	`duration__n_500`	control	treatment-duration coefficient: bias, coverage and SE calibration at n = 500	coverage interval lies below nominal or clears the declared floor	bias -0.0033, coverage 0.8625 to 0.9234, SE ratio 0.8638	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 -0.000607, coverage 0.9265 to 0.9699, SE ratio 0.9920	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 -0.5509 to -0.4817	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 -0.4902 to -0.4844	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 -0.5600 to -0.4904	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 -0.4801 to -0.4710	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 -0.0049 to 0.0046, margin 0.0145	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 -0.0500 to -0.0403, margin 0.0148	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 0.0550, 0.0363 to 0.0792	pass

The property study uses the exact censored binary support law. Its regimen truths and influence curves are derived independently before the fixed projection is applied. The targeting control is a separately implemented backward recursion with every fluctuation removed.

Measured values and declared margins#

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`	30	property cells passing
`property_cells_total`	30	property cells reported
`max_standardized_bias`	0.0376	largest primary standardized bias
`min_coverage`	0.9337	lowest primary coverage
`max_margin_utilization`	0.000370	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:projection_displacement`	0.2500	minimum projection displacement

Limits#

The evidence covers four plans and two time points on one censored binary law.
Projection weights, plan durations, and the identity link are fixed before fitting.
The row covers ordinary n_folds=1 longitudinal MSM targeting only.
The cross-fitted longitudinal MSM projection covers the cross-fitted fit.
Intervals are pointwise Wald intervals rather than simultaneous regions. The default-band study measures the default band of this study’s subject fit.
The mechanisms are supplied exactly in the primary comparison.
Survival, competing risks, observation weights, and clustering are outside this row.