Repeated-outcome longitudinal TMLE#

This study validates LTMLE with outcome=Repeated(["Y1", "Y2", "Y3"]). A repeated outcome is measured at every node and is not absorbing. The parameter is the mean \(E[Y_h^d]\) for each plan \(d\) and horizon \(h\). The section repeated outcomes states the estimator and its exact-law evidence. The declared comparator is R ltmle 1.3-0 with survivalOutcome = FALSE.

Its runs are pending as TRACKER Study queue item 14(c). This page holds Python property evidence, with comparator runs pending.

The study reports each unique parameter once. The dynamic plan equals the always-treated plan at the first node, so its mean and contrast at \(t = 1\) duplicate those of always. A fast test checks that identity. The duplicates are not cells of this study.

What was compared#

The R column states the declared design of the pending comparator runs.

setting	`cleverly`	R `ltmle`
datasets	2,400 censored panels of 3,000 rows from `make_longitudinal_repeated`, with three nodes	the identical rows
fit	one fit with `outcome=Repeated(["Y1", "Y2", "Y3"])`	one fit of the prefix through \(Y_h\) for each horizon \(h\). The earlier \(Y_s\) are covariates
plans	`never`, `always`, and a rule that treats at node 1 and then treats exactly when the previous outcome was positive	the same three plans
reported parameters	the mean of each plan at \(t = 1, 2, 3\), and the contrasts of the other two plans against `never`	the same, 13 unique cells
mechanism	the generating treatment and censoring probabilities, read through a declared design width, with `g_bounds` from the study constant	a numeric `gform` with the same probabilities and bounds
sequential regressions	follower-stratified quasibinomial regressions on the history at each node	the same family, with a main-terms `Qform` on the same history at each block
fitting	in sample, `n_folds = 1`	in sample
intervals	pointwise 95% Wald from the influence curve	`variance.method = "ic"`
initial estimate	not published. A single-fold fit exposes no untargeted plug-in	a second call with `gcomp = TRUE`, the sequential regression with no fluctuation

The cleverly rows publish no initial estimate. The first node’s regression of the targeted chain is a different quantity from the R column. The R plug-in is the value a targeting step must move away from.

Each truth is an exact enumeration of the binary law (cleverly.datasets.repeated_truth). tests/studies/ltmle_repeated_law.py repeats the enumeration on the observed-data law.

The baseline covariate \(W\) takes the values 0, 1 and 2. The law has depth two. The treatment, censoring and outcome at node \(t\) read \(Y_{t-2}\). The covariate \(L_t\) reads \(L_{t-1}\), and the outcome \(Y_t\) also reads \(L_{t-1}\). These terms act at \(t = 3\) only. The first node’s regression on \(W\) has one design column for three cells, so its fluctuation is not zero.

At \(t = 3\) the smallest cumulative mechanism product among the units that follow a plan is 0.025, for the rule. Every plan keeps at least 11% of the units to the last node. At most 2.2% of the followers have a cumulative product below 0.05.

Accuracy against known truth#

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
three-time-point law with a three-valued baseline covariate, a repeated binary outcome that reads the two previous nodes, and monotone censoring	`ate_regimen[always vs never @ t=1]`	difference in the mean of the repeated outcome between the plans “treat at every time” against “treat at no time” at horizon t = 1	`cleverly`	-0.000963 to 0.000622	0.9442	0.9791	pass
three-time-point law with a three-valued baseline covariate, a repeated binary outcome that reads the two previous nodes, and monotone censoring	`ate_regimen[always vs never @ t=2]`	difference in the mean of the repeated outcome between the plans “treat at every time” against “treat at no time” at horizon t = 2	`cleverly`	-0.0012 to 0.000960	0.9529	1.0090	pass
three-time-point law with a three-valued baseline covariate, a repeated binary outcome that reads the two previous nodes, and monotone censoring	`ate_regimen[always vs never @ t=3]`	difference in the mean of the repeated outcome between the plans “treat at every time” against “treat at no time” at horizon t = 3	`cleverly`	-0.0012 to 0.0021	0.9483	0.9930	pass
three-time-point law with a three-valued baseline covariate, a repeated binary outcome that reads the two previous nodes, and monotone censoring	`ate_regimen[treat then treat after an outcome vs never @ t=2]`	difference in the mean of the repeated outcome between the plans “treat at the first time, then treat at each later time after an outcome of one” against “treat at no time” at horizon t = 2	`cleverly`	-0.0013 to 0.000903	0.9567	1.0266	pass
three-time-point law with a three-valued baseline covariate, a repeated binary outcome that reads the two previous nodes, and monotone censoring	`ate_regimen[treat then treat after an outcome vs never @ t=3]`	difference in the mean of the repeated outcome between the plans “treat at the first time, then treat at each later time after an outcome of one” against “treat at no time” at horizon t = 3	`cleverly`	-0.0018 to 0.0018	0.9387	0.9625	pass
three-time-point law with a three-valued baseline covariate, a repeated binary outcome that reads the two previous nodes, and monotone censoring	`ey_regimen[always @ t=1]`	mean of the repeated outcome under the plan treat at every time at horizon t = 1	`cleverly`	-0.000482 to 0.000368	0.9496	0.9964	pass
three-time-point law with a three-valued baseline covariate, a repeated binary outcome that reads the two previous nodes, and monotone censoring	`ey_regimen[always @ t=2]`	mean of the repeated outcome under the plan treat at every time at horizon t = 2	`cleverly`	-0.0011 to 0.000149	0.9454	0.9993	pass
three-time-point law with a three-valued baseline covariate, a repeated binary outcome that reads the two previous nodes, and monotone censoring	`ey_regimen[always @ t=3]`	mean of the repeated outcome under the plan treat at every time at horizon t = 3	`cleverly`	-0.000869 to 0.0012	0.9387	0.9786	pass
three-time-point law with a three-valued baseline covariate, a repeated binary outcome that reads the two previous nodes, and monotone censoring	`ey_regimen[never @ t=1]`	mean of the repeated outcome under the plan treat at no time at horizon t = 1	`cleverly`	-0.000552 to 0.000779	0.9508	0.9786	pass
three-time-point law with a three-valued baseline covariate, a repeated binary outcome that reads the two previous nodes, and monotone censoring	`ey_regimen[never @ t=2]`	mean of the repeated outcome under the plan treat at no time at horizon t = 2	`cleverly`	-0.0012 to 0.000485	0.9433	1.0014	pass
three-time-point law with a three-valued baseline covariate, a repeated binary outcome that reads the two previous nodes, and monotone censoring	`ey_regimen[never @ t=3]`	mean of the repeated outcome under the plan treat at no time at horizon t = 3	`cleverly`	-0.0015 to 0.0010	0.9529	0.9863	pass
three-time-point law with a three-valued baseline covariate, a repeated binary outcome that reads the two previous nodes, and monotone censoring	`ey_regimen[treat then treat after an outcome @ t=2]`	mean of the repeated outcome under the plan treat at the first time, then treat at each later time after an outcome of one at horizon t = 2	`cleverly`	-0.0013 to 0.000135	0.9475	1.0089	pass
three-time-point law with a three-valued baseline covariate, a repeated binary outcome that reads the two previous nodes, and monotone censoring	`ey_regimen[treat then treat after an outcome @ t=3]`	mean of the repeated outcome under the plan treat at the first time, then treat at each later time after an outcome of one at horizon t = 3	`cleverly`	-0.0014 to 0.000993	0.9325	0.9647	pass

Agreement with the canonical implementation#

Python property evidence, with comparator runs pending. The R ltmle 1.3-0 runs have not run on the declared draws, so this study publishes no paired verdict. The change that registers them adds the paired verdicts and the parity gate test_cleverly_reproduces_r_targeting_move_in_every_cell.

Theory properties#

property	cell	role	what was tested	what must hold	measured	result
`double_robustness`	`always_t3__both_wrong`	control	mean under always at horizon t = 3, repeated binary outcome: outcome regression of cell means on the baseline covariate alone, and intercept-only treatment and censoring mechanisms	bias interval must fall entirely outside the margin, and inside it around the exact limit of that fit, with the reported standard error still on the scale of the empirical spread	bias 0.0223 to 0.0260, margin 0.0068, SE ratio 1.2594, bias around its limit -0.0023 to 0.0014, margin 0.0068	pass
`double_robustness`	`always_t3__mechanism_correct`	positive	mean under always at horizon t = 3, repeated binary outcome: 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 -0.0024 to 0.0011, margin 0.0062, SE ratio 0.9739	pass
`double_robustness`	`always_t3__outcome_correct`	positive	mean under always at horizon t = 3, repeated binary outcome: 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 -0.0031 to 0.000245, margin 0.0061, SE ratio 1.1849	pass
`double_robustness`	`always_vs_never_t3__both_wrong`	control	contrast of always against never at horizon t = 3, repeated binary outcome: outcome regression of cell means on the baseline covariate alone, and intercept-only treatment and censoring mechanisms	bias interval must fall entirely outside the margin, and inside it around the exact limit of that fit, with the reported standard error still on the scale of the empirical spread	bias 0.0747 to 0.0801, margin 0.0099, SE ratio 1.1883, bias around its limit -0.0043 to 0.0012, margin 0.0099	pass
`double_robustness`	`always_vs_never_t3__mechanism_correct`	positive	contrast of always against never at horizon t = 3, repeated binary outcome: 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 -0.0028 to 0.0028, margin 0.0101, SE ratio 1.0117	pass
`double_robustness`	`always_vs_never_t3__outcome_correct`	positive	contrast of always against never at horizon t = 3, repeated binary outcome: 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 -0.0045 to 0.000955, margin 0.0099, SE ratio 1.0961	pass
`double_robustness`	`never_t3__both_wrong`	control	mean under never at horizon t = 3, repeated binary outcome: outcome regression of cell means on the baseline covariate alone, and intercept-only treatment and censoring mechanisms	bias interval must fall entirely outside the margin, and inside it around the exact limit of that fit, with the reported standard error still on the scale of the empirical spread	bias -0.0553 to -0.0513, margin 0.0072, SE ratio 1.1182, bias around its limit -0.000913 to 0.0031, margin 0.0072	pass
`double_robustness`	`never_t3__mechanism_correct`	positive	mean under never at horizon t = 3, repeated binary outcome: 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 -0.0029 to 0.0016, margin 0.0080, SE ratio 1.0258	pass
`double_robustness`	`never_t3__outcome_correct`	positive	mean under never at horizon t = 3, repeated binary outcome: 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 -0.0019 to 0.0025, margin 0.0079, SE ratio 1.0069	pass
`double_robustness`	`rule_t3__both_wrong`	control	mean under the rule that treats after an outcome at horizon t = 3, repeated binary outcome: outcome regression of cell means on the baseline covariate alone, and intercept-only treatment and censoring mechanisms	bias interval must fall entirely outside the margin, and inside it around the exact limit of that fit, with the reported standard error still on the scale of the empirical spread	bias -0.0224 to -0.0190, margin 0.0063, SE ratio 0.9095, bias around its limit -0.0019 to 0.0016, margin 0.0063	pass
`double_robustness`	`rule_t3__mechanism_correct`	positive	mean under the rule that treats after an outcome at horizon t = 3, repeated binary outcome: 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 -0.0024 to 0.0019, margin 0.0079, SE ratio 0.9913	pass
`double_robustness`	`rule_t3__outcome_correct`	positive	mean under the rule that treats after an outcome at horizon t = 3, repeated binary outcome: 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 -0.0021 to 0.0019, margin 0.0073, SE ratio 0.7386	pass
`double_robustness`	`rule_vs_never_t3__both_wrong`	control	contrast of the rule against never at horizon t = 3, repeated binary outcome: outcome regression of cell means on the baseline covariate alone, and intercept-only treatment and censoring mechanisms	bias interval must fall entirely outside the margin, and inside it around the exact limit of that fit, with the reported standard error still on the scale of the empirical spread	bias 0.0299 to 0.0352, margin 0.0096, SE ratio 1.0343, bias around its limit -0.0038 to 0.0014, margin 0.0096	pass
`double_robustness`	`rule_vs_never_t3__mechanism_correct`	positive	contrast of the rule against never at horizon t = 3, repeated binary outcome: 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 -0.0026 to 0.0035, margin 0.0111, SE ratio 1.0223	pass
`double_robustness`	`rule_vs_never_t3__outcome_correct`	positive	contrast of the rule against never at horizon t = 3, repeated binary outcome: 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 -0.0034 to 0.0026, margin 0.0109, SE ratio 0.8856	pass
`full_history_necessity`	`always_t3__full_history`	positive	mean under always at horizon t = 3, repeated binary outcome: an outcome regression of cell means on the full history, with intercept-only mechanisms, so the regression alone carries the estimate	bias interval inside the equivalence margin	bias -0.000845 to 0.000921, margin 0.0030	pass
`full_history_necessity`	`always_t3__markov_history`	control	mean under always at horizon t = 3, repeated binary outcome: the same fit with the outcome regression on the previous node only, [W, Y_{t-1}, L_t], and the same intercept-only mechanisms	bias interval outside the margin, and inside it around the exact limit of the previous-node fit	bias 0.0204 to 0.0223, margin 0.0032, bias around its limit -0.0010 to 0.000904, margin 0.0032	pass
`full_history_necessity`	`always_vs_never_t3__full_history`	positive	contrast of always against never at horizon t = 3, repeated binary outcome: an outcome regression of cell means on the full history, with intercept-only mechanisms, so the regression alone carries the estimate	bias interval inside the equivalence margin	bias -0.0011 to 0.0018, margin 0.0048	pass
`full_history_necessity`	`always_vs_never_t3__markov_history`	control	contrast of always against never at horizon t = 3, repeated binary outcome: the same fit with the outcome regression on the previous node only, [W, Y_{t-1}, L_t], and the same intercept-only mechanisms	bias interval outside the margin, and inside it around the exact limit of the previous-node fit	bias 0.0693 to 0.0722, margin 0.0049, bias around its limit -0.0011 to 0.0018, margin 0.0049	pass
`full_history_necessity`	`never_t3__full_history`	positive	mean under never at horizon t = 3, repeated binary outcome: an outcome regression of cell means on the full history, with intercept-only mechanisms, so the regression alone carries the estimate	bias interval inside the equivalence margin	bias -0.0014 to 0.000824, margin 0.0038	pass
`full_history_necessity`	`never_t3__markov_history`	control	mean under never at horizon t = 3, repeated binary outcome: the same fit with the outcome regression on the previous node only, [W, Y_{t-1}, L_t], and the same intercept-only mechanisms	bias interval outside the margin, and inside it around the exact limit of the previous-node fit	bias -0.0505 to -0.0484, margin 0.0035, bias around its limit -0.0014 to 0.000680, margin 0.0035	pass
`full_history_necessity`	`rule_t3__full_history`	positive	mean under the rule that treats after an outcome at horizon t = 3, repeated binary outcome: an outcome regression of cell means on the full history, with intercept-only mechanisms, so the regression alone carries the estimate	bias interval inside the equivalence margin	bias -0.000663 to 0.0015, margin 0.0037	pass
`full_history_necessity`	`rule_t3__markov_history`	control	mean under the rule that treats after an outcome at horizon t = 3, repeated binary outcome: the same fit with the outcome regression on the previous node only, [W, Y_{t-1}, L_t], and the same intercept-only mechanisms	bias interval outside the margin, and inside it around the exact limit of the previous-node fit	bias -0.0270 to -0.0251, margin 0.0032, bias around its limit -0.000623 to 0.0013, margin 0.0032	pass
`full_history_necessity`	`rule_vs_never_t3__full_history`	positive	contrast of the rule against never at horizon t = 3, repeated binary outcome: an outcome regression of cell means on the full history, with intercept-only mechanisms, so the regression alone carries the estimate	bias interval inside the equivalence margin	bias -0.000855 to 0.0023, margin 0.0053	pass
`full_history_necessity`	`rule_vs_never_t3__markov_history`	control	contrast of the rule against never at horizon t = 3, repeated binary outcome: the same fit with the outcome regression on the previous node only, [W, Y_{t-1}, L_t], and the same intercept-only mechanisms	bias interval outside the margin, and inside it around the exact limit of the previous-node fit	bias 0.0220 to 0.0248, margin 0.0047, bias around its limit -0.000710 to 0.0021, margin 0.0047	pass
`interval_calibration`	`always_t1__estimated_mechanisms`	positive	mean under always at horizon t = 1, repeated binary outcome: n = 8,000, in sample, main-terms logistic mechanisms estimated, quasibinomial GLM outcome regressions, every horizon	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9433 to 0.9560, SE ratio 0.9852 to 1.0279	pass
`interval_calibration`	`always_t2__estimated_mechanisms`	positive	always-treat risk at horizon two: n = 8,000, in sample, main-terms logistic mechanisms estimated, quasibinomial GLM outcome regressions, every horizon	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9414 to 0.9543, SE ratio 0.9757 to 1.0168	pass
`interval_calibration`	`always_t3__continuous_gaussian`	positive	mean under always at horizon t = 3, repeated binary outcome: n = 8,000, the three-value outcome law, least-squares outcome regressions, main-terms logistic mechanisms estimated	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9393 to 0.9524, SE ratio 0.9738 to 1.0138	pass
`interval_calibration`	`always_t3__crossfit_boosted`	positive	mean under always at horizon t = 3, repeated binary outcome: n = 8,000, five folds, main-terms logistic mechanisms estimated, a depth-2 booster of 50 rounds with an L2 penalty for both outcome learners	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9406 to 0.9536, SE ratio 0.9811 to 1.0218	pass
`interval_calibration`	`always_t3__crossfit_glm`	positive	mean under always at horizon t = 3, repeated binary outcome: n = 8,000, five folds, main-terms logistic mechanisms estimated, quasibinomial GLM outcome regressions	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9372 to 0.9505, SE ratio 0.9800 to 1.0222	pass
`interval_calibration`	`always_t3__estimated_mechanisms`	positive	mean under always at horizon t = 3, repeated binary outcome: n = 8,000, in sample, main-terms logistic mechanisms estimated, quasibinomial GLM outcome regressions, every horizon	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9396 to 0.9527, SE ratio 0.9744 to 1.0143	pass
`interval_calibration`	`always_t3__noise_control`	control	mean under always at horizon t = 3, repeated binary outcome: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8230 to 0.8446, SE ratio 0.6879 to 0.7162	pass
`interval_calibration`	`always_t3__shrunken_se_control`	control	mean under always at horizon t = 3, repeated binary outcome: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8153 to 0.8373, SE ratio 0.6821 to 0.7106	pass
`interval_calibration`	`always_vs_never_t1__estimated_mechanisms`	positive	contrast of always against never at horizon t = 1, repeated binary outcome: n = 8,000, in sample, main-terms logistic mechanisms estimated, quasibinomial GLM outcome regressions, every horizon	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9361 to 0.9496, SE ratio 0.9616 to 1.0010	pass
`interval_calibration`	`always_vs_never_t2__estimated_mechanisms`	positive	contrast of always against never at horizon t = 2, repeated binary outcome: n = 8,000, in sample, main-terms logistic mechanisms estimated, quasibinomial GLM outcome regressions, every horizon	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9426 to 0.9554, SE ratio 0.9803 to 1.0200	pass
`interval_calibration`	`always_vs_never_t3__continuous_gaussian`	positive	contrast of always against never at horizon t = 3, repeated binary outcome: n = 8,000, the three-value outcome law, least-squares outcome regressions, main-terms logistic mechanisms estimated	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9414 to 0.9543, SE ratio 0.9796 to 1.0200	pass
`interval_calibration`	`always_vs_never_t3__crossfit_boosted`	positive	contrast of always against never at horizon t = 3, repeated binary outcome: n = 8,000, five folds, main-terms logistic mechanisms estimated, a depth-2 booster of 50 rounds with an L2 penalty for both outcome learners	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9402 to 0.9532, SE ratio 0.9821 to 1.0225	pass
`interval_calibration`	`always_vs_never_t3__crossfit_glm`	positive	contrast of always against never at horizon t = 3, repeated binary outcome: n = 8,000, five folds, main-terms logistic mechanisms estimated, quasibinomial GLM outcome regressions	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9417 to 0.9545, SE ratio 0.9747 to 1.0145	pass
`interval_calibration`	`always_vs_never_t3__estimated_mechanisms`	positive	contrast of always against never at horizon t = 3, repeated binary outcome: n = 8,000, in sample, main-terms logistic mechanisms estimated, quasibinomial GLM outcome regressions, every horizon	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9435 to 0.9562, SE ratio 0.9808 to 1.0221	pass
`interval_calibration`	`always_vs_never_t3__noise_control`	control	contrast of always against never at horizon t = 3, repeated binary outcome: 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 0.8477, SE ratio 0.6960 to 0.7257	pass
`interval_calibration`	`always_vs_never_t3__shrunken_se_control`	control	contrast of always against never at horizon t = 3, repeated binary outcome: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8146 to 0.8365, SE ratio 0.6868 to 0.7153	pass
`interval_calibration`	`never_t1__estimated_mechanisms`	positive	mean under never at horizon t = 1, repeated binary outcome: n = 8,000, in sample, main-terms logistic mechanisms estimated, quasibinomial GLM outcome regressions, every horizon	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9373 to 0.9507, SE ratio 0.9601 to 1.0001	pass
`interval_calibration`	`never_t2__estimated_mechanisms`	positive	mean under never at horizon t = 2, repeated binary outcome: n = 8,000, in sample, main-terms logistic mechanisms estimated, quasibinomial GLM outcome regressions, every horizon	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9419 to 0.9548, SE ratio 0.9841 to 1.0251	pass
`interval_calibration`	`never_t3__continuous_gaussian`	positive	mean under never at horizon t = 3, repeated binary outcome: n = 8,000, the three-value outcome law, least-squares outcome regressions, main-terms logistic mechanisms estimated	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9459 to 0.9583, SE ratio 0.9864 to 1.0274	pass
`interval_calibration`	`never_t3__crossfit_boosted`	positive	mean under never at horizon t = 3, repeated binary outcome: n = 8,000, five folds, main-terms logistic mechanisms estimated, a depth-2 booster of 50 rounds with an L2 penalty for both outcome learners	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9441 to 0.9567, SE ratio 0.9877 to 1.0297	pass
`interval_calibration`	`never_t3__crossfit_glm`	positive	mean under never at horizon t = 3, repeated binary outcome: n = 8,000, five folds, main-terms logistic mechanisms estimated, quasibinomial GLM outcome regressions	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9422 to 0.9550, SE ratio 0.9726 to 1.0129	pass
`interval_calibration`	`never_t3__estimated_mechanisms`	positive	mean under never at horizon t = 3, repeated binary outcome: n = 8,000, in sample, main-terms logistic mechanisms estimated, quasibinomial GLM outcome regressions, every horizon	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9458 to 0.9582, SE ratio 0.9768 to 1.0176	pass
`interval_calibration`	`never_t3__noise_control`	control	mean under never at horizon t = 3, repeated binary outcome: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8165 to 0.8384, SE ratio 0.6824 to 0.7105	pass
`interval_calibration`	`never_t3__shrunken_se_control`	control	mean under never at horizon t = 3, repeated binary outcome: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8157 to 0.8376, SE ratio 0.6837 to 0.7121	pass
`interval_calibration`	`rule_t2__estimated_mechanisms`	positive	mean under the rule that treats after an outcome at horizon t = 2, repeated binary outcome: n = 8,000, in sample, main-terms logistic mechanisms estimated, quasibinomial GLM outcome regressions, every horizon	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9418 to 0.9547, SE ratio 0.9909 to 1.0329	pass
`interval_calibration`	`rule_t3__continuous_gaussian`	positive	mean under the rule that treats after an outcome at horizon t = 3, repeated binary outcome: n = 8,000, the three-value outcome law, least-squares outcome regressions, main-terms logistic mechanisms estimated	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9396 to 0.9527, SE ratio 0.9666 to 1.0077	pass
`interval_calibration`	`rule_t3__crossfit_boosted`	positive	mean under the rule that treats after an outcome at horizon t = 3, repeated binary outcome: n = 8,000, five folds, main-terms logistic mechanisms estimated, a depth-2 booster of 50 rounds with an L2 penalty for both outcome learners	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9396 to 0.9527, SE ratio 0.9885 to 1.0307	pass
`interval_calibration`	`rule_t3__crossfit_glm`	positive	mean under the rule that treats after an outcome at horizon t = 3, repeated binary outcome: n = 8,000, five folds, main-terms logistic mechanisms estimated, quasibinomial GLM outcome regressions	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9409 to 0.9538, SE ratio 0.9794 to 1.0214	pass
`interval_calibration`	`rule_t3__estimated_mechanisms`	positive	mean under the rule that treats after an outcome at horizon t = 3, repeated binary outcome: n = 8,000, in sample, main-terms logistic mechanisms estimated, quasibinomial GLM outcome regressions, every horizon	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9370 to 0.9504, SE ratio 0.9642 to 1.0039	pass
`interval_calibration`	`rule_t3__noise_control`	control	mean under the rule that treats after an outcome at horizon t = 3, repeated binary outcome: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8173 to 0.8391, SE ratio 0.6858 to 0.7153	pass
`interval_calibration`	`rule_t3__shrunken_se_control`	control	mean under the rule that treats after an outcome at horizon t = 3, repeated binary outcome: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8125 to 0.8346, SE ratio 0.6748 to 0.7025	pass
`interval_calibration`	`rule_vs_never_t2__estimated_mechanisms`	positive	contrast of the rule against never at horizon t = 2, repeated binary outcome: n = 8,000, in sample, main-terms logistic mechanisms estimated, quasibinomial GLM outcome regressions, every horizon	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9466 to 0.9589, SE ratio 0.9877 to 1.0282	pass
`interval_calibration`	`rule_vs_never_t3__continuous_gaussian`	positive	contrast of the rule against never at horizon t = 3, repeated binary outcome: n = 8,000, the three-value outcome law, least-squares outcome regressions, main-terms logistic mechanisms estimated	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9425 to 0.9552, SE ratio 0.9862 to 1.0270	pass
`interval_calibration`	`rule_vs_never_t3__crossfit_boosted`	positive	contrast of the rule against never at horizon t = 3, repeated binary outcome: n = 8,000, five folds, main-terms logistic mechanisms estimated, a depth-2 booster of 50 rounds with an L2 penalty for both outcome learners	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9437 to 0.9563, SE ratio 0.9875 to 1.0286	pass
`interval_calibration`	`rule_vs_never_t3__crossfit_glm`	positive	contrast of the rule against never at horizon t = 3, repeated binary outcome: n = 8,000, five folds, main-terms logistic mechanisms estimated, quasibinomial GLM outcome regressions	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9431 to 0.9558, SE ratio 0.9793 to 1.0201	pass
`interval_calibration`	`rule_vs_never_t3__estimated_mechanisms`	positive	contrast of the rule against never at horizon t = 3, repeated binary outcome: n = 8,000, in sample, main-terms logistic mechanisms estimated, quasibinomial GLM outcome regressions, every horizon	SE-ratio and coverage intervals inside the calibration bands	coverage 0.9421 to 0.9549, SE ratio 0.9741 to 1.0141	pass
`interval_calibration`	`rule_vs_never_t3__noise_control`	control	contrast of the rule against never at horizon t = 3, repeated binary outcome: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8255 to 0.8469, SE ratio 0.6952 to 0.7249	pass
`interval_calibration`	`rule_vs_never_t3__shrunken_se_control`	control	contrast of the rule against never at horizon t = 3, repeated binary outcome: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8182 to 0.8400, SE ratio 0.6821 to 0.7102	pass
`outcome_history_necessity`	`always_t3__history_free`	control	mean under always at horizon t = 3, repeated binary outcome: an end-of-study fit with no earlier outcome in any history, cell-mean outcome regressions: the earlier outcomes are confounders, and the fit leaves them out of the outcome regression and both mechanisms	bias interval outside the margin, and inside it around the coarse g-formula	bias 0.0163 to 0.0201, margin 0.0065, bias around its limit -0.000997 to 0.0029, margin 0.0065	pass
`outcome_history_necessity`	`always_t3__repeated`	positive	mean under always at horizon t = 3, repeated binary outcome: the repeated-outcome fit, mechanisms estimated, GLM outcome regressions	bias interval inside the equivalence margin	bias -0.000740 to 0.0028, margin 0.0059	pass
`outcome_history_necessity`	`always_vs_never_t3__history_free`	control	contrast of always against never at horizon t = 3, repeated binary outcome: an end-of-study fit with no earlier outcome in any history, cell-mean outcome regressions: the earlier outcomes are confounders, and the fit leaves them out of the outcome regression and both mechanisms	bias interval outside the margin, and inside it around the coarse g-formula	bias 0.0570 to 0.0627, margin 0.0095, bias around its limit -0.0028 to 0.0029, margin 0.0095	pass
`outcome_history_necessity`	`always_vs_never_t3__repeated`	positive	contrast of always against never at horizon t = 3, repeated binary outcome: the repeated-outcome fit, mechanisms estimated, GLM outcome regressions	bias interval inside the equivalence margin	bias -0.0021 to 0.0034, margin 0.0093	pass
`outcome_history_necessity`	`never_t3__history_free`	control	mean under never at horizon t = 3, repeated binary outcome: an end-of-study fit with no earlier outcome in any history, cell-mean outcome regressions: the earlier outcomes are confounders, and the fit leaves them out of the outcome regression and both mechanisms	bias interval outside the margin, and inside it around the coarse g-formula	bias -0.0438 to -0.0395, margin 0.0073, bias around its limit -0.0013 to 0.0030, margin 0.0073	pass
`outcome_history_necessity`	`never_t3__repeated`	positive	mean under never at horizon t = 3, repeated binary outcome: the repeated-outcome fit, mechanisms estimated, GLM outcome regressions	bias interval inside the equivalence margin	bias -0.0018 to 0.0026, margin 0.0073	pass
`repeated_outcome_necessity`	`always_t2__absorbing_reading`	control	always-treat risk at horizon two: an end-of-study fit of the maximum of the outcomes through the horizon, cell means with a GLM on an unseen cell	bias interval outside the margin, and inside it around the maximum’s own truth	bias 0.1123 to 0.1131, margin 0.0020, bias around its limit -0.000626 to 0.000169, margin 0.0020	pass
`repeated_outcome_necessity`	`always_t2__repeated`	positive	always-treat risk at horizon two: the repeated-outcome fit, mechanisms estimated, GLM outcome regressions	bias interval inside the equivalence margin	bias -0.000606 to 0.000107, margin 0.0018	pass
`repeated_outcome_necessity`	`always_t3__absorbing_reading`	control	mean under always at horizon t = 3, repeated binary outcome: an end-of-study fit of the maximum of the outcomes through the horizon, cell means with a GLM on an unseen cell	bias interval outside the margin, and inside it around the maximum’s own truth	bias 0.1648 to 0.1660, margin 0.0032, bias around its limit -0.000676 to 0.000586, margin 0.0032	pass
`repeated_outcome_necessity`	`always_t3__repeated`	positive	mean under always at horizon t = 3, repeated binary outcome: the repeated-outcome fit, mechanisms estimated, GLM outcome regressions	bias interval inside the equivalence margin	bias -0.000343 to 0.000828, margin 0.0030	pass
`repeated_outcome_necessity`	`never_t2__absorbing_reading`	control	mean under never at horizon t = 2, repeated binary outcome: an end-of-study fit of the maximum of the outcomes through the horizon, cell means with a GLM on an unseen cell	bias interval outside the margin, and inside it around the maximum’s own truth	bias 0.1406 to 0.1416, margin 0.0025, bias around its limit -0.000640 to 0.000349, margin 0.0025	pass
`repeated_outcome_necessity`	`never_t2__repeated`	positive	mean under never at horizon t = 2, repeated binary outcome: the repeated-outcome fit, mechanisms estimated, GLM outcome regressions	bias interval inside the equivalence margin	bias -0.000646 to 0.000323, margin 0.0025	pass
`repeated_outcome_necessity`	`never_t3__absorbing_reading`	control	mean under never at horizon t = 3, repeated binary outcome: an end-of-study fit of the maximum of the outcomes through the horizon, cell means with a GLM on an unseen cell	bias interval outside the margin, and inside it around the maximum’s own truth	bias 0.1769 to 0.1781, margin 0.0030, bias around its limit -0.000798 to 0.000386, margin 0.0030	pass
`repeated_outcome_necessity`	`never_t3__repeated`	positive	mean under never at horizon t = 3, repeated binary outcome: the repeated-outcome fit, mechanisms estimated, GLM outcome regressions	bias interval inside the equivalence margin	bias -0.000917 to 0.000528, margin 0.0037	pass
`repeated_outcome_necessity`	`rule_t2__absorbing_reading`	control	mean under the rule that treats after an outcome at horizon t = 2, repeated binary outcome: an end-of-study fit of the maximum of the outcomes through the horizon, cell means with a GLM on an unseen cell	bias interval outside the margin, and inside it around the maximum’s own truth	bias 0.1123 to 0.1132, margin 0.0022, bias around its limit -0.000590 to 0.000257, margin 0.0022	pass
`repeated_outcome_necessity`	`rule_t2__repeated`	positive	mean under the rule that treats after an outcome at horizon t = 2, repeated binary outcome: the repeated-outcome fit, mechanisms estimated, GLM outcome regressions	bias interval inside the equivalence margin	bias -0.000592 to 0.000214, margin 0.0021	pass
`repeated_outcome_necessity`	`rule_t3__absorbing_reading`	control	mean under the rule that treats after an outcome at horizon t = 3, repeated binary outcome: an end-of-study fit of the maximum of the outcomes through the horizon, cell means with a GLM on an unseen cell	bias interval outside the margin, and inside it around the maximum’s own truth	bias 0.2234 to 0.2245, margin 0.0028, bias around its limit -0.000733 to 0.000354, margin 0.0028	pass
`repeated_outcome_necessity`	`rule_t3__repeated`	positive	mean under the rule that treats after an outcome at horizon t = 3, repeated binary outcome: the repeated-outcome fit, mechanisms estimated, GLM outcome regressions	bias interval inside the equivalence margin	bias -0.000594 to 0.000724, margin 0.0034	pass
`rule_necessity`	`rule_t2__declared`	positive	mean under the rule that treats after an outcome at horizon t = 2, repeated binary outcome: the declared covariate-dependent rule assigns both treatment arms	bias interval inside the equivalence margin	bias -0.0011 to 0.000817, margin 0.0042	pass
`rule_necessity`	`rule_t2__reversed_rule`	control	mean under the rule that treats after an outcome at horizon t = 2, repeated binary outcome: the same fit treats after a zero outcome instead of after an outcome of one	bias interval must fall entirely outside the margin, and inside it around the reversed rule’s own truth	bias -0.0570 to -0.0551, margin 0.0040, bias around its limit -0.000339 to 0.0015, margin 0.0040	pass
`rule_necessity`	`rule_t3__declared`	positive	mean under the rule that treats after an outcome at horizon t = 3, repeated binary outcome: the declared covariate-dependent rule assigns both treatment arms	bias interval inside the equivalence margin	bias -0.0017 to 0.0017, margin 0.0073	pass
`rule_necessity`	`rule_t3__reversed_rule`	control	mean under the rule that treats after an outcome at horizon t = 3, repeated binary outcome: the same fit treats after a zero outcome instead of after an outcome of one	bias interval must fall entirely outside the margin, and inside it around the reversed rule’s own truth	bias -0.0608 to -0.0578, margin 0.0064, bias around its limit -0.0014 to 0.0015, margin 0.0064	pass
`simultaneous_coverage`	`all_reported__pointwise_joint_control`	control	every parameter the calibration fit reports: the pointwise 95% intervals of the same fits, read jointly	joint coverage upper endpoint must fall below the nominal rate	joint coverage 0.5830 to 0.6114	pass
`simultaneous_coverage`	`all_reported__simultaneous_band`	positive	every parameter the calibration fit reports: 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 0.9339 to 0.9476	pass
`small_sample_calibration`	`always_t3__continuous_gaussian`	diagnostic	mean under always at horizon t = 3, repeated binary outcome: the in-sample continuous calibration arm at n = 2,000	reported, no verdict	SE ratio 0.8986 to 0.9744	reported
`small_sample_calibration`	`always_t3__estimated_mechanisms`	diagnostic	mean under always at horizon t = 3, repeated binary outcome: the in-sample binary calibration arm at n = 2,000	reported, no verdict	SE ratio 0.9470 to 1.0289	reported
`small_sample_calibration`	`always_vs_never_t3__continuous_gaussian`	diagnostic	contrast of always against never at horizon t = 3, repeated binary outcome: the in-sample continuous calibration arm at n = 2,000	reported, no verdict	SE ratio 0.9102 to 0.9898	reported
`small_sample_calibration`	`always_vs_never_t3__estimated_mechanisms`	diagnostic	contrast of always against never at horizon t = 3, repeated binary outcome: the in-sample binary calibration arm at n = 2,000	reported, no verdict	SE ratio 0.9254 to 1.0030	reported
`small_sample_calibration`	`never_t3__continuous_gaussian`	diagnostic	mean under never at horizon t = 3, repeated binary outcome: the in-sample continuous calibration arm at n = 2,000	reported, no verdict	SE ratio 0.9200 to 0.9998	reported
`small_sample_calibration`	`never_t3__estimated_mechanisms`	diagnostic	mean under never at horizon t = 3, repeated binary outcome: the in-sample binary calibration arm at n = 2,000	reported, no verdict	SE ratio 0.9088 to 0.9852	reported
`small_sample_calibration`	`rule_t3__continuous_gaussian`	diagnostic	mean under the rule that treats after an outcome at horizon t = 3, repeated binary outcome: the in-sample continuous calibration arm at n = 2,000	reported, no verdict	SE ratio 0.9261 to 1.0087	reported
`small_sample_calibration`	`rule_t3__estimated_mechanisms`	diagnostic	mean under the rule that treats after an outcome at horizon t = 3, repeated binary outcome: the in-sample binary calibration arm at n = 2,000	reported, no verdict	SE ratio 0.9789 to 1.0658	reported
`small_sample_calibration`	`rule_vs_never_t3__continuous_gaussian`	diagnostic	contrast of the rule against never at horizon t = 3, repeated binary outcome: the in-sample continuous calibration arm at n = 2,000	reported, no verdict	SE ratio 0.9162 to 1.0025	reported
`small_sample_calibration`	`rule_vs_never_t3__estimated_mechanisms`	diagnostic	contrast of the rule against never at horizon t = 3, repeated binary outcome: the in-sample binary calibration arm at n = 2,000	reported, no verdict	SE ratio 0.9588 to 1.0429	reported
`targeting_necessity`	`always_t3__targeted`	positive	mean under always at horizon t = 3, repeated binary outcome: the estimator fluctuates a misspecified outcome model, so targeting does all the adjusting	bias interval inside the equivalence margin	bias 0.000520 to 0.0037, margin 0.0062	pass
`targeting_necessity`	`always_t3__untargeted`	control	mean under always at horizon t = 3, repeated binary outcome: the iterated cell means on the baseline covariate among the units that followed the plan, with no fluctuation step	bias interval must fall entirely outside the margin, and inside it around the baseline-only plug-in limit	bias 0.0247 to 0.0282, margin 0.0068, bias around its limit 0.000122 to 0.0036, margin 0.0068	pass
`targeting_necessity`	`always_vs_never_t3__targeted`	positive	contrast of always against never at horizon t = 3, repeated binary outcome: the estimator fluctuates a misspecified outcome model, so targeting does all the adjusting	bias interval inside the equivalence margin	bias -0.000067 to 0.0051, margin 0.0101	pass
`targeting_necessity`	`always_vs_never_t3__untargeted`	control	contrast of always against never at horizon t = 3, repeated binary outcome: the iterated cell means on the baseline covariate among the units that followed the plan, with no fluctuation step	bias interval must fall entirely outside the margin, and inside it around the baseline-only plug-in limit	bias 0.0779 to 0.0831, margin 0.0100, bias around its limit -0.0010 to 0.0041, margin 0.0100	pass
`targeting_necessity`	`never_t3__targeted`	positive	mean under never at horizon t = 3, repeated binary outcome: the estimator fluctuates a misspecified outcome model, so targeting does all the adjusting	bias interval inside the equivalence margin	bias -0.0024 to 0.0016, margin 0.0078	pass
`targeting_necessity`	`never_t3__untargeted`	control	mean under never at horizon t = 3, repeated binary outcome: the iterated cell means on the baseline covariate among the units that followed the plan, with no fluctuation step	bias interval must fall entirely outside the margin, and inside it around the baseline-only plug-in limit	bias -0.0558 to -0.0522, margin 0.0070, bias around its limit -0.0015 to 0.0021, margin 0.0070	pass
`targeting_necessity`	`rule_t3__targeted`	positive	mean under the rule that treats after an outcome at horizon t = 3, repeated binary outcome: the estimator fluctuates a misspecified outcome model, so targeting does all the adjusting	bias interval inside the equivalence margin	bias -0.000839 to 0.0031, margin 0.0076	pass
`targeting_necessity`	`rule_t3__untargeted`	control	mean under the rule that treats after an outcome at horizon t = 3, repeated binary outcome: the iterated cell means on the baseline covariate among the units that followed the plan, with no fluctuation step	bias interval must fall entirely outside the margin, and inside it around the baseline-only plug-in limit	bias -0.0224 to -0.0193, margin 0.0059, bias around its limit -0.000531 to 0.0025, margin 0.0059	pass
`targeting_necessity`	`rule_vs_never_t3__targeted`	positive	contrast of the rule against never at horizon t = 3, repeated binary outcome: the estimator fluctuates a misspecified outcome model, so targeting does all the adjusting	bias interval inside the equivalence margin	bias -0.0013 to 0.0043, margin 0.0109	pass
`targeting_necessity`	`rule_vs_never_t3__untargeted`	control	contrast of the rule against never at horizon t = 3, repeated binary outcome: the iterated cell means on the baseline covariate among the units that followed the plan, with no fluctuation step	bias interval must fall entirely outside the margin, and inside it around the baseline-only plug-in limit	bias 0.0308 to 0.0355, margin 0.0090, bias around its limit -0.0016 to 0.0030, margin 0.0090	pass

The property families draw from their own streams of make_longitudinal_repeated. Their draws differ from the primary draws. The table above lists every cell. This section states what each family tests.

family	what it tests	size
`interval_calibration`	pointwise coverage and SE ratio inside the calibration bands. The cells are in-sample with estimated mechanisms, cross-fitted GLM, cross-fitted boosted trees, and a continuous outcome. The in-sample cells cover every horizon, \(t = 1, 2, 3\). A shrunken-SE control and a noise control must fail the same bands	\(n\) = 8,000, 8,000 replications
`small_sample_calibration`	the in-sample and continuous arms refitted at \(n\) = 2,000. The role is `diagnostic`. The table reports these cells and no rule gates them	\(n\) = 2,000, 2,000 replications
`double_robustness`	`mechanism_correct`: bias inside the margin when the outcome regression reads \(W\) alone and the mechanisms are correct. `outcome_correct`: bias inside the margin when the outcome regression reads the full history and the mechanisms read \(W\) alone. `both_wrong`: the control with a \(W\) regression and intercept-only mechanisms must miss the truth and reach its exact limit. A static plan has no fluctuation in this fit. The rule has one, and it moves the limit by 0.046 empirical standard deviations of the rule’s mean at \(t = 3\) (0.031 for its contrast). Replications cannot separate that move from the plug-in limit, because the 99% bias interval at 1,400 replications is 0.069 standard deviations wide on each side. The claim that the rule’s fluctuation moves the fit rests on `test_the_limit_functional_reproduces_cleverly_on_a_sample`, which matches a fit to the limit functional on one sample to \(10^{-10}\)	\(n\) = 2,000, 1,400 replications
`targeting_necessity`	the targeted fit against a plug-in with no fluctuation. The plug-in must move the estimate by 0.25 empirical standard deviations of the targeted arm and reach the exact \(W\)-only g-formula	\(n\) = 2,000, 1,600 replications
`simultaneous_coverage`	the default band over all 15 reported parameters, against its pointwise control	\(n\) = 8,000, 8,000 replications
`rule_necessity`	the declared rule against its reverse, which treats after a zero outcome. The reversed rule must miss the declared rule’s truth and reach the truth of the reversed rule	\(n\) = 2,000, 2,000 replications
`repeated_outcome_necessity` (N1)	an absorbing reading of the outcome, which fits \(\max_{s \le h} Y_s\). The control must miss the truth and reach the cumulative-maximum limit. Its outcome learner is a cell-means regression with a GLM fallback on an unseen cell	\(n\) = 8,000, 2,800 replications
`outcome_history_necessity` (N2)	a fit that leaves every \(Y_s\) out of the history, so the earlier outcomes act as confounders. The control must miss the truth and reach the coarse g-formula limit	\(n\) = 2,000, 1,200 replications
`full_history_necessity` (N3)	an outcome regression on the full history against one on \([W, Y_{t-1}, L_t]\), both with intercept-only mechanisms. The control must miss the truth and reach the exact limit of the single-fold fit	\(n\) = 8,000, 1,200 replications

The in-sample cells use \(n\) = 8,000. At \(n\) = 2,000 the standard error runs a few percent below the sampling spread for the rule. The ratio moves toward 1 at the \(1/n\) rate of in-sample fitting. The module tests/studies/ltmle_repeated_properties.py records the pilots behind these sizes. The small_sample_calibration cells report the shortfall at \(n\) = 2,000.

The calibration budget is the smallest multiple of 400 replications at which the 28 gated cells pass together with probability 0.95. The probability pools the cells of each arm. It takes the arm mean from the pilot and adds the noise of that mean. It sets the spread between cells at its 80% upper bound of 0.0059. The probability is 0.952 at 8,000 replications.

The sizes of the other families come from their bias checks. Each family has its own mean. The spread between cells is at its 80% upper bound of 0.0325. Each family takes the smallest multiple of 200 replications at which it passes with probability 0.991, the sixth root of 0.95. The six families then pass together with probability 0.961. A reading that gives every family one mean gives 0.994 at these sizes.

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`	2400	primary replications per scenario
`n`	3000	observations per primary replication
`independent_tests_total`	13	implementation-estimand tests against the truth
`independent_tests_passed`	13	of those, passing
`paired_tests_total`	0	paired comparisons with R `ltmle`
`paired_tests_passed`	0	of those, passing
`property_cells_total`	97	repeated-sampling property cells, reported cells excluded
`property_cells_passed`	97	cells whose own and family verdicts pass
`property_cells_reported`	10	cells reported and not gated, as the `small_sample_calibration` diagnostics
`max_standardized_bias`	0.0426	largest absolute primary bias in empirical standard deviations
`min_coverage`	0.9325	lowest measured primary-study coverage
`min_se_ratio_ci_lower`	0.9286	lowest bootstrap primary SE-ratio endpoint
`max_se_ratio_ci_upper`	1.0674	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
`margin:calibration_coverage_upper`	0.9800	calibration-cell coverage band, upper limit
`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
`margin:calibration_se_ratio_upper`	1.0700	calibration-cell SE-ratio band, upper limit
`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
`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:shrunken_se_factor`	0.7000	factor the shrunken-SE control multiplies the standard error by
`margin:necessity_displacement`	0.2500	smallest paired displacement of a necessity control, in empirical standard deviations
`margin:targeting_displacement`	0.2500	smallest displacement of the untargeted plug-in from the targeted fit, in empirical standard deviations of the targeted arm

Limitations#

limit	what it means for use
one law with three nodes and a binary outcome	the continuous outcome has one calibration arm, and the rate in \(n\) is not measured
the comparator runs are pending	the `targeting_necessity` family carries the claim that targeting moves the estimate. The pending parity gate `test_cleverly_reproduces_r_targeting_move_in_every_cell` adds a check against R. In each cell the root-mean-square difference from R must stay below 1% of R’s root-mean-square targeting move. No replication may differ from R by more than 1% of R’s mean standard error
the declared comparison fixes the mechanisms	the paired margins alone cannot separate an untargeted plug-in from the targeted fit, because the plug-in is nearly unbiased on this law. The comparison does not claim parity for learned treatment or censoring models
the in-sample interval is mildly anticonservative at \(n\) = 2,000	the `small_sample_calibration` cells report the shortfall and no rule gates them. The gated calibration cells draw \(n\) = 8,000. The primary cells draw \(n\) = 3,000 and answer to the coverage floor
the cross-fitted learners are one GLM and one shallow booster	the study does not cover a larger learner library or a different fold count
the absorbing, history-free and Markov controls read exact limits of this law	they show that the repeated reading, the earlier outcomes and the depth of the outcome regression matter here. They do not bound the difference on another law
positivity is bounded away from zero, not absent	at \(t = 3\) the smallest cumulative mechanism product among followers is 0.025, and at most 2.2% of the followers fall below 0.05. No cell validates active truncation or near-positivity behavior
the second-order dependence acts at node 3 only	the law has three nodes, so the depth-two terms act at the last node. No cell reads a longer history
no recurrent-event or truncation-by-death law	the parameter is the mean in a population where censoring is prevented

The causal interpretation requires consistency, sequential exchangeability, longitudinal positivity, and conditionally independent censoring.

Reproduction#

The fixture README gives the regeneration commands. The manifest records the pending comparator, the declared R sources, the seeds and the artifact hashes. The replications and property results carry every published row.