Vector modified treatment policies

This study validates modified treatment policies on a vector treatment. A vector holds a binary flag and one or two doses. The point TMLE fits it with policies=, and LTMLE fits it as a node. Both read the ratio from the joint density that a JointPolicy induces (ratio="density") or from a stacked classifier (ratio="classifier"). The vector treatments and vector nodes sections state the estimators, their conditions and their exact-law evidence.

Status of this page. The study is registered, and its Python phases have run. The 20 truth tests and the 36 property cells pass (performance-tests.csv, properties.csv). The property rows were fitted at commit 8b0e5b9b on two machines. Provenance revisions records that run. The evidence label is “Python property evidence, with comparator runs pending”. The evidence has two parts.

part	what it is	status
Python property evidence	exact laws with closed-form efficiency bounds, interval calibration with two controls, double robustness in four nuisance configurations, and three necessity pairs	complete: 20 of 20 truth tests and 36 of 36 property cells pass. It decides whether the method ships
lmtp comparator runs	the pinned lmtp 1.5.4 fits the same samples with vector trt nodes, and the study pairs the results	pending. The study record sets pending_reference="lmtp", so no paired row is published. The runs register in a later change, and a finding in them does not block the method. The fast-tier witnesses and the Python property cells carry the claim until then

The publication policy is reporting. A red cell stays red on this page and in the red-cell ledger, with the owner row that names it.

What was compared

The four scenarios draw 1,000 samples of 2,000 rows each.

scenario	law and plans	pairing
point_vector_mixed	the point law, TMLE, treatment \((F, D)\) with a binary flag and a dose. The plans are natural course, dose up and flag scaled. The ratio is the oracle induced density, which lmtp receives with both shifted components	margin-read: TMLE and lmtp target by different constructions
point_vector_categorical	the categorical law, TMLE, treatment \((A_1, A_2)\) with two binary components. The plans are natural course, a2 off and a1 on. The ratio is the discrete formula on the exact joint probabilities	margin-read
longitudinal_vector_mixed	the two-node law, LTMLE at one fold, nodes \((F_1, D_1)\) and \((F_2, D_2)\). The plans are the point plans at both nodes. Each node’s ratio (node_ratio) and shifted components go to lmtp	exact under a gap of \(10^{-6}\)
longitudinal_vector_ordered_classifier	the ordered law as one LTMLE node \((F, D_1, D_2)\). The plans are natural course, ordered and full. Each side fits its own stacked classifier	the parity rule below

setting	cleverly	R lmtp
datasets	1,000 samples of 2,000 rows per scenario	the identical rows
density ratio	an oracle pooled-hazard density with ceil(320 (n / 2000)^(2/3)) bins (320 at n = 2,000), and the exact joint probabilities on the categorical law	the same ratio array, read from the replicate data
outcome regressions	correctly specified logistic models: main terms plus the squared doses, or the saturated joint levels on the categorical law	SL.glm on the same columns
classifier of the ordered scenario	an unpenalized logistic regression of the stack label on one feature, the plan’s true log ratio, which the learner computes from each stacked row	the R twin SL.logratio.<plan>, with mtp = TRUE
intervals	pointwise 95% Wald from the influence curve	the same, from lmtp’s influence curve

The comparator adapter is tests/canonical/lmtp_vector_adapter.R. It builds an LmtpTask whose treatment nodes are vectors, such as trt = list(c("F", "D")). With mtp = FALSE, lmtp 1.5.4 multiplies each ratio by the indicator that no component moved (R/density_ratios.R, lines 58 to 61). That indicator is zero on every row that a plan moves, so the adapter sets mtp = TRUE.

The parity check requires that the share of positive lmtp ratios equal the package’s share. Under mtp = TRUE both shares are 1, because neither classifier predicts 0. Under mtp = FALSE the lmtp share falls to the share of unmoved rows, and the check fails.

The ordered pair reads by three outcomes, declared before any run. Every ratio gap below \(10^{-6}\) makes the pair an exactness witness. Every relative gap below \(10^{-4}\), with a diagnosed and named numerical cause, makes it a margin-read pair. Any other reading blocks the declaration until the cause is found.

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
two nodes, each a binary flag and a truncated-normal dose	ate_regimen[dose up vs natural course]	difference in mean outcome between the plans “add 0.5 to each dose, capped at 5.5” against “leave the observed treatment mechanism unchanged”	cleverly point TMLE and LTMLE under modified treatment policies on a vector treatment	-0.000489 to 0.000894	0.9510	1.0294	pass
two nodes, each a binary flag and a truncated-normal dose	ate_regimen[flag scaled vs natural course]	difference in mean outcome between the plans “set each flag to one, and multiply the dose by 1.25 where the natural flag is one” against “leave the observed treatment mechanism unchanged”	cleverly point TMLE and LTMLE under modified treatment policies on a vector treatment	-0.0017 to 0.0016	0.9510	0.9928	pass
two nodes, each a binary flag and a truncated-normal dose	ey_regimen[dose up]	mean outcome under the plan add 0.5 to each dose, capped at 5.5	cleverly point TMLE and LTMLE under modified treatment policies on a vector treatment	-0.000840 to 0.0015	0.9520	1.0018	pass
two nodes, each a binary flag and a truncated-normal dose	ey_regimen[flag scaled]	mean outcome under the plan set each flag to one, and multiply the dose by 1.25 where the natural flag is one	cleverly point TMLE and LTMLE under modified treatment policies on a vector treatment	-0.0019 to 0.0020	0.9500	0.9572	pass
two nodes, each a binary flag and a truncated-normal dose	ey_regimen[natural course]	mean outcome under the plan leave the observed treatment mechanism unchanged	cleverly point TMLE and LTMLE under modified treatment policies on a vector treatment	-0.000810 to 0.0010	0.9440	0.9994	pass
one node of a binary flag and two ordered doses, its ratio a fitted stacked classifier	ate_regimen[full vs natural course]	difference in mean outcome between the plans “set the flag to one where the first dose is at least 3, scale the first dose where the flag is one, and move the second dose as in ordered” against “leave the observed treatment mechanism unchanged”	cleverly point TMLE and LTMLE under modified treatment policies on a vector treatment	-0.000951 to 0.000852	0.9430	0.9776	pass
one node of a binary flag and two ordered doses, its ratio a fitted stacked classifier	ate_regimen[ordered vs natural course]	difference in mean outcome between the plans “multiply the first dose by 1.25, then add 0.5 to the second where the first is at most 3” against “leave the observed treatment mechanism unchanged”	cleverly point TMLE and LTMLE under modified treatment policies on a vector treatment	-0.000909 to 0.000863	0.9510	1.0058	pass
one node of a binary flag and two ordered doses, its ratio a fitted stacked classifier	ey_regimen[full]	mean outcome under the plan set the flag to one where the first dose is at least 3, scale the first dose where the flag is one, and move the second dose as in ordered	cleverly point TMLE and LTMLE under modified treatment policies on a vector treatment	-0.0015 to 0.000988	0.9540	1.0021	pass
one node of a binary flag and two ordered doses, its ratio a fitted stacked classifier	ey_regimen[natural course]	mean outcome under the plan leave the observed treatment mechanism unchanged	cleverly point TMLE and LTMLE under modified treatment policies on a vector treatment	-0.0011 to 0.000618	0.9660	1.0554	pass
one node of a binary flag and two ordered doses, its ratio a fitted stacked classifier	ey_regimen[ordered]	mean outcome under the plan multiply the first dose by 1.25, then add 0.5 to the second where the first is at most 3	cleverly point TMLE and LTMLE under modified treatment policies on a vector treatment	-0.0015 to 0.000984	0.9510	1.0332	pass
a point treatment of two binary components	ate_policy[a1 on vs natural course]	difference in means under the modified treatment policies “set the first component to one” against “leave the observed treatment mechanism unchanged”	cleverly point TMLE and LTMLE under modified treatment policies on a vector treatment	-0.0011 to 0.000736	0.9580	1.0219	pass
a point treatment of two binary components	ate_policy[a2 off vs natural course]	difference in means under the modified treatment policies “set the second component to zero where the first is one” against “leave the observed treatment mechanism unchanged”	cleverly point TMLE and LTMLE under modified treatment policies on a vector treatment	-0.0010 to 0.000573	0.9630	1.0436	pass
a point treatment of two binary components	ey_policy[a1 on]	mean under the modified treatment policy set the first component to one	cleverly point TMLE and LTMLE under modified treatment policies on a vector treatment	-0.0020 to 0.000659	0.9560	1.0125	pass
a point treatment of two binary components	ey_policy[a2 off]	mean under the modified treatment policy set the second component to zero where the first is one	cleverly point TMLE and LTMLE under modified treatment policies on a vector treatment	-0.0019 to 0.000513	0.9530	1.0340	pass
a point treatment of two binary components	ey_policy[natural course]	mean under the modified treatment policy leave the observed treatment mechanism unchanged	cleverly point TMLE and LTMLE under modified treatment policies on a vector treatment	-0.0013 to 0.000438	0.9450	1.0094	pass
a point treatment of one binary flag and one truncated-normal dose	ate_policy[dose up vs natural course]	difference in means under the modified treatment policies “add 0.5 to each dose, capped at 5.5” against “leave the observed treatment mechanism unchanged”	cleverly point TMLE and LTMLE under modified treatment policies on a vector treatment	-0.000560 to 0.000410	0.9520	1.0031	pass
a point treatment of one binary flag and one truncated-normal dose	ate_policy[flag scaled vs natural course]	difference in means under the modified treatment policies “set each flag to one, and multiply the dose by 1.25 where the natural flag is one” against “leave the observed treatment mechanism unchanged”	cleverly point TMLE and LTMLE under modified treatment policies on a vector treatment	-0.0012 to 0.0010	0.9410	0.9825	pass
a point treatment of one binary flag and one truncated-normal dose	ey_policy[dose up]	mean under the modified treatment policy add 0.5 to each dose, capped at 5.5	cleverly point TMLE and LTMLE under modified treatment policies on a vector treatment	-0.0011 to 0.0010	0.9450	0.9889	pass
a point treatment of one binary flag and one truncated-normal dose	ey_policy[flag scaled]	mean under the modified treatment policy set each flag to one, and multiply the dose by 1.25 where the natural flag is one	cleverly point TMLE and LTMLE under modified treatment policies on a vector treatment	-0.0015 to 0.0015	0.9320	0.9518	pass
a point treatment of one binary flag and one truncated-normal dose	ey_policy[natural course]	mean under the modified treatment policy leave the observed treatment mechanism unchanged	cleverly point TMLE and LTMLE under modified treatment policies on a vector treatment	-0.000838 to 0.000958	0.9420	0.9938	pass

Agreement with the canonical implementation

Python property evidence, with comparator runs pending. The lmtp runs have not run, so this study publishes no paired row, and equivalence.csv is empty. The change that registers the runs adds the paired table here.

Theory properties

property	cell	role	what was tested	what must hold	measured	result
double_robustness	longitudinal_vector_density__both_correct	positive	the two-node flag-and-dose contrast, its ratios the oracle joint induced density: 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 -0.0013 to 0.0011, margin 0.0051, SE ratio 0.9877	pass
double_robustness	longitudinal_vector_density__both_wrong	control	the two-node flag-and-dose contrast, its ratios the oracle joint induced density: 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 0.0615 to 0.0644, margin 0.0062, SE ratio 1.4677	pass
double_robustness	longitudinal_vector_density__outcome_correct	positive	the two-node flag-and-dose contrast, its ratios the oracle joint induced density: 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 -0.000657 to 0.0022, margin 0.0061, SE ratio 1.4254	pass
double_robustness	longitudinal_vector_density__outcome_wrong	positive	the two-node flag-and-dose contrast, its ratios the oracle joint induced density: the outcome regression is misspecified and the joint density is the oracle, read from 320 oracle bins per dose	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.000841 to 0.0016, margin 0.0053, SE ratio 1.0045	pass
double_robustness	longitudinal_vector_ordered_density__both_correct	positive	the ordered two-dose node contrast, its ratio the oracle joint induced density: 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 -0.000267 to 0.000990, margin 0.0027, SE ratio 0.9983	pass
double_robustness	longitudinal_vector_ordered_density__both_wrong	control	the ordered two-dose node contrast, its ratio the oracle joint induced density: 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 0.0483 to 0.0499, margin 0.0034, SE ratio 1.1535	pass
double_robustness	longitudinal_vector_ordered_density__outcome_correct	positive	the ordered two-dose node contrast, its ratio the oracle joint induced density: 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 -0.000550 to 0.000694, margin 0.0027, SE ratio 1.4222	pass
double_robustness	longitudinal_vector_ordered_density__outcome_wrong	positive	the ordered two-dose node contrast, its ratio the oracle joint induced density: the outcome regression is misspecified and the joint density is the oracle, read from 320 oracle bins per dose	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.000515 to 0.000796, margin 0.0028, SE ratio 0.9779	pass
double_robustness	point_vector_ordered_density__both_correct	positive	the ordered two-dose contrast read as a point treatment, its ratio the oracle joint induced density: 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 -0.000793 to 0.000461, margin 0.0027, SE ratio 0.9984	pass
double_robustness	point_vector_ordered_density__both_wrong	control	the ordered two-dose contrast read as a point treatment, its ratio the oracle joint induced density: 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 0.0195 to 0.0206, margin 0.0024, SE ratio 1.6662	pass
double_robustness	point_vector_ordered_density__outcome_correct	positive	the ordered two-dose contrast read as a point treatment, its ratio the oracle joint induced density: 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 -0.000210 to 0.000754, margin 0.0021, SE ratio 1.8336	pass
double_robustness	point_vector_ordered_density__outcome_wrong	positive	the ordered two-dose contrast read as a point treatment, its ratio the oracle joint induced density: the outcome regression is misspecified and the joint density is the oracle, read from 320 oracle bins per dose	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.000550 to 0.000738, margin 0.0028, SE ratio 1.0138	pass
image_necessity	ordered_crossed__declared_image	positive	the ordered-law node contrast of the crossed plan against the natural course: the joint density evaluates the flag map at each preimage’s natural first dose, with the outcome misspecified	bias interval inside the equivalence margin	bias -0.000744 to 0.000720, margin 0.0022	pass
image_necessity	ordered_crossed__shifted_read_control	control	the ordered-law node contrast of the crossed plan against the natural course: the same fit evaluates the flag map at the value’s own first dose	bias interval must fall entirely outside the margin	bias 0.0105 to 0.0122, margin 0.0027	pass
interval_calibration	longitudinal_vector_classifier__correctly_specified	positive	the two-node flag-and-dose contrast, its ratios a classifier on the true log ratio: both nuisances are correctly specified with an independently computed efficiency bound	SE ratio and coverage intervals both inside their calibration bands, with both efficiency-ratio intervals inside their bands	coverage 0.9439 to 0.9553, SE ratio 0.9806 to 1.0159, empirical efficiency ratio 0.9742 to 1.0093, reported efficiency ratio 0.9877 to 0.9915	pass
interval_calibration	longitudinal_vector_classifier__noise_control	control	the two-node flag-and-dose contrast, its ratios a classifier on the true log ratio: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8164 to 0.8360, SE ratio 0.6842 to 0.7099, empirical efficiency ratio 1.3936 to 1.4462, reported efficiency ratio 0.9877 to 0.9915	pass
interval_calibration	longitudinal_vector_classifier__shrunken_se_control	control	the two-node flag-and-dose contrast, its ratios a classifier on the true log ratio: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8152 to 0.8349, SE ratio 0.6864 to 0.7114, empirical efficiency ratio 0.9738 to 1.0094, reported efficiency ratio 0.6914 to 0.6941	pass
interval_calibration	longitudinal_vector_density__correctly_specified	positive	the two-node flag-and-dose contrast, its ratios the oracle joint induced density: both nuisances are correctly specified with an independently computed efficiency bound	SE ratio and coverage intervals both inside their calibration bands, with both efficiency-ratio intervals inside their bands	coverage 0.9422 to 0.9540, SE ratio 0.9813 to 1.0196, empirical efficiency ratio 0.9753 to 1.0127, reported efficiency ratio 0.9919 to 0.9962	pass
interval_calibration	longitudinal_vector_density__noise_control	control	the two-node flag-and-dose contrast, its ratios the oracle joint induced density: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8193 to 0.8392, SE ratio 0.6891 to 0.7158, empirical efficiency ratio 1.3885 to 1.4417, reported efficiency ratio 0.9919 to 0.9962	pass
interval_calibration	longitudinal_vector_density__shrunken_se_control	control	the two-node flag-and-dose contrast, its ratios the oracle joint induced density: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8207 to 0.8405, SE ratio 0.6861 to 0.7136, empirical efficiency ratio 0.9750 to 1.0139, reported efficiency ratio 0.6943 to 0.6973	pass
interval_calibration	longitudinal_vector_ordered_classifier__correctly_specified	positive	the ordered two-dose node contrast, its ratio a classifier on the true log ratio: both nuisances are correctly specified with an independently computed efficiency bound	SE ratio and coverage intervals both inside their calibration bands, with both efficiency-ratio intervals inside their bands	coverage 0.9423 to 0.9538, SE ratio 0.9771 to 1.0140, empirical efficiency ratio 0.9793 to 1.0157, reported efficiency ratio 0.9914 to 0.9941	pass
interval_calibration	longitudinal_vector_ordered_classifier__noise_control	control	the ordered two-dose node contrast, its ratio a classifier on the true log ratio: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8194 to 0.8389, SE ratio 0.6878 to 0.7123, empirical efficiency ratio 1.3938 to 1.4435, reported efficiency ratio 0.9914 to 0.9941	pass
interval_calibration	longitudinal_vector_ordered_classifier__shrunken_se_control	control	the ordered two-dose node contrast, its ratio a classifier on the true log ratio: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8204 to 0.8399, SE ratio 0.6843 to 0.7097, empirical efficiency ratio 0.9792 to 1.0154, reported efficiency ratio 0.6940 to 0.6959	pass
interval_calibration	longitudinal_vector_ordered_density__correctly_specified	positive	the ordered two-dose node contrast, its ratio the oracle joint induced density: 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 0.9406 to 0.9491, SE ratio 0.9817 to 1.0073, empirical efficiency ratio 0.9900 to 1.0160, reported efficiency ratio 0.9964 to 0.9984	pass
interval_calibration	longitudinal_vector_ordered_density__noise_control	control	the ordered two-dose node contrast, its ratio the oracle joint induced density: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8215 to 0.8355, SE ratio 0.6914 to 0.7101, empirical efficiency ratio 1.4047 to 1.4424, reported efficiency ratio 0.9964 to 0.9984	pass
interval_calibration	longitudinal_vector_ordered_density__shrunken_se_control	control	the ordered two-dose node contrast, its ratio the oracle joint induced density: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8207 to 0.8348, SE ratio 0.6871 to 0.7053, empirical efficiency ratio 0.9900 to 1.0161, reported efficiency ratio 0.6975 to 0.6989	pass
interval_calibration	point_vector_classifier__correctly_specified	positive	the flag-and-dose point contrast, its ratio a classifier on the true log ratio: both nuisances are correctly specified with an independently computed efficiency bound	SE ratio and coverage intervals both inside their calibration bands, with both efficiency-ratio intervals inside their bands	coverage 0.9418 to 0.9534, SE ratio 0.9711 to 1.0064, empirical efficiency ratio 0.9932 to 1.0293, reported efficiency ratio 0.9987 to 1.0006	pass
interval_calibration	point_vector_classifier__noise_control	control	the flag-and-dose point contrast, its ratio a classifier on the true log ratio: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8192 to 0.8387, SE ratio 0.6845 to 0.7110, empirical efficiency ratio 1.4057 to 1.4603, reported efficiency ratio 0.9987 to 1.0007	pass
interval_calibration	point_vector_classifier__shrunken_se_control	control	the flag-and-dose point contrast, its ratio a classifier on the true log ratio: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8165 to 0.8361, SE ratio 0.6793 to 0.7048, empirical efficiency ratio 0.9928 to 1.0301, reported efficiency ratio 0.6991 to 0.7005	pass
interval_calibration	point_vector_density__correctly_specified	positive	the flag-and-dose point contrast, its ratio the oracle joint induced density: 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 0.9440 to 0.9514, SE ratio 0.9806 to 1.0041, empirical efficiency ratio 0.9961 to 1.0202, reported efficiency ratio 0.9998 to 1.0008	pass
interval_calibration	point_vector_density__noise_control	control	the flag-and-dose point contrast, its ratio the oracle joint induced density: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8249 to 0.8374, SE ratio 0.6969 to 0.7132, empirical efficiency ratio 1.4025 to 1.4353, reported efficiency ratio 0.9998 to 1.0007	pass
interval_calibration	point_vector_density__shrunken_se_control	control	the flag-and-dose point contrast, its ratio the oracle joint induced density: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8151 to 0.8279, SE ratio 0.6866 to 0.7023, empirical efficiency ratio 0.9970 to 1.0198, reported efficiency ratio 0.6998 to 0.7005	pass
jacobian_necessity	ordered_full__declared_jacobian	positive	the ordered-law node contrast of the full plan against the natural course: the joint density multiplies each preimage by the inverse derivative of the D1 scale, the only non-unit Jacobian, with the outcome misspecified	bias interval inside the equivalence margin	bias -0.0010 to 0.000778, margin 0.0027	pass
jacobian_necessity	ordered_full__jacobian_drop_control	control	the ordered-law node contrast of the full plan against the natural course: the same fit drops the inverse derivative of the D1 scale	bias interval must fall entirely outside the margin	bias 0.0045 to 0.0064, margin 0.0028	pass
preimage_necessity	flag_scaled__declared_preimage	positive	the two-node contrast of the flag-scaled plan against the natural course: the joint density sums over every categorical preimage, with the outcome misspecified	bias interval inside the equivalence margin	bias -0.0010 to 0.0024, margin 0.0053	pass
preimage_necessity	flag_scaled__preimage_cut_control	control	the two-node contrast of the flag-scaled plan against the natural course: the same fit keeps only the preimages in the value’s own categorical cell	bias interval must fall entirely outside the margin	bias 0.0758 to 0.0806, margin 0.0073	pass

The truths and the efficiency bounds come from the exact laws and their closed forms (tests/studies/vector_mtp_common.py). The quadrature is checked at \(10^{-10}\).

family	cells	replications and size
interval_calibration	six positives: point TMLE, two-node LTMLE and the ordered law as one LTMLE node, each on the density route and on the classifier route. Each positive has a shrunken-SE control and a noise control	24,000, 9,600 and 19,200 replications on the density routes, and 10,000 on each classifier route, at n = 2,000. The design test sizes them
double_robustness	point TMLE on the ordered law under full, two-node LTMLE under flag scaled, and the ordered law as one LTMLE node under full. Each runs both_correct, outcome_correct, outcome_wrong and the both_wrong control	2,000 replications at n = 2,000
preimage_necessity	two-node LTMLE under flag scaled, wrong outcome regression. The declared fit against the same draw with the preimage sum cut to the observed categorical cell	1,000 pairs at n = 2,000
jacobian_necessity	the ordered law under full, wrong outcome regression. The declared fit against the same draw with the Jacobian of the D1 scale dropped. That scale is the only map whose Jacobian is not 1	1,000 pairs at n = 2,000
image_necessity	the ordered law under crossed. The declared fit against the same draw with the flag map read at the observed second value of \(D_1\) instead of the natural value of the preimage	1,000 pairs at n = 2,000

Each necessity control removes one term that can vanish at the truth, so each needs a nonzero witness. With a wrong outcome regression the targeting step carries the whole correction, and each defect then moves the estimate. The crossed plan exists because the full plan is blind to the flag map’s read of \(D_1\). The exact-law shift between the declared and the defective map is 0.000 under full and 1.243 SD under crossed (tests/unit/test_vector_mtp_design.py). A control passes only when its pair moves by more than the declared displacement.

The design test sizes every replication count before the run. It requires the product of the pass probabilities of the six calibration positives and their controls to reach 0.90. The product holds with every probed number moved one Monte Carlo standard error toward failing.

Measured values and declared margins

Names beginning margin: are thresholds declared before the run. Everything else is measured from the committed results and checked at the precision printed.

quantity	value	source
replicates	1000	paired replications
n	2000	observations per paired replication
independent_tests_passed	20	truth tests passing
independent_tests_total	20	truth tests reported
paired_tests_passed	0	paired comparisons passing
paired_tests_total	0	paired comparisons reported
property_cells_passed	36	property cells passing
property_cells_total	36	property cells reported
max_standardized_bias	0.0464	largest primary standardized bias
min_coverage	0.9320	lowest primary coverage
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	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:shrunken_se_factor	0.7000	negative-control SE multiplier
margin:efficiency_ratio_lower	0.9000	exact-EIF ratio lower bound
margin:efficiency_ratio_upper	1.1000	exact-EIF ratio upper bound
margin:necessity_displacement	0.1000	minimum Jacobian and image necessity displacement
margin:preimage_displacement	0.1000	minimum preimage necessity displacement
margin:subject_failure_bound	0.0100	declared share of replications a cell may lose to a recordable fitting failure
margin:subject_failures_allowed	10	the same bound, in primary replications

Limits

limit	what it means for use
three laws with a binary flag and at most two doses, at n = 2,000	the evidence covers these laws and this size. The study has no size ladder, so it claims no rate
the density is an oracle	the density route reads the generating pooled-hazard probabilities at a bin count that grows as \(n^{2/3}\). The default estimated density is not tested here
the classifier is a one-feature oracle	the classifier route reads the correctly specified logistic model of the plan’s log ratio. The study says nothing about the rate or the interval of a learned classifier
the density of \(D_2\) and the map of \(D_2\) may read the shifted \(D_1\) without a failing cell	those defects move the outcome-wrong positives by 0.42 and -0.19 SD only. The fast-tier mutation witnesses in tests/unit/test_vector_density_mutations.py carry them
the point pairs compare two constructions	TMLE targets with the covariate submodel over the observed and the policy values, and lmtp with a ratio-weighted intercept. The pairs read under the default margins and are not exactness checks
the lmtp comparison is pending	the agreement table is empty until the reference runs register. The Python cells carry the claim until then
a policy must be known and fixed	a policy learned from the sample, or one that depends on the law, is outside this evidence