Clustered point-treatment CV-TMLE

This study validates cluster-robust inference for cross-fitted point-treatment TMLE. It uses the binary-outcome clustered law with ten observations per cluster. Each replication draws 200 clusters of that size. A shared hidden effect modifies the treatment effect without changing treatment assignment. The population ATE is 0.10405 and the population risk ratio is 1.2513.

The law’s arm-by-latent coefficient is declared against a measurement, not chosen by hand. The rule matches the influence-curve design effect of the Gaussian clustered law this row replaces. A pilot measures 1.954 for the binary law and 1.949 for the Gaussian one. clustered_dgp records the rule, the pilot, and the seeds. The outcome intraclass correlation is a different quantity, and it is larger. It is a sample statistic rather than a population one, so it needs its condition: 0.319 at n = 200,000 and 0.313 at n = 20,000, both at seed 11, against a design effect near two. tests/unit/test_datasets.py reads the smaller figure.

What was compared

setting	cleverly	R lmtp 1.5.4
construction	stacked point-treatment CV-TMLE	lmtp_tmle internals with the supplied folds
folds	five grouped folds, balanced on nothing	the identical rowwise assignment
treatment mechanism	exact propensity from the law	the identical exact density ratio
outcome regression	unpenalized logistic regression, C=1e6	SL.glm for a binomial outcome
independent unit	cluster identifier, aggregated as cluster sums	the same identifier, aggregated by ife as cluster means
ATE inference	joint difference influence curve	subtraction of the two ife arm objects
intervals	pointwise 95% Wald	pointwise 95% Wald

Both runners reject a fold assignment that splits a cluster. The Python runner writes its realized assignment beside each sampled row. The R adapter retains that assignment without rebuilding it.

cleverly fits an unpenalized logistic regression because SL.glm is unpenalized. A default penalty would make the two working models differ before targeting ran.

The split this row is conditional on

Every replication of this row uses one grouped partition. The estimator’s random_state is the fixed integer 0. Every draw emits the same cluster labels in the same order. The partition therefore depends on nothing a draw varies, so all 800 primary and 2,400 property replications share one byte-identical assignment. canonical_clustered_tmle.assert_the_declared_fixed_partition refuses a fit that ran under any other assignment, and tests/unit/test_clustered_tmle_method_study.py states the claim on two different draws.

The partition is external in the sense the split law requires: it reads nothing from the sample. The published coverage is conditional on it. The coverage is not an average over the split distribution. A reader who repartitions gets a different split, and this row does not measure the extra variability that adds.

What supports a grouped split

One reviewed source covers the partition and nothing else. Wang, Park, Small and Li (2024), Journal of the American Statistical Association 119(548):2959-2971, partition the clusters at random into parts of roughly equal size, and prove a cross-fitted result under that partition. Section 4.2 and Theorem 4(b) carry it. Both locators are a section number and a theorem number, read in the NIHMS author manuscript (PMC11795269), which carries the published volume, issue, pages and DOI. References gives the entry in full.

Their estimator is an AIPW-type estimator with cluster-level treatment, not this package’s TMLE with row-level treatment. The source supports the split law. It does not prove the estimator here.

The rest is the package’s own estimating-equation argument. It treats clusters as the independent units, and it needs four conditions: clusters independent of one another, a bounded random cluster size, no interference between clusters, and the usual remainder rates on the nuisances. With clusters in place of rows, the argument is Zheng and van der Laan (2011), Theorem 2. The design in this row satisfies all four conditions by construction, with every cluster of ten rows. The unequal-size study covers sizes from 2 to 18.

This study is the empirical witness for that argument, and it is the only one. No source read here proves the estimator is valid under clustering.

The two implementations aggregate the influence curve differently. cleverly sums the rowwise values inside each cluster. Pinned ife takes the variance of the cluster means. The two formulas agree only when every cluster holds the same number of rows. This design fixes that number at ten. The clustered inference audit records the pinned ife behavior and the archive it pins.

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
binary-outcome law with ten rows per cluster and shared effect modification	ate	average treatment effect	cleverly clustered point-treatment CV-TMLE	-0.0056 to 0.000556	0.9525	0.9721	pass
binary-outcome law with ten rows per cluster and shared effect modification	ate	average treatment effect	R lmtp	-0.0056 to 0.000558	0.9537	0.9737	pass
binary-outcome law with ten rows per cluster and shared effect modification	ey0	counterfactual mean under no treatment	cleverly clustered point-treatment CV-TMLE	-0.000789 to 0.0025	0.9500	1.0306	pass
binary-outcome law with ten rows per cluster and shared effect modification	ey0	counterfactual mean under no treatment	R lmtp	-0.000841 to 0.0024	0.9513	1.0305	pass
binary-outcome law with ten rows per cluster and shared effect modification	ey1	counterfactual mean under treatment	cleverly clustered point-treatment CV-TMLE	-0.0049 to 0.0015	0.9437	0.9841	pass
binary-outcome law with ten rows per cluster and shared effect modification	ey1	counterfactual mean under treatment	R lmtp	-0.0049 to 0.0015	0.9425	0.9837	pass

Agreement with the canonical implementation

Comparator-backed: a paired comparison with lmtp 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
binary-outcome law with ten rows per cluster and shared effect modification	ate	average treatment effect	-0.000005	0.000986	1.0031	-0.0012	different estimator	0.0020 vs 0.0500	equivalent
binary-outcome law with ten rows per cluster and shared effect modification	ey0	counterfactual mean under no treatment	0.000049	0.0183	1.0016	-0.0013	different estimator	0.0026 vs 0.0500	equivalent
binary-outcome law with ten rows per cluster and shared effect modification	ey1	counterfactual mean under treatment	0.000044	0.0084	1.0006	0.0012	different estimator	0.0021 vs 0.0500	equivalent

Theory properties

property	cell	role	what was tested	what must hold	measured	result
clustered_inference	cluster_robust	positive	five-fold point-treatment TMLE with cluster-robust ATE inference	SE-ratio and coverage intervals both stay inside their calibration bands	coverage 0.9346 to 0.9585, SE ratio 0.9698 to 1.0455, paired coverage gain 0.0908 to 0.1229	pass
clustered_inference	iid_control	control	the identical rows, point estimates, and influence curves treated as independent	the SE-ratio upper endpoint must not exceed the declared IID-control ceiling	coverage 0.8212 to 0.8600, SE ratio 0.6909 to 0.7454, paired coverage gain 0.0908 to 0.1229	pass

The property study fits each sample once. Its IID control reuses the same rows, ATE estimate, and rowwise influence curve. Only the variance aggregation changes. This pairing isolates the cluster covariance calculation from nuisance fitting and targeting.

Measured values

Names beginning margin: are thresholds declared before the run. The remaining values come from the committed artifacts. The documentation gate checks every printed value.

quantity	value	source
replicates	800	primary replications
n	2000	observations per replication
independent_tests_total	6	implementation-estimand truth tests
independent_tests_passed	6	truth tests passing
paired_tests_total	3	paired implementation tests
paired_tests_passed	3	paired tests passing
property_cells_total	2	clustered-inference cells
property_cells_passed	2	property cells passing
max_standardized_bias	0.0748	largest absolute standardized bias
min_coverage	0.9425	lowest primary coverage
min_coverage_ci_lower	0.9179	lowest primary 99% coverage endpoint
max_margin_utilization	0.0183	largest share of the paired margin used
properties[clustered_inference/cluster_robust]:coverage	0.9475	cluster-robust ATE coverage
properties[clustered_inference/cluster_robust]:se_ratio	1.0055	cluster-robust ATE SE ratio
properties[clustered_inference/cluster_robust]:se_ratio_ci_lower	0.9698	cluster-robust SE-ratio lower endpoint
properties[clustered_inference/cluster_robust]:se_ratio_ci_upper	1.0455	cluster-robust SE-ratio upper endpoint
properties[clustered_inference/iid_control]:coverage	0.8413	IID-control ATE coverage
properties[clustered_inference/iid_control]:se_ratio	0.7176	IID-control ATE SE ratio
properties[clustered_inference/iid_control]:se_ratio_ci_upper	0.7454	IID-control SE-ratio upper endpoint
properties[clustered_inference/cluster_robust]:coverage_gain_ci_lower	0.0908	paired coverage-gain lower endpoint
margin:confidence_level	0.9900	confidence level for Monte Carlo intervals
margin:alpha	0.0500	nominal size of the estimator’s own intervals
margin:nominal_coverage	0.9500	nominal coverage those intervals claim
margin:bootstrap_replicates	10000	resamples behind every bootstrap interval
margin:standardized_bias	0.2500	bias equivalence margin, in empirical standard deviations
margin:coverage_floor	0.9000	validity floor the exact coverage lower endpoint must clear
margin:over_coverage_ceiling	0.9900	above this, coverage is conservative rather than invalid
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:calibration_se_ratio_lower	0.9300	cluster-robust SE-ratio lower limit
margin:calibration_se_ratio_upper	1.0700	cluster-robust SE-ratio upper limit
margin:calibration_coverage_lower	0.9200	cluster-robust coverage lower limit
margin:calibration_coverage_upper	0.9800	cluster-robust coverage upper limit
margin:type_i_ceiling	0.1000	largest size the one-sided type-I bound may establish
margin:paired_difference	0.1500	paired similarity margin, in pooled empirical standard deviations
margin:rmse_noninferiority	1.1000	largest RMSE ratio the one-sided upper bound may reach
margin:coverage_noninferiority	-0.0250	smallest coverage difference the one-sided lower bound may reach
margin:calibration_noninferiority	0.0500	largest excess SE-calibration error the upper bound may reach
margin:minimum_power	0.8000	rejection lower bound the power control must clear
margin:root_n_slope	-0.5000	the contraction rate root-n asymptotics predict
margin:root_n_slope_lower	-0.6250	accepted slope band, lower limit
margin:root_n_slope_upper	-0.3750	accepted slope band, upper limit
margin:excluded_slope	-0.2500	the slower rate the interval must exclude
margin:iid_control_se_ceiling	0.8000	IID-control SE-ratio ceiling
margin:clustered_coverage_gain	0.0300	paired coverage-gain floor

Limitations

limitation	what it means for use
One cluster size and one dependence law	The row validates clusters of ten under shared effect modification. The unequal-size study covers informative cluster size
Equal cluster sizes	cleverly aggregates cluster sums and pinned ife aggregates cluster means, as the clustered inference audit records. The two formulas agree only at equal sizes, so the row does not establish parity for unbalanced clusters. No pinned comparator cross-fits with cluster sums at unequal sizes
200 clusters and a normal reference	At 200 clusters the intervals use the normal reference. The few-cluster study measures the \(t\) reference below 40 clusters
Binary outcome and binary treatment	The row does not establish continuous outcomes, multi-valued treatments, missing outcomes, or longitudinal treatment
One fixed partition	Every replication uses the one grouped assignment described above. The coverage is conditional on it, and the row does not measure the variability a repartition adds
One five-fold split	The row does not establish repeated, fold-evaluated, or fold-specific targeting
Exact treatment mechanism	The comparison isolates targeting and inference. It does not compare learned propensity models
Pointwise identity-scale intervals	The row does not cover simultaneous bands, bootstrap intervals, ratios, weights, or strata. The default-band study measures the default band of this study’s subject fit.
Main-effects outcome learners	The row does not establish flexible learner-library parity under clustering

Reproduction

The fixture README gives smoke and full commands. The manifest records the container, runner, adapter, harness, configuration, and result-determining Python modules.