Three Routes to One Answer: Reconciling AIPW, TMLE, and Double Machine Learning for Applied Researchers

1.1 Why an applied researcher reaches for a doubly-robust estimator

• Flexible machine-learning fits can reintroduce first-order bias into plug-in estimates, leaving conventional confidence intervals prone to undercoverage.The doubly robust approaches replace that first-order nuisance term with a second-order remainder when nuisance estimation rates multiply faster than n^-1/2.Source: §1.1 · p. 2

1.2 Three literatures, one estimator

• Different terminology and package defaults can make AIPW, TMLE, and DML estimates hard to compare, despite their shared efficient influence function.The tutorial treats their theoretical equivalence as established background and focuses on finite-sample choices such as folds and propensity truncation.Source: §1.2 · pp. 2–3

1.3 What this tutorial does

• The tutorial derives all three routes from one efficient influence function, applies shared nuisance fits on open NHEFS data, reconciles major R packages, and provides a reporting checklist.Its contribution is a worked reconciliation, not a new estimator or theorem.Source: §1.3 · p. 3
• The tutorial places overlap diagnosis ahead of estimator choice because the three routes agree closely with overlap but degrade together without it.It presents no estimator as the winner.Source: §1.3 · p. 3

2.0.1 The motivating question

• The motivating question is how much weight change is caused by quitting smoking, rather than by differences between people who quit and those who continue.The tutorial uses this question as its running example.Source: §2.0.1 · p. 4

2.0.2 The data

• The complete-case NHEFS sample contains 1,566 people—403 quitters and 1,163 continuing smokers—and targets the ATE in that analytic population.The exposure is quitting between 1971 and 1982; the outcome is weight change in kilograms, and loss to follow-up is not separately modeled.Source: §2.0.2 · p. 4

2.0.3 The causal story: a hypothesized DAG

• The hypothesized DAG treats baseline covariates as common causes of quitting and weight change, making them the adjustment set.These include age, sex, race, education, baseline weight, smoking intensity and duration, physical activity, and alcohol use; the DAG encodes assumptions, not findings.Source: §2.0.3 · p. 5, Figure 1 · p. 5

2.0.4 The estimand and what it takes to identify it

• The target is the mean difference in weight change if everyone quit rather than continued smoking, identified under consistency, exchangeability, and positivity.Consistency and exchangeability are not testable from the data alone; positivity is partly checkable through overlap.Source: §2.0.4 · pp. 5–6

2.0.5 Roadmap

• The walkthrough fixes the question, data, and estimand, diagnoses overlap, fits nuisance functions, builds a shared influence function, and compares estimator routes.Later steps examine cross-fitting, benchmark calibration, positivity failure, software packages, and reporting.Source: §2.0.5 · p. 6, Table 2 · p. 7

3 Step 1 — Diagnose overlap first

• Positivity is inspectable in the observed treatment distribution, so it should be checked before choosing AIPW, TMLE, or DML.Consistency and no unmeasured confounding instead require subject-matter reasoning.Source: §3 · p. 7

3.1 What positivity is, and why it decides feasibility

• Positivity asks whether people across covariate profiles could plausibly be treated or untreated; absent comparisons leave estimates dependent on outcome-model extrapolation.No estimator can manufacture the missing comparison.Source: §3.1 · p. 7

3.2 Reading the NHEFS overlap diagnostic

• NHEFS propensity densities share the [0.044, 0.759] range without mass near 0 or 1, indicating good common support.Because the diagnostic uses the estimated propensity from the same Super Learner, it assesses practical rather than structural positivity.Source: §3.2 · p. 8, Figure 2 · p. 8

3.3 Truncation, and why weak overlap hurts every route equally

• Report propensity distributions by exposure, the fraction outside a prespecified truncation window, and the truncation rule with its rationale.NHEFS propensities already fall within [0.044, 0.759], so truncating to [0.025, 0.975] affects zero of 1,566 observations; the bounds cap inverse-probability weights at 40.Source: §3.2 · p. 8, §3.3 · pp. 8–9
• As g(X) approaches 0 or 1, weak overlap worsens the efficiency bound and finite-sample remainder for AIPW, TMLE, and DML alike.All three share exposure to small g(X)(1 − g(X)); inactive truncation in NHEFS leaves the ATE target unchanged.Source: §3.3 · p. 9

3.4 When the diagnostic fails: a preview

• In LaLonde-PSID, barely overlapping propensity densities mean trimming discards 84–89% of the sample, and the remainder cannot pin down the target.Without overlap, no estimator recovers the effect from the data without additional modeling assumptions.Source: §3.4 · pp. 9–10
• NHEFS’s doubly robust estimates agree within 0.10 kg, consistent with the canonical 3.4–3.5 kg benchmark.This agreement is presented after the overlap diagnostic establishes that NHEFS clears the positivity bar.Source: §3.4 · pp. 9–10

4.1 A single Super Learner for both nuisances

• One shared Super Learner library estimates both the propensity score and outcome regression, and the unchanged fits are passed to every estimator.The library combines candidate learners by cross-validated performance; its convex metalearner uses non-negative weights that sum to one.Source: §4 · p. 10, §4.1 · pp. 10–11

4.2 The two knobs that actually move the answer

• With library and folds fixed, AIPW versus TMLE shifts the estimate about 0.01 kg, while fold scheme shifts it 0.10 kg and library or implementation shifts it 0.17 kg.Across the engine, estimates span 3.32 kg (full-sample AIPW) to 3.42 kg (DC-TMLE at r = 100); software estimates span 3.32–3.49 kg.Source: Table 4 · p. 11, §4.2 · p. 11
• Cross-fitting addresses own-observation bias by fitting nuisances on other folds and evaluating the influence function out-of-fold.The tutorial uses five outer folds for cross-fit estimators and three splits for double-cross-fit variants; propensity scores are truncated to [0.025, 0.975].Source: §4.2 · p. 12

4.3 Fair by construction

• Because all estimators use identical out-of-fold nuisance fits, residual differences at a fixed fold scheme reflect their combination step rather than a luckier nuisance fit.The shared fits also fix the second-order remainder across estimators; differences across fold schemes remain a separate source of variation.Source: §4.3 · p. 12

5.1 The estimand and the g-formula

• The target is the average weight-change effect of quitting smoking, identified under consistency, no unmeasured confounding, and positivity.Positivity requires each covariate profile to have a nonzero chance of either treatment level.Source: §5.1 · p. 13

5.2 Two nuisances, two singly-robust baselines

• On NHEFS, g-computation returns 3.27 kg and IPTW returns 3.19 kg; each baseline is consistent only when its own outcome or propensity model is correct.Both use shared Super Learner nuisances, with propensity scores truncated to [0.025, 0.975].Source: §5.2 · p. 14

5.3 The efficient influence function, term by term

• The efficient influence function combines the outcome contrast with a propensity-weighted residual correction, and its empirical mean-zero equation underlies the estimators.The correction weights each observed outcome-model residual by inverse propensity, with opposite signs for treated and untreated observations.Source: §5.3 · pp. 14–15
• AIPW returns 3.32 kg (95% CI 2.50, 4.14) and TMLE returns 3.33 kg (95% CI 2.50, 4.15), near the 3.4–3.5 kg benchmark.The doubly robust correction brings the singly robust baselines, 3.27 and 3.19 kg, to 3.32–3.33 kg.Source: §5.3 · p. 15

5.4 Why the correction repairs both baselines: double robustness

• The correction is doubly robust: its bias vanishes if either nuisance model is correct, because the bias factors into propensity and outcome-model errors.For flexible machine-learning nuisances, cross-fitting controls the empirical-process term; the second-order remainder is a product of nuisance errors.Source: §5.3 · p. 15, §5.4 · p. 16
• AIPW, TMLE, and DML are asymptotically equivalent and efficient under the product-rate condition, a shared theoretical property rather than a tutorial result.They implement the common score through a one-step average, a targeting fluctuation, or out-of-fold evaluation.Source: §5.4 · pp. 16–17

6 Step 4 — Two combination rules, three fitting schemes (one score)

• The six estimators cross two combination rules with three nuisance-fitting schemes; AIPW and DML share the one-step score, while TMLE targets the outcome model.All use the same efficient influence function and solve the same estimating equation.Source: §6 · p. 17, Table 7 · p. 18

6.1 Route 1 — AIPW: the one-step rule, full sample

• AIPW adds the empirical mean of the estimated influence function to the g-computation plug-in, using the propensity-weighted residual correction.Its doubly-robust construction can use either nuisance model to correct the other.Source: §6.1 · p. 18

6.2 Route 2 — TMLE: the other combination rule, full sample

• TMLE fluctuates the fitted outcome model until the empirical influence-function mean is zero, then takes a range-respecting plug-in estimate.The fluctuation is propensity-informed.Source: §6.2 · p. 19

6.3 Route 3 — DML: the one-step score again, fit out of fold

• DML is the one-step AIPW estimator with propensity and outcome nuisances fit out of fold, rather than a separate combination rule.Cross-fitting separates nuisance training observations from the observations used to score them.Source: §6.3 · pp. 19–20

6.4 They converge on NHEFS

• 3.32–3.42 kg: the six doubly-robust NHEFS estimates span 0.10 kg, and every interval covers the canonical 3.4–3.5 kg benchmark.The span is much narrower than any one confidence interval.Source: §6.4 · p. 20, Figure 4 · p. 21

6.5 The same machinery on a binary outcome: RHC

• +0.043 to +0.049: the four RHC cross-fit and double-cross-fit estimates span about 0.006 on the risk-difference scale, while full-sample AIPW is +0.033.The full-sample AIPW correction is under-sized with in-sample overfitting; RHC has no ground truth, so the agreement is an internal-consistency check.Source: §6.5 · p. 22

6.6 A Rosetta Stone for three literatures

• The three literatures use different names, notation, references, and software for the same mathematical objects.The tutorial lines up these correspondences in a Rosetta Stone table.Source: §6.6 · pp. 24–25, Table 8 · p. 25

7.1 Reusing the same rows to fit and to score inflates your confidence

• Cross-fitting keeps each observation out of its own nuisance-model training data, preventing own-observation residual deflation from making standard errors too small.With flexible ML nuisances, in-sample residuals can make intervals too narrow and coverage fall below nominal.Source: §7.1 · p. 26

7.2 A calibration check where the truth is known

• 0.99: cross-fit DML and CV-TMLE coverage exceeded full-sample AIPW (0.90) and TMLE (0.93), with double-cross-fit coverage between them (0.95, 0.96).The AIPW-to-DML coverage gap was resolved across the 50 realizations; full-sample under-coverage against nominal was not resolved.Source: Figure 6 · p. 27, §7.2 · pp. 27–28
• At n = 747, cross-fit intervals were about a third wider (mean width 1.34 versus 1.04), while bias and RMSE differences remained within Monte-Carlo error.The full-sample AIPW-to-DML coverage gap was resolved, but the TMLE-to-CV-TMLE gap was borderline.Source: §7.2 · pp. 27–28
• At n = 747, double-cross-fit AIPW and TMLE had higher bias (−0.25, −0.23) and RMSE (≈0.70), as disjoint-fold fitting left each nuisance about a third of the data.The passage presents double cross-fitting as a large-sample tool; at this sample size the extra split is a handicap, not an advantage.Source: §7.2 · p. 28
• For the IHDP coverage calculation, the 50 simulated realizations—not the 20 fold partitions within each—are the independent Monte-Carlo units.Coverage should be reported with the replicate count and clustered Monte-Carlo standard error; repeated splitting would change the estimator rather than merely measure single-split performance.Source: §7.2 · pp. 27–29

7.3 Repeating the split: how many repetitions are enough?

• By roughly 30–50 repetitions, all four RHC split-based estimates settled; double-cross-fit estimates were about 0.0025 higher than their single-split values.Single-cross-fit estimates changed by less than 0.0003, and interval widths stayed essentially flat as repetitions increased.Source: §7.3 · p. 29

8 Step 6 — Sensitivity and failure modes (positivity)

• The point estimate stabilizes by about 30–50 repetitions, while the 95% confidence-interval width barely changes.Double-cross-fit estimates drift about 0.0025 above their single-split values; single-cross-fit estimates return to theirs.Source: Figure 7 · p. 30
• Diagnose overlap before choosing among AIPW, TMLE, and DML: their shared influence function links close estimates under good overlap and joint degradation under positivity failure.Figure 8 contrasts the good-overlap and near-disjoint propensity distributions.Source: §8 · p. 30, Figure 8 · p. 32, §8.2 · p. 33

8.1 Where everything breaks: the LaLonde/NSW–PSID positivity failure

• −$1,284 for cross-fit AIPW and −$5,186 for CV-TMLE miss the $1,794 reference on the untrimmed LaLonde/NSW–PSID sample; cross-fitting does not rescue the positivity failure.The pooled ATE is not identified from these data without extrapolation; held-out TMLE corrections can amplify the failure.Source: §8.1 · pp. 30–31, Table 9 · p. 31
• 84–89% of units are discarded by overlap trimming, which changes the target population rather than recovering the pooled effect.The intervals span the $1,794 directional reference only after widening to roughly $2,000, and the reference concerns a different population.Source: §8.1 · pp. 30–31, Table 10 · p. 31

8.2 Two sensitivity checks on NHEFS (good overlap)

• Under NHEFS's good overlap, changing the propensity-truncation bound barely moves the estimate because none of the 1,566 propensities approach 0 or 1.Tightening or loosening the bound leaves the point estimate within the 0.10 kg spread across the six doubly robust estimators.Source: §8.2 · p. 33, Table 11 · p. 33
• 3.32–3.49 kg is the NHEFS estimate envelope across learner libraries and packages; the spread tracks nuisance and fold choices, not the estimator label.With good overlap, the answer is robust to estimator, library, and truncation within a fraction of one standard error.Source: §8.2 · pp. 33–34

9.1 A reporting checklist for doubly-robust and double-ML analyses

• The reporting checklist records the estimand and identification assumptions, shared learner library, cross-fitting, propensity truncation, overlap diagnosis, trimming sensitivity, estimator rationale, and standard-error source.Its rows are choices readers need to reproduce or audit the estimate.Source: §9.1 · p. 34, Table 12 · p. 35
• The analytic standard error supports Wald intervals for doubly robust and double-ML estimators, whereas the singly robust baselines lack a comparable valid interval here.A sandwich interval would require a correctly specified parametric model, and a naive bootstrap does not remove first-order regularization bias.Source: §9.1 · pp. 34–35

9.2 Software reconciliation: one estimand, five implementations

• 3.32–3.49 kg is the range across five implementations of the NHEFS estimand, with the spread tracking nuisance libraries, fold counts, and truncation rather than estimator labels.After matching the library and truncation and averaging over 20 cross-fits, the hand-coded engine, tmle, and AIPW each return 3.39 kg, agreeing within 0.01 kg.Source: Table 13 · p. 36, §9.2 · p. 36

9.3 Exercises

• The exercises test truncation sensitivity, learner-library ablation, cross-fitting and coverage, repeated splits, and package reconciliation using the shared NHEFS and IHDP setup.The package exercise asks which aligned setting closes most of the gap; the learner library usually does.Source: §9.3 · pp. 37–38

10 Discussion

• 3.32–3.42 kg is the range of six doubly robust NHEFS estimates, and every interval covers the canonical 3.4–3.5 kg benchmark.The same convergence appears on the RHC and IHDP examples; the shared-nuisance comparison demonstrates that estimator labels matter less than holding nuisances and folds fixed.Source: §10 · p. 38

10.1 What actually matters: diagnose overlap first

• Diagnose overlap before estimating: NHEFS groups overlap across propensity scores, but LaLonde/PSID estimates span −$819 to −$14,629 and trimming discards 84–89% of the sample.The pooled effect outside common support is not identified from the data; trimming changes the target rather than recovering it.Source: §10.1 · p. 38

10.2 Cross-fitting buys valid inference, not a different estimand

• On IHDP, cross-fit DML and CV-TMLE cover 0.99 versus 0.90 for full-sample AIPW and 0.93 for full-sample TMLE, with about a third more interval width.The coverage gap is resolved for AIPW→DML, while full-sample under-coverage against nominal is not; cross-fitting's role is inferential, not a change in estimand.Source: §10.2 · p. 39

10.3 Reconciling the software: same estimand, different defaults

• 3.32–3.49 kg is the NHEFS range across implementations, and matching settings brings the hand-coded engine, tmle, and AIPW to 3.39 kg within 0.01 kg.DoubleML and tmle3 use learner ecosystems or folds that are not matched learner-for-learner in this reconciliation.Source: §10.3 · p. 39, §10.4 · p. 39

10.4 A short guide: which route when

• Route choice concerns finite-sample behavior and diagnostics, not the estimand, and the comparison is a guide rather than a ranking.The guidance assumes overlap has already been checked.Source: §10.4 · p. 39, Table 14 · p. 40

10.5 Relation to prior work

• The tutorial extends earlier comparative and single-estimator guides by unifying three routes, sharing nuisance fits and folds, grading against benchmarks, and reconciling software.It supplies an end-to-end implementation alongside code-free reviews and single-estimator walkthroughs.Source: §10.5 · p. 40

10.6 Extensions: time-to-event outcomes

• For time-to-event outcomes, AIPW/one-step, TMLE, and DML target survival and risk measures while adding a censoring model alongside outcome-hazard and propensity models.The workflow remains overlap diagnosis followed by a doubly robust route and cross-fitting for flexible learners.Source: §10.6 · p. 40
• For survival analyses, the tutorial points readers to Talbot et al. [2025] and Rytgaard et al. [2023], and to the survtmle and concrete [Chen et al., 2025] packages.Source: §10.6 · p. 41

10.7 Further reading

• Further reading spans causal inference and the g-formula [2020], Super Learner and targeted learning [2007], semiparametric methods [2023], and DML and cross-fitting [2018].The section also points to step-by-step tutorials for individual routes and time-to-event outcomes.Source: §10.7 · p. 41

10.8 Limitations

• The real-data demonstration covers only NHEFS continuous weight change and RHC binary mortality; unrepresented confounding structures could yield a different qualitative pattern.The IHDP known-truth evidence is pedagogical, not evidence of external validity for a particular effect.Source: §10.8 · p. 41
• IHDP bias contrasts come from one generating process at one sample size; pairwise bias intervals overlap, making the observed ordering suggestive at most.Cross-fit DML bias is −0.15 (MCSE ≈0.07), versus full-sample AIPW at −0.18 (MCSE ≈0.07).Source: §10.8 · p. 41

10.9 Conclusion

• Shared-library, identical-fold estimates agree within 0.10 kg on NHEFS, packages span a matched-defaults band, and IHDP bias differences are within Monte Carlo error.The conclusion recommends diagnosing overlap first, using cross-fitting with flexible learners, and reporting Monte Carlo standard errors before interpreting bias or coverage differences.Source: §10.9 · p. 41
• The paper frames the operative question as whether the effect is identified and inference is valid, rather than which estimator to choose.The equivalence is asymptotic, with finite-sample departures depending on positivity and cross-fitting.Source: §10.8 · p. 41, §10.9 · p. 42

• Code, data retrieval scripts, and a browsable reproduction of the analyses are available in the companion repositories.Exact reproduction requires the pinned package versions in renv.lock and sessionInfo(), since Super Learner ensemble weights are version-sensitive.Source: Data Availability Statement · p. 42

• The research received no specific grant from any funding agency.Source: Funding · p. 42