Why Cross-Validate a Targeted Estimator?
TMLE combines an outcome model with a targeting step designed around the causal parameter of interest. When the outcome and treatment models are fit flexibly, however, evaluating those fits on the same observations used to train them can make their apparent performance optimistic and complicate estimation. Cross-validated TMLE separates nuisance fitting from validation-fold evaluation: for each fold, nuisance models are fit on the other folds, then their predictions are targeted using the held-out fold.
This arrangement permits data-adaptive nuisance fits, including Super Learner fits, while ensuring that each observation's initial nuisance predictions come from a fit that did not use that observation's outcome. Targeting is still an essential part of the method: the held-out outcomes are used in the fold-specific targeting step. Cross-validation does not make the entire estimator independent of those outcomes, nor does it by itself guarantee valid inference.
Setup and Definition
Let the observed data be independent and identically distributed observations \(O_i=(W_i,A_i,Y_i)\), where \(W_i\) is a vector of pre-treatment covariates, \(A_i\in\{0,1\}\) is treatment, and \(Y_i\) is the outcome. The target is the average treatment effect (ATE), \(\psi_0=E\{Y(1)-Y(0)\}\). Identification from observed data requires consistency, conditional exchangeability \(Y(a)\mathrel{\perp} A\mid W\), and positivity: the conditional treatment probabilities are strictly between zero and one for covariate values in the target population. For stable estimation, analyses commonly require probabilities bounded away from zero and one, or otherwise justify a restricted target population.
Define the outcome regression \(Q_0(a,W)=E(Y\mid A=a,W)\) and treatment mechanism \(g_0(1\mid W)=P(A=1\mid W)\). For any fitted versions \(Q\) and \(g\), the ATE clever covariate is \(H_g(A,W)=A/g(1\mid W)-(1-A)/\{1-g(1\mid W)\}\). When \(Y\) is binary, the outcome regression takes values in \([0,1]\), and a logistic fluctuation keeps updated predictions in that range.
In fold \(k\), write the nuisance fits as \(Q_{-k}\) and \(g_{-k}\). For a binary outcome, the usual one-dimensional logistic fluctuation is
The fold-specific parameter \(\epsilon_k\) is estimated by logistic maximum likelihood with the initial prediction as an offset. Equivalently, when a finite maximum exists, it solves the score equation
To form the ATE estimate, evaluate the updated regression under both treatment values for every validation observation. The resulting estimate is
For unequal fold sizes, the formula automatically weights each fold by its share of the total sample. Some implementations use a shared fluctuation across validation predictions or other closely related CV-TMLE constructions. The fold-specific version above is the focus here; an analysis should state which construction it uses rather than treating all implementations as identical.
Why the Targeting Step Works
At the true data-generating distribution, the efficient influence function for the ATE is
Here the nuisance functions are the true \(Q_0\) and \(g_0\), and \(\psi_0\) is the true ATE. This expression is not generally a mean-zero efficient influence function when arbitrary fitted nuisance estimates are substituted for the true functions. With fitted nuisances, the corresponding expression is a plug-in estimating function; its mean need not be zero.
The targeting equation uses the residual component of this influence-function structure. A short algebraic identity shows how it aligns the plug-in estimate with an augmented estimate that uses the fold's fitted treatment mechanism. Define the fold's augmented estimating expression by averaging the targeted outcome contrast plus \(H_{g_{-k}}(A_i,W_i)\{Y_i-Q^*_{-k}(A_i,W_i)\}\). The second term sums to zero within the fold by the targeting equation. Therefore the fold's augmented expression equals its targeted plug-in contrast average, provided the fluctuation score is solved as specified.
This explains the role of targeting: it adjusts the outcome predictions along a specified fluctuation so that a sample analogue of the residual score is zero. It does not establish that the nuisance models are correct. Under appropriate regularity conditions, identification assumptions, positivity, and sufficiently accurate nuisance estimation, the targeting and orthogonality structure can support a consistent, asymptotically linear estimator. Cross-fitting helps control some empirical-process difficulties from flexible nuisance fitting, but it does not remove the need for suitable remainder-rate conditions or other assumptions for inference.
Worked Example: A Logistic Targeting Step in One Fold
Consider a hypothetical validation fold of four observations. Suppose the training data produced \(g_{-k}(1\mid W)=0.5\) for each person, and initial outcome predictions \(Q_{-k}(1,W)=0.40\) and \(Q_{-k}(0,W)=0.30\). The validation fold has two treated outcomes, 1 and 0, and two control outcomes, both 0. These fitted treatment probabilities are positive for both arms, so the clever covariate is defined: it is \(2\) for treated observations and \(-2\) for controls.
Because the initial predictions are constant in this small illustration, the updated observed-treatment predictions are \(q_1=\operatorname{expit}\{\operatorname{logit}(0.40)+2\epsilon_k\}\) for treated observations and \(q_0=\operatorname{expit}\{\operatorname{logit}(0.30)-2\epsilon_k\}\) for controls. Substitute the four outcomes and clever covariates into the score:
Thus the score requires \(q_1-q_0=0.5\). Solving the one-dimensional logistic equation gives approximately \(\widehat\epsilon_k=0.488\), \(q_1\approx0.639\), and \(q_0\approx0.139\). Substitution checks the score: \(2-4(0.639)+4(0.139)\approx0.000\), up to rounding. The targeted predictions for the two treatment levels are approximately 0.639 and 0.139 for each covariate value, so this fold's plug-in ATE is \(0.639-0.139=0.500\), in outcome-probability units. This is a deliberately tiny hypothetical fold; its estimate is not intended to be precise.
Combining Folds and Choosing the Fluctuation
Each fold contributes a targeted contrast for every person in that fold. The overall estimate weights those contrasts by the number of people they represent, rather than averaging fold means equally when fold sizes differ. The example below isolates that arithmetic; the fold-specific predictions are assumed already targeted.
Worked Example: Weighting Unequal Validation Folds
Suppose a hypothetical analysis has two validation folds. In fold 1, three targeted individual contrasts are 0.05, 0.15, and 0.25. Their sum is 0.45. In fold 2, two contrasts are 0.20 and 0.40, with sum 0.60. The estimate across all five people is
Equivalently, the fold means are \(0.45/3=0.150\) and \(0.60/2=0.300\), and weighting them by fold size gives \(\{3(0.150)+2(0.300)\}/5=0.210\). An unweighted average of the two fold means would be 0.225, which gives the smaller fold too much weight and is not the stated estimator.
For continuous outcomes, a common fluctuation is additive: \(Q^*_{-k,\epsilon_k}(A,W)=Q_{-k}(A,W)+\epsilon_k H_{g_{-k}}(A,W)\). If the score is solved by least squares and the denominator is nonzero, the fold-specific solution is
This formula follows by substituting the additive update into the targeting score: the updated score is the numerator minus \(\epsilon_k\) times the denominator, so setting it to zero yields the ratio. Link functions and fluctuation choices must respect the outcome's support; an additive update can be inappropriate for bounded outcomes if it produces predictions outside the allowed range.
Worked Example: An Additive Fluctuation for a Continuous Outcome
In a hypothetical fold, suppose the four clever covariates are \(2,2,-2,-2\), and the corresponding initial-model residuals \(Y_i-Q_{-k}(A_i,W_i)\) are \(2,1,1,0\). Then the numerator is \(2(2)+2(1)-2(1)-2(0)=4\), while the denominator is \(2^2+2^2+(-2)^2+(-2)^2=16\). Therefore \(\widehat\epsilon_k=4/16=0.25\). The updated score is \(4-(0.25)(16)=0\), as required.
For any covariate value in this fold, the treatment-1 prediction increases by \(0.25(2)=0.50\), while the treatment-0 prediction decreases by \(0.25(-2)=-0.50\). The predicted treatment contrast therefore increases by \(0.50-(-0.50)=1.00\) outcome units relative to its initial contrast. Whether this is substantively plausible depends on the outcome scale and the fitted predictions; a large update is a reason to inspect the fit, not an automatic proof of an error.
Assumptions, Diagnostics, and Failure Modes
Cross-validated TMLE relies on the causal identification assumptions and on data and estimation conditions appropriate to the target and inference procedure. In particular, positivity matters for the clever covariate: when estimated treatment probabilities are near zero, inverse probabilities can be large, making the targeting score unstable. Probability truncation can limit extreme values, but changes the fitted mechanism and may introduce bias; it should be prespecified or justified and reported.
Check the analysis fold by fold. Confirm that every observation appears in one validation fold, nuisance predictions are generated without using that observation's outcome for nuisance fitting, and treatment probabilities are finite and in the intended range. Check that each fluctuation converged and that its score residual is close to zero relative to the scale of the terms in the score. Review the size and direction of \(\widehat\epsilon_k\), updated predictions, fold-specific ATEs, and their sensitivity to the fold assignment. Large updates, separation in a logistic fluctuation, predictions at probability boundaries, or unusually different fold estimates can signal weak positivity, sparse treatment patterns, a poor nuisance fit, or an unstable small-fold targeting step.
A fold-specific targeting score may have no finite logistic maximum, for example when the validation outcomes are separated in the direction of the clever covariate. A software convergence message is not enough to establish that targeting succeeded: inspect the actual score residual and predictions. Possible responses include using a justified regularized fluctuation, changing the fold design, or revisiting the nuisance fits and positivity. Such changes must be documented because they can alter the estimator.
How It Differs From Alternatives
| Approach | How it uses the data | Main distinction |
|---|---|---|
| Cross-validated TMLE | Fits nuisance functions outside each validation fold, then targets within the fold. | Produces targeted plug-in predictions with fold-specific nuisance fits and updates. |
| Cross-fitted AIPW | Fits nuisance functions outside each validation fold, then averages an augmented score. | Uses the augmented estimating expression directly rather than obtaining the estimate from targeted plug-in predictions. |
| Non-cross-validated TMLE | Fits nuisance functions and targets using the full sample. | Can be simpler, but flexible nuisance fitting and evaluation use the same observations. |
Cross-validated TMLE is not automatically superior in every finite sample. Fold-specific targeting can be unstable when folds are small, and computational cost increases with repeated nuisance fits. Cross-fitting also cannot repair unmeasured confounding, lack of positivity, a poorly defined estimand, or nuisance fits that fail to learn the relevant functions. The choice between TMLE and AIPW depends on the analysis goals and implementation; both require appropriate identification and estimation conditions. For inference, use a variance procedure justified for the cross-validated estimator rather than assuming that cross-validation alone validates a standard error.
Common Mistake
- Wrong reasoning: “The held-out fold was not used anywhere in fitting, so its outcomes are independent of the whole fold-specific estimator.” The fold outcomes are used to estimate \(\epsilon_k\) in the targeting step.
- Correction: Nuisance predictions are out-of-fold, but targeting deliberately uses validation outcomes. Cross-fitting limits adaptive reuse in nuisance fitting; it does not make the targeted estimator independent of the validation data.
- Wrong reasoning: “The fitted-nuisance version of the efficient influence function must have mean zero.”
- Correction: The efficient influence function displayed above is defined at the true distribution. With arbitrary fitted nuisance functions, the analogous expression is an estimating function and need not have mean zero. The targeting score is a sample equation imposed by the procedure, not a claim that every fitted-nuisance expression is a true efficient influence function.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In a fold-specific logistic fluctuation, which data are used to fit the nuisance functions, and which data estimate the fold's fluctuation parameter?
- Two validation folds have sizes 4 and 6 and fold-specific ATE estimates 0.10 and 0.20. What is the combined estimate?
- Why does a treatment probability close to zero create a diagnostic concern for the clever covariate?
- Does solving the fold-specific targeting score prove that the outcome model is correctly specified? Explain.
- State precisely which nuisance functions appear in the efficient influence function for the ATE.
Solutions. (1) The nuisance functions are fit outside the fold; its validation observations are used to estimate the fluctuation parameter. (2) The weighted estimate is \(\{4(0.10)+6(0.20)\}/10=0.160\). (3) The inverse probability in the treated component can become large, increasing the magnitude and instability of the clever covariate. (4) No. The score equation is an imposed targeting condition; it does not prove correct specification of \(Q\) or \(g\). (5) At the true distribution, the efficient influence function uses \(Q_0\), \(g_0\), and \(\psi_0\), not arbitrary fitted nuisance functions.