TMLE Algorithm for the ATE with Binary Outcomes

From Initial Fits to a Targeted ATE

For a binary outcome, TMLE turns initial predictions into a causal-effect estimate through a defined sequence: fit an outcome model and a propensity model, update the outcome predictions using the observed outcomes, and average the updated contrast between treatment levels. The update is designed to target the estimate of interest; it is not simply a second, unrelated outcome-model fit.

Let $A\in\{0,1\}$ denote treatment, $Y\in\{0,1\}$ the outcome, and $W$ the measured covariates for each of $n$ individuals. Write $Q_0(a,W)=E(Y\mid A=a,W)$ for the true conditional outcome mean and $g_0(W)=P(A=1\mid W)$ for the true propensity score. Their fitted versions are $\hat Q(a,W)$ and $\hat g(W)$. The target is the average treatment effect (ATE), the population average difference between the potential outcomes under treatment and control.

Definition: Under consistency, conditional exchangeability of treatment and the potential outcomes given $W$, and positivity, the ATE is identified by $$ \psi_0=E\{Q_0(1,W)-Q_0(0,W)\}. $$ The binary-outcome TMLE estimator described here replaces the true conditional means with targeted predictions and averages their difference over the observed covariate values.

The assumptions have substantive roles. Consistency says each person’s observed outcome equals the potential outcome under the treatment actually received. Conditional exchangeability says that, given $W$, treatment assignment is independent of the potential outcomes. Positivity requires $0<g_0(W)<1$ for covariate patterns in the target population. These are identification assumptions, not consequences of fitting the models. The basic algorithm also treats the observed individuals as independent draws from the target population; dependence requires appropriate design and inference methods.

The fitted propensity score must be strictly between zero and one where it is used, and the initial outcome predictions must be strictly between zero and one so that their logits are finite. A fitted probability of exactly zero or one cannot be directly entered into the logistic fluctuation. Extreme but nonzero values can also cause unstable updates.

The Logistic Targeting Step

The algorithm uses the clever covariate from binary-treatment TMLE and the logistic fluctuation. For each individual, evaluate the clever covariate at the observed treatment. Fit a one-parameter logistic model with the initial outcome prediction as an offset and this covariate as the sole predictor. There is no additional intercept: the coefficient $\hat\epsilon$ is the fluctuation parameter.

Targeting update: Define $$ \hat H(A,W)=\frac{A}{\hat g(W)}-\frac{1-A}{1-\hat g(W)}. $$ Estimate $\hat\epsilon$ from the logistic fluctuation with offset $\operatorname{logit}\hat Q(A,W)$ and predictor $\hat H(A,W)$. For each treatment level $a$, form $$ \hat Q^*(a,W)=\operatorname{expit}\left\{\operatorname{logit}\hat Q(a,W)+\hat\epsilon\left(\frac{a}{\hat g(W)}-\frac{1-a}{1-\hat g(W)}\right)\right\}, $$ where $\operatorname{expit}(x)=1/(1+\exp(-x))$. The final estimator is $$ \hat\psi_{\mathrm{TMLE}}=\frac{1}{n}\sum_{i=1}^{n}\{\hat Q^*(1,W_i)-\hat Q^*(0,W_i)\}. $$

The fluctuation parameter is fitted using each person’s observed treatment and outcome. Once it is estimated, however, the updated prediction is computed twice for every person: once setting $a=1$, and once setting $a=0$. The estimator averages those two counterfactual predictions’ difference over the empirical covariate distribution. It does not average only over treated people or only over control people.

Why the Fluctuation Solves a Score Equation

The logistic fluctuation is chosen so its fitted coefficient solves a specific score equation. At a candidate value $\epsilon$, write $\hat Q_\epsilon(A,W)$ for the fitted probability under the observed treatment after the logistic update. The score for the one-dimensional logistic coefficient, apart from a constant scaling factor, is

U(\epsilon)=\sum_{i=1}^{n}\hat H(A_i,W_i)\{Y_i-\hat Q_\epsilon(A_i,W_i)\}.

The fitted coefficient makes $U(\hat\epsilon)=0$, subject to numerical tolerance. To see why a finite solution is unique when one exists, differentiate the score. The derivative of a logistic probability with respect to its linear predictor is $\hat Q_\epsilon(1-\hat Q_\epsilon)$, so

\frac{dU(\epsilon)}{d\epsilon} =-\sum_{i=1}^{n}\hat H(A_i,W_i)^2\hat Q_\epsilon(A_i,W_i)\{1-\hat Q_\epsilon(A_i,W_i)\}<0.

The inequality holds when at least one clever covariate is nonzero and all fitted probabilities are strictly between zero and one. Thus the score decreases strictly with $\epsilon$, so it can have at most one finite root. In practice the root is found by fitting the specified logistic model or by numerical root-finding. Failure to converge or a coefficient that grows without settling is a warning, not a reason to report an arbitrary update.

Worked Example: A Four-Person Targeting Step

Consider a hypothetical randomized sample of four individuals. Two received treatment and both had $Y=1$; two received control and had outcomes $Y=1$ and $Y=0$. Suppose the initial fitted outcome probability is $0.50$ under either treatment level for every person, and the fitted propensity is $0.50$. Randomization supports conditional exchangeability here, and the fitted propensity satisfies positivity. The targeting calculation uses these supplied initial predictions.

For treated individuals, $\hat H=1/0.50=2$; for controls, $\hat H=-1/(1-0.50)=-2$. Let $q=\hat Q_\epsilon(1,W)=\operatorname{expit}(2\epsilon)$. Since $\hat Q_\epsilon(0,W)=\operatorname{expit}(-2\epsilon)=1-q$, the score is

U(\epsilon)=2\{(1-q)+(1-q)\}-2\{q+(q-1)\}=6-8q.

Setting the score equal to zero gives $6-8q=0$, so $q=0.75$. Therefore $$ \hat\epsilon=\frac{\operatorname{logit}(0.75)}{2} =\frac{\log(0.75/0.25)}{2} =\frac{1.098612}{2}=0.5493. $$

The targeted predictions are $\hat Q^*(1,W)=\operatorname{expit}(2\times0.5493)=0.7500$ and $\hat Q^*(0,W)=\operatorname{expit}(-2\times0.5493)=0.2500$. Since the four individuals share the same covariate pattern in this illustration, the final estimate is $$ \hat\psi_{\mathrm{TMLE}}=\frac{1}{4}\sum_{i=1}^{4}(0.7500-0.2500)=0.5000. $$

The score check is direct: the treated residuals sum to $0.25+0.25=0.50$, contributing $2(0.50)=1.00$; the control residuals are $0.75$ and $-0.25$, contributing $-2(0.50)=-1.00$. The total is zero. The example demonstrates the arithmetic, but four observations cannot establish that the causal assumptions hold in a real application.

Computing Updated Predictions and the Final Contrast

The same estimated $\hat\epsilon$ is applied to both treatment-level predictions. The sign and size of the clever covariate differ by treatment level, so it is incorrect to update only the prediction corresponding to the treatment actually received and then form a contrast from those observed-treatment predictions.

Worked Example: Updating Both Predictions at One Covariate Pattern

Suppose a hypothetical fitted model gives $\hat g(W_i)=0.50$, initial control and treatment predictions $\hat Q(0,W_i)=0.30$ and $\hat Q(1,W_i)=0.40$, and the fitted fluctuation coefficient is $\hat\epsilon=0.1215$. The initial predictions are interior probabilities and the propensity satisfies positivity for this pattern.

For the control prediction, the clever covariate is $-1/(1-0.50)=-2$. Hence $$ \hat Q^*(0,W_i)=\operatorname{expit}\{\operatorname{logit}(0.30)-2(0.1215)\} =\operatorname{expit}(-0.847298-0.2430) =\operatorname{expit}(-1.090298) =0.2516. $$

For the treated prediction, the clever covariate is $1/0.50=2$. Thus $$ \hat Q^*(1,W_i)=\operatorname{expit}\{\operatorname{logit}(0.40)+2(0.1215)\} =\operatorname{expit}(-0.405465+0.2430) =\operatorname{expit}(-0.162465) =0.4595. $$

The targeted contrast at this covariate pattern is $0.4595-0.2516=0.2079$, or 20.79 percentage points in predicted outcome probability. This is one covariate-specific contrast, not generally the ATE. If all individuals in a sample had these same targeted predictions, their average contrast would also be 0.2079; otherwise each individual’s contrast must be calculated and the contrasts averaged.

Worked Example: Standardizing Targeted Predictions

Consider a hypothetical sample with four individuals in two covariate patterns, $W=L$ and $W=H$, with two individuals in each pattern. Suppose the fitted models yield $\hat g(W)=0.50$ in both patterns and the logistic fluctuation gives $\hat\epsilon=0$. The targeted predictions are therefore the initial predictions. Assume those predictions are $0.20$ under control and $0.40$ under treatment for $W=L$, and $0.50$ under control and $0.70$ under treatment for $W=H$. These interior probabilities and propensity scores meet the numerical requirements for the update; causal interpretation still relies on identification assumptions.

For each of the two individuals with $W=L$, the contrast is $0.40-0.20=0.20$. For each of the two with $W=H$, it is $0.70-0.50=0.20$. The sample standardization is $$ \hat\psi_{\mathrm{TMLE}} =\frac{2(0.20)+2(0.20)}{4} =\frac{0.80}{4} =0.2000. $$

If instead the two $W=H$ contrasts were $0.10$ while the two $W=L$ contrasts remained $0.20$, the estimate would be $$ \frac{2(0.20)+2(0.10)}{4}=0.1500. $$ This illustrates why the final estimate is an average over the observed covariate distribution: the prevalence of each covariate pattern affects the marginal ATE estimate.

Practical Checks, Alternatives, and Limitations

Before interpreting a result, check the computation as well as the assumptions. Confirm that the fluctuation uses the observed-treatment clever covariate and the initial outcome logit as an offset. Confirm that the same fitted coefficient is used to predict both treatment levels. Finally, evaluate the score at the fitted coefficient: a value far from zero indicates a fitting or numerical problem.

Inspect propensity scores. Values close to zero or one make the clever covariate large, so a small number of observations can have substantial influence. Such values may signal limited overlap or model problems; they do not disappear merely because a logistic update converges.
Inspect outcome predictions and convergence. Initial predictions at exactly zero or one make the logit undefined. Extreme predictions, separation, or failure of the fluctuation fit can produce unstable or non-finite updates. Any probability adjustment used to avoid boundary predictions should be documented because it changes the numerical procedure.
Check the final predictions and estimate. Binary-outcome predictions should remain between zero and one, and the ATE contrast should lie between $-1$ and $1$. These range checks are necessary but do not establish correct model fitting or causal identification.
Check the target population. The empirical average standardizes to the observed sample’s covariate distribution. If the intended target population differs, the averaging distribution must be changed accordingly.

TMLE is a plug-in procedure: after targeting, it averages predictions that are valid probabilities. AIPW instead combines outcome predictions with weighted residual corrections, while outcome regression alone averages fitted outcome contrasts without this targeting step. These estimators have different finite-sample behavior; no single numerical result is guaranteed to be best in every data set. TMLE does not repair unmeasured confounding, lack of positivity, or poor nuisance fits. Truncating extreme propensity scores can reduce variability but changes the procedure and may introduce bias.

Common mistake: “The treatment effect for each person is the observed outcome minus the prediction under the treatment they did not receive.” This is not the TMLE calculation: an individual’s two potential outcomes are not both observed. Instead, predict each person’s outcome under both treatment levels using the targeted model, subtract those two predictions, and average the contrasts. The result is an estimate of a population average under the causal assumptions, not a directly observed individual effect.

Fit the initial nuisance models.
Obtain $\hat Q(a,W_i)$ for both treatment levels and $\hat g(W_i)$. Check that the probabilities needed for the update are interior and assess overlap.

Fit the fluctuation.
Use the observed outcome as the response, the initial observed-treatment logit as offset, and $\hat H(A_i,W_i)$ as the sole predictor. Obtain $\hat\epsilon$ and check the score equation.

Update both potential-outcome predictions.
For every $i$, calculate $\hat Q^*(1,W_i)$ and $\hat Q^*(0,W_i)$ using the same $\hat\epsilon$.

Standardize the contrast.
Compute $\hat Q^*(1,W_i)-\hat Q^*(0,W_i)$ for every individual, then average the $n$ contrasts to obtain the ATE estimate.

Check Your Understanding

Answer each question, then compare your work with the solutions.

With $\hat g(W_i)=0.25$, what are the clever covariates for a treated and a control individual?
Why must the targeting coefficient be applied to both treatment-level predictions?
If the average targeted contrast in a sample of 100 is 0.08, what does the point estimate mean in context?
Name two numerical diagnostics to check after fitting the logistic fluctuation.
Why does a successful fluctuation fit not establish that the estimate has a causal interpretation?

Solutions. (1) For treatment, $\hat H=1/0.25=4$. For control, $\hat H=-1/(1-0.25)=-1.333$ to four significant figures. (2) The final estimate compares predicted outcomes under both treatment levels for every observed covariate pattern; updating only the observed-treatment prediction would not form that standardized contrast. (3) The estimated average treatment effect is an 0.08 increase in outcome probability, or 8 percentage points, for the target population represented by the sample, subject to the causal assumptions. (4) Check whether the fitted fluctuation score is approximately zero and whether the model converged to a finite coefficient; also inspect propensity scores for values near zero or one. (5) Model convergence is a computational property. Causal interpretation also requires consistency, conditional exchangeability, positivity, and appropriate sampling and modeling conditions.