TMLE Targeting Equations and the Efficient Influence Function

Why the Targeting Equation Matters

A targeted maximum likelihood estimate (TMLE) is a substitution estimate: it updates an estimated outcome regression and then averages the resulting predicted treatment contrasts. The targeting step is designed to align that plug-in estimate with the efficient influence function (EIF) for the average treatment effect (ATE). This alignment is more than a computational detail: it supplies an estimating equation that supports a first-order efficiency result under suitable conditions.

The aim here is to connect three objects: the logistic fluctuation score, the empirical EIF equation, and the ATE estimator’s first-order expansion. The focus is the ATE with a binary treatment and a binary outcome.

Definition: Let \(O=(W,A,Y)\) denote an observed unit, where \(W\) is a vector of pretreatment covariates, \(A\in\{0,1\}\) is treatment, and \(Y\in\{0,1\}\) is the outcome. Write \(Q_0(a,w)=E(Y\mid A=a,W=w)\) for the true outcome regression and \(g_0(w)=P(A=1\mid W=w)\) for the true propensity score. Under consistency, conditional exchangeability, and positivity, the ATE is \(\psi_0=E\{Q_0(1,W)-Q_0(0,W)\}\).

Consistency means the observed outcome equals the potential outcome under the received treatment. Conditional exchangeability means treatment is independent of the potential outcomes given \(W\). Positivity requires \(0<g_0(W)<1\) almost surely. These assumptions identify the ATE from the observed-data distribution; they do not ensure that a particular fitted model is correct.

For working estimates \(\hat Q\) and \(\hat g\), define the clever covariate \(\hat H(A,W)=A/\hat g(W)-(1-A)/\{1-\hat g(W)\}\). The EIF evaluated at working functions \(Q,g\) and a candidate ATE value \(\psi\) is \(D^*(Q,g,\psi)(O)=H_g(A,W)\{Y-Q(A,W)\}+Q(1,W)-Q(0,W)-\psi\). At the true functions and true ATE, this EIF has mean zero and its variance is the semiparametric efficiency bound.

From the Logistic Fluctuation to the Score Equation

Start with an initial outcome-regression estimate \(\hat Q^0(a,w)\), whose predictions lie strictly between zero and one. A one-dimensional logistic fluctuation adds the clever covariate as an offset:

\operatorname{logit}\{\hat Q^\epsilon(A,W)\} = \operatorname{logit}\{\hat Q^0(A,W)\} + \epsilon\hat H(A,W).

Here \(\epsilon\) is estimated by maximizing the Bernoulli log-likelihood for the observed outcomes, with the initial prediction as an offset. Differentiating that log-likelihood with respect to \(\epsilon\) gives the score \(\sum_{i=1}^n \hat H(A_i,W_i)\{Y_i-\hat Q^\epsilon(A_i,W_i)\}\). Thus, if a finite maximum is attained at an interior value, its score is zero. The targeted regression is \(\hat Q^*=\hat Q^{\hat\epsilon}\). In practice, a numerical solution may leave a small residual rather than exactly zero.

Key equation: When the fluctuation score is solved, the targeted regression satisfies the empirical targeting equation \(\mathbb P_n[\hat H(A,W)\{Y-\hat Q^*(A,W)\}]=0\), where \(\mathbb P_n f=n^{-1}\sum_{i=1}^n f(O_i)\). The resulting substitution estimate is \(\hat\psi=\mathbb P_n\{\hat Q^*(1,W)-\hat Q^*(0,W)\}\).

Worked Example: Solving a Logistic Targeting Equation

Consider four hypothetical observations. All have \(\hat g(W)=0.5\) and initial predictions \(\hat Q^0(1,W)=\hat Q^0(0,W)=0.5\). Two observations are treated, with outcomes 1 and 0; two are controls, both with outcome 0. For treated observations \(\hat H=2\), and for controls \(\hat H=-2\).

The fluctuation gives \(\hat Q^\epsilon(1,W)=\operatorname{expit}(2\epsilon)\) and \(\hat Q^\epsilon(0,W)=\operatorname{expit}(-2\epsilon)\). Let \(q=\operatorname{expit}(2\epsilon)\), so \(\operatorname{expit}(-2\epsilon)=1-q\). The unnormalised score is

2\{(1-q)+(0-q)\}+(-2)\{(0-(1-q))+(0-(1-q))\} =6-8q.

Setting this equal to zero gives \(q=6/8=0.75\), and hence \(\hat\epsilon=\operatorname{logit}(0.75)/2=0.5493\). The updated predictions are 0.75 under treatment and 0.25 under control. The treated score contributions sum to \(2(0.25)+2(-0.75)=-1\); the control contributions sum to \((-2)(-0.25)+(-2)(-0.25)=1\). The total is zero, as required.

Every observation has the same predicted contrast, so the plug-in ATE is \(0.75-0.25=0.50\), in probability units. Its empirical EIF is centered: the residual part averages to zero by the targeting equation, and the contrast minus the estimate averages to zero by the definition of the plug-in estimate.

Why the Empirical EIF Is Centered

The targeting equation makes the residual component of the empirical EIF zero. The plug-in definition makes its contrast component zero as well. More explicitly, use the targeted estimates and define \(\hat D(O)=\hat H(A,W)\{Y-\hat Q^*(A,W)\}+\hat Q^*(1,W)-\hat Q^*(0,W)-\hat\psi\). Then

\mathbb P_n\hat D = \underbrace{\mathbb P_n[\hat H\{Y-\hat Q^*(A,W)\}]}_{0} + \underbrace{\mathbb P_n[\hat Q^*(1,W)-\hat Q^*(0,W)]-\hat\psi}_{0} =0.

This is an exact algebraic statement when the score equation is solved exactly. It does not say that the estimated EIF equals the true EIF observation by observation, nor by itself prove consistency or efficiency. Those require conditions on the nuisance estimates and the data-generating process.

Worked Example: Checking EIF Centering with Unequal Clever Covariates

Suppose a hypothetical targeted fit for three observations has the following observed treatment, propensity estimate, outcome, and fitted outcome under the received treatment. The corresponding fitted contrasts are also shown.

Observation	\(A\)	\(\hat g(W)\)	\(Y\)	\(\hat Q^*(A,W)\)	\(\hat Q^(1,W)-\hat Q^(0,W)\)
1	1	0.25	1	0.75	0.55
2	0	0.50	0	0.25	0.45
3	1	0.50	0	0.75	0.45

The clever covariates are \(4,-2,2\), respectively. The three weighted residuals are \(4(1-0.75)=1\), \((-2)(0-0.25)=0.5\), and \(2(0-0.75)=-1.5\). Their sum is zero, so the empirical targeting equation holds. The plug-in estimate is \(\hat\psi=(0.55+0.45+0.45)/3=1.45/3=0.4833\).

The three empirical EIF values are \(1+0.55-0.4833=1.0667\), \(0.5+0.45-0.4833=0.4667\), and \(-1.5+0.45-0.4833=-1.5333\), up to rounding. Their sum is zero. The unequal clever covariates change the individual residual contributions, but not the centering argument.

Second-Order Remainder and First-Order Efficiency

Centering alone is not the efficiency result. The key additional fact is that, under suitable nuisance-estimation conditions, the difference between the targeted estimate and the empirical mean of the true EIF is smaller than the usual \(n^{-1/2}\) sampling scale. The product structure of the remainder explains why errors in both nuisance functions can be tolerated.

For the following identity, use the targeted estimates \(\hat Q^*,\hat g\), and suppose the targeting equation is exact. Write \(d_a(W)=\hat Q^*(a,W)-Q_0(a,W)\) for \(a=0,1\). The true conditional means of the residuals are \(E\{Y-\hat Q^*(1,W)\mid A=1,W\}=-d_1(W)\) and \(E\{Y-\hat Q^*(0,W)\mid A=0,W\}=-d_0(W)\). Weighting these residuals by the estimated clever covariate gives

E[\hat H\{Y-\hat Q^*(A,W)\}\mid W] = -\frac{g_0(W)}{\hat g(W)}d_1(W) + \frac{1-g_0(W)}{1-\hat g(W)}d_0(W).

Add the conditional plug-in contrast error \(d_1(W)-d_0(W)\). Combining like terms yields the second-order remainder

R_2(\hat Q^*,\hat g) = E\left[ \{\hat g(W)-g_0(W)\} \left\{ \frac{\hat Q^*(1,W)-Q_0(1,W)}{\hat g(W)} + \frac{\hat Q^*(0,W)-Q_0(0,W)}{1-\hat g(W)} \right\} \right].

This derivation shows the cancellation: if \(\hat g=g_0\), the first-order outcome-regression errors cancel in the sum; if \(\hat Q^*=Q_0\), the remainder is zero as well. The displayed expression also makes clear why estimated propensities near zero or one can magnify the remainder.

Worked Example: Calculating the Remainder

In a hypothetical single covariate stratum, suppose \(g_0=0.40\), \(\hat g=0.50\), \(\hat Q^*(1)-Q_0(1)=0.10\), and \(\hat Q^*(0)-Q_0(0)=-0.05\). The remainder contribution for this stratum is

(0.50-0.40) \left\{ \frac{0.10}{0.50} + \frac{-0.05}{1-0.50} \right\} = 0.10(0.20-0.10) = 0.010.

The outcome errors are measured on the probability scale, and the contribution is on the ATE probability scale. If either nuisance error were zero, this product contribution would be zero. Across multiple covariate values, the remainder is the population average of the corresponding contributions.

To see the first-order expansion, let \(D_0=D^*(Q_0,g_0,\psi_0)\). Since \(\mathbb P_n\hat D=0\), take its population expectation and add and subtract the empirical fluctuation. The preceding conditional calculation gives

\hat\psi-\psi_0 = (\mathbb P_n-P_0)D_0 + R_2(\hat Q^*,\hat g) + (\mathbb P_n-P_0)(\hat D-D_0),

where \(P_0 f=E\{f(O)\}\). This identity uses the plug-in definition of \(\hat\psi\) and the targeting equation; it is not an assumption that the fitted nuisance functions are correct.

A standard sufficient route to first-order efficiency is that the remainder is \(o_p(n^{-1/2})\), the empirical fluctuation of \(\hat D-D_0\) is \(o_p(n^{-1/2})\), and the usual regularity conditions hold. In particular, positivity must keep the denominators controlled, and the nuisance estimates must converge sufficiently well. With cross-fitting, suitable product-rate conditions can control the remainder while reducing restrictions from overfitting; without cross-fitting, additional empirical-process conditions may be needed. Under these conditions,

\hat\psi-\psi_0 = (\mathbb P_n-P_0)D_0+o_p(n^{-1/2}).

This is the asymptotic linear representation with the EIF. In a regular model where \(D_0\) is the efficient influence function, it establishes first-order efficiency: the estimator has the leading sampling behavior associated with the semiparametric efficiency bound. It does not make finite-sample performance automatic, and it does not replace checking positivity, nuisance fit, or the conditions supporting the expansion.

Common Mistake: Treating Centering as a Proof of Efficiency

A common mistake is to argue that because the targeted empirical EIF averages to zero, the TMLE must be efficient. The zero is a property of the fitted estimating equation and plug-in construction. Efficiency additionally requires that the estimated EIF be sufficiently close, in the relevant empirical sense, to the true EIF, and that the product remainder be negligible at the root-\(n\) scale. If propensity estimates approach zero or one, or nuisance errors converge too slowly, those requirements can fail even when the numerical targeting equation is solved.

Key takeaway: The logistic fluctuation solves the empirical clever-covariate residual equation. Combined with the plug-in definition, this centers the empirical fitted EIF. First-order efficiency follows only when the second-order remainder and the empirical fluctuation from replacing the true EIF with its fitted version are both negligible at the root-\(n\) scale.

Check Your Understanding

Use the equations and assumptions above to answer these questions.

Why does differentiating the logistic fluctuation likelihood produce the empirical targeting equation?
Which two terms make the empirical fitted EIF average to zero after targeting?
In the remainder expression, what happens if the propensity estimate equals the true propensity score?
Why can propensities near zero or one undermine the efficiency argument even if the score equation is solved?
What additional requirements, beyond empirical centering, support a first-order efficient ATE estimate?