TMLE Versus Untargeted Sequential Regression

Why Compare These Estimators?

Both longitudinal TMLE and untargeted sequential regression begin with models for the outcome process and use them to estimate a counterfactual mean. Their difference is not simply that one uses regression and the other does not: both use sequential outcome regressions. The distinction is that TMLE updates those regressions so their residuals satisfy estimating equations linked to the efficient influence function. The untargeted plug-in estimator does not generally impose those equations.

That calibration matters for inference and for the estimator’s response to errors in nuisance fits. It does not guarantee that the outcome models or treatment mechanism are correct, nor does it repair a positivity failure. The comparison below uses a fixed, finite treatment horizon and makes the assumptions explicit.

Setup and the Untargeted Plug-In Estimate

Let \(O=(W,A_0,S_1,A_1,\ldots,S_K,A_K,Y)\) denote one observed record. Here \(W\) is baseline information; \(S_t\) is the measured history just before treatment \(A_t\); and \(Y\) is the terminal outcome. The history \(S_t\) contains \(W\) and the observed treatment and covariate history available before \(A_t\). Let \(g_t(a\mid s)\) specify the intervention’s distribution of treatment at time \(t\), and let \(\pi_t(a\mid s)=P(A_t=a\mid S_t=s)\) be the observed treatment mechanism.

For a fixed regime \(g=(g_0,\ldots,g_K)\), define \(Q_{K+1}=Y\). Working backward, define \(Q_t(s,a)\) as the conditional expectation of the next intervention-averaged outcome, given the current history and treatment. Write \(\bar Q_t(s)\) for its average under the intervention. This recursion defines the target \(\psi=E[\bar Q_0(W)]\).

For continuous treatments, the sum is replaced by an integral. Given fitted regressions \(\hat Q_t\), untargeted sequential regression applies the same backward recursion using those fits and reports \(P_n\hat{\bar Q}_0(W)\), where \(P_n\) denotes the sample average. In other words, it predicts each person’s outcome under the intervention and averages those predictions. It does not, by construction, adjust the fitted regressions to make the efficient-score equations hold.

The Efficient-Score Calibration

For clarity, define the cumulative clever covariate \(H_t\) as the product of intervention probabilities divided by observed treatment probabilities through time \(t\). The longitudinal efficient influence function (EIF) for this intervention mean has one weighted residual for each treatment time, plus a baseline term. Its residual compares the next intervention-averaged prediction with the regression for the treatment actually observed.

At the final time, the term \(\bar Q_{K+1}(S_{K+1})\) means \(Y\). For a deterministic regime, \(g_j(A_j\mid S_j)\) is zero unless the observed treatment agrees with the regime. The EIF has mean zero at the true distribution under the identification assumptions and the usual regularity conditions for the statistical model.

Proof. Form the fitted EIF by replacing \(Q_t\), \(\bar Q_t\), and \(\psi\) in the display above by their fitted versions and \(\hat\psi\). Averaging that expression gives

Each summand in the sum is zero by the stated calibration condition. The last two terms cancel because \(\hat\psi=P_n\hat{\bar Q}_0(W)\). Therefore \(P_n\hat D^*=0\). This is a finite-sample algebraic identity, not a claim that \(\hat D^*\) equals the true EIF. \(\square\)

A standard targeting step obtains these equations by fitting a fluctuation submodel for each sequential regression. For a binary outcome, a common choice is a logistic fluctuation with the current regression as an offset and the relevant clever covariate as a covariate. The score equation for its fluctuation parameter is the weighted residual equation above. The regressions are updated backward so each step uses the already updated next-time prediction. The exact fluctuation depends on the outcome support; it must preserve that support.

Worked Comparisons

What Calibration Does—and Does Not—Establish

Under suitable regularity conditions, empirical EIF calibration supports an asymptotically linear estimator. A precise way to see its role is to state the conditions needed for that conclusion rather than treating the finite-sample score equation as a guarantee of efficiency.

Derivation. Average the assumed representation and use the calibrated equation:

Rearranging gives \(\hat\psi-\psi=P_nD^*+P_nR_n\). The assumed bound on the remainder yields the stated expansion. Since the observations are independent and identically distributed and \(D^*\) has finite variance, the classical central limit theorem applies to \(P_nD^*\). The crucial qualification is the remainder assumption: empirical score calibration alone does not prove it. In practice, controlling the remainder requires adequate nuisance estimation, positivity, and appropriate conditions for the estimation procedure, including cross-fitting where needed. \(\square\)

An untargeted plug-in estimator can still be consistent, and it can sometimes have good large-sample properties. For example, if its sequential regressions consistently estimate the required conditional means under the identification assumptions, the plug-in recursion can estimate the intervention mean. But there is no general finite-sample reason for its weighted residuals to satisfy the EIF equations. Conversely, targeting an incorrect set of nuisance fits does not automatically make the target estimate consistent or efficient. The earlier material on product-rate conditions gives the relevant type of control needed for asymptotic linearity.

Diagnostics, Alternatives, and Limitations

The main practical comparison is between standardization through fitted outcome regressions and score-calibrated standardization. The untargeted estimate is direct and easy to inspect as an average of predicted outcomes. TMLE adds a principled update tied to the EIF and, for bounded outcomes with an appropriate fluctuation, can keep predictions within the outcome range. Neither feature removes the need to examine the data and assumptions. In particular, the update may be unstable when cumulative clever covariates are extreme or the fluctuation approaches a boundary.

Useful checks include confirming that the intervention and observed treatment probabilities use the same treatment coding, verifying the time order in every \(H_t\), checking the range of each updated prediction, and calculating the empirical weighted residual at every time point. For a converged targeting fit, those residual averages should be close to zero (up to numerical tolerance). Also compare initial and updated predictions: a very large change may signal weak positivity, separation, or a poorly behaved fluctuation rather than successful learning from abundant information.

• Mistake: “TMLE is efficient because its fitted EIF averages to zero.” That is not enough. The fitted EIF must also approximate the true EIF and the remaining bias must be negligible at the required rate.
• Mistake: “An untargeted estimate is always biased.” Not so. Correctly estimated sequential regressions can yield a consistent plug-in estimate. The distinction is the absence of automatic efficient-score calibration.
• Mistake: “Targeting fixes positivity.” It does not create observations for treatment-history combinations that are absent or nearly absent. Extreme clever covariates can make the update unstable.
• Mistake: “A bounded estimate is necessarily well calibrated.” Respecting the outcome bounds is useful, but it does not establish consistency, adequate uncertainty quantification, or a small bias.

Longitudinal inverse-probability weighting is another standard alternative, but it estimates the intervention mean through treatment weights rather than through this sequential-regression plug-in construction. The contrast here is narrower: targeting adds EIF-based calibration to sequential regression, while an untargeted plug-in estimate reports the initial recursion. The choice between methods depends on nuisance estimation, support, stability, and the inferential goal—not on a universal ranking.