TMLE Versus Longitudinal Inverse-Probability Weighting

One Regime, Two Estimation Strategies

Longitudinal TMLE and longitudinal inverse-probability weighting (IPW) can target the same quantity: the mean outcome that would be observed if everyone followed a specified treatment regime. Their difference is not the causal question, but how they use the observed data to estimate its answer. IPW reweights observed treatment histories to represent the regime. TMLE combines sequential outcome regressions with treatment weights and targets the resulting plug-in estimate through fluctuation steps.

This comparison assumes the longitudinal setup familiar from the earlier material on sequential nuisance functions and empirical EIF equations. It makes the shared target and the estimating equations explicit, so that the role of each nuisance model is visible.

Definition: Observe independent and identically distributed records with baseline covariates \(L_0\), treatments \(A_0,\ldots,A_K\), time-varying covariates \(L_1,\ldots,L_K\), and outcome \(Y\). Let \(H_t=(L_0,A_0,L_1,\ldots,A_{t-1},L_t)\) be the history available before treatment \(A_t\). A regime \(d=(d_0,\ldots,d_K)\) assigns treatment \(d_t(H_t)\) at each time. The regime-specific mean is \(\psi_d=E[Y^d]\), where \(Y^d\) is the outcome under adherence to \(d\).

For the identification and comparisons below, assume consistency, sequential exchangeability of each treatment given its observed history, and positivity: \(P(A_t=d_t(H_t)\mid H_t)>0\) on histories that can arise under the regime. Assume also that the outcome is integrable. Censoring is omitted here; if present, it requires its own observation mechanism in the nuisance models and weights.

Write \(g_t(a\mid h)=P(A_t=a\mid H_t=h)\). Define the sequential outcome regressions backward by \(Q_{K+1}=Y\) and \(Q_t(h_t)=E[Q_{t+1}(H_{t+1})\mid H_t=h_t,A_t=d_t(h_t)]\). Then \(\psi_d=E[Q_0(L_0)]\). For an observed record, define the cumulative regime weight \(W_t=\prod_{s=0}^{t} I\{A_s=d_s(H_s)\}/g_s\{d_s(H_s)\mid H_s\}\). These definitions distinguish the two nuisance ingredients: \(g_t\) describes treatment assignment, and \(Q_t\) describes future outcomes under the regime.

The Estimating Equations

The basic IPW estimator is the empirical weighted outcome mean \(\hat\psi_{\mathrm{IPW}}=P_n(W_KY)\), where \(P_n\) denotes the sample average. Its estimating equation is \(P_n(W_KY-\psi)=0\). Thus, once estimated treatment probabilities are inserted, the estimator’s direct nuisance dependence is on the treatment models. It does not use a fitted outcome regression to correct the weighted mean.

Longitudinal TMLE begins with estimates of the sequential \(Q_t\) and treatment probabilities. It updates the \(Q_t\) estimates using fluctuations whose covariates involve the corresponding cumulative treatment weights. Its final estimate is a plug-in mean, \(P_n\hat Q_0^*\), based on the targeted regressions. The targeting steps are designed to make the empirical efficient-influence-function equation approximately zero (or exactly zero when the chosen fluctuation and optimization achieve that equation). In the nonparametric model with known treatment rules and no censoring, the influence-function form is

D^*(O)=Q_0(L_0)-\psi_d+ \sum_{t=0}^{K}W_t\{Q_{t+1}(H_{t+1})-Q_t(H_t)\}.

Here each \(Q_t\) is evaluated at the regime treatment for that time, as in its definition. The residual at time \(t\) compares the next-stage regression with the current-stage regression. TMLE uses these residuals, weighted by the probability of following the regime through time \(t\), to target the plug-in estimate. Consequently, it depends on both the treatment mechanism and the sequential outcome regressions.

Key distinction: IPW estimates the regime mean by weighting observed outcomes. TMLE estimates it by targeting sequential outcome regressions, using treatment weights to construct the targeting steps. The treatment probabilities matter to both; the outcome regressions provide an additional route to accuracy and, under suitable conditions, consistency for TMLE.

Why the Weighted Residuals Matter

The following identity explains how the treatment mechanism connects the influence-function equation to the target. It also clarifies why adding the weighted residuals changes the nuisance dependence.

Result: Under consistency, sequential exchangeability, positivity, and integrability, if the treatment probabilities in \(W_t\) are the true probabilities, then for any integrable sequence of functions \(Q_0,\ldots,Q_{K+1}\) of the appropriate histories, with \(Q_{K+1}=Y\), \(E[Q_0+\sum_{t=0}^{K}W_t(Q_{t+1}-Q_t)]=\psi_d\).

Proof. For any integrable function \(f(H_{t+1})\), conditioning on the observed history and treatment shows that weighting by \(I\{A_t=d_t(H_t)\}/g_t\{d_t(H_t)\mid H_t\}\) removes the treatment assignment probability. Applying this argument successively at times \(0,\ldots,t\) gives \(E[W_t f(H_{t+1})]=E_d[f(H_{t+1})]\), where \(E_d\) denotes expectation under the regime distribution. The same argument applies to a function of \(H_t\), so \(E[W_t Q_t(H_t)]=E_d[Q_t(H_t)]\).

It follows that the expectation on the left side of the result is \(E_d[Q_0]+\sum_{t=0}^{K}\{E_d[Q_{t+1}]-E_d[Q_t]\}=E_d[Q_{K+1}]\). Because \(Q_{K+1}=Y\), this is \(E[Y^d]=\psi_d\). The sum telescopes, proving the identity. \(\square\)

There is a second useful case. If the sequential regressions are correct, then for each \(t\), \(E[Q_{t+1}-Q_t\mid H_t,A_t=d_t(H_t)]=0\). Multiplying by the regime indicator and its history-measurable weight gives \(E[W_t(Q_{t+1}-Q_t)]=0\). Also, correctness of \(Q_0\) gives \(E[Q_0]=\psi_d\). Thus the influence-function expression has mean zero even when the treatment probabilities used in its weights are not the true probabilities, provided the relevant weighted terms are well-defined. These population identities describe why TMLE can benefit from either nuisance component; they do not imply that arbitrary finite-sample fits automatically yield a consistent estimate.

Worked Comparisons

Worked Example: A Two-Time IPW Contribution

Consider two treatment times, \(K=1\), and a regime that assigns \(A_0=1\) and \(A_1=0\). For a participant with \(g_0(1\mid H_0)=0.5\), \(g_1(0\mid H_1)=0.25\), and observed treatments \(A_0=1,A_1=0\), the cumulative weight is

W_1=\frac{1}{0.5}\frac{1}{0.25}=2\times4=8.

If this participant has \(Y=1\), their contribution to the numerator of \(P_n(W_1Y)\) is \(8\times1=8\). If the sample contains four participants, the contribution to the IPW mean is \(8/4=2\); other participants’ contributions must still be added. The contribution can exceed one even though the outcome is binary. This is not an arithmetic error: it is a warning that a single record with a large weight can dominate a finite-sample estimate.

Worked Example: Different Finite-Sample Estimates

Suppose there are six observations, three in each of two baseline strata \(X=0\) and \(X=1\). The target regime sets treatment to one. In each stratum use the known treatment probability \(g(1\mid X)=0.5\). The treated outcomes are \(0.2\) in stratum \(X=0\), and \(0.8,0.6\) in stratum \(X=1\); the other three observations are untreated. Let the initial outcome regression predictions be \(Q(0)=0.3\) and \(Q(1)=0.7\).

The IPW estimate is the weighted sum of treated outcomes divided by all six observations:

\hat\psi_{\mathrm{IPW}} =\frac{2(0.2)+2(0.8)+2(0.6)}{6} =\frac{3.2}{6} \approx 0.533.

The corresponding one-time augmented estimating equation uses the plug-in mean plus weighted residuals. The plug-in part is \(\{3(0.3)+3(0.7)\}/6=0.5\). The weighted residual sum is \(2(0.2-0.3)+2(0.8-0.7)+2(0.6-0.7)=-0.2\), so its contribution to the sample mean is \(-0.2/6\). The augmented estimate is therefore \(0.5-0.2/6\approx0.467\). The estimates differ because the observed treatment counts and outcomes do not balance exactly. TMLE uses targeted regressions rather than simply reporting this initial augmented estimate, but the calculation shows how outcome predictions enter the estimating equation.

Worked Example: A Correct Outcome Model Can Offset a Wrong Weight Model

Let \(X\) be binary with \(P(X=0)=P(X=1)=0.5\). Treatment is randomized with true probability \(g(1\mid X)=0.5\) in both strata. Suppose the true treated-outcome means are \(Q(0)=0.2\) and \(Q(1)=0.8\). The regime mean is \(\psi_d=0.5(0.2)+0.5(0.8)=0.5\).

Now suppose the fitted treatment probabilities are wrong: \(g^*(1\mid0)=0.25\) and \(g^*(1\mid1)=0.75\). The expectation of the resulting IPW score is

E\left[\frac{I(A=1)Y}{g^*(1\mid X)}\right] =\frac{1}{2}\frac{0.5}{0.25}(0.2) +\frac{1}{2}\frac{0.5}{0.75}(0.8) =0.2+0.266\overline{6} =0.466\overline{6}.

The IPW expectation is not the target. In contrast, if \(Q(X)\) is correct, the augmented score has expectation \(E[Q(X)+I(A=1)\{Y-Q(X)\}/g^*(1\mid X)]\). Within each stratum, \(E[Y-Q(X)\mid A=1,X]=0\), so the weighted residual has expectation zero even with \(g^*\). The score expectation is then \(E[Q(X)]=0.5\), the target. This demonstrates the possible protection from a correct outcome regression; it does not make a badly estimated treatment model harmless for every TMLE algorithm or every target.

Assumptions, Diagnostics, and Trade-Offs

The causal assumptions are shared: neither estimator repairs a failure of consistency, sequential exchangeability, or positivity. Their statistical vulnerabilities differ. IPW’s identifying formula requires correct treatment probabilities and can be highly variable when regime adherence is rare. TMLE also uses treatment probabilities, so extreme weights can destabilize its targeting steps. Its outcome regressions can improve precision and provide a consistency route when they are correct, but that benefit depends on adequate sequential fits and a targeting procedure that successfully uses them.

Feature	Longitudinal IPW	Longitudinal TMLE
Main estimating construction	Weighted observed outcome \(P_n(W_KY)\)	Targeted plug-in mean \(P_n\hat Q_0^*\)
Direct nuisance dependence	Treatment probabilities	Treatment probabilities and sequential outcome regressions
Finite-sample behavior	Can be dominated by a few large cumulative weights	Can be more stable and respects outcome bounds with an appropriate fluctuation, but targeting can still be unstable
Typical failure signal	Extreme weights, low effective sample size, sensitivity to weight truncation	Extreme clever covariates, failed or boundary updates, poor sequential predictions

In practice, inspect the distribution of the cumulative weights, the largest weights, the proportion of observations following the regime, and whether fitted probabilities approach zero. For TMLE, also inspect the sequential predictions and whether the fluctuation converged without boundary estimates. A bounded plug-in estimate is not evidence that positivity holds or that the nuisance models are adequate. Conversely, an IPW estimate outside the outcome range can signal instability, but a range-respecting estimate is not proof of validity.

Weight truncation can reduce variance but changes the effective weighting scheme and may introduce bias; it is not a general cure for lack of positivity. TMLE is not uniformly superior: poor outcome regressions, failed targeting, or weak overlap can still yield unreliable results. When both methods are reported, differences between them are useful diagnostics of model sensitivity, not a test that identifies which estimate is correct. The earlier material on product-rate conditions and cross-fitting is relevant when making large-sample claims with flexible nuisance fits.

Common Mistake

Pitfall: “TMLE uses weights too, so it is just IPW with a different label.” The reasoning is incorrect because the weights play different roles. IPW multiplies the observed outcome to form its estimate; TMLE uses weights in targeting sequential outcome regressions and reports a targeted plug-in mean. The correction is to compare both the estimating equation and the nuisance dependence, not merely to note that both contain treatment probabilities.

Check Your Understanding

Answer each question, then compare with the solutions below.

For a one-time regime, an observation follows the regime and has estimated treatment probability \(0.2\). What is its IPW weight?
Which nuisance models enter the basic IPW estimator directly, and which enter longitudinal TMLE?
If the treatment probabilities are correct, what makes the weighted residual terms telescope to the regime mean?
Why can a correct outcome regression help when the treatment probabilities are misspecified?
Name two diagnostics that can reveal instability in a longitudinal analysis.

Solutions. (1) The weight is \(1/0.2=5\). (2) Basic IPW directly uses treatment probabilities; TMLE uses treatment probabilities and sequential outcome regressions. (3) Correct weighting changes each weighted expectation into an expectation under the regime distribution; summing the successive differences cancels intermediate terms and leaves \(E_d[Y]\). (4) With a correct regression, the conditional residual \(Y-Q(X)\) has mean zero among observations following the regime, so its weighted expectation is zero and the plug-in expectation remains the target. (5) Examples include very large cumulative weights, few observations following the regime, probabilities near zero, a low effective sample size, and failed or boundary targeting updates.