Sequential Double Robustness Conditions

Why Robustness Is Sequential

In a longitudinal study, information is collected over several decision points. At each point, treatment depends on the observed history, and later outcomes may be missing because of dropout. A regime-specific mean is therefore supported by a chain of treatment and censoring probabilities, alongside a chain of conditional outcome regressions.

The robustness question is not simply whether “the outcome model” or “the weight model” is correct. In the recursively targeted longitudinal estimator, each stage contributes its own bias term. At a given stage, that term disappears if the corresponding outcome regression is correct, or if the treatment-and-censoring probability at that stage is correct. This stage-by-stage condition permits some outcome regressions to be wrong, provided the mechanisms at those stages are correct.

Setup and Conditions

Consider decision times $t=0,\ldots,K$. Let $H_t$ denote the measured history available just before the decision at time $t$, and let $d_t(H_t)$ be the treatment specified by the regime. Let $B_t=1$ indicate both that the person receives the regime-specified treatment at time $t$ and that the person remains uncensored through the next measurement. Histories after that interval are denoted $H_{t+1}$. The terminal outcome is $Y$, with $H_{K+1}$ containing the information needed to define it.

Write $g_t(H_t)$ for the conditional probability of $B_t=1$ given the observed history among people still under follow-up. It is the product of the probability of receiving $d_t(H_t)$ and the probability of remaining uncensored, with the latter conditional on the relevant treatment and history. The cumulative probability through stage $t$ is $G_t=\prod_{j=0}^{t}g_j(H_j)$.

Definition: The true sequential outcome regressions are defined backwards by $Q_{K+1}=Y$ and $$ Q_t(H_t)=E[Q_{t+1}(H_{t+1})\mid H_t,\ B_t=1],\qquad t=K,\ldots,0. $$ The target is $\psi=E[Q_0(H_0)]$. For working regressions $q_t$, set $q_{K+1}=Y$. For working mechanism probabilities $\hat g_t$, set $\hat G_t=\prod_{j=0}^{t}\hat g_j(H_j)$.

This representation presumes consistency, no interference, and sequential exchangeability: conditional on the measured history at each decision, treatment assignment and censoring contain no further information about the relevant future potential outcomes. It also requires sequential positivity: along histories occurring under the regime, the probabilities of the specified treatment and continued observation are positive. For the weighted expressions below, assume $g_t$ and $\hat g_t$ are positive on the relevant support, and the displayed expectations exist. The treatment and censoring mechanisms need not be modeled separately for the algebra: correctness of their product $g_t$ suffices. Correctly specifying both components is a standard sufficient way to obtain that product.

The Sequential Bias Identity

The estimating-equation representation associated with longitudinal targeting has the following population form: $$ \Psi(q,\hat g)=E[q_0(H_0)]+\sum_{t=0}^{K}E\left[\frac{I(B_0=\cdots=B_t=1)}{\hat G_t}\{q_{t+1}(H_{t+1})-q_t(H_t)\}\right]. $$ This expression uses weighted residuals at successive stages and a plug-in starting value $q_0$. Longitudinal TMLE targets the regressions through fluctuations tied to these residuals and reports a plug-in estimate. The population identity below gives the robustness condition; in a finite sample, the targeted plug-in value and this estimating-equation expression need not be algebraically identical.

Let $E_d$ denote expectation under the regime’s intervention distribution, and define $r_t=G_t/\hat G_t$, with $r_{-1}=1$. For any integrable function $f(H_{t+1})$, the change-of-measure identity is $$ E\left[\frac{I(B_0=\cdots=B_t=1)}{\hat G_t}f(H_{t+1})\right] =E_d[r_t f(H_{t+1})]. $$ Indeed, the observed probability of following and remaining uncensored through stage $t$ is $G_t$; dividing by $\hat G_t$ leaves the likelihood ratio $G_t/\hat G_t$ relative to the regime distribution. This identity requires the positivity and integrability conditions stated above.

Theorem: Suppose the sequential identification and positivity assumptions hold, and the expectations in the estimating equation are finite. Put $\delta_t=q_t-Q_t$ for $t=0,\ldots,K$, and $\delta_{K+1}=0$. Then $$ \Psi(q,\hat g)-\psi =E_d[(1-r_0)\delta_0] +\sum_{t=1}^{K}E_d[(r_{t-1}-r_t)\delta_t]. $$ Consequently, $\Psi(q,\hat g)=\psi$ if, at every stage $t$, either $q_t=Q_t$ almost surely under the regime distribution or $\hat g_t=g_t$ almost surely on the relevant history support.

Proof. Apply the change-of-measure identity to each weighted residual. The population expression becomes $$ \Psi(q,\hat g)=E_d[q_0]+\sum_{t=0}^{K}E_d[r_t(q_{t+1}-q_t)]. $$ Because $Q_t(H_t)=E_d[Q_{t+1}(H_{t+1})\mid H_t]$, and $r_t$ is a function of the history through $H_t$, each term $E_d[r_t(Q_{t+1}-Q_t)]$ is zero. Also $E_d[Q_0]=\psi$. Subtract this true-regression version of the expression, and substitute $q_t=Q_t+\delta_t$: $$ \Psi(q,\hat g)-\psi =E_d[\delta_0]+\sum_{t=0}^{K}E_d[r_t(\delta_{t+1}-\delta_t)]. $$ Collect the coefficient of each $\delta_t$. The coefficient for $\delta_0$ is $1-r_0$; for $\delta_t$, $1\le t\le K$, it is $r_{t-1}-r_t$; and the terminal term vanishes because $\delta_{K+1}=0$. This proves the identity. If $q_t=Q_t$, its stage’s term is zero. If $\hat g_t=g_t$, then $r_t=r_{t-1}$, so its term is zero. Thus every term vanishes under the stated stagewise condition. $\square$

For $t\geq1$, $r_t=r_{t-1}g_t/\hat g_t$. Therefore a correct mechanism at stage $t$ cancels the coefficient of that stage’s regression error, even when earlier mechanism models are wrong. Conversely, a correct regression at stage $t$ cancels that stage’s contribution even if its mechanism model is wrong. This is the specific sense in which the robustness is sequential.

Worked Examples

Worked Example: Correct Mechanisms at Every Stage

Suppose there are two decision stages, $t=0,1$, and both working mechanism probabilities are correct. Then $r_0=r_1=1$. Let the working regressions have errors $\delta_0=0.10$ and $\delta_1=-0.20$ on the regime support. The bias identity gives $$ (1-r_0)\delta_0+(r_0-r_1)\delta_1 =(1-1)(0.10)+(1-1)(-0.20)=0. $$ The working regressions are both wrong, but their errors do not bias the population estimating-equation target under these correct mechanisms. This conclusion requires the positivity and identification conditions; it does not imply that the estimate is precise.

Worked Example: Different Correct Components at Different Stages

Again take two stages. Suppose the mechanism is correct at stage 0, so $r_0=1$. At stage 1, let $g_1=0.40$ and $\hat g_1=0.20$, giving $r_1=r_0(0.40/0.20)=2$. Let the stage-0 regression error be $\delta_0=0.30$, while the stage-1 regression is correct, so $\delta_1=0$. The two bias contributions are $$ (1-r_0)\delta_0+(r_0-r_1)\delta_1 =(1-1)(0.30)+(1-2)(0)=0. $$ The first contribution vanishes because stage 0’s mechanism is correct; the second vanishes because stage 1’s regression is correct. Correctness can differ by stage rather than requiring one entire nuisance-model family to be correct.

Worked Example: A Stage Where Both Components Are Wrong

With one decision stage, suppose $g_0=0.50$ but $\hat g_0=0.25$, so $r_0=0.50/0.25=2$. Suppose the true regression mean is $Q_0=0.40$ and the working value is $q_0=0.60$, so $\delta_0=0.20$. The bias is $$ (1-r_0)\delta_0=(1-2)(0.20)=-0.20. $$ For a direct check, the population estimating-equation value is $0.60+(0.50/0.25)(0.40-0.60)=0.20$, which is $0.20$ below the target $0.40$. The stage has both a regression error and a mechanism error, so the stagewise condition provides no protection.

Using the Condition in Longitudinal TMLE

The theorem describes a population robustness condition, not a guarantee that every numerical TMLE algorithm will achieve it automatically. To apply it to fitted nuisance functions, their limits must satisfy the stated correctness pattern, the targeting steps must be successfully carried out, and the relevant empirical averages must behave like their population counterparts. For a consistency claim with estimated nuisance functions, one typically requires convergence to the stated limits and suitable control of estimation error; merely fitting a model or solving a sample score equation does not establish those conditions.

In particular, if both the treatment and censoring probabilities are correctly specified at a stage, their product $g_t$ is correct there, so that stage’s regression error is removed from the bias identity. If only one component is correct, the product need not be correct: an incorrect censoring probability, for example, may offset a correct treatment probability. The algebraic condition concerns the combined probability, not the separate labels attached to its factors.

The condition also does not repair a violation of sequential exchangeability, a failure of positivity, or an outcome-regression error at a stage where the mechanism is misspecified. Nor does it promise low variance. Very small estimated probabilities can produce unstable cumulative weights even when the probability models are correctly specified in principle.

Diagnostics, Alternatives, and Common Mistakes

Diagnostics should be organized by stage. Examine the fitted treatment probabilities and censoring probabilities on regime-relevant histories, their product, and the resulting cumulative probabilities. Extremely small $\hat g_t$ or $\hat G_t$ signals potential instability. Check that the recursive regressions use the intended histories and that each targeting step was executed. These checks can reveal implementation or support problems, but observed data cannot prove that an untestable mechanism or regression model is exactly correct.

Approach	Robustness feature	Important limitation
Outcome-regression plug-in	Uses the sequential regressions directly	Does not generally correct misspecified regressions with weighting
Inverse-probability weighting	Uses treatment and censoring mechanisms	Can be unstable with small cumulative probabilities
Longitudinal TMLE	Targets regressions using mechanism-based clever covariates; admits stagewise robustness conditions	Still relies on identification, positivity, appropriate nuisance limits, and successful targeting

Common mistake: “Double robust” means that the whole outcome-regression sequence must be correct, or the whole treatment-and-censoring sequence must be correct. That is a valid broad sufficient condition, but it misses the sequential result. The bias identity shows the more refined condition: at each stage, either that stage’s regression is correct or that stage’s combined mechanism probability is correct. The reverse mistake is to assume any single correct model anywhere protects every other stage. It does not; the condition must cover every stage’s bias term.

Key takeaway: For longitudinal TMLE, inspect robustness stage by stage. A regression error at time $t$ is neutralized when the treatment-and-censoring probability $g_t$ is correct; if both are wrong at that stage, the bias identity leaves a contribution that may not vanish.

Check Your Understanding

Answer each question, then check the brief solutions below.

At a stage $t\geq1$, what condition makes the coefficient $r_{t-1}-r_t$ equal zero?
With one decision, $g_0=0.60$, $\hat g_0=0.30$, and $\delta_0=-0.10$, calculate the bias contribution.
Suppose the treatment probability is correct at a stage but the censoring probability is wrong. Is the combined mechanism $g_t$ necessarily correct?
Does the stagewise condition establish robustness to a violation of sequential exchangeability? Explain.

Solutions. (1) It is zero when $r_t=r_{t-1}$; since $r_t=r_{t-1}g_t/\hat g_t$, a correct stage-$t$ combined mechanism, $\hat g_t=g_t$, is sufficient. (2) Here $r_0=0.60/0.30=2$, so the contribution is $(1-2)(-0.10)=0.10$. (3) No. The product of the treatment and censoring probabilities can be wrong if one factor is wrong. (4) No. The robustness identity assumes identification conditions, including sequential exchangeability; it concerns nuisance-model errors under those assumptions.