Efficient Influence Function for a Regime-Specific Mean

Why the Influence Function Has Several Residuals

A longitudinal regime-specific mean depends on a sequence of covariate distributions and outcome regressions, not just on the final outcome model. Its efficient influence function therefore has a baseline term and one residual term for each treatment decision. The residuals measure successive prediction errors; their weights account for the chance of observing the regime’s treatment choices and remaining uncensored long enough to see each error.

The derivation below uses a fixed regime $d=(d_0,\ldots,d_K)$, where $d_t(H_t)$ specifies the treatment at decision time $t$ as a function of the history available then. It gives the influence function under a nonparametric model for the observed-data distribution, with treatment and censoring treated as nuisance mechanisms.

Notation and Assumptions

Let $W$ denote baseline covariates. At decision time $t=0,\ldots,K$, let $H_t=(W,L_1,\ldots,L_t,A_0,\ldots,A_{t-1})$ be the history immediately before treatment $A_t$; in particular, $H_0=W$. Let $R_0=1$, and let $R_{t+1}=1$ indicate that an individual uncensored at time $t$ remains observed through the next covariate measurement, or through the outcome measurement when $t=K$. Thus censoring is monotone, and treatment at $t$ is observed when $R_t=1$. The terminal outcome is $Y$, observed when $R_{K+1}=1$.

For an individual following the regime through time $t$, define the treatment probability $g_t(d_t(H_t)\mid H_t)=P(A_t=d_t(H_t)\mid H_t,R_t=1)$. Define the conditional probability of remaining observed by $q_{t+1}(H_t,d_t(H_t))=P(R_{t+1}=1\mid R_t=1,H_t,A_t=d_t(H_t))$. These probabilities are evaluated at regime-consistent histories. Write $I_t=1$ when the observed treatments from time 0 through $t$ agree with the regime, and $I_t=0$ otherwise.

Definition: Let $Q_{K+1}=Y$. For $t=K,\ldots,0$, define the sequential outcome regressions recursively by $$ Q_t(H_t)=E\!\left[Q_{t+1}\mid H_t,\ A_t=d_t(H_t),\ R_{t+1}=1\right]. $$ For $t<K$, $Q_{t+1}$ is evaluated at the next history, which includes $L_{t+1}$; for $t=K$, it is the terminal outcome. The regime-specific mean is $\psi=E[Q_0(W)]$.

The recursion is interpreted under consistency, sequential exchangeability of treatment and censoring given the recorded history, and positivity. In particular, each regime action must have positive conditional probability, and each required censoring probability must be positive. For the result below, assume probabilities are bounded away from zero and that the weighted residual terms have finite second moments. Also assume a regular, differentiable-in-quadratic-mean submodel for the observed-data likelihood, so its score decomposition and the corresponding differentiation under conditional expectations are valid.

The Efficient Influence Function

Theorem: Under the stated assumptions, the efficient influence function for $\psi$ in the nonparametric observed-data model is $$ D^*(O)=Q_0(W)-\psi+ \sum_{t=0}^{K} \frac{I_tR_{t+1}} {\displaystyle\prod_{s=0}^{t}g_s(d_s(H_s)\mid H_s)\,q_{s+1}(H_s,d_s(H_s))} \left(Q_{t+1}-Q_t(H_t)\right). $$ Each residual is evaluated only for individuals whose treatment history follows the regime and who remain observed through that residual’s time point.

The first term varies across baseline covariate values. The $t$th residual term corrects the prediction $Q_t$ using the next observed covariate or outcome. Its numerator indicator selects individuals who actually followed the regime so far and were not censored; its denominator reverses the probabilities of those events. The product in that denominator is cumulative: a residual late in follow-up requires every preceding treatment and censoring event needed to observe it.

Derivation of the Representation

A useful first check is that the displayed function has mean zero. The baseline term has expectation $E[Q_0]-\psi=0$. For a fixed $t$, condition on a regime-consistent history and the regime treatment at $t$. Conditional on remaining observed, $Q_t$ is the expectation of $Q_{t+1}$. The residual therefore has conditional mean zero. Multiplying by the indicators and inverse probabilities changes the distribution over histories, but not that conditional mean. Hence $E[D^*(O)]=0$.

To verify the influence-function property, factor the observed-data likelihood into the baseline distribution, treatment mechanisms, censoring mechanisms, covariate-transition distributions among those observed, and the terminal outcome distribution among those observed. A regular score is the sum of the corresponding conditional scores. The parameter $\psi$ does not depend on the treatment or censoring mechanisms, so its derivative in those score directions is zero. Its derivative in the baseline and transition-distribution directions is represented by the baseline and residual terms, respectively.

Baseline distribution.
Changing the distribution of $W$ changes $E[Q_0(W)]$. Its derivative in a mean-zero baseline score direction $S_W$ is $E[(Q_0(W)-\psi)S_W]$.

Covariate transitions and outcome.
At time $t<K$, changing the conditional distribution of $L_{t+1}$ changes the next regression $Q_{t+1}$; at $t=K$, changing the outcome distribution changes $Y$. The respective conditional derivative is the score-weighted residual $Q_{t+1}-Q_t$.

Recover the regime distribution.
In the observed data, the required treatment and censoring events occur with probabilities $g_s$ and $q_{s+1}$. Multiplying by their inverse cumulative product converts the observed-data expectation of each residual contribution to its expectation along the regime.

Check nuisance directions.
Each residual is conditionally centered, so its inner product with treatment and censoring scores is zero. The baseline term is centered as well. Thus the displayed function represents the parameter derivative for every score direction.

More explicitly, for the $t$th transition or outcome score, iterated expectation gives the inner product with the weighted residual as the derivative of the corresponding conditional mean in the regime-specific recursion. The inverse factors cancel the conditional probabilities of receiving the regime treatment and remaining observed at times $0,\ldots,t$; the remaining expectation is precisely over the regime histories. Scores for treatment and censoring contribute zero because the parameter is invariant to those mechanisms and the residuals are centered conditional on the histories at which those scores are defined. The score decomposition therefore yields $E[D^*(O)S]=\dot\psi$ for every regular score $S$. In the nonparametric observed-data model, this mean-zero gradient in the tangent space is the canonical gradient, hence the efficient influence function. $\square$

Worked Examples

Worked Example: One Decision with Censoring

Suppose there is one decision ($K=0$), no baseline variation, and the regime is $A_0=1$. Among those uncensored, let $P(A_0=1)=0.4$, and let the probability of remaining observed after $A_0=1$ be $q_1=0.8$. Suppose $Q_0=E[Y\mid A_0=1,R_1=1]=0.6$. The parameter is $\psi=0.6$.

For an observed individual with $A_0=1$, $R_1=1$, and $Y=1$, the baseline term is $Q_0-\psi=0$. The residual is $Y-Q_0=1-0.6=0.4$, and the cumulative probability in the denominator is $0.4(0.8)=0.32$. Thus $$ D^*(O)=0+\frac{0.4}{0.32}=1.25. $$ For an otherwise comparable individual with $Y=0$, the residual is $0-0.6=-0.6$, so $D^*(O)=-0.6/0.32=-1.875$. These are individual influence-function values, not estimates of the mean by themselves. Their weighted residual contribution averages to zero in expectation because $Q_0$ is the conditional mean among regime-treated individuals who remain observed.

Worked Example: Two Decisions Without Censoring

Let $K=1$, with no baseline variation and no censoring. The regime sets $A_0=1$ and $A_1=0$. Among those with $A_0=1$, let $P(L_1=1)=0.5$. Suppose $Q_1(L_1=0)=0.2$ and $Q_1(L_1=1)=0.8$. Then $$ Q_0=0.5(0.2)+0.5(0.8)=0.5,\qquad \psi=0.5. $$ Let both regime treatment probabilities be $0.5$. For a person with $A_0=1,L_1=1,A_1=0,Y=1$, both regime indicators equal one.

At $t=0$, the denominator is $g_0=0.5$, and the residual is $Q_1-Q_0=0.8-0.5=0.3$. The contribution is $0.3/0.5=0.6$. At $t=1$, the denominator is $g_0g_1=0.5(0.5)=0.25$, and the residual is $Y-Q_1=1-0.8=0.2$. This contribution is $0.2/0.25=0.8$. The baseline term is $0.5-0.5=0$, so $$ D^*(O)=0+0.6+0.8=1.4. $$ The first residual corrects the prediction across the next covariate; the second corrects the terminal outcome prediction. The second correction uses both treatment probabilities because both decisions must match the regime.

Worked Example: A Small Censoring Probability

Return to a single decision and suppose $Q_0=0.4$, $\psi=0.4$, $g_0=0.8$, and $q_1=0.25$. For an uncensored regime-treated person with $Y=1$, the residual is $1-0.4=0.6$, the denominator is $0.8(0.25)=0.2$, and the weighted residual is $0.6/0.2=3$. Thus $D^*=0+3=3$.

If instead $q_1=0.05$ while the other values stay fixed, the denominator becomes $0.8(0.05)=0.04$, and the contribution becomes $0.6/0.04=15$. The arithmetic is correct, but the larger contribution signals a potential variance problem: a small number of observed individuals represent many who were censored. The calculation does not establish that the estimate is stable or that the censoring model is adequate.

Diagnostics, Alternatives, and Limitations

The influence function helps diagnose where an analysis may be fragile. Inspect the distributions of the cumulative weights, especially their upper tail, and tabulate the regime treatment and censoring probabilities across relevant histories. Large weights, sparse histories, or probabilities near zero suggest weak practical positivity. Also check the sequential outcome regressions: a good fit for the final outcome alone does not ensure that every $Q_t$ is well estimated. These checks can reveal instability, but they cannot verify exchangeability or establish that all confounders were measured.

The inverse-probability-weighted estimating equation is a main alternative: it emphasizes weighting and can be less reliant on outcome-regression extrapolation, but its variability can be severe when cumulative probabilities are small. The longitudinal g-formula emphasizes sequential outcome regression; it can be more stable with strong prediction, but depends heavily on those regressions. The efficient influence function displays both ingredients in one representation and provides the efficiency benchmark under the stated model. It does not remove the need for identification assumptions, adequate nuisance estimation, or positivity.

A frequent error is to include only the terminal residual $Y-Q_K$, weighted by the probability of the full treatment and censoring history. That omits the residuals for earlier covariate transitions, whose distributions also change under the regime. The correction is to include one residual at each time, $Q_{t+1}-Q_t$, with the cumulative probability through that time. Another error is to treat the baseline term as an additional inverse-weighted residual; it is simply $Q_0(W)-\psi$.

Check Your Understanding

Answer each question using the definitions and influence-function expression above.

With no censoring and two decisions, what treatment probabilities appear in the denominator of the residual at $t=0$, and at $t=1$?
In the one-decision example, calculate the influence-function value for a regime-treated, uncensored person with $Y=0$.
Why can a small censoring probability make an influence-function-based estimate unstable?
Which component of the influence function reflects variation in baseline covariates?

Solutions. (1) At $t=0$, the denominator contains $g_0$; at $t=1$, it contains $g_0g_1$, because both decisions must follow the regime. (2) The baseline term is zero and the residual is $-0.6$; dividing by $0.4(0.8)=0.32$ gives $-1.875$. (3) Its inverse increases the weight on the relatively few individuals remaining observed, which can produce large contributions and high variance. (4) The baseline component is $Q_0(W)-\psi$.