Subgroup-Specific Marginal Effects with LTMLE

Subgroup Targets and Why the Denominator Matters

A marginal effect averages over a specified population. Sometimes the scientific question concerns a baseline-defined part of that population: for example, the effect of a treatment regime among patients whose baseline risk score exceeds a prespecified threshold. The target is still marginal over covariates within that subgroup; it is not an effect conditional on every covariate value, nor an effect for a hypothetical covariate distribution different from the subgroup observed in the study.

Let $W$ denote baseline covariates, and let $B$ be a fixed, measurable subset of their support. Write $I_B=I(W\in B)$ and $p_B=P(W\in B)$, with $p_B>0$. There are $K$ treatment times. At time $t$, $H_t$ is the observed history just before treatment $A_t$, and $g=(g_1,\ldots,g_K)$ is a specified deterministic treatment regime, with $g_t(H_t)$ the treatment assigned by the regime. Let $Y^g$ be the outcome under that regime.

Definition: The subgroup-specific regime mean is $$ \psi_{B,g}=E[Y^g\mid W\in B] =\frac{\theta_{B,g}}{p_B}, \qquad \theta_{B,g}=E[I_B Y^g]. $$ For two regimes $g_1$ and $g_0$, the subgroup treatment contrast is $$ \Delta_B=\psi_{B,g_1}-\psi_{B,g_0}. $$ The distribution being averaged over is the baseline covariate distribution conditional on $W\in B$.

Identification requires consistency, sequential exchangeability, and positivity. Consistency means that a person whose observed treatment history follows $g$ has observed outcome $Y=Y^g$. Sequential exchangeability means that, at each time, conditional on the observed history just before treatment, treatment assignment is independent of the relevant future counterfactual outcomes under the regimes of interest. Positivity requires $P(A_t=g_t(H_t)\mid H_t)>0$ wherever histories relevant to the target occur with positive probability among subgroup members. The subgroup itself must have positive probability. These assumptions identify the regime mean from the observed longitudinal distribution; they do not guarantee that it can be estimated precisely if the probabilities are very small.

Longitudinal Regression and the Subgroup Influence Function

For a compact expression, let $H_{t+1}$ be the history after treatment $A_t$ and the next covariate information, with $H_{K+1}$ including the history needed to predict the final outcome. Define $Q_{K+1}(H_{K+1})=E[Y\mid H_{K+1}]$. Recursively, for $t=K,\ldots,1$, define $$ Q_t(H_t)=E[Q_{t+1}(H_{t+1})\mid H_t,A_t=g_t(H_t)]. $$ Then $Q_1(W)=E[Y^g\mid W]$, and $\psi_{B,g}=E[Q_1(W)\mid W\in B]$. Write $$ C_t=\prod_{s=1}^t\frac{I\{A_s=g_s(H_s)\}}{\pi_s}, \qquad \pi_s=P(A_s=g_s(H_s)\mid H_s). $$ The longitudinal residual contribution is $$ R_g=\sum_{t=1}^K C_t\{Q_{t+1}(H_{t+1})-Q_t(H_t)\} +C_K\{Y-Q_{K+1}(H_{K+1})\}. $$

The subgroup mean’s influence function is the full-population regime influence function reweighted by subgroup membership and centred at the subgroup mean. The centring and the subgroup normalisation are both essential.

Theorem: Under the identification assumptions above, and regularity conditions for the longitudinal model, the efficient influence function for $\psi_{B,g}$ in the nonparametric observed-data model is $$ D_{B,g}(O)=\frac{I_B}{p_B}\{Q_1(W)-\psi_{B,g}+R_g\}. $$ Equivalently, the influence function for the numerator $\theta_{B,g}$ is $$ D_{\theta}(O)=I_B\{Q_1(W)+R_g\}-\theta_{B,g}, $$ and that for $p_B$ is $D_p(O)=I_B-p_B$.

Derivation. The longitudinal regime-mean influence function has baseline term $Q_1(W)-E[Q_1(W)]$ and the sequential residual terms $R_g$. For the numerator, the target is $E[I_B Y^g]$, so the corresponding baseline term is multiplied by $I_B$, and the whole expression must be centred by $\theta_{B,g}$. This gives $D_\theta=I_B\{Q_1+R_g\}-\theta_{B,g}$. In particular, it is not $I_B\{Q_1-\psi_{B,g}+R_g\}$: that expression omits the fluctuation in subgroup membership represented by the numerator’s centring.

The ratio rule for a parameter $\theta/p$ gives influence function $(D_\theta-\psi D_p)/p$, where $\psi=\theta/p$. Substituting the two influence functions yields

\begin{aligned} \frac{D_\theta-\psi_{B,g}D_p}{p_B} &=\frac{I_B(Q_1+R_g)-\theta_{B,g} -\psi_{B,g}(I_B-p_B)}{p_B}\\ &=\frac{I_B(Q_1-\psi_{B,g}+R_g)}{p_B}, \end{aligned}

because $\theta_{B,g}=p_B\psi_{B,g}$. This establishes the stated subgroup influence function. Its expectation is zero: the baseline term has conditional mean zero within $B$, and each sequential residual has conditional mean zero under the appropriate regime history. The factor $I_B/p_B$ makes explicit that only subgroup members contribute, with the subgroup probability providing the normalisation. $\square$

Targeting and Estimating a Contrast

For each regime, longitudinal TMLE estimates the sequential outcome regressions, updates them using treatment-mechanism information, and averages the resulting baseline prediction over the observed subgroup. With fitted quantities denoted by hats, the plug-in estimate is

\hat\psi_{B,g} =\frac{\sum_{i=1}^n I(W_i\in B)\hat Q_{1,g}(W_i)} {\sum_{i=1}^n I(W_i\in B)}.

A subgroup-targeted update can use the subgroup-restricted clever covariates $I(W\in B)C_t$, or the analysis can be carried out in the subgroup sample with the corresponding conditional target. The former keeps the full data available for nuisance fitting while targeting the subgroup parameter; in either implementation, the final average is over subgroup members. A standard full-population update is not automatically a targeted update for every subgroup mean. The update and the final plug-in average must correspond to the same target.

For two regimes, estimate each mean and subtract: $\hat\Delta_B=\hat\psi_{B,g_1}-\hat\psi_{B,g_0}$. Its influence function is $D_{B,g_1}-D_{B,g_0}$, provided both targets are evaluated in the same observed population. An estimated standard error can be formed from the sample variance of the estimated contrast influence values divided by $n$, followed by a square root. This inference requires the usual conditions for asymptotic linearity, including suitable nuisance-estimation rates and appropriate handling of cross-fitting when flexible learners are used.

Worked Example: Averaging a Regime Prediction Within a Subgroup

Suppose four of five study participants satisfy $W\in B$. Under a specified regime, the updated baseline predictions for those four are $0.70,\ 0.50,\ 0.80,\ 0.60$. The fifth participant is outside $B$, with prediction $0.40$. Calculate the subgroup estimate.

The subgroup size is $n_B=4$, so the empirical subgroup mean is $$ \hat\psi_{B,g}=\frac{0.70+0.50+0.80+0.60}{4} =\frac{2.60}{4}=0.65. $$

The prediction $0.40$ is excluded: including it would average over the full sample, not over the empirical distribution conditional on $W\in B$. The target is a mean, not the subgroup’s unnormalised sum $2.60$.

Worked Example: Checking the Numerator Ratio Correction

Suppose $p_B=0.40$, and the regime prediction is constant at $Q_1(W)=0.60$, with no residual contribution, so $\psi_{B,g}=0.60$ and $\theta_{B,g}=0.40(0.60)=0.24$. Verify the baseline influence-function calculation for an observation with $I_B=1$, and then for one with $I_B=0$.

For a subgroup member, $D_\theta=1(0.60)-0.24=0.36$ and $D_p=1-0.40=0.60$. The ratio influence value is $$ \frac{D_\theta-\psi_{B,g}D_p}{p_B} =\frac{0.36-0.60(0.60)}{0.40}=0. $$ For a nonmember, $D_\theta=0-0.24=-0.24$ and $D_p=0-0.40=-0.40$, giving $$ \frac{-0.24-0.60(-0.40)}{0.40}=0. $$ This is correct: if every baseline prediction is the same and there are no residual contributions, the subgroup mean has no first-order variation.

If one incorrectly used $I_B(Q_1-\psi_{B,g})$ as the numerator influence function, it would be zero for both types of observation. Applying the ratio rule would then produce nonzero values for subgroup membership changes, even though the actual numerator $p_B\psi_{B,g}$ changes with $p_B$. The correct numerator influence function includes the centring term $-\theta_{B,g}$.

Worked Example: A Subgroup Treatment Contrast

Among the same baseline-defined subgroup, suppose the updated predictions under regime $g_1$ for three members are $0.72,\ 0.64,\ 0.80$, and under regime $g_0$ they are $0.52,\ 0.60,\ 0.55$. Calculate both subgroup means and the contrast.

The first mean is $$ \hat\psi_{B,g_1}=\frac{0.72+0.64+0.80}{3} =\frac{2.16}{3}=0.72. $$ The second is $$ \hat\psi_{B,g_0}=\frac{0.52+0.60+0.55}{3} =\frac{1.67}{3}\approx0.5567. $$ Therefore, $$ \hat\Delta_B=0.72-0.5567\approx0.1633. $$ The estimated mean outcome under $g_1$ is about 0.163 higher than under $g_0$ on the outcome’s scale. Both means use the same three subgroup members; averaging the predictions over different covariate distributions would not estimate this contrast.

Diagnostics, Alternatives, and Limitations

Check the analysis at the level of the target, the sequential fits, and the final estimate. Confirm that subgroup membership is defined using baseline information and that the denominator is the number of subgroup members. Inspect estimated probabilities of following each regime at every time among relevant histories, as well as the resulting cumulative clever covariates. Very large values or few observations matching a regime signal practical positivity problems; targeted updates can become unstable even when formal positivity holds. Check boundary predictions, convergence of the targeting steps, and whether the empirical efficient-influence-function equations are close to zero. For a contrast, also verify that both regime estimates use the same subgroup and compatible nuisance fits.

A common alternative is to fit an outcome regression with treatment-by-subgroup interactions and report an interaction-derived effect. That approach can impose a particular model for heterogeneity and does not by itself ensure that each regime mean is averaged over the subgroup’s empirical covariate distribution. Longitudinal inverse-probability weighting estimates a related target by weighting observed histories, but can be highly variable when cumulative treatment probabilities are small. LTMLE combines sequential outcome regression and treatment-mechanism information in a targeted plug-in estimate; it does not eliminate the need for positivity or adequate nuisance estimation.

Subgroup estimates can be imprecise when $p_B$ is small, because only a fraction of the sample contributes directly to the target. If $B$ is selected after examining outcomes or treatment effects, the fixed-subgroup influence function and ordinary standard errors do not account for that selection. A data-adaptive subgroup requires methods that address the selection procedure. Finally, a baseline subgroup effect does not describe a subgroup defined by post-treatment variables, for which conditioning can change the causal target and introduce bias.

Key takeaway: Define the subgroup mean as a ratio, target the conditional mean within the subgroup, and retain the numerator’s centring by $\theta_{B,g}$ when deriving influence functions. The ratio correction is what produces the properly normalised subgroup influence function.

Check Your Understanding

Answer each question, then compare with the solutions below.

If $p_B=0.25$ and $\theta_{B,g}=0.15$, what is $\psi_{B,g}$?
Why is $I_B\{Q_1-\psi_{B,g}+R_g\}$ not the numerator influence function?
Three subgroup predictions under one regime are $0.30,\ 0.60,\ 0.90$. What is the plug-in subgroup mean?
If a subgroup is defined using a covariate measured after treatment begins, what key feature of this tutorial’s target no longer holds?
Name one diagnostic for practical positivity problems in subgroup LTMLE.

Solutions. (1) $\psi_{B,g}=\theta_{B,g}/p_B=0.15/0.25=0.60$. (2) It omits the numerator’s centring term $-\theta_{B,g}$; the correct expression is $I_B(Q_1+R_g)-\theta_{B,g}$, which accounts for variation in subgroup membership. (3) $(0.30+0.60+0.90)/3=0.60$. (4) The subgroup is no longer defined solely by baseline covariates, so it is not the baseline-defined subgroup target considered here. (5) Inspect cumulative clever covariates or estimated probabilities of following the regime; very large cumulative weights indicate instability.

Subgroup-Specific Marginal Effects with LTMLE

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