Regime-Specific Risks at Fixed Follow-Up Times

What a Fixed-Time Risk Measures

A regime-specific risk answers a focused question: under a specified treatment regime, what proportion of the target population would experience the event by a chosen follow-up time? The time point matters. A risk by the end of year one and a risk by the end of year three are different estimands, even when they are calculated from the same longitudinal model.

Here, \(A_k\) is treatment at time \(k\), \(L_k\) is the covariate history available for treatment decisions, and \(D_{k+1}\) indicates an event in the interval after time \(k\). Let \(g\) denote a treatment regime that specifies treatment at each time as a function of the history. The event is absorbing: after it occurs, the individual is no longer at risk of a first event. For a fixed horizon \(t\), define \(Y_t^g\) to indicate whether the event has occurred by that horizon under regime \(g\).

The parametric g-formula estimates this risk by simulating covariate and event histories under \(g\), then averaging the resulting event indicators. The earlier material on the longitudinal g-formula gives the identification argument and the factorization that support this calculation. In brief, interpretation as a causal risk relies on consistency, sequential exchangeability, positivity for the histories generated under the regime, and correctly specified conditional models (or suitable alternatives that yield valid estimates). Simulation itself does not establish these assumptions.

From a Regime to an Estimated Risk

Suppose the follow-up consists of \(K\) intervals, and the requested horizon \(t\) is no later than \(K\). A fitted parametric g-formula uses models for covariate evolution and the event process. For each target individual, it preserves the observed baseline covariates, assigns treatment according to \(g\), and generates later covariates and event outcomes in their time order. It records \(Y_t^g\), not merely the event status in the final interval.

Let \(X_i\) be the baseline covariates for target individual \(i\), and let \(\hat\theta\) collect the fitted model parameters. With \(M\) simulated histories per individual, the direct estimator is

The simulated event indicator is one if any event occurred by \(t\), and zero otherwise. Thus a participant who has an event in an early interval contributes one at every later horizon in that same simulated history. Participants without an event by \(t\) contribute zero. The averaging is over both target individuals and simulated histories; it does not mean that each simulated history is a new study participant.

Why Averaging the Indicators Targets Risk

The key operation is marginalization over the histories that would arise under the regime. Let \(H_t^g\) denote the covariate history generated under \(g\) up to the fixed horizon. Conditional on that history, the event indicator has a conditional probability \(P(Y_t^g=1\mid H_t^g)\). The law of total expectation gives the following identity.

To verify the identity, write the indicator as a random variable and condition on the history. Since \(Y_t^g\) is binary, its conditional expectation is its conditional probability of equaling one. Therefore

The outer expectation averages over the regime-specific distribution of histories. Averaging event probabilities over the observed histories instead would generally use the wrong distribution when treatment changes covariate evolution. This is why the simulation must generate covariates under the regime, rather than simply predict outcomes for each person's observed covariates.

For a finite target sample, a second useful fact concerns simulation error. Hold the baseline covariates and fitted parameters fixed, and suppose the \(M\) histories generated for each individual are independent draws from the fitted model under \(g\). Then the simulation average converges to the fitted model's empirical-target risk as \(M\) grows. In detail, define \(q_i=P_{\hat\theta}(Y_{it}^g=1\mid X_i)\). Each \(Y_{it}^{g,(m)}\) is a Bernoulli variable with mean \(q_i\), so the strong law of large numbers gives \(M^{-1}\sum_m Y_{it}^{g,(m)}\xrightarrow{a.s.}q_i\). Since \(n\) is fixed and finite, averaging those \(n\) convergent quantities gives

This is a statement about Monte Carlo approximation conditional on the fitted models and target sample. It does not establish that the fitted empirical-target risk equals the true population causal risk.

Worked Calculations

Conditional Probabilities, Diagnostics, and Limitations

When available, the conditional-probability version of the estimator replaces a final Bernoulli event draw with its fitted probability, while still averaging over simulated covariate histories. For a generated history \(H_{it}^{g,(m)}\), let \(q_{it}^{(m)}=P_{\hat\theta}(Y_t^g=1\mid H_{it}^{g,(m)},X_i)\). Averaging \(q_{it}^{(m)}\) instead of \(Y_{it}^{g,(m)}\) has the same conditional expectation, by the identity proved above, but removes variation from the final event draw. It does not remove variation from simulated covariate histories, model fitting, or a mismatch between the model and the data-generating process.

A fixed-time risk is often easier to interpret than a collection of interval-specific conditional probabilities: it answers directly how many people out of a target population would have experienced an event by a stated time under the regime. But the estimate can be unreliable if the fitted event or covariate models are poor, if the regime leads to histories with little data support, or if the simulation implementation fails to follow the intended time ordering.

Useful checks include confirming that the requested horizon is measured from the correct time origin; checking that event indicators are absorbing after a first event; and comparing simulated covariate and event frequencies with predictions from the fitted models. Inspect whether simulated histories under the regime contain implausible values or combinations. A simulated estimate outside \([0,1]\) is impossible for a risk and signals a calculation error. Large changes across independent simulation runs may indicate that \(M\) is too small, while stability across runs does not rule out model misspecification or violations of causal assumptions.

When analytic integration is manageable, integrating the fitted conditional probabilities can avoid some simulation noise. Forward simulation is useful when the longitudinal history is too complex for convenient integration, and it naturally accommodates treatment-confounder feedback. Neither approach automatically fixes incorrect models or unsupported regimes. Also, a risk at one fixed time is only one summary: it does not display how risk evolves at other times. The interpretation must name the horizon and regime, and should not be generalized to unreported follow-up times without further calculation.

Common Mistake

Wrong reasoning: “The event probability in the last interval is 0.20, so the risk by the end of follow-up is 20%.” This ignores events that occurred in earlier intervals and can also ignore that only those still event-free can experience a first event later.

Correction: Construct the indicator of any event by the specified horizon for each simulated history, then average those indicators. In the two-interval example above, the second-interval conditional probability was 0.29, but the risk by the end of interval two was 0.361 because 10% had already experienced an event in the first interval.