Translating Fitted Hazards into Absolute Treatment Effects

From a Hazard Model to an Absolute Effect

A fitted hazard model does not directly report how many people survive under each treatment strategy, or how much survival time a strategy adds over a clinically relevant horizon. Those quantities require a translation step: obtain the hazard predicted by the marginal structural model (MSM) for each regime, reconstruct survival under that regime, and then compare the resulting curves. This tutorial focuses on that translation, not on fitting the weighted hazard model.

Let \(g\) index a treatment regime, let \(T^g\) be the event time that would be observed under that regime, and define \(S_g(t)=P(T^g>t)\). In applications, \(g=1\) and \(g=0\) may label two regimes of interest. The fitted MSM supplies a predicted hazard for each regime; the survival and absolute-effect measures below are functions of those predictions. Their causal interpretation requires that the regimes, treatment assignment, and outcome are well defined; consistency, sequential exchangeability, and positivity hold; and the MSM is correctly specified for the contrasts being estimated. Follow-up must also be complete through the horizon, or censoring must be handled by a valid method. A hazard fit alone does not establish these conditions.

The sign convention matters. A positive \(\Delta_S(t)\) means higher survival under regime 1, whereas a negative event-risk difference means fewer events under regime 1. A positive RMST contrast means more expected event-free time accumulated before \(\tau\) under regime 1. Always report the regimes, horizon, and contrast direction.

Reconstructing Survival from the Fitted Hazard

For a discrete-time fit, let \(q_{gk}\) be the predicted probability of the event in interval \(k\), conditional on being event-free at its start, under regime \(g\). If \(S_g(k)\) denotes survival through interval \(k\), then \(S_g(0)=1\) and each interval multiplies the previous survival by the probability of not having the event. Thus,

For continuous time, let \(h_g(t)\) be the predicted hazard and \(\Lambda_g(t)=\int_0^t h_g(u)\,du\) its cumulative hazard. The survival function is

These are two versions of the same survival reconstruction principle, as established in survival analysis. In a Cox MSM with regime-specific linear predictor \(\eta_g\), \(h_g(t)=h_0(t)\exp(\eta_g)\), so \(\Lambda_g(t)=\exp(\eta_g)\Lambda_0(t)\) and \(S_g(t)=\exp\{-\exp(\eta_g)\Lambda_0(t)\}\). The fitted baseline cumulative hazard and the regime-specific predictor must both be used; a hazard ratio alone is not enough to determine an absolute survival probability.

Justification. In discrete time, \(0\leq 1-q_{gk}\leq 1\); multiplying factors in this range cannot increase a nonnegative product or make it exceed 1. In continuous time, nonnegative integrable \(h_g\) implies \(\Lambda_g(t)\geq0\), and \(\Lambda_g(t)\) is non-decreasing. Therefore \(0<\exp\{-\Lambda_g(t)\}\leq1\) and this exponential is non-increasing in \(t\). These properties provide basic checks on reconstructed curves. \(\square\)

RMST and Why the Horizon Matters

RMST converts an entire survival curve up to \(\tau\) into expected time alive and event-free during that period. Its formula follows by writing \(\min(T^g,\tau)=\int_0^\tau I(T^g>t)\,dt\). Since the integrand is nonnegative and bounded by 1, Tonelli's theorem allows expectation and integration to be interchanged:

This identity holds for nonnegative event times and finite \(\tau\); no assumption that the mean event time is finite is needed. For a discrete-time outcome with unit intervals, the corresponding restricted mean through \(K\) intervals is \(\sum_{k=0}^{K-1}S_g(k)\), where \(S_g(k)\) is survival at the start of interval \(k+1\). With unequal intervals of lengths \(d_k\), the grouped-time analogue \(\sum_k d_k S_g(t_{k-1})\) treats survival as constant within each interval. It is therefore an approximation or a convention for grouped event times, not an exact continuous-time RMST unless that within-interval survival behavior is justified.

Interpretation, Diagnostics, and Limitations

Absolute effects answer questions that a hazard ratio does not: what is the survival probability at a specified time, how large is the event-risk difference, or how much event-free time is gained before a specified horizon? These effects depend on the underlying survival level and generally vary with time. Even under proportional hazards, a constant hazard ratio can correspond to a small absolute difference when events are rare and a larger difference when baseline risk is higher. RMST adds a horizon-specific summary, but changing \(\tau\) changes the estimand.

Useful checks follow directly from the reconstruction. Predicted discrete hazards must lie in \([0,1]\); a continuous hazard should be nonnegative; survival must begin at 1 and never increase. Check that the baseline cumulative hazard and regime-specific predictors correspond to the desired treatment regimes and time scale. Compare the estimated curves with observed follow-up support: where few weighted individuals remain at risk, the tail may be unstable even if the fitted curve looks smooth. Large or highly variable weights, sparse events, and extrapolation beyond supported follow-up can make absolute contrasts unreliable.

A common mistake is to multiply a Cox hazard ratio by a baseline event risk, or to interpret \(1-\mathrm{HR}\) as a risk reduction. Neither operation gives a regime-specific survival probability. The correction is to obtain each regime's predicted hazard (including the baseline hazard), integrate or accumulate it to form survival, and then take the desired contrast. Another error is reporting an event-risk difference with the survival-difference sign; writing both definitions explicitly prevents this.

The main alternatives answer related but distinct questions. A hazard ratio summarizes relative instantaneous event rates under the model. A fixed-time risk difference compares event probabilities at one time. A survival difference gives the same contrast with reversed sign, while an RMST contrast averages survival over a chosen horizon and has a time-unit interpretation. Direct weighted Kaplan–Meier estimates can describe survival without the same parametric hazard specification, but may not encode the desired regime contrast in the same way. Whichever approach is used, uncertainty should reflect the full estimation procedure: resampling or an appropriate variance calculation should propagate uncertainty into the survival or RMST contrast, rather than reporting only uncertainty for the hazard coefficient.

These translations do not repair a misspecified MSM or unsupported treatment regimes. They also do not make a hazard contrast collapsible into a risk contrast. The absolute effects inherit the assumptions and limitations of the fitted hazards, and their interpretation is restricted to the population and follow-up horizon those hazards represent.