Contrasting Regime-Specific Survival Curves

What a Survival Contrast Measures

A fitted survival curve describes the probability of remaining event-free through a given time under a treatment regime. A contrast between two such curves answers a different question from either curve alone: at a chosen follow-up time, how much does estimated event-free survival differ between the regimes? The answer depends on the time and on the order in which the regimes are subtracted or divided. State both whenever reporting a contrast.

Let \(g\) and \(h\) denote two treatment regimes, and let \(S_g(t)\) and \(S_h(t)\) be their survival probabilities through time \(t\) in the target population. In a fitted weighted Cox marginal structural model, write the regime-specific hazard as \(\lambda^g(t)=\lambda_0(t)\exp\{\eta(g;\beta)\}\), where \(\lambda_0(t)\) is the common baseline hazard and \(\eta(g;\beta)\) is the model’s linear predictor for regime \(g\). Define \(q_g=\exp\{\eta(g;\beta)\}\). The fitted cumulative hazard and survival curve are

Here \(\widehat{\Lambda}_0(t)\) is the weighted Breslow estimate of the baseline cumulative hazard. The curves are plug-in predictions from the fitted model; this tutorial concerns contrasts between their point estimates at specified times.

A difference is an absolute contrast, often reported in percentage points. A ratio is a relative contrast between survival probabilities. Neither should be reported without the follow-up time and regime order. In particular, a positive survival difference corresponds to a negative difference in cumulative event risk: since \(F_g(t)=1-S_g(t)\), \(F_g(t)-F_h(t)=-\Delta_{g,h}(t)\).

Deriving the Contrasts from a Weighted Cox Fit

The calculations follow directly from the fitted model. For clarity, let \(\widehat{\Lambda}_0(t)\) be abbreviated as \(\widehat{\Lambda}_0\) at a fixed time, and let \(\hat q_g=\exp\{\eta(g;\hat\beta)\}\). The fitted regime-specific survivals are \(\exp(-\widehat{\Lambda}_0\hat q_g)\) and \(\exp(-\widehat{\Lambda}_0\hat q_h)\). Therefore the fitted pointwise contrasts are

Derivation. By the definition of the ratio and the fitted survival formula,

This identity shows why a constant hazard multiplier ratio does not imply a constant survival ratio. The latter also depends on how much cumulative baseline hazard has accrued by time \(t\). At \(t=0\), when the cumulative hazard is zero, both fitted survival curves equal one and both contrasts equal one for the ratio and zero for the difference. \(\square\)

The direction of the survival contrast is also determined by the fitted multipliers. This gives a useful check on calculations.

Proof. The exponential function is strictly increasing, so \(x\mapsto\exp(-x)\) is strictly decreasing. Since \(\widehat{\Lambda}_0(t)\geq0\) and \(\hat q_g<\hat q_h\), multiplication gives \(\widehat{\Lambda}_0(t)\hat q_g\leq\widehat{\Lambda}_0(t)\hat q_h\). Applying the decreasing function yields \(\exp\{-\widehat{\Lambda}_0(t)\hat q_g\}\geq\exp\{-\widehat{\Lambda}_0(t)\hat q_h\}\). If the cumulative hazard is positive, both inequalities are strict. Subtracting the survivals gives the stated sign of the difference. \(\square\)

Worked Pointwise Comparisons

Assumptions, Diagnostics, and Alternatives

These calculations are arithmetic once fitted curves are available, but a causal interpretation requires more than arithmetic. In the usual longitudinal-treatment setting, consistency and sequential exchangeability are needed to identify regime-specific outcomes from observed data; positivity is needed so the relevant treatment choices have support. Correctly specified treatment weights (and, when used, censoring weights), an adequate marginal structural Cox model for the regimes being compared, and appropriate handling of censoring are also required. If these conditions fail, a precisely calculated contrast can still be a poor estimate of the intended causal quantity.

Check the inputs before interpreting the result. The estimated baseline cumulative hazard should be nonnegative and nondecreasing, and the fitted survival probabilities should lie between zero and one and be nonincreasing over time. For a model with a common baseline hazard, the curve ordering should agree with the ordering of the fitted multipliers; a reversed order suggests an implementation, coding, or reporting error. Also inspect whether the chosen time is within meaningful follow-up and whether weighted risk sets contain adequate information. Extreme or unstable weights, sparse late risk sets, and strong dependence on weight truncation can make contrasts unreliable even when the displayed numbers look plausible.

Pointwise contrasts are not whole-curve summaries. A difference at one time does not describe when the curves separated, and a ratio at one time does not summarize overall event experience. The main alternatives depend on the question: present both regime-specific curves to show their pattern over follow-up, or use a prespecified summary over an interval when the scientific target is an overall time-based comparison. A weighted Kaplan–Meier analysis may be useful as a less model-dependent descriptive comparison under its own identification and censoring assumptions, whereas the weighted Cox fit imposes a proportional-hazards structure on the regime-specific hazards. Neither approach repairs inadequate treatment support or invalid weights.

Do not treat ordinary confidence intervals for each curve as if the two curves were independent when forming uncertainty for their difference or ratio: both estimates come from the same fitted analysis. A point estimate can be computed as above, but valid uncertainty for the contrast must account for their joint estimation. The contrast is also sensitive to the chosen follow-up time, so that time should be scientifically motivated rather than selected after inspecting which point gives the largest separation.