The hazard ratio compares instantaneous risks among those who have already survived to time \(t\). Because treatment can change who survives up to \(t\), the sets being compared are post-treatment selected, so the hazard ratio generally lacks a simple causal interpretation even with no unmeasured confounding.
Let \(T^{(a)}\) denote the potential event time under treatment \(a\in\{0,1\}\). The (cause-specific) hazard under \(a\) is \[
\lambda_a(t)
= \lim_{\Delta t \downarrow 0}
\frac{\Pr\{t \le T^{(a)} < t+\Delta t \mid T^{(a)} \ge t\}}{\Delta t},
\] and the (time-specific) hazard ratio is \[
\mathrm{HR}(t) \;=\; \frac{\lambda_1(t)}{\lambda_0(t)}.
\]
Two issues make \(\mathrm{HR}(t)\) non-causal:
Conditioning on a post-treatment variable. The risk set at time \(t\) is those with \(T^{(a)}\!\ge t\). If treatment affects early survival, then \[
\{T^{(1)} \ge t\} \;\text{and}\; \{T^{(0)} \ge t\}
\] include different compositions of individuals because of treatment itself. Thus, even if \((T^{(0)},T^{(1)}) \perp A \mid L\) (no unmeasured confounding given \(L\)), the event \(\{T^{(a)}\ge t\}\) is not independent of \(A\). Comparing \(\lambda_1(t)\) vs \(\lambda_0(t)\) is therefore a comparison of selected subpopulations at time \(t\), not a clean “effect of treatment on outcome.”
Non-collapsibility. Even without confounding, marginal and conditional hazard ratios differ: \[
\frac{\lambda_1(t)}{\lambda_0(t)}
\;\neq\;
\sum_{\ell} w_\ell \,\frac{\lambda_1(t\mid L=\ell)}{\lambda_0(t\mid L=\ell)} ,
\] so the hazard ratio does not aggregate to a population-average causal contrast. One can observe \(\mathrm{HR}(t)\!\approx\!1\) while survival probabilities differ meaningfully across arms.
Survival Probability Causal Effect
A more transparent target is the survival probability causal effect (often called SPCE): \[
\Delta_S(t) \;=\; S_1(t) - S_0(t),
\qquad
S_a(t) \;=\; \Pr\{T^{(a)} > t\}.
\]
\(\Delta_S(t)\) is a collapsible - the difference in the probability of being event-free at time \(t\) if everyone were treated vs if everyone were untreated. - We can compute \(\Delta_S(t)\) using propensity score weighted Kaplan–Meier estimation. - Extensions is possible to incorporate informative censoring and competing risks.
1. Hands on tutorial 1 on fitting PS-weighted marginal Cox and PS-weighted KM.
Mathematically, we will aim at estimating SPCE. - the survival probability causal effect (SPCE) at a chosen time \(t\), i.e. \[
\Delta_S(t) \equiv S_1(t) - S_0(t)
\] where \(S_a(t)=Pr\{T^{(a)}>t\}\) is the counterfactual survival probability under treatment \(a, a\in \{0,1\}\).
Causal assumptions:
Stable unit treatmnet assumption (no multiple versions of treatment and no interference)
Consistency (i.e., observed outcome is consistent with their potential outcome under the treatment they actually received),
Positivity (i.e., non-zero probability of receiving any of the treatment levels),
Conditional exchangeability (also known as no unmeasured confounding).
