Interpreting Marginal Structural Hazard Ratios

What a Weighted Cox Treatment Coefficient Represents

A weighted Cox model can summarize a causal contrast between treatment regimes, but the coefficient is not automatically a causal effect merely because weights were used. Its interpretation depends on what population the weights represent, whether treatment and censoring are adequately handled, and whether a proportional-hazards model is an appropriate description of the regime-specific hazards.

To make the target precise, consider a baseline binary treatment \(A\in\{0,1\}\), baseline covariates \(L\), and event time \(T\). Let \(T^a\) be the event time that would occur under treatment regime \(a\). Define \(S^a(t)=P(T^a>t)\). When the survival function is differentiable and positive, the marginal hazard under regime \(a\) is

“Marginal” means that the hazard is defined for the population under the intervention \(A=a\), rather than conditional on a particular value of baseline covariates. It is still conditional on survival to time \(t\): the people contributing to the hazard at \(t\) are those who would remain event-free through \(t\) under that regime.

This is a regime-level comparison, not a statement that every individual’s hazard is multiplied by the same amount. Nor is \(\exp(\beta)\) a ratio of cumulative risks or survival probabilities. Those distinctions matter even when the Cox model fits well.

Assumptions Behind the Causal Interpretation

For a baseline treatment, an inverse-probability treatment weight can be written \(W=1/P(A\mid L)\). In a treatment-specific weighted analysis, the corresponding weight is \(W^a=I(A=a)/P(A=a\mid L)\), where \(I(\cdot)\) is an indicator. These weights reweight each treatment group to represent the baseline covariate distribution of the target population. Interpreting the resulting hazards causally requires more than calculating these weights.

• Consistency: if a person receives \(A=a\), their observed event time equals \(T^a\). The treatment regimes must also be defined sufficiently clearly that this equality has a meaningful interpretation.
• Conditional exchangeability: \((T^0,T^1)\) is independent of \(A\) conditional on \(L\). In words, after conditioning on the measured baseline covariates, treatment assignment carries no further information about the potential event times.
• Positivity: \(P(A=a\mid L=l)>0\) for each regime of interest and covariate value \(l\) represented in the target population. Very small probabilities can make weights unstable even when they are technically positive.
• Appropriate censoring adjustment: censoring must be absent through the period of interest, or conditional independent censoring and censoring positivity must hold given the variables used to construct valid censoring weights. Treating censored people as if they had remained event-free is not a valid substitute.
• Correct marginal model: the regime-specific hazards must satisfy the stated proportional-hazards form over the time interval for which a constant hazard-ratio interpretation is claimed.
• Valid weight construction and estimation: the weights must target the intended population and account for the relevant treatment and censoring mechanisms. Inference also requires appropriate handling of the weighted estimation procedure.

The treatment and censoring assumptions identify the regime-specific event distributions; the proportional-hazards assumption gives the treatment coefficient its constant-ratio interpretation. One does not imply the other. Weighting cannot repair unmeasured confounding, a poorly defined intervention, or a wrong marginal hazard model.

Why Weighting Identifies Regime-Specific Hazards

Proof. Write \(p_a(L)=P(A=a\mid L)\). First condition on \(L\), and then on treatment assignment:

The last equality follows from conditional exchangeability and consistency: among people with \(A=a\) and covariates \(L\), the observed \(T\) has the conditional distribution of \(T^a\). Taking expectations over \(L\) gives the stated identity. In particular, use \(g(T)=I(T\geq t)\) to identify the regime-specific size of the event-free risk set, and use event indicators over short intervals to identify regime-specific event increments. Their ratio, in the limit defining the hazard, identifies \(\lambda^a(t)\). With censoring, the same argument requires valid censoring adjustment for the full-data risk sets and event increments. \(\square\)

This identity explains the role of treatment weights: they recover regime-specific distributions by correcting the treatment groups’ covariate composition. A weighted Cox fit then imposes a particular model on the identified marginal hazards; it is not the weights alone that guarantee a constant hazard ratio.

From a Hazard Ratio to Survival: A Derivation

A constant marginal hazard ratio implies a specific relationship between the regime-specific survival curves. This is useful for interpretation, but it also shows why the hazard ratio is not itself a survival difference.

Derivation. For an absolutely continuous survival distribution, \(S^a(0)=1\) and the hazard satisfies \(\lambda^a(t)=-\frac{d}{dt}\log S^a(t)\) almost everywhere. Integrating gives

Under the model, \(\lambda^1(u)=\exp(\beta)\lambda^0(u)\). Therefore,

The survival difference \(S^1(t)-S^0(t)\) depends on time and on the baseline survival curve, even when the hazard ratio is constant. The derivation gives no general equality between a hazard ratio and a risk ratio. \(\square\)

What the Weights Change—and What They Do Not

A small example shows how weighting changes the population represented by each treatment group. Suppose half the target population is high risk and half low risk. Treatment probabilities are \(0.8\) for high-risk people and \(0.2\) for low-risk people. Among treated people, the population masses from the two strata are \(0.5(0.8)=0.4\) and \(0.5(0.2)=0.1\). Their inverse-probability weights are \(1/0.8=1.25\) and \(1/0.2=5\), respectively. The weighted masses are \(0.4(1.25)=0.5\) and \(0.1(5)=0.5\).

For controls, the stratum masses are \(0.5(0.2)=0.1\) and \(0.5(0.8)=0.4\). Their weights are \(1/0.2=5\) and \(1/0.8=1.25\), yielding weighted masses \(0.1(5)=0.5\) and \(0.4(1.25)=0.5\). Thus both weighted groups represent a population split equally between the two strata.

Diagnostics, Alternatives, and Common Pitfalls

Several checks help reveal when the coefficient’s intended interpretation is in doubt. Examine treatment probabilities and the distribution of weights; near-zero treatment probabilities can produce a small number of observations with disproportionate influence. Check whether weighting achieves balance in measured baseline covariates. Inspect regime-specific survival estimates and weighted event patterns over time for substantial departures from a constant hazard contrast. These checks can reveal problems, but none proves the absence of unmeasured confounding or proves that the marginal proportional-hazards model is true.

A common mistake is to read \(\exp(\hat{\beta})=0.65\) as “treatment lowers each person’s risk by 35%.” The calculation \(1-0.65=0.35\) is correct, but the conclusion is not: the coefficient compares marginal hazards among the survivors under each regime at each time, not individual risks or cumulative probabilities. Another mistake is to assume that a weighted fit is automatically marginal and causal. Weighting addresses measured treatment selection only under the stated exchangeability, positivity, consistency, and weight-construction conditions; it does not guarantee the proportional-hazards model.

The main alternative summaries answer different questions. Regime-specific survival curves show survival probabilities over time, and their difference at a chosen time gives an absolute survival contrast. A risk difference or risk ratio at a specified time compares cumulative event probabilities by then. These measures can be more directly useful for decisions, but they do not replace the hazard-ratio estimand when the scientific question concerns the instantaneous event rate among those still at risk. Comparing all these summaries can clarify the practical meaning of a fitted model.

Finally, a single coefficient is a useful summary only when its constant-ratio assumption is defensible over the time interval of interest. If the relative hazards vary materially, \(\exp(\hat{\beta})\) should not be reported as a constant causal hazard ratio. A fitted coefficient may still summarize the data under the chosen model, but that summary does not establish that the model’s time-constant interpretation holds.