How would you check the proportional hazards assumption for a treatment indicator in a Cox model?

## What is being assumed Proportional hazards says the hazard ratio between two covariate profiles is the same at every time point. For a binary treatment indicator, `h(t | treated) / h(t | control) = exp(b)` with no `t` on the right-hand side. The assumption fails when the effect changes over the follow-up — a retention intervention that strongly suppresses churn during the first month and does nothing afterwards is the classic case. ## Check 1: Schoenfeld residuals At each observed event time, the Schoenfeld residual for a covariate is the covariate value of the subject who actually had the event, minus the weighted average of that covariate across the risk set, weighted by each member's fitted risk score. Intuitively, it asks: at this moment, was the subject who failed more or less treated than the model expected, given who was still at risk? If the effect is constant, these residuals contain no time trend and scatter around zero. If treatment protects strongly early and not later, early residuals sit below zero (failures come disproportionately from controls) and later residuals drift up toward zero. Scaling the residuals by an estimate of the variance turns them into a direct estimate of the time-varying coefficient `b(t)`, so the plot can be read as the coefficient path. Fit a smoother through it: flat means proportional, a rising or falling line means not. The associated test — commonly attributed to Grambsch and Therneau — tests whether the slope of the scaled residuals against a time transform is zero, per covariate and globally. ## Check 2: log-minus-log curves Under proportional hazards, `S_treated(t) = S_control(t)^HR`. Take `-log` of both sides to get cumulative hazards related by a factor of `HR`, then take `log` again: `log(-log S_treated(t)) = log(HR) + log(-log S_control(t))` So plotting `log(-log S(t))` against time (or log time) for each arm should give two curves separated by a constant vertical distance of `log(HR)` — parallel, whatever shape they have. Curves that converge, diverge, or cross are direct visual evidence of a violation. Crossing is the most severe case: treatment helps early and hurts later, or vice versa, and no single hazard ratio can represent it. This check works cleanly for categorical covariates with a decent number of events per level; for continuous covariates you would have to bin, which weakens it. ## Check 3: fit the alternative The most direct test is to fit what the alternative claims. Add a term for treatment interacted with a function of time — `log t` is a common choice, or a piecewise indicator for period — so the coefficient becomes `b(t) = b + g * log t`. If `g` is indistinguishable from zero, the constant-effect model is not contradicted. This has the advantage of producing an interpretable description of the violation rather than just a warning: it tells you how the effect moves. ## Reading the result like a practitioner - **Power scales with events.** A test on 50,000 churn events will flag departures far too small to change any decision. A test on 40 events will pass almost anything. Always look at the residual plot alongside the p-value and ask how large the drift in the coefficient is over the period that matters. - **A violated assumption is not automatically fatal.** If proportionality fails, the single fitted coefficient still estimates a kind of average of the time-varying log hazard ratio across the observed follow-up, weighted by where the events fell. That average is a real quantity, but it is specific to your study's follow-up window, so it should not be quoted as if it were a stable property of the treatment. - **Check the covariates you care about first.** A mild violation on a nuisance adjuster matters much less than one on the treatment indicator whose effect is the headline result. - **Beware misspecification masquerading as non-proportionality.** A continuous covariate entered linearly when its true effect is curved, or an omitted strong predictor, can produce residual trends that look like a PH violation. Check the functional form before rewriting the model around time-varying effects. ## What a good answer sounds like Name at least two checks, say what each would look like under the null, and add the judgment layer: quantify the size of the departure, decide whether it changes the conclusion over the horizon the business cares about, and only then reach for a remedy.