In a Cox model where region violates proportional hazards, when do you stratify rather than model a time-varying effect?

## The situation A Cox model of subscriber churn adjusts for region, and the diagnostic says region's hazard ratio is not constant over follow-up. Nothing is broken about the other coefficients yet, but the model as written asserts something false about region. There are three defensible responses and they answer different questions. ## Option 1: stratify on region A stratified Cox model splits subjects into strata and lets each stratum have its own arbitrary baseline hazard `h0s(t)`, while sharing one coefficient vector across strata: `h(t | x, stratum s) = h0s(t) * exp(b'x)` The partial likelihood becomes a product over strata, with risk sets formed inside each stratum. Because each region's baseline is unrestricted, region can affect the hazard in any time-varying way at all and the assumption cannot be violated by it. What you pay: - **No estimate for region.** There is no coefficient, no hazard ratio, no test. The variable has been adjusted away, not measured. If a stakeholder asks how much region matters, the model cannot answer. - **Efficiency loss with thin strata.** Comparisons happen only within strata. A stratum containing three events contributes almost nothing, and dozens of small strata can meaningfully widen every other interval. - **Proportionality is still assumed for everything else.** Stratifying on region does nothing for a treatment indicator whose own effect fades. - **Interactions become implicit.** If a covariate's effect genuinely differs by region, a shared coefficient still forces one number; stratification frees the baseline, not the slopes. Stratification is the right call when region is a nuisance adjuster with a handful of levels and adequate events in each, and nobody needs a regional effect size. ## Option 2: model the time-varying effect Make the coefficient a function of time. Two practical forms: - **Interaction with a time transform.** Add `region x log(t)` so the log hazard ratio is `b + g*log(t)`. Compact and testable, but it forces a specific functional shape on the drift. - **Piecewise-constant effects.** Split follow-up into pre-specified windows — days 0-30 and 31 onward — and estimate a separate hazard ratio in each. Less elegant, far easier to explain, and it maps directly onto a business statement: the effect was concentrated in the first month. This is the choice when the variable is the finding rather than the background. It converts a violated assumption into a result: not a nuisance to be absorbed, but a description of how the effect evolves. The cost is more parameters, decisions about window boundaries that must be pre-specified rather than chosen after seeing the data, and a report nobody can summarise in a single number. ## Option 3: keep the single coefficient, and caveat it When proportionality fails mildly, the fitted coefficient still estimates an average of the time-varying log hazard ratio over the observed follow-up, weighted by when events occurred. That is a genuine quantity and it may be exactly what a decision needs — but it is a property of your study window, not of the world. If you take this path, say so explicitly, give the follow-up length the average covers, and never let it be quoted as a stable long-run effect or extrapolated to a longer horizon. ## A fourth framing worth naming If several covariates violate proportionality at once, the model form itself may be wrong for the data. An accelerated failure time model makes covariates stretch or compress the time axis rather than scale the hazard — `log T = b'x + error` — and sometimes fits such data more naturally, at the price of committing to a distribution for the error term. Raising this shows you know proportional hazards is a modelling choice, not a law. ## The organisational part The deciding judgment at lead level is procedural as much as statistical. Write into the analysis plan, before looking at results, which covariates get proportionality checks, what magnitude of drift triggers a remedy, and which remedy applies to which kind of variable. Otherwise the team runs the check, sees a borderline result, and picks whichever fix produces the more agreeable headline — and every stratification or window boundary chosen after the fact quietly inflates the false-positive rate of the conclusion built on it. ## The answer to give Stratify nuisance variables you do not need to quantify. Model the time-varying effect for variables whose behaviour is the result. Keep the average only when it is small, honest and window-labelled. Decide the rule in advance.