9.3 What to do when hazards are not proportional
Several remedies are available, and the choice depends on whether the covariate’s effect is of interest.
- Stratify by a categorical covariate that violates the assumption and is not of primary interest (the stratified model of the previous section). Here, stratifying by cell type removes its violation; the other coefficients are still estimated.
- Split the follow-up into periods and estimate a separate coefficient in each: with R’s survSplit, each subject’s record is cut at the chosen times and the covariate is replaced by its product with period indicators. The model becomes one with time-dependent coefficients, still a Cox model, fitted with the counting-process form Surv(start, stop, event).
- Model the change smoothly, \(\beta (t) = \beta + \gamma g(t)\), for example with \(g(t) = \log t\); in R the tt() term does this. A test of \(\gamma = 0\) is another test of proportional hazards.
- Use a model that does not assume proportional hazards, such as the accelerated failure time models of the next section, or report survival at fixed times or a restricted mean.
Example 9.3.1. Splitting the veteran follow-up at \(100\) days, ten Karnofsky points multiply the hazard by \(0.65\) in the first \(100\) days (\(p < 10^{-6}\)) and by \(1.04\) afterwards (\(p = 0.69\)). With \(\beta (t) = \beta + \gamma \log t\) instead, \(\hat \gamma = 0.0135\) (\(p = 0.002\)): the coefficient \(-0.081 + 0.0135\log t\) is \(-0.081\) at one day, \(-0.019\) at \(100\) days and \(0\) at about \(400\) days (Software 9.4.3). Both say the same thing as Figure 9: the performance score measures the patient’s state at entry, which predicts the near future well and the distant future poorly, a common reason for non-proportional hazards with baseline measurements.
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