Cox Proportional Hazards › Disadvantages

The most serious disadvantage is the proportional-hazards assumption itself. The entire model depends on the claim that $\exp(\beta^\top x)$ multiplies the hazard by the same amount at every point in time. If that claim fails, for example, if a treatment is protective early and harmful later, a fitted coefficient can become a population- and follow-up-dependent summary, but their interpretation as a constant hazard ratio becomes misleading, and the survival predictions that follow from the model will be systematically off. We will discuss the standard diagnostics in detail later, because in practice it is the assumption that fails, not the mathematics.

Its relative-hazard interpretation must also be separated from absolute survival and from causal effects. An exponentiated Cox coefficient gives the multiplier of the instantaneous hazard for a one-unit covariate increase, holding the other covariates fixed. The hazard is a rate, not a probability. It says nothing directly about how much survival probability changes, because that depends on the cumulative baseline hazard and on how far into the future you are looking. Reporting "a hazard ratio of 0.5" as if it were "half as likely to die" is a common and consequential misinterpretation.

Finally, the model is not free of practical limitations. The simple model assumes noninformative censoring conditional on the included covariates. If dropout still depends on prognosis after conditioning on those covariates, the fitted coefficients can be biased. It assumes the log hazard is linear in the included covariates; nonlinear effects require their own functional-form checks. Our treatment of tied event times uses approximations. Efron accounts for fractional depletion of the tied-event weight in coarsened continuous times. The recording process and risk-set composition also matter when choosing a tie method. It also does not give you an absolute risk prediction unless you re-estimate the baseline hazard, which is an extra step you may be tempted to skip.