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Novice stats question here:

When I run a cox model over daily survival data as a function of 3 covariates, the results:

Fixed coefficients
             coef exp(coef)    se(coef)      z       p
var1  -0.02306065 0.9772032 0.004961884  -4.65 3.4e-06
var2  -0.11755938 0.8890877 0.007519678 -15.63 0.0e+00
var3  0.01145542 1.0115213 0.032680554   0.35 7.3e-01

show that var1 and var2 increase survival time and the effects are statistically significant. I further interpret this to mean that an additional unit increase in var1 reduces the daily hazard by a factor of 0.977 on average -- that is, by 2.3 percent. Similarly, each unit increase in var2 reduces the hazard by a factor of 0.889, or 11.1 percent.

My question is: can I convert these interpretations into something like

  1. each unit increase in var1 increases mean survival time by n days on average, or
  2. each unit increase in var2 increases the probability that survival will exceed 2 days by p%

?

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The advantage in Cox regression--that you don't need to specify a form for the baseline hazard as a function of time--becomes a disadvantage in this case.

You can't make general statements about mean survival times or survival probabilities at specific times from the coefficients of a Cox model unless you also specify the corresponding baseline hazard. This isn't particularly difficult, as survival software will typically provide ways to make predictions based on the model and the empirical baseline hazard that the model estimated. But it's not so simple as you had hoped.

Even with parametric survival models there often won't be a simple formula depending just on the covariate values. In those models the full set of coefficients (including those estimated for the parameters of the baseline survival function) would contain the needed information.

If you prefer to think in terms of relative times to failure instead of instantaneous hazards, you might consider instead an accelerated failure time (AFT) parametric model based for example on a Weibull or lognormal distribution. Except with a Weibull you would no longer be talking about hazard ratios as other AFT baseline functions don't follow proportional hazards. If the model fits well, however, the coefficients for the covariates are easier to interpret in terms of changes of expected event times.

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You cannot derive absolute changes or differences in time directly from the hazard ratios. You can create fitted mortality or survival curves and consider differences in survival or mortality at a common time point or compare the time until a common cumulative mortality or survival is encountered.

While reductions in hazard are commonly referred to as percentages, these ratios represent multiplicative change in risks that are often non- linear. I recommend avoiding a percentage interpretation.

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