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For a linear regression plugging different X values to the fitted model gives us the different predicted Y values, for a logistic regression we get the different log odds: what does the output of the CoxPH model mean for different X values?

What is the intuitive interpretation of the resulting value?

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Following up on the link provided by @mkt in a comment, after the coefficient vector $\beta$ has been determined, the hazard as a function of time, $h(t)$, for a new case with predictor values $X^T$ is:

$$ \ln h(t) = \ln h_{0}(t) + \beta_{1} X_{1} + \dots + \beta_{p} X_{p} = \ln h_{0}(t) + X^T \bf{\beta}. $$

where $h_0(t)$ is the baseline hazard over time.

In a linear or logistic regression the linear predictor $X^T \beta$ can be written to include the intercept and provide the predicted outcome value (linear regression) or log-odds (logistic regression) for the new case.

The equivalent of the intercept in a Cox regression is an empirical baseline hazard not constant in time. That's the hazard as a function of time if all predictor values were 0, a situation that often does not represent a real-world possibility. It's usually more informative to examine the difference in linear-predictor values for two realistic scenarios; that difference is the difference in log hazard between the scenarios, and exponentiating that difference gives the hazard ratio between them.

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  • $\begingroup$ Since we're interested in the relative hazard between two new cases does that mean that a predicted hazard of only one case does not give us any interpretable information? $\endgroup$ – Metrician Sep 30 '19 at 12:27
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    $\begingroup$ @Metrician you can use the empirical baseline hazard $h_0(t)$ along with the value of the linear predictor $X^T \beta$ for a case to get the estimated hazard over time for that case, and thus you can plot estimated survival probability over time, along with standard errors etc, for that case if you wish.That survival curve is interpretable, insofar as the baseline hazard is representative of the population in question. $\endgroup$ – EdM Sep 30 '19 at 14:07
  • $\begingroup$ To get the estimated hazard over time for the case do we simply plug different time points {1, 2, 3, ... } into X ? $\endgroup$ – Metrician Sep 30 '19 at 19:43
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    $\begingroup$ @Metrician $X$ in the above represents the set of predictor values, not time values. You need to extract the baseline hazard $h_0(t)$ from the Cox model and then use the formula above to calculate the hazard as a function of time for a specific case with a particular value of $X$. It's simplest to use freely available and vetted software tools for this purpose, such as the survfit() function in the R survival package, to show the integrated survivor function. (In a Cox PH model the hazard is 0 between observed event times, so the integrated hazard over time is more interpretable.) $\endgroup$ – EdM Sep 30 '19 at 20:18
  • $\begingroup$ Oh I see now. Thanks a lot $\endgroup$ – Metrician Sep 30 '19 at 22:20

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