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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?
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.
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.)
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