# Can we compute predicted cumulative incidence after fitting a Cox model?

After fitting a Cox regression, we can compute the predicted survival curve S(t) e.g. in R:

survfit(formula, newdata, ...)

where formula is a coxph object. With the KM estimate, cumulative incidence is 1-S(t). Can we also do the same thing with the predicted S(t) from the Cox model to get the predicted cumulative incidence? I'm not sure if this is appropriate or whether it's better to use KM estimate for cumulative incidence.

• The KM is a fully non-parametric method, whereas smoothed baseline hazard estimates exploit the proportional hazard assumption to avail prediction across a range of predictors. For categorical covariates, you can create KM curves for each combination of covariates if there is adequate subjects and failures in each group. Nov 4, 2022 at 22:00
• Thanks @AdamO. However, I wanted to know if what I suggested is valid? Nov 5, 2022 at 11:40

With at most one event possible per individual, the cumulative distribution of events over time, $$F(t)$$, is simply $$F(t)=1-S(t)$$ by definition. That has nothing to do with Kaplan-Meier versus Cox models. As that Wikipedia page says:

Sometimes complementary cumulative distribution functions are called survival functions in general.

So "survival functions" can be defined in many contexts outside of what you might typically consider "survival analysis."

The choice between Kaplan-Meier and Cox-modeled survival/cumulative-distribution functions is thus based on what you want to display. Kaplan-Meier curves are closer to raw data, but if restricted to subsets of covariate values they lose power. Cox models employ all the data and can be used for survival-curve predictions at any set of covariate values, but you display a predicted survival or cumulative-distribution curve.

In R, the plot.survfit() function has a fun argument that leads to display of the cumulative distribution over time (with fun="F") or other transformations of a survival function. In particular, if an individual can have multiple events you might choose fun="cumhaz" to plot the cumulative hazard of events.

• Thank you. I would have thought that predicted survival or cumulative incidence would be preferable to account for other covariates like in all multiple regression models (assuming the model is correctly specified). Nov 6, 2022 at 14:28
• In the same way, I don't really get the idea of crude incidence. Surely a regression model that accounts for other effects to yield adjusted (predicted) incidence is preferable? Nov 6, 2022 at 14:31
• @user167591 a lot depends on the nature of the study. In a randomized trial the Kaplan-Meier curves are often preferred, as the randomization ideally controls for covariates and you don't need to make assumptions like proportional hazards to display the results. Predicted curves can be useful in observational studies insofar as the model is correct. Thus you also need to document the quality of the model carefully when you display predicted curves.
– EdM
Nov 6, 2022 at 15:27
• Thank you @EdM, that makes sense. Really sorry just one more thing, is it possible to get the incidence rate (not cumulative) after fitting the Cox model e.g. number per 1000 person years etc.? I know how to get the crude incidence rate or the adjusted one from a Poisson model using person time as offset, but assumed such an adjusted incidence rate could also be obtained after fitting in a Cox model? Nov 6, 2022 at 17:07
• Oh...I guess we get from the relationship: CI = 1 - exp(-IR x T) ? Nov 6, 2022 at 17:17