# Estimate the cumulative baseline hazard function from a Cox model coefficients

CoxPh model described as $h(t|X)=h_{0}(t)e^{\beta X}$ where $h_{0}(t)$ is the baseline hazard function and is non-parametric. If you have the following model,

require(survival) surv_object <- Surv(time = ovarian$futime, event = ovarian$fustat) fit.coxph <- coxph(surv_object ~ rx + resid.ds + age_group + ecog.ps, data = ovarian) (fit.coxph)

The cumulative hazard function of the above model $H_{0}(t)$ can be obtained in this manner cumulativeHazard<- basehaz(fit.coxph) and the smooth baseline hazard function can also be obtained using the method described by Royston

But, assuming you neither have the access to the fitted model (in this case fit.coxph) nor the data used in modeling (in this case ovarian), instead you have access to the model's exponentiated coefficients $\beta$ (hazard ratios), sample size $n$ (in this case n=26), and the number of events $\delta$ (in this case $\delta$=12). Typically, this is what is found in the results of the most published papers(data is never published)

To my question, how can one estimate/approximate the cumulative hazard function $H_{0}(t)$ or even baseline hazard function $h_{0}(t)$ from the available information.

Many thanks

• The baseline hazard function and the cumulative baseline hazard function are functions in time $t$. Do you have any information about the mean censoring times and/or mean event times? – Nussig Sep 15 '18 at 13:34
• Do you have information about the estimated survival probabilities? The cumulative hazard is the negative log of the survival probabilities, i.e. $H(t)=-log(S(t))$. Thus, if you have $\hat{S}(t)$, you can get the estimate $\hat{H}(t)=-log(\hat{S}(t))$. – Nussig Sep 15 '18 at 13:37
• @Nussig Yes, mean censoring time is provided sometimes. For example, using the ovarian data, the mean event time is 351.3 from mean(ovarian$futime[ovarian$fustat==1]) – Smock Sep 15 '18 at 13:47
• I was thinking along that lines that if you have no covariates, or maybe just two groups, you can estimate the baseline hazard function under the assumption of an exponential model by dividing the number of events by the total exposure time. – Nussig Sep 15 '18 at 14:20
• @Nussig could you kindly elaborate this comment in the answer section. It is a useful thought – Smock Sep 15 '18 at 14:36

## 1 Answer

The Cox proportional hazards model does not estimate the baseline hazard. This was originally the aim of being semi-parametric (i.e., in a proportional hazards model distributional assumptions for the event times are "hidden" in the baseline hazard).

To get an estimate of the cumulative hazard function (and then also of survival probabilities) typically the Breslow estimator is used. This a non-parametric estimator that similarly to the Kaplan-Meier estimator is calculated based on when the true events occurred and the number of subjects at risk at these time points. There are also many other alternatives proposed in the literature, such as the smooth baseline hazard function of Royston that you mentioned.

Coming back to your question, and based on what was described above, it is not possible to estimate non-parametrically/flexibly the cumulative baseline hazard without having access to the time points at which the events occurred. As Nussig suggested, one possibility to get something would be to assume an exponential distribution under which the baseline hazard is constant. However, this is a very restrictive parametric assumption that is often not realistic.