# 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