The definition of the Cox proportional hazard model is:
h(t) = h0(t)exp(βX)
Given an estimated time t and all values of conditional variables (x 1, x 2, ..., x n ), the survival function is:
S(t) = S0 (t)exp(βx)
S 0(t), the baseline survival function, is composed of the survival probabilities at times t i. Three estimators often used to estimate these survival probabilities are:
- Breslow estimator
- Nelson-Aalen-Breslow estimator
- Kalbfleisch and Prentice estimator
The first two estimators can be used with Efron ties modification.
The CoxSurvFit function uses the Nelson-Aalen-Breslow estimator with Efron ties modification for baseline function estimation.
The Nelson-Aalen estimator of the integrated hazard is:
In 1972, Breslow suggested estimating the survival function as:
In 1984, Cox and Oakes described a simpler estimator that extends the Nelson-Aalen estimate of the cumulative hazard to the case of covariates:
where the sum is over the risk set Ri. The cumulative hazard and survival functions are then estimated as:
and:
With Efron ties modification:
