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 The PHREG Procedure

## Survival Distribution Estimates for the Cox Model

Two estimators of the survivor function are available: one is the product-limit estimate and the other is based on the empirical cumulative hazard function.

### Product-Limit Estimates

Let Ci denote the set of individuals censored in the half-open interval [ti , ti+1), where t0=0 and .Let denote the censoring times in [ti , ti+1); l ranges over Ci . The likelihood function for all individuals is given by
where D0 is empty. The likelihood L is maximized by taking S0(t)=S0(ti+0) for and allowing the probability mass to fall only on the observed event times t1, , tk. By considering a discrete model with hazard contribution at ti, you take , where . Substitution into the likelihood function produces
If you replace with estimated from the partial likelihood function and then maximize with respect to , , , the maximum likelihood estimate of becomes a solution of
When only a single failure occurs at ti, can be found explicitly. Otherwise, an iterative solution is obtained by the Newton method.

The estimated baseline cumulative hazard function is

where is the estimated baseline survivor function given by

For details, refer to Kalbfleisch and Prentice (1980). For a given realization of the explanatory variables ,the product-limit estimate of the survival function at is

### Empirical Cumulative Hazards Function Estimates

Let be a given realization of the explanatory variables. The empirical cumulative hazard function estimate at is
The variance estimator of is given by the following (Tsiatis 1981):
where is the estimated covariance matrix of and

The empirical cumulative hazard function (CH) estimate of the survivor function for is

### Confidence Intervals for the Survivor Function

Let and correspond to the product-limit (PL) and empirical cumulative hazard function (CH) estimates of the survivor function for , respectively. Both the standard error of log() and the standard error of log() are approximated by , which is the square root of the variance estimate of ; refer to Kalbfleish and Prentice (1980, p. 116). By the delta method, the standard errors of and are given by
respectively. The standard errors of log[-log()] and log[-log()] are given by
respectively.

Let be the upper percentile point of the standard normal distribution. A confidence interval for the survivor function is given in the following table.
 Method CLTYPE Confidence Limits LOG PL LOG CH LOGLOG PL LOGLOG CH NORMAL PL NORMAL CH

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