7.00.02 - Background - Aster Analytics

Teradata Aster® Analytics Foundation User GuideUpdate 2

Product
Aster Analytics
Release Number
7.00.02
Release Date
September 2017
Content Type
Programming Reference
User Guide
Publication ID
B700-1022-700K
Language
English (United States)

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: