The productlimit estimator was developed by Kaplan and Meier in the 1950s and estimates the survival

1

• The product-limit estimator was developed by
  Kaplan and Meier in the 1950s and estimates the
  survival function in the presence of right
  censoring. It is sometimes called the
  Kaplan-Meier estimator.
• Suppose Y1,Y2,Yn are right-censored by t1,t2,tn
  . We observe (Zi,?i), where Zimin(Yi,ti), where
  the deltas are the censoring variables - assume
  there are no tied observations so Z(1)lt Z(2)lt lt
  Z(n) .
• To construct the estimator, first divide the
  support 0, Z(n)) into disjoint intervals
  Ij(Z(j-1), Z(j) , j1,2,,n Z(0)0
• Then the risk set at time u , R(u), is the set of
  indices of subjects still alive and under
  observation at time u- that is at time just prior
  to u.

2

• Nj number at risk elements in R(Z(j))
• Dj number of deaths at Z(j) (0 or 1) and
• to estimate this conditional probability we have
• The product of such estimates gives an estimate
  of the survival function, called the Kaplan-Meier
  product-limit estimator

3

• If u is fixed and there are no ties in the
  observed, possibly right-censored data, then the
  Kaplan-Meier product-limit (PL) estimator is
• In the case of tied observations, we have
  definition 6.3

4

• here t(j) are the ordered times of failure
  (death), Njnumber at risk at time t(j) (this
  means the number that have neither failed nor
  been censored prior to t(j)). Dj died at time
  t(j). Then write
• the quantity in brackets is the conditional
  probability of surviving to time tj1 given that
  one has survived to time tj. For tltt1, the
  estimated survival time is 0. If there are no
  censored times above tk, then the est. survival
  time is 0. When there are censored times above
  tk, then the est. surv. time is undefined.

5

• Use PROC LIFETEST in SAS to get the K-M P-L
  estimates and the corresponding estimated
  survival curves
• proc lifetest methodkm plots(s)
• time ycensor(0) strata group run quit
• Here, y is the survival variable censor is the
  censoring variable group is the explanatory
  variable containing two values that divides the
  subjects into two groups.
• HW On pg.110ff, do 6.2 (well do most of this
  in class today), 6.7 (b), 6.19 (a). On pg. 99,
  use SAS to analyze this data from example 5.3
  (youll have to type it in). Well look at your
  answers next time.