# Nt1320 Unit 4

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To explain this estimator take the following example: Let S(t) be the probability that a member from a given population will have a lifetime exceeding time, t. For a sample of size N from this population, let the observed times until death of the N sample members be t_1≤t_2≤t_3≤〖…t〗_N Corresponding to each ti is ni, the number "at risk" just prior to time ti, and di, the number of deaths at time ti.
Note that the intervals between events are typically not uniform. For example, a small data set might begin with 7 cases. Suppose the survivals of these seven patients (sorted by length of years) are: 1, 2+, 3+, 4, 5+,10, 12+. Note that the "+" signs mean that the patient was still alive at the end of his or her follow-up but has since dropped out of follow-up. In other words, these are the censored patients. This data can be put into a table as …show more content…

During the interval two patients were censored (2+ and 3+) so that at the end of the interval four patients were still at risk. Since the interval ends with the death of one of those, the chance of surviving the interval is estimated as 3/4. Also notice that at the start of the next interval (4 through 10 years), only three patients were at risk due to the death at the end of the interval. The actual curve plotted from this computation is shown in Fig. 2.3. As shown on the plot, small vertical tick-marks indicate losses, where a patient 's survival time has been right-censored. Fig. 2.3 Kaplan-Meier curve

The Kaplan–Meier estimator is the nonparametric maximum likelihood estimate of S(t), where the maximum is taken over the set of all piecewise constant survival curves with breakpoints at the event times ti. It is a product of the form
S(t)=∏_(t_i of an (observed)