Life Table Analysis

The most straightforward way to describe the survival in a sample is to compute the Life Table. The life table technique is one of the oldest methods for analyzing survival (failure time) data (for example, see Berkson and Gage, 1950; Cutler and Ederer, 1958; Gehan, 1969). This table can be thought of as an enhanced frequency distribution table. The distribution of survival times is divided into a certain number of intervals. For each interval you can compute the number and proportion of cases or objects that entered the respective interval alive, the number and proportion of cases that failed in the respective interval (that is, number of terminal events, or number of cases that died), and the number of cases that were lost or censored in the respective interval.

Based on those numbers and proportions, several additional statistics can be computed. See also Weibull and Reliability/Failure Time Analysis Overview in the Process Analysis module.

Number of Cases at Risk
Number of Cases at Risk is the number of cases that entered the respective interval alive, minus half of the number of cases lost or censored in the respective interval.
Proportion Failing
Proportion Failing is computed as the ratio of the number of cases failing in the respective interval, divided by the number of cases at risk in the interval.
Proportion Surviving
Proportion Surviving is computed as one minus the proportion failing.
Cumulative Proportion Surviving (Survival Function)
Cumulative Proportion Surviving is the cumulative proportion of cases surviving up to the respective interval. Since the probabilities of survival are assumed to be independent across the intervals, this probability is computed by multiplying out the probabilities of survival across all previous intervals. The resulting function is also called the survivorship or survival function.
Probability Density
Probability Density is the estimated probability of failure in the respective interval, computed per unit of time, that is:

Fi = (Pi-Pi+1) /hi

In this formula, Fi is the respective probability density in the ith interval, Pi is the estimated cumulative proportion surviving at the beginning of the ith interval (at the end of interval i-1), Pi+1 is the cumulative proportion surviving at the end of the ith interval, and hi is the width of the respective interval.

Hazard Rate
The hazard rate (the term was first used by Barlow, 1963) is defined as the probability per time unit that a case that has survived to the beginning of the respective interval fails in that interval. Specifically, it is computed as the number of failures per time units in the respective interval, divided by the average number of surviving cases at the midpoint of the interval.
Median Survival Time
Median Survival Time is the survival time at which the cumulative survival function is equal to 0.5. Other percentiles (25th and 75th percentile) of the cumulative survival function can be computed accordingly.
Note: The 50th percentile (median) for the cumulative survival function is usually not the same as the point in time up to which 50% of the sample survived. This would only be the case if there were no censored observations prior to this time.
Required Sample Sizes
In order to arrive at reliable estimates of the three major functions (survival, probability density, and hazard) and their standard errors at each time interval the minimum recommended sample size is 30.