The `km.outcomes`

function is part of the conf package. The
Kaplan-Meier product-limit estimator (KMPLE) is used to estimate the
survivor function for a data set of positive values in the presence of
right censoring^{1}. The `km.outcomes`

function
generates a matrix with all possible combinations of observed failures
and right-censored values and the resulting support values for the
Kaplan-Meier product-limit estimator for a sample of size \(n\). The function has only the sample size
\(n\) as its argument.

The `km.outcomes`

function is accessible following
installation of the `conf`

package:

```
install.packages("conf")
library(conf)
```

The KMPLE is a nonparametric estimate of the survival function from a data set of lifetimes that includes right-censored observations and is used in a variety of application areas. For simplicity, we will refer to the object of interest generically as the item and the event of interest as the failure.

Let \(n\) denote the number of items
on test. For a given \(n\), there are
\(2^{n+1} -1\) different possible
outcomes (failure times or censoring times) for observing an experiment
at a specific time of interest. For any combination of failure or
censored times at a specific time, the KMPLE can be calculated. The
KMPLE of the survival function \(S(t)\)
is given by \[
\hat{S}(t) =
\prod\limits_{i:t_i \leq t}\left( 1 - \frac{d_i}{n_i}\right),
\] for \(t \ge 0\), where \(t_1, \, t_2, \, \ldots, \, t_k\) are the
times when at least one failure is observed (\(k\) is an integer between 1 and \(n\), which is the number of distinct
failure times in the data set), \(d_1, \, d_2,
\, \ldots, \, d_k\) are the number of failures observed at times
\(t_1, \, t_2, \, \ldots, \, t_k\), and
\(n_1, \, n_2, \, \ldots, \, n_k\) are
the number of items at risk just prior to times \(t_1, \, t_2, \, \ldots, \, t_k\). It is
common practice to have the KMPLE “cut off” after the largest time
recorded if it corresponds to a right-censored observation^{2}. The KMPLE drops to
zero after the largest time recorded if it is a failure; the KMPLE is
undefined, however, after the largest time recorded if it is a
right-censored observation.

The support values are calculated for each number of observed events between times 0 and the observation time, listed in column \(l\) for each combination of failure times or censoring times up to that time. The columns labeled as \(d1, d2, ..., dn\) list a 0 if the event corresponds to a censored observation and a 1 if the event corresponds to a failure.

The support values are listed numerically in the \(S(t)\) column, and in order to keep the support values as exact fractions, the numerators and denominators are stored separately in the output columns named \(num\) and \(den\).

To illustrate a simple case, consider the KMPLE for the experiment when there are \(n = 4\) items on test.

Let’s consider an experiment where failures occur at times \(t = 1\) and \(t = 3\), and right censorings occur at times \(t = 2\) and \(t = 4\). In this setting, the KMPLE is

\[\begin{equation*} \hat{S}(t) = \begin{cases} 1 & \qquad 0 \le t < 1 \\ \left(1 - \frac{1}{4}\right) = \frac{3}{4} & \qquad 1 \leq t < 3 \\ \left(1 - \frac{1}{4}\right) \left(1 - \frac{1}{2}\right) = \frac{3}{8} & \qquad 3 \leq t < 4 \\ \text{NA} & \qquad t \geq 4, \end{cases} \end{equation*}\]

where NA indicates that the KMPLE is undefined^{3}.

```
library(conf)
# display the outcomes and KMPLE for n = 4 items on test
= 4
n km.outcomes(n)
#> l d1 d2 d3 d4 S(t) num den
#> [1,] 0 -1 -1 -1 -1 1.0000000 1 1
#> [2,] 1 0 NA NA NA 1.0000000 1 1
#> [3,] 1 1 NA NA NA 0.7500000 3 4
#> [4,] 2 0 0 NA NA 1.0000000 1 1
#> [5,] 2 1 0 NA NA 0.7500000 3 4
#> [6,] 2 0 1 NA NA 0.6666667 2 3
#> [7,] 2 1 1 NA NA 0.5000000 6 12
#> [8,] 3 0 0 0 NA 1.0000000 1 1
#> [9,] 3 1 0 0 NA 0.7500000 3 4
#> [10,] 3 0 1 0 NA 0.6666667 2 3
#> [11,] 3 1 1 0 NA 0.5000000 6 12
#> [12,] 3 0 0 1 NA 0.5000000 1 2
#> [13,] 3 1 0 1 NA 0.3750000 3 8
#> [14,] 3 0 1 1 NA 0.3333333 2 6
#> [15,] 3 1 1 1 NA 0.2500000 6 24
#> [16,] 4 0 0 0 0 NA NA NA
#> [17,] 4 1 0 0 0 NA NA NA
#> [18,] 4 0 1 0 0 NA NA NA
#> [19,] 4 1 1 0 0 NA NA NA
#> [20,] 4 0 0 1 0 NA NA NA
#> [21,] 4 1 0 1 0 NA NA NA
#> [22,] 4 0 1 1 0 NA NA NA
#> [23,] 4 1 1 1 0 NA NA NA
#> [24,] 4 0 0 0 1 0.0000000 0 1
#> [25,] 4 1 0 0 1 0.0000000 0 4
#> [26,] 4 0 1 0 1 0.0000000 0 3
#> [27,] 4 1 1 0 1 0.0000000 0 12
#> [28,] 4 0 0 1 1 0.0000000 0 2
#> [29,] 4 1 0 1 1 0.0000000 0 8
#> [30,] 4 0 1 1 1 0.0000000 0 6
#> [31,] 4 1 1 1 1 0.0000000 0 24
```

If we observe the experiment at time \(t_0 = 2.5\), \(\hat{S}(t) = 3/4\) and is represented by row 5 where \(l = 2\) events have occurred: one failure \(d1=1\) and one censored item \(d2=0\). Notice that \(d3\) and \(d4\) are NA since they have not been observed yet. If instead, we choose \(t_0 = 4.5\), \(\hat{S}(t) = \text{NA}\) and is represented by row 21 where \(l = 4\) events have occurred: first was a failure \(d1=1\), second was a censored item \(d2=0\), third was a failure \(d3=1\), and the fourth and last item was a censored \(d4 = 0\).

Looking at the above output from the Specific Example, the first row corresponds to choosing a time value \(t_0\) that satisfies \(0 < t_0 < 1\), which is associated with an observation time prior to the occurrence of an observed failure or censoring time. That is, \(l = 0\) events have occurred, and -1’s are listed to represent these initialized values. All \(n\) items are on test and \(\hat{S}(t)= 1\).

For the second row, \(l=1\) event has occurred and that event is a censored item \(d1 = 0\). We have not observed any of the other items so they are listed as NA’s.

The third row shows the case when only \(l=1\) event has occurred and that event is a failure; that is, \(d1 = 1\). Again, we have not observed any of the other items so they are listed as NA’s.

For more information on how the \(\hat{S}(t)\) values are generated, please
refer to the vignette titled *km.support* which is available via
the link on the conf package
webpage.

In addition, the functions `km.pmf`

and
`km.surv`

, which are also part of the conf package, have
dependencies on `km.outcomes`

.

Kaplan, E. L., and Meier, P. (1958), “Nonparametric Estimation from Incomplete Observations,” Journal of the American Statistical Association, 53, 457–481.↩︎

Kalbfleisch, J. D., and Prentice, R. L. (2002), The Statistical Analysis of Failure Time Data (2nd ed.), Hoboken, NJ: Wiley.↩︎

Qin Y., Sasinowska H. D., Leemis L. M. (2023), “The Probability Mass Function of the Kaplan–Meier Product–Limit Estimator,” The American Statistician, 77 (1), 102–110.↩︎