# km.outcomes

#### Yuxin Qin yqin08@wm.edu, Lawrence Leemis leemis@math.wm.edu, Heather Sasinowska hdsasinowska@wm.edu

#### 2024-05-05

## Introduction

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 censoring1. 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.

## Installation Instructions

The `km.outcomes` function is accessible following installation of the `conf` package:

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

## Details

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 observation2. 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 \(den\).

## Examples

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

### Specific Example

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 undefined3.

    library(conf)
    #  display the outcomes and KMPLE for n = 4 items on test
    n = 4
    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 \(d4 = 0\).

### General Example

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.

## Package Notes

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`.

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

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

3. 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.↩︎