The `km.pmf`

function is part of the conf package. The
function calculates the probability mass function for the support values
of Kaplan and Meier’s product–limit estimator^{1}. 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. The `km.pmf`

function generates the probability
mass function for the support values of the KMPLE for a particular
sample size `n`

, probability of observing a failure
`h`

at the time of interest expressed as the cumulative
probability `perc`

associated with \(X = \min(T, C)\), where \(T\) is the failure time and \(C\) is the censoring time under a
random-censoring scheme.

The `km.pmf`

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. 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 (NA), however, after the largest time recorded if it is a
right-censored observation.

The support values (calculated in `km.support`

) are
calculated from \(\hat{S}(t)\) at any
\(t \ge 0\) for all possible outcomes
of an experiment with \(n\) items on
test.

The function `km.pmf`

calculates the probability for each
support value and produces a plot of the probabilities. The probability
of NA, the event that the last time recorded is a right-censored
observation, is also calculated and plotted at the arbitrary position of
s = 1.1.

`n`

sample size

`h`

probability of observing a failure; that is, P(X =
T)

`perc`

cumulative probability associated with X = (T,
C)

`plot`

option to plot the probability mass function
(default is TRUE)

`sep`

option to show the breakdown of the probability for
each support value (see function `km.outcomes`

for details on
the breakdown) (default is TRUE)

`xfrac`

option to label support values on the x-axis as
exact fractions (default is TRUE)

`cex.lollipop`

size of the dots atop the spikes

The following section provides various examples for the usage of
`km.pmf`

.

Qin et al.^{3} derived the probability mass function of
the KMPLE for one particular setting where there are \(n = 3\) items on test, the failure times
\(T_1,T_2\) and \(T_3\) and the censoring times \(C_1,C_2\) and \(C_3\) both follow an exponential(1)
distribution. The fixed time of interest is \(t_0 = -\ln(1/2)/2\), which is the median of
\(X = \min(T, C)\), where \(T\) is the failure time and \(C\) is the censoring time under a
random-censoring scheme. Therefore, `perc = 0.5`

.

In this case, since failure and censoring times have the same
exponential distribution, they are equally likely to occur; that is,
`h = 1/2`

.

For this example, `km.pmf`

is called with the arguments
`n = 3`

, `h = 0.5`

, and `perc = 0.5`

.
The optional defaults are used for this example. Two columns of output
are produced: the support values and their probabilities. In addition,
by default, a plot of the probability mass function is created with hash
marks on the point mass lines to show the breakdown of the probability
for each of the support values for the different possible outcomes (see
function `km.outcomes`

for details).

```
library(conf)
# display the probability mass function
km.pmf(n=3, h = 1/2, perc = 0.5)
#> Additional delay due to printing hash marks.
#> S P
#> 1 0.0000000 0.06250
#> 2 0.3333333 0.09375
#> 3 0.5000000 0.09375
#> 4 0.6666667 0.28125
#> 5 1.0000000 0.40625
#> 6 NA 0.06250
```

In other experiments, it may be more reasonable to expect that we
have a higher or lower chance of censoring. In this example we will
start with a higher rate of right-censoring, so a lower chance that we
will observe a failure. Set `h = 1/3`

. We may also want to
look at different values of \(t_0\).
Here we will choose the 75th percentile of \(X
= \min(T, C)\); that is, `perc = 0.75`

. In addition,
we will remove the hash marks, use decimals on the \(x\)-axis, and increase the size of the dot
on top of the point masses.

```
# display the probability mass function with lower failure rate (higher censoring rate)
km.pmf(n=3, h = 1/3, perc = 0.75, sep = FALSE, xfrac = FALSE, cex.lollipop = 2)
#> S P
#> 1 0.0000000 0.140625
#> 2 0.3333333 0.046875
#> 3 0.5000000 0.093750
#> 4 0.6666667 0.140625
#> 5 1.0000000 0.296875
#> 6 NA 0.281250
```

For comparison purposes, the following plot and probability mass
function have an earlier time of interest (`perc = 0.35`

) so
it is less likely that all of the observations have occurred making the
probability of survival less likely to be 0 or NA, which occur at the
end of the experiment. We can also see with the higher rate of censoring
that early in the experiment there is a high probability of survival
equal to 1. This point mass represents all possible censored items
before a failure at the 35th percentile of \(X
= \min(T, C)\).

```
# display the probability mass function with an earlier time of interest
km.pmf(n=3, h = 1/3, perc = 0.35, sep = FALSE, xfrac = FALSE, cex.lollipop = 2)
#> S P
#> 1 0.0000000 0.01429167
#> 2 0.3333333 0.02654167
#> 3 0.5000000 0.05308333
#> 4 0.6666667 0.20095833
#> 5 1.0000000 0.67654167
#> 6 NA 0.02858333
```

In addition, instead of examining a different time of interest, we
can look at a higher failure rate. For the following plot and
probability mass function, we set `h = 2/3`

. We return to the
75th percentile (`perc = 0.75`

). With a lower probability of
censoring, we have smaller probabilities at 1 and NA than in the first
plot of this example. There is lower probability of 1 because we are
more likely to observe at least one failure, and there is lower
probability of NA (and higher probability of 0) because we are more
likely (with probability 2/3) to observe a failure for the last item on
test.

```
# display the probability mass function with a higher probability of failure (lower censoring rate)
km.pmf(n=3, h = 2/3, perc = 0.75, sep = FALSE, xfrac = FALSE, cex.lollipop = 2)
#> S P
#> 1 0.0000000 0.281250
#> 2 0.3333333 0.187500
#> 3 0.5000000 0.093750
#> 4 0.6666667 0.187500
#> 5 1.0000000 0.109375
#> 6 NA 0.140625
```

Since all possible outcomes and probabilities are calculated, large
sample size `n`

may affect the speed of the function. Due to
CPU and memory limitations, `n`

is limited to values from 1
to 23. For a larger sample size `n`

, it is recommended to set
`sep = FALSE`

to remove the hash marks to reduce the delay in
rendering the plot, and `xfrac = FALSE`

(removing the exact
fractions from the \(y\)-axis) and
`cex.lollipop = 0.01`

(making tiny dots on top of the point
masses) for a better visual effect.

```
# display the probability mass function with a sample size of 8
km.pmf(8, 1/2, 0.75, sep = FALSE, xfrac = FALSE, cex.lollipop = 0.01)
#> S P
#> 1 0.0000000 0.050056458
#> 2 0.1250000 0.002085686
#> 3 0.1428571 0.002085686
#> 4 0.1458333 0.002085686
#> 5 0.1500000 0.002085686
#> 6 0.1562500 0.002085686
#> 7 0.1666667 0.004171371
#> 8 0.1714286 0.002085686
#> 9 0.1750000 0.002085686
#> 10 0.1785714 0.002085686
#> 11 0.1822917 0.002085686
#> 12 0.1875000 0.004171371
#> 13 0.1904762 0.002085686
#> 14 0.1944444 0.002085686
#> 15 0.2000000 0.004171371
#> 16 0.2083333 0.004171371
#> 17 0.2142857 0.004171371
#> 18 0.2187500 0.004171371
#> 19 0.2222222 0.002085686
#> 20 0.2250000 0.002085686
#> 21 0.2285714 0.002085686
#> 22 0.2333333 0.002085686
#> 23 0.2343750 0.002085686
#> 24 0.2380952 0.002085686
#> 25 0.2430556 0.002085686
#> 26 0.2500000 0.015295029
#> 27 0.2571429 0.002085686
#> 28 0.2625000 0.002085686
#> 29 0.2666667 0.002085686
#> 30 0.2678571 0.002085686
#> 31 0.2734375 0.002085686
#> 32 0.2777778 0.002085686
#> 33 0.2812500 0.002085686
#> 34 0.2857143 0.011123657
#> 35 0.2916667 0.011123657
#> 36 0.3000000 0.011123657
#> 37 0.3125000 0.011123657
#> 38 0.3214286 0.002085686
#> 39 0.3281250 0.002085686
#> 40 0.3333333 0.018075943
#> 41 0.3428571 0.009037971
#> 42 0.3500000 0.009037971
#> 43 0.3571429 0.009037971
#> 44 0.3645833 0.009037971
#> 45 0.3750000 0.024564743
#> 46 0.3809524 0.006952286
#> 47 0.3888889 0.006952286
#> 48 0.4000000 0.015990257
#> 49 0.4166667 0.015990257
#> 50 0.4285714 0.022479057
#> 51 0.4375000 0.022479057
#> 52 0.4444444 0.006952286
#> 53 0.4500000 0.013441086
#> 54 0.4571429 0.006952286
#> 55 0.4666667 0.006952286
#> 56 0.4687500 0.013441086
#> 57 0.4761905 0.006952286
#> 58 0.4861111 0.006952286
#> 59 0.5000000 0.048279762
#> 60 0.5142857 0.013441086
#> 61 0.5250000 0.013441086
#> 62 0.5333333 0.006952286
#> 63 0.5357143 0.013441086
#> 64 0.5468750 0.013441086
#> 65 0.5555556 0.006952286
#> 66 0.5625000 0.013441086
#> 67 0.5714286 0.025800705
#> 68 0.5833333 0.025800705
#> 69 0.6000000 0.032289505
#> 70 0.6250000 0.035173416
#> 71 0.6428571 0.013441086
#> 72 0.6562500 0.013441086
#> 73 0.6666667 0.025800705
#> 74 0.6857143 0.018848419
#> 75 0.7000000 0.018848419
#> 76 0.7142857 0.021732330
#> 77 0.7291667 0.021732330
#> 78 0.7500000 0.036134720
#> 79 0.8000000 0.018848419
#> 80 0.8333333 0.021732330
#> 81 0.8571429 0.022693634
#> 82 0.8750000 0.022876740
#> 83 1.0000000 0.022891998
#> 84 NA 0.050056458
```

For more information on how the \(\hat{S}(t)\) values are generated, please
refer to the vignette titled *km.support*.

For more information on the hash marks generated on the plot, please
refer to the vignette titled *km.outcomes*.

In addition, `km.pmf`

calls the functions
`km.support`

and `km.outcomes`

.

These functions and vignettes are both available via the link on the conf package webpage.

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.↩︎