EMPIRICAL CDF PLOT

EMPIRICAL CDF PLOT

Graphics Command

Generates a empirical cumulative distribution function (cdf) plot.

In reliability analysis, many data sets consists of a set of failure times, which may be truncated at some limit value. The cumulative distribution function (CDF) is defined as:
F(t) = prob(T < t)

where T is the lifetime of a randomly selected unit.

A empirical cdf plot is a plot of the empirical CDF versus failure time. The failure time is plotted on the horizontal axis. At each failure time, the following two points are calculated and plotted on the vertical axis:

\( y_1 = \frac{i-1}{n} \)
\( y_2 = \frac{i}{n} \)


with n and i denoting the number of data points and the rank of the failure time, respectively. The last failure time only calculates one of these points.

When all of the points are connected, a staircase type plot results. The vertical step is constant for the failure times, while the lengths of the horizontal steps are determined by the distances between failure times.

This plot is closely related to the TAIL AREA PLOT and the KAPLAN MEIER PLOT.

EMPIRICAL CDF PLOT <y>             <SUBSET/EXCEPT/FOR qualification>
where <y> is a response variable containing failure times;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

EMPIRICAL CDF PLOT <y> <x>             <SUBSET/EXCEPT/FOR qualification>
where <y> is a response variable containing counts;
            <x> is a response variable containing failure times;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This synatx is used for binned data. That is, contains the failure times and contains the number of occurences corresponding to each failure time.

EMPIRICAL CDF PLOT Y1
EMPIRICAL CDF PLOT Y1 SUBSET TAG > 1
EMPIRICAL CDF PLOT Y X

You can optionally generate approximate confidence intervals for the empirical cdf plot. To generate the confidence interval, enter
SET EMPIRICAL CDF PLOT CONFIDENCE LIMITS ON
To turn off the confidence limits, enter
SET EMPIRICAL CDF PLOT CONFIDENCE LIMITS OFF

These confidence intervals are based on a binomial approximation to the normal distribution (Higgins, pp. 15-16)

\[ \hat{F}(x) = \phi(1 - \alpha/2) \sqrt{\frac{\hat{F} (1 - \hat{F})}{n}} \]

with \( \phi \) and \( \alpha \) denoting the percent point function of the standard normal distribution and the confidence level, respectively.

By default, 95% confidence levels are generated. To specify a different confidence level, enter the command

LET ALPHA = <value>

where common values for <value> are 0.90 or 0.99.

None

None

LINES	= Sets the type for plot lines.
CHARACTERS	= Sets the type for plot characters.
TAIL AREA PLOT	= Generates a tail area plot.
KAPLAN MEIER PLOT	= Generates a Kaplan Meier plot.
... HAZARD PLOT	= Generates a hazard plot.
PROBABILITY PLOT	= Generates a probability plot.
PLOT	= Generates a data or function plot.

Higgins (2004), "Introduction to Modern Nonparametric Statistics," Duxbury Press, pp. 15-16.

Reliability

1998/08
2023/05: Added support for approximate confidence intervals

SKIP 25
READ HAHN.DAT MILES  TAG
TITLE EMPIRICAL CDF PLOT OF HAHN.DAT
XLIMITS 0 150000
Y1LABEL CDF
X1LABEL FAILURE TIME
EMPIRICAL CDF PLOT MILES
    

. Step 1:   Read some data
.
skip 25
read weibbury.dat y
skip 0
.
. Step 2:   Set some plot control
.
case asis
title case asis
tic mark label case asis
label case asis
.
line solid dotted dotted
line color black blue blue
.
ylimits 0 1
.
. Step 3:   Generate plot
.
set empirical cdf plot confidence limits on
empirical cdf plot y