Learn more about Statistics Toolbox

ecdf - Empirical cumulative distribution function

Syntax

[f,x] = ecdf(y)
[f,x,flo,fup] = ecdf(y)
ecdf(...)
ecdf(ax,...)
[...] = ecdf(y,param1,val1,param2,val2,...)

Description

[f,x] = ecdf(y) calculates the Kaplan-Meier estimate of the cumulative distribution function (cdf), also known as the empirical cdf. y is a vector of data values. f is a vector of values of the empirical cdf evaluated at x.

[f,x,flo,fup] = ecdf(y) also returns lower and upper confidence bounds for the cdf. These bounds are calculated using Greenwood's formula, and are not simultaneous confidence bounds.

ecdf(...) without output arguments produces a plot of the empirical cdf.

ecdf(ax,...) plots into axes ax instead of gca.

[...] = ecdf(y,param1,val1,param2,val2,...) specifies additional parameter/value pairs chosen from the following:

Parameter	Value
'censoring'	Boolean vector of the same size as x. Elements are 1 for observations that are right-censored and 0 for observations that are observed exactly. Default is all observations observed exactly.
'frequency'	Vector of the same size as x containing nonnegative integer counts. The jth element of this vector gives the number of times the jth element of x was observed. Default is 1 observation per element of x.
'alpha'	Value between 0 and 1 for a confidence level of 100(1-alpha)%. Default is alpha=0.05 for 95% confidence.
'function'	Type of function returned as the f output argument, chosen from 'cdf' (default), 'survivor', or 'cumulative hazard'.
'bounds'	Either 'on' to include bounds, or 'off' (the default) to omit them. Used only for plotting.

Examples

Generate random failure times and random censoring times, and compare the empirical cdf with the known true cdf:

y = exprnd(10,50,1); % Random failure times exponential(10)
d = exprnd(20,50,1); % Drop-out times exponential(20)
t = min(y,d); % Observe the minimum of these times
censored = (y>d); % Observe whether the subject failed

% Calculate and plot empirical cdf and confidence bounds
[f,x,flo,fup] = ecdf(t,'censoring',censored);
stairs(x,f,'LineWidth',2)
hold on
stairs(x,flo,'r:','LineWidth',2)
stairs(x,fup,'r:','LineWidth',2)

% Superimpose a plot of the known population cdf
xx = 0:.1:max(t);
yy = 1-exp(-xx/10);
plot(xx,yy,'g-','LineWidth',2)
legend('Empirical','LCB','UCB','Population',...
       'Location','SE')
hold off

References

[1] Cox, D. R., and D. Oakes. Analysis of Survival Data. London: Chapman & Hall, 1984.

See Also

cdfplot, ecdfhist

dwtest

ecdfhist

© 1984-2009- The MathWorks, Inc.    -   Site Help   -   Patents   -   Trademarks   -   Privacy Policy   -   Preventing Piracy   -   RSS