Statistics Toolbox

ecdf

Empirical (Kaplan-Meier) cumulative distribution function

Syntax

• [f,x] = ecdf(y)
  [f,x,flo,fup] = ecdf(y)
  [...] = ecdf(y,'param1',value1,'param2',value2,...)
  

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(y,'param1',value1,'param2',value2,...) specifies additional parameter name-value pairs chosen from the following:

'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 non-negative 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'.

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 are exponential(10)
  d = exprnd(20,50,1); % Drop-out times are exponential(20)
  t = min(y,d);        % Observe the minimum of these times
  censored = (y>d);    % Observe also whether the subject failed
  
  % Calculate and plot the empirical cdf and confidence bounds
  [f,x,flo,fup] = ecdf(t,'censoring',censored);
  stairs(x,f);
  hold on;
  stairs(x,flo,'r:'); stairs(x,fup,'r:');
  
  % Superimpose a plot of the known true cdf
  xx = 0:.1:max(t); yy = 1-exp(-xx/10); plot(xx,yy,'g-')
  hold off;
  
  
  

See Also

cdfplot

References

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

dummyvar

errorbar