Hazard and Survivor Functions for Different Groups

This example shows how to estimate and plot the cumulative hazard and survivor functions for different groups.

Step 1. Load and organize sample data.

Load the sample data.

load('readmissiontimes.mat')

The data has readmission times of patients with information on their gender, age, weight, smoking status, and censorship. This is simulated data.

Create a matrix of readmission times and censoring for each gender.

female = [ReadmissionTime(Sex==1),Censored(Sex==1)];
male = [ReadmissionTime(Sex==0),Censored(Sex==0)];

Step 2. Estimate and plot cumulative distribution function for each gender.

Plot the Kaplan-Meier estimate of the cumulative distribution function for female and male patients.

figure()
ecdf(gca,female(:,1),'Censoring',female(:,2));
hold on
[f,x] = ecdf(male(:,1),'Censoring',male(:,2));
stairs(x,f,'--r')
hold off
legend('female','male','Location','SouthEast')

Step 3. Plot survivor functions.

Compare the survivor functions for female and male patients.

figure()
ax1 = gca;
ecdf(ax1,female(:,1),'Censoring',female(:,2),'function','survivor');
hold on
[f,x] = ecdf(male(:,1),'Censoring',male(:,2),'function','survivor');
stairs(x,f,'--r')
legend('female','male')

This figure shows that readmission times are shorter for male patients than female patients.

Step 4. Fit Weibull survivor functions.

Fit Weibull distributions to readmission times of female and male patients.

pd = fitdist(female(:,1),'wbl','Censoring',female(:,2))

pd = 

  WeibullDistribution

  Weibull distribution
    A = 12.5593   [10.749, 14.6745]
    B = 1.99834   [1.56489, 2.55185]

pd2 = fitdist(male(:,1),'wbl','Censoring',male(:,2))

pd2 = 

  WeibullDistribution

  Weibull distribution
    A = 4.63991   [3.91039, 5.50551]
    B = 1.94422   [1.48496, 2.54552]

pd2 = fitdist(male(:,1),'wbl','Censoring',male(:,2))

pd2 = 

  WeibullDistribution

  Weibull distribution
    A = 4.63991   [3.91039, 5.50551]
    B = 1.94422   [1.48496, 2.54552]

Plot the Weibull survivor functions for female and male patients on estimated survivor functions.

plot(0:1:25,1-cdf('wbl',0:1:25,12.5593,1.99834),'-.')
plot(0:1:25,1-cdf('wbl',0:1:25,4.63991,1.94422),':r')
hold off
legend('Festimated','Mestimated','FWeibull','MWeibull')

Weibull distribution provides a good fit for the data.

Step 5. Estimate cumulative hazard and fit Weibull cumulative hazard functions.

Estimate the cumulative hazard function for the genders and fit Weibull cumulative hazard functions.

figure()
[f,x] = ecdf(female(:,1),'Censoring',female(:,2),...
'function','cumhazard');
plot(x,f)
hold on
plot(x,cumsum(pdf(pd,x)./(1-cdf(pd,x))),'-.')
[f,x] = ecdf(male(:,1),'Censoring',male(:,2),...
'function','cumhazard');
plot(x,f,'--r')
plot(x,cumsum(pdf(pd2,x)./(1-cdf(pd2,x))),':r')
legend('Festimated','FWeibull','Mestimated','MWeibull',...
'Location','North')