To succesfully complete the following task use the following shiny app.

Task 1

Use this shiny app to fill the following table with the symbols:

This symbol will represent your opinion on the choice of this particular two-sample statistic in the case of violation null hypothesis described by the plot of hazard functions.

Using the app find at least one example (distributions of survival times \(T_1\), \(T_2\) and their censoring distributions) where some choice of weight function is inappropriate (low power under evident H0 violation) while some other weight function rejects obvious violation of null hypothesis.


Statistic Weight function \(G(\rho, \gamma)\)
Logrank (Mantel) \(1\) \(G(0,0)\)
Prentice-Wilcoxon \(\widehat{S}(t-)\) \(G(1,0)\)
\(\left[1-\widehat{S}(t-)\right]\) \(G(0, 1)\)
Fleming-Harrington \(\rho = \gamma = 1\) \(G(1, 1)\)


The rest of this exercise is focused on real data analyses with different hazard shapes.

Homework assignment for Tasks 2-5

Compare survival distributions between the two groups of interest in the following datasets. Conduct the analysis according to the same steps as in the previous exercise.

  1. Calculate and plot survival function.

  2. Calculate and plot Nelson-Aalen estimators of cumulative hazard function.

  3. Plot smoothed hazard functions.

  4. Perform logrank, Prentice-Wilcoxon, \(G(0,1)\) or some particular \(G(\rho,\gamma)\) tests using functions survdiff or FHtestrcc. Interpret the results. Discuss the choice of weight function with respect to the power of the resulting test.

Since combining Exercises 5+6 could be time-consuming for you to do in one week, choose only 2 of the following 4 tasks to be solve together with homework from Exercise 5!


Task 2+3

Activate the dataset veteran by calling data(veteran). It includes 137 observations and eight variables. The observations are lung cancer patients. The variable time contains survival time (in days) since the start of treatment. The variable status includes the event indicator (1 = death, 0 = censoring). The variable trt distinguishes two different treatments (1 = standard, 2 = experimental). The treatment was randomly assigned to the patients.


Task 4

Return to Exercise 3 Task 1, the nurshome dataset. Perform the comparison of survival distributions of subpopulations of your choice. Try to find some interesting division.


You should also be able to perform some simulation study for yourself:

Task 5 (simulation study)

Let the censoring distribution in both samples be exponential with the rate \(\lambda=0.2\) (the expectation is \(1/\lambda=5\)), independent of survival. Put both samples in a single dataframe, introducing a variable distinguishing which sample each observation comes from.

Compare survival distributions between the two groups using the generated data. Conduct the analysis according to the steps 1–4 and display the true survival and hazard functions in the plots generated at steps 1–3 of the analysis.