Suppose the survival (and censoring) times of two groups of 6 individuals are as follows:

Group 1:    15  26  15   5   2   6  

Group 2:   14  31+ 25   9  29+ 30+

Use both logrank and Wilcoxon test (the simplest version introduced in the lecture) to test the difference in the survival times. Note that both test statistics can be expressed as , where 

and

Tasks:

1. To calculate the test statistics, create a table with columns , where the columns contain the values of for logrank statistic and contain the corresponding values for the Wilcoxon statistic. Note that the expressions for and for the Wilcoxon test are actually a bit simpler than those for logrank test (some components cancel out).

2. Calculate the column sums to get the values and   for both statistics and then find the values of the two statistics. Also find the corresponding p-values. (Using, for instance, the function "pchisq" in R).  

Thus, you need to submit:

1) The table - either copied to the text box or uploaded in a separate file (can be Excel, pdf, Word or any other format). Round numbers to max 3 decimal digits. 

2) The logrank statistic and its p-value (either in the text box or in the same file as the table)

3) The Wilcoxon statistic and its p-value

4) Your overall conclusion - are the survival times significantly different?

Suppose time to event has a distribution with survival function:

1. Explain, why it is a valid survival function (if you think it is not valid, provide also an explanation).
2. Find median survival. 
3. Derive a hazard function for that random variable. 
4. Is hazard increasing, decreasing or does it have a finite nonzero minimum or maximum?

You may provide the answers in the text box, but you may also provide a handwritten solution and take a photo of it, and upload.