This problem makes use of the following table.

You can create the table in R as follows:

df <- data.frame(observation = rep(c(26.5, 37.2, 57.3, 90.8, 20.2, 89.8)),
                 censoring = rep(c(1, 1, 1, 0, 0, 0)),
                 covariate = rep(c(0.1, 11, -0.3, 2.8, 1.8, 0.4)), stringsAsFactors = T)

Questions

1. Create two groups of observations. In Group 1, \(X < 2\), whereas in Group 2, \(X \geq 2\). Use the ifelse() and as.factor() functions to add a new column to the dataframe, named group, containing factor levels "Group 1" and "Group 2".
   Plot the Kaplan-Meier survival curves corresponding to the two groups. Store the model in the variable fit.km. Be sure to label the curves so that it is clear which curve corresponds to which group. By eye, does there appear to be a difference between the two groups’ survival curves? Answer the question below.
   
   
   • MC1 : Which group do you expect to have a significantly higher risk than the other group?
     • 1: Group 1
     • 2: Group 2
     • 3: There is no significant difference visible between both groups
       
       
2. Fit Cox’s proportional hazards model, using the group indicator as a covariate, store the model in the variable fit.cox. Inspect the output of the model and answer the following question.
   
   
   • MC2 :
     A) The risk associated with group 2 is 0.34 times the risk associated with group 1
     B) There is no evidence the true coefficient value is non-zero
     • 1: Both statements are true
     • 2: Both statements are false
     • 3: A is true, B is false
     • 4: A is false, B is true
       
       
3. Recall that in the case of a single binary covariate, the log-rank test statistic should be identical to the score statistic for the Cox model. Conduct a log-rank test to determine whether there is a difference between the survival curves for the two groups. Store the result in the variable logrank.test. Verify that the log-rank test statistic equals the score statistic for the Cox model and answer the following question.
   
   
   • MC3 : How does the p-value for the log-rank test statistic compare to the p-value for the score statistic for the Cox model from question 2? (ignore any roundings)
     • 1: They are equal
     • 2: The p-value for log-rank is higher
     • 3: The p-value for log-rank is lower

NOTE: the outputs of logrank.test and summary(fit.cox) return rounded \(p\)-values. In order to compare the exact \(p\)-values, inspect the appropriate attributes of the objects.

Assume that:

• The ISLR2 library has been loaded.
• The survival library has been loaded.