Applied

This exercise focuses on the brain tumor data, which is included in the ISLP library.

(a) Plot the Kaplan-Meier survival curve with ± 1 standard error bands, using the KaplanMeierFitter() estimator in the lifelines package.
(b) Draw a bootstrap sample of size n = 88 from the pairs ( yi, δi ), and compute the resulting Kaplan-Meier survival curve. Repeat this process B = 200 times. Use the results to obtain an estimate of the standard error of the Kaplan-Meier survival curve at each timepoint. Compare this to the standard errors obtained in (a).
(c) Fit a Cox proportional hazards model that uses all of the predictors to predict survival. Summarize the main findings.
(d) Stratify the data by the value of ki . (Since only one observation has ki==40 , you can group that observation together with the observations that have ki==60 .) Plot Kaplan-Meier survival curves for each of the five strata, adjusted for the other predictors.

This exercise makes use of the data in Table 11.4.

(a) Create two groups of observations. In Group 1, X < 2, whereas in Group 2, X ≥ 2. Plot the Kaplan-Meier survival curves corresponding to the two groups. 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?
(b) Fit Cox’s proportional hazards model, using the group indicator as a covariate. What is the estimated coefficient? Write a sentence providing the interpretation of this coefficient, in terms of the hazard or the instantaneous probability of the event. Is there evidence that the true coefficient value is non-zero?
(c) Recall from Section 11.5.2 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. 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 (b)?

12 Unsupervised Learning

Most of this book concerns supervised learning methods such as regression and classification. In the supervised learning setting, we typically have access to a set of p features X 1 , X 2 , . . . , Xp , measured on n observations, and a response Y also measured on those same n observations. The goal is then to predict Y using X 1 , X 2 , . . . , Xp .

This chapter will instead focus on unsupervised learning , a set of statistical tools intended for the setting in which we have only a set of features X 1 , X 2 , . . . , Xp measured on n observations. We are not interested in prediction, because we do not have an associated response variable Y . Rather, the goal is to discover interesting things about the measurements on X 1 , X 2 , . . . , Xp . Is there an informative way to visualize the data? Can we discover subgroups among the variables or among the observations? Unsupervised learning refers to a diverse set of techniques for answering questions such as these. In this chapter, we will focus on two particular types of unsupervised learning: principal components analysis , a tool used for data visualization or data pre-processing before supervised techniques are applied, and clustering , a broad class of methods for discovering unknown subgroups in data.

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