Survival analysis with high-dimensional covariates

Abstract

In recent years, breakthroughs in biomedical technology have led to a wealth of data in which the number of features (for instance, genes on which expression measurements are available) exceeds the number of observations (e.g. patients). Sometimes survival outcomes are also available for those same observations. In this case, one might be interested in (a) identifying features that are associated with survival (in a univariate sense), and (b) developing a multivariate model for the relationship between the features and survival that can be used to predict survival in a new observation. Due to the high dimensionality of this data, most classical statistical methods for survival analysis cannot be applied directly. Here, we review a number of methods from the literature that address these two problems.

Figure 1

Estimated FDR for Cox scores, modified Cox scores, and LPC scores are shown for the renal cell carcinoma data set of Zhao et al.

Figure 2

Hierarchical clustering of the patients is shown on the left. Kaplan–Meier survival curves for the two largest subgroups defined by hierarchical clustering are shown on the right; the p-value for the log-rank test is 0.0102.

Figure 3

For the Zhao et al. data, predictors obtained via PC regression and SPC on the training set were used to define three subgroups on the test set. For these subgroups, Kaplan–Meier survival curves and p-values for the log rank test statistic are shown.

Figure 4

For the Zhao et al. data, the y-axes show the average value of 2(l(X_test β̂_train, y_test, δ_test)− l(0, y_test, δ_test)) across cross-validation folds; a large value indicates a good fit on independent data. The notation l(γ, y, δ) indicates the log partial likelihood of the Cox model with outcome (y, δ) and predictor γ. Scout(2, 1) is not shown in this figure because it involves two tuning parameters.