Lasso tree for cancer staging with survival data.

The tumor-node-metastasis staging system has been the lynchpin of cancer diagnosis, treatment, and prognosis for many years. For meaningful clinical use, an orderly grouping of the T and N categories into a staging system needs to be defined, usually with respect to a time-to-event outcome. This can be reframed as a model selection problem with respect to features arranged on a partially ordered two-way grid, and a penalized regression method is proposed for selecting the optimal grouping. Instead of penalizing the L1-norm of the coefficients like lasso, in order to enforce the stage grouping, we place L1 constraints on the differences between neighboring coefficients. The underlying mechanism is the sparsity-enforcing property of the L1 penalty, which forces some estimated coefficients to be the same and hence leads to stage grouping. Partial ordering constraints is also required as both the T and N categories are ordinal. A series of optimal groupings with different numbers of stages can be obtained by varying the tuning parameter, which gives a tree-like structure offering a visual aid on how the groupings are progressively made. We hence call the proposed method the lasso tree. We illustrate the utility of our method by applying it to the staging of colorectal cancer using survival outcomes. Simulation studies are carried out to examine the finite sample performance of the selection procedure. We demonstrate that the lasso tree is able to give the right grouping with moderate sample size, is stable with regard to changes in the data, and is not affected by random censoring.