Time-dependent tree-structured survival analysis with unbiased variable selection through permutation tests

Abstract

Incorporating time-dependent covariates into tree-structured survival analysis (TSSA) may result in more accurate prognostic models than if only baseline values are used. Available time-dependent TSSA methods exhaustively test every binary split on every covariate; however, this approach may result in selection bias toward covariates with more observed values. We present a method that uses unbiased significance levels from newly proposed permutation tests to select the time-dependent or baseline covariate with the strongest relationship with the survival outcome. The specific splitting value is identified using only the selected covariate. Simulation results show that the proposed time-dependent TSSA method produces tree models of equal or greater accuracy as compared to baseline TSSA models, even with high censoring rates and large within-subject variability in the time-dependent covariate. To illustrate, the proposed method is applied to data from a cohort of bipolar youths to identify subgroups at risk for self-injurious behavior.

Keywords: bipolar disorder; permutation test; recursive partitioning; repeated measures; variable selection.

Figure 1

Proportion (95% CI) of 200 trees that selected the correct covariate for split 1. This node split on a continuous, time-dependent covariate. Solid line: time-dependent tree. Dashed line: baseline tree.

Figure 2

Mean bias (95% CI) of the selected cut-point at split 1 among trees with the correct variable selected at split 1. Solid line: time-dependent tree. Dashed line: baseline tree.

Figure 3

Proportion (95% CI) of trees that selected the correct variable for split 2, conditional on having selected the correct variable for split 1. Solid line: time-dependent tree. Dashed line: baseline tree.

Figure 4

Time-dependent discrimination index (AUC) and 95% CI based on an independent sample of N = 200. A value of 1 represents perfect discrimination between individuals with different survival outcomes; a value of .5 is equivalent to discrimination based only on chance. Note that the scale ranges only from .5 to 1. Solid line: time-dependent tree. Dashed line: baseline tree.

Figure 5

Time-dependent tree-structured survival analysis model. Each terminal node of the tree shows estimated time to 50% events and 25% events, along with 95% confidence intervals. For continuous variables, observations ≤ the cut point are sent to the left child node. For the categorical variable, observations with no first degree relatives with a mood disorder are sent to the left child node. Abbreviations: % MDE = % Weeks in a major depressive episode, % MIX = % Weeks with mixed mood symptoms, % SUD = % Weeks with a substance use disorder, % PST = % Hours spent in inpatient or outpatient psychosocial treatment, FDRs with Mood dx = First degree relatives with a history of mood disorders.