PERFORMANCE GUARANTEES FOR INDIVIDUALIZED TREATMENT RULES

Abstract

Because many illnesses show heterogeneous response to treatment, there is increasing interest in individualizing treatment to patients [11]. An individualized treatment rule is a decision rule that recommends treatment according to patient characteristics. We consider the use of clinical trial data in the construction of an individualized treatment rule leading to highest mean response. This is a difficult computational problem because the objective function is the expectation of a weighted indicator function that is non-concave in the parameters. Furthermore there are frequently many pretreatment variables that may or may not be useful in constructing an optimal individualized treatment rule yet cost and interpretability considerations imply that only a few variables should be used by the individualized treatment rule. To address these challenges we consider estimation based on l(1) penalized least squares. This approach is justified via a finite sample upper bound on the difference between the mean response due to the estimated individualized treatment rule and the mean response due to the optimal individualized treatment rule.

Fig 1

Plots for: the conditional mean function Q₀(X; A) (left), Q₀(X; A) and the associated best wavelet fit when J_n = 8 (middle), and Q₀(X; A) and the associated best wavelet fit when J_n = 128 (right) (example 4).

Fig 2

Comparison of the l₁-PLS based method with the OLS method and the PP method (examples 1 – 4): Plots for medians and median absolute deviations (MAD) of the Value of the estimated decision rules (top panels) and the number of variables (terms) needed for treatment assignment (including the main treatment effect term, bottom panels) over 1000 samples versus sample size on the log scale. The black dash-dotted line in each plot on the first row denotes the Value of the optimal treatment rule, V (d₀), for each example. (n = 32; 64; 128; 256; 512; 1024. The corresponding numbers of basis functions in example 4 are J_n = 8; 16; 32; 64; 64; 128).