Because many illnesses show heterogeneous response to treatment, there is increasing interest in individualizing treatment to patients (Insel 2009). An individualized treatment rule is a decision rule that recommends treatment according to patient characteristics. Assuming high clinical outcomes are favorable, we consider the use of clinical trial data in the construction of an individualized treatment rule leading to highest mean outcome. 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. To deal with the computational difficulty, we consider estimation based on minimization of a quadratic loss.;This dissertation investigates model selection and l 1 penalization techniques aiming to improve the quality of the quadratic loss minimization method. Note that 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 treatment rule. In the first approach we consider the use of an l1 penalty in addition to the quadratic loss. Furthermore, although the quadratic minimization approach reduces the computational difficulty, it may deviate from the goal of estimating the best individualized treatment rule since a different loss function is used. In the second approach, we consider the use of model selection techniques, where a treatment rule is obtained by minimizing the quadratic loss within each model and then a model is selected by maximizing the original objective function. To justify these two approaches, we provide finite sample upper bounds on the difference between the mean outcome due to the estimated individualized treatment rule and the mean response due to the optimal individualized treatment rule.
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