  Learning an individualized dose rule in personalized medicine is a
challenging statistical problem. Existing methods often suffer from the curse
of dimensionality, especially when the decision function is estimated
nonparametrically. To tackle this problem, we propose a dimension reduction
framework that effectively reduces the estimation to a lower-dimensional
subspace of the covariates. We exploit that the individualized dose rule can be
defined in a subspace spanned by a few linear combinations of the covariates,
leading to a more parsimonious model. The proposed framework does not require
the inverse probability of the propensity score under observational studies due
to a direct maximization of the value function. This distinguishes us from the
outcome weighted learning framework, which also solves decision rules directly.
Under the same framework, we further propose a pseudo-direct learning approach
that focuses more on estimating the dimensionality-reduced subspace of the
treatment outcome. Parameters in both approaches can be estimated efficiently
using an orthogonality constrained optimization algorithm on the Stiefel
manifold. Under mild regularity assumptions, the results on the asymptotic
normality of the proposed estimators are established, respectively. We also
derive the consistency and convergence rate for the value function under the
estimated optimal dose rule. We evaluate the performance of the proposed
approaches through extensive simulation studies and a warfarin pharmacogenetic
dataset.
