  We provide additional statistical background for the methodology developed in
the clinical analysis of knee osteoarthritis in "A Precision Medicine Approach
to Develop and Internally Validate Optimal Exercise and Weight Loss Treatments
for Overweight and Obese Adults with Knee Osteoarthritis" (Jiang et al. 2020).
Jiang et al. 2020 proposed a pipeline to learn optimal treatment rules with
precision medicine models and compared them with zero-order models with a
Z-test. The model performance was based on value functions, a scalar that
predicts the future reward of each decision rule. The jackknife (i.e.,
leave-one-out cross validation) method was applied to estimate the value
function and its variance of several outcomes in IDEA. IDEA is a randomized
clinical trial studying three interventions (exercise (E), dietary weight loss
(D), and D+E) on overweight and obese participants with knee osteoarthritis. In
this report, we expand the discussion and justification with additional
statistical background. We elaborate more on the background of precision
medicine, the derivation of the jackknife estimator of value function and its
estimated variance, the consistency property of jackknife estimator, as well as
additional simulation results that reflect more of the performance of jackknife
estimators. We recommend reading Jiang et al. 2020 for clinical application and
interpretation of the optimal ITR of knee osteoarthritis as well as the overall
understanding of the pipeline and recommend using this article to understand
the underlying statistical derivation and methodology.
