Abstract
Different from the standard treatment discovery framework which is used for finding single treatments for a homogenous group of patients, personalized medicine involves finding therapies that are tailored to each individual in a heterogeneous group. In this paper, we propose a new semiparametric additive single-index model for estimating individualized treatment strategy. The model assumes a flexible and nonparametric link function for the interaction between treatment and predictive covariates. We estimate the rule via monotone B-splines and establish the asymptotic properties of the estimators. Both simulations and an real data application demonstrate that the proposed method has a competitive performance.
Highlights
Where X is a p-dimensional covariate vector and may contain 1 as the intercept, βT X is a single index and both μ and ψ are unknown functions
We note that as sample size increases, the mean squared error of the single index coefficient and estimated link function for three methods decreases, the percentage of making correct decisions (PCD) increases and the estimated value function gets closer to the true value function
We report mean of estimated single index coefficient biases, mean squared errors of estimated single index coefficients, mean squared errors of estimated link functions, PCD and Val over 1000 replications with their empirical standard deviations one line below
Summary
Note that model (1) remains the same if we replace ψ(x) by ψ(rx) for any r > 0. A natural estimate of β is obtained by minimizing the least square, given as n i=1. We can set Nj(βT X) = βT X as a linear function in (1) and compute the ordinary least squares (OLS) estimator for β. Step 2: Given the initial estimates of the index values {Zi = β( )T Xi, i = 1, · · · , n}, minimize over ξ by solving the followng quadratic programming (QP) problem: min ξ. This problem can be solved using the nonlinear least squares (NLS) algorithm. Go to Step 2 and iterate until convergence, i.e. for a small ε > 0, which takes value 1e-3 in our numerical studies. The estimated single index β could change at each step, the knots change in the iteration. 5 to 10 knots will be sufficient to have very good results
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