Abstract
We consider a general resource-allocation problem, namely, to maximize a mean outcome given a cost constraint, through the choice of a treatment rule that is a function of an arbitrary fixed subset of an individual’s covariates.
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Notes
- 1.
We are abusing notation here for the sake of convenience by using \(\varPsi (\cdot )\) to denote the mapping both from the full distribution to \(\mathbb {R}^d\) and from the relevant components to \(\mathbb {R}^d\).
- 2.
The nuisance parameters are those components \(g_0\) of the efficient influence curve\(D^*(Q_0,g_0)\) that \(\varPsi (Q_0)\) does not depend on.
- 3.
It is not hard to extend this model to incorporate uncertainty in E(A|W, Z) for calculating \(c_T(Z,W)\), and thus estimating \(c_T(Z,W)\) from the data, given fixed functions \(c_Z, \ c_A\). There is a correction term that gets added to the efficient influence curve.
- 4.
We are only making this assumption for the sake of easing notation. We can forgo this assumption by introducing notation; i.e., \(Z=l(V)\) is the lower cost intent-to-treat value for a stratum defined by covariates V.
- 5.
The \(U'_Y\) term is an exogenous r.v. whose purpose is for sampling binary Y with mean \(\tilde{f}_Y(W,Z,A,\tilde{U}_Y)\).
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Toth, B., van der Laan, M. (2018). Targeted Learning of Optimal Individualized Treatment Rules Under Cost Constraints. In: Peace, K., Chen, DG., Menon, S. (eds) Biopharmaceutical Applied Statistics Symposium . ICSA Book Series in Statistics. Springer, Singapore. https://doi.org/10.1007/978-981-10-7820-0_1
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