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 \(\tilde{f}_Y(W,Z,A,\tilde{U}_Y)\).
References
Angrist, J. D., & Krueger, A. B. (1991). Does compulsory school attendance affect schooling and earnings? The Quarterly Journal of Economics, 106(4), 979–1014.
Angrist, J. D., Imbens, G. W., & Rubin, D. B. (1996). Identification of causal effects using instrumental variables. Journal of the American Statistical Association, 91, 444–471.
Brookhart, M. A., & Schneeweiss, S. (2007). Preference-based instrumental variable methods for the estimation of treatment effects. International Journal of Biostatistics, 3(1), 1–14.
Brookhart, M. A., Rassen, J. A., & Schneeweiss, S. (2010). Instrumental variable methods in comparative safety and effectiveness research. Pharmacoepidemiology and Drug Safety, 19(6), 537–554.
Chakraborty, B., & Moodie, E. E. (2013). Statistical methods for dynamic treatment regimes. Berlin Heidelberg New York: Springer.
Chakraborty, B., Laber, E., & Zhao, Y. (2013). Inference for optimal dynamic treatment regimes using an adaptive m-out-of-n bootstrap scheme. Biometrics, 69(3), 714–723.
Chesney, M. A. (2006). The elusive gold standard. Future perspectives for HIV adherence assessment and intervention. Journal of Acquired Immune Deficiency Syndromes, 43(1), S149–155.
Editors: National Research Council (US) Committee on A Framework for Developing a New Taxonomy of Disease. (2011). Toward precision medicine: Building a knowledge network for biomedical research and a new taxonomy of disease. National Academies Press (US), Washington DC.
Gruber, S., & van der Laan, M. (2010). A targeted maximum likelihood estimator of a causal effect on a bounded continuous outcome. International Journal of Biostatistics, 6(1). Article 26.
Imbens, G. W., & Angrist, J. D. (1994). Identification and estimation of local average treatment effects. Econometrica, 62, 467–475.
Karp, R. (1972). Reducibility among combinatorial problems. New York Berlin Heidelberg: Springer.
Luedtke, A., & van der Laan, M. (2016a). Optimal individualized treatments in resource-limited settings. International Journal of Biostatistics, 12(1), 283–303.
Luedtke, A., & van der Laan, M. (2016b). Statistical inference for the mean outcome under a possibly non-unique optimal treatment strategy. Annals of Statistics, 44(2), 713–742.
Newey, W. (1990). Semiparametric efficiency bounds. Journal of Applied Econometrics, 5(2), 99–135.
Pearl, J. (2000). Causality: Models, reasoning, and inference. Cambridge University Press.
Robins, J. M. (2004). Optimal structural nested models for optimal sequential decisions. in Proceeding of the Second Seattle Symposium in Biostatistics (Vol. 179, pp. 189–326).
Toth, B. (2016). Targeted learning of individual effects and individualized treatments using an instrumental variable. PhD dissertation, U.C. Berkeley.
Toth, B., & van der Laan, M. (2016). TMLE for marginal structural models based on an instrument. U.C. Berkeley Division of Biostatistics Working Papers Series, working paper 350.
van der Laan, M., & Rose, S. (2011). Targeted learning: Causal inference for observational and experimental data. New York: Springer.
van der Laan, M., Rubin, D. (2006). Targeted maximum likelihood learning. International Journal of Biostatistics, 2(1). Article 11.
van der Laan, M., Polley, E. C. & Hubbard, A. (2007). Super learner. Statistical Applications in Genetics and Molecular Biology, 6(1). Article 25.
Zhang, B., Tsiatis, A., Davidian, M., Zhang, M., & Laber, E. (2012). A robust method for estimating optimal treatment regimes. Biometrics, 68, 1010–1018.
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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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