zbMATH — the first resource for mathematics

Performance guarantees for individualized treatment rules. (English) Zbl 1216.62178
Summary: Because many illnesses show heterogeneous response to treatment, there is increasing interest in individualizing treatment to patients [T. R. Insel, Arch. Gen. Psychiatry 66, 128–133 (2009)]. An individualized treatment rule is a decision rule that recommends treatment according to patient characteristics. We consider the use of clinical trial data in the construction of an individualized treatment rule leading to highest mean response. This is a difficult computational problem because the objective function is the expectation of a weighted indicator function that is nonconcave in the parameters. Furthermore, there are frequently many pretreatment variables that may or may not be useful in constructing an optimal individualized treatment rule, yet cost and interpretability considerations imply that only a few variables should be used by the individualized treatment rule. To address these challenges, we consider estimation based on $$l_1$$-penalized least squares. This approach is justified via a finite sample upper bound on the difference between the mean response due to the estimated individualized treatment rule and the mean response due to the optimal individualized treatment rule.

MSC:
 62P10 Applications of statistics to biology and medical sciences; meta analysis 92C50 Medical applications (general) 62C99 Statistical decision theory 62N02 Estimation in survival analysis and censored data 62H99 Multivariate analysis 65C60 Computational problems in statistics (MSC2010)
Full Text:
References:
 [1] Bartlett, P. L. (2008). Fast rates for estimation error and oracle inequalities for model selection. Econometric Theory 24 545-552. · Zbl 1284.62583 · doi:10.1017/S0266466608080225 [2] Bartlett, P. L., Jordan, M. L. and McAuliffe, P. L. (2006). Convexity, classification, and risk bounds. J. Amer. Statist. Assoc. 101 138-156. · Zbl 1118.62330 · doi:10.1198/016214505000000907 · miranda.asa.catchword.org [3] Bickel, P. J., Ritov, Y. and Tsybakov, A. B. (2009). Simultaneous analysis of lasso and Dantzig selector. Ann. Statist. 37 1705-1732. · Zbl 1173.62022 · doi:10.1214/08-AOS620 [4] Bunea, F., Tsybakov, A. and Wegkamp, M. (2007). Sparsity oracle inequalities for the Lasso. Electron. J. Stat. 1 169-194 (electronic). · Zbl 1146.62028 · doi:10.1214/07-EJS008 [5] Cai, T., Tian, L., Lloyd-Jones, D. M. and Wei, L. J. (2008). Evaluating subject-level incremental values of new markers for risk classification rule. Working Paper 91, Harvard Univ. Biostatistics Working Paper Series. · Zbl 1322.62306 [6] Cai, T., Tian, L., Uno, H., Solomon, S. D. and Wei, L. J. (2010). Calibrating parametric subject-specific risk estimation. Biometrika 97 389-404. · Zbl 1205.62161 · doi:10.1093/biomet/asq012 [7] Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences , 2nd ed. Lawrence Erlbaum Associates, Hillsdale, NJ. · Zbl 0747.62110 [8] Donoho, D. L. and Johnstone, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 425-455. JSTOR: · Zbl 0815.62019 · doi:10.1093/biomet/81.3.425 · links.jstor.org [9] Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. J. Amer. Statist. Assoc. 96 1348-1360. JSTOR: · Zbl 1073.62547 · doi:10.1198/016214501753382273 · links.jstor.org [10] Feldstein, M. L., Savlov, E. D. and Hilf, R. (1978). A statistical model for predicting response of breast cancer patients to cytotoxic chemotherapy. Cancer Res. 38 2544-2548. [11] Insel, T. R. (2009). Translating scientific opportunity into public health impact: A strategic plan for research on mental illness. Arch. Gen. Psychiatry 66 128-133. [12] Ishigooka, J., Murasaki, M. Miura, S. and The Olanzapine Late-Phase II Study Group (2011). Olanzapine optimal dose: Results of an open-label multicenter study in schizophrenic patients. Psychiatry and Clinical Neurosciences 54 467-478. [13] Keller, M. B., McCullough, J. P., Klein, D. N., Arnow, B., Dunner, D. L., Gelenberg, A. J., Markowitz, J. C., Nemeroff, C. B., Russell, J. M., Thase, M. E., Trivedi, M. H. and Zajecka, J. (2000). A comparison of nefazodone, the cognitive behavioral-analysis system of psychotherapy, and their combination for the treatment of chronic depression. N. Engl. J. Med. 342 1462-1470. [14] Kent, D. M., Hayward, R. A., Griffith, J. L., Vijan, S., Beshansky, J. R., Califf, R. M. and Selker, H. P. (2002). An independently derived and validated predictive model for selecting patients with myocardial infarction who are likely to benefit from tissue plasminogen activator compared with streptokinase. Am. J. Med. 113 104-111. [15] Koltchinskii, V. (2009). Sparsity in penalized empirical risk minimization. Ann. Inst. H. Poincaré Probab. Statist. 45 7-57. · Zbl 1168.62044 · doi:10.1214/07-AIHP146 · eudml:78023 [16] Lesko, L. J. (2007). Personalized medicine: Elusive dream or imminent reality? Clin. Pharmacol. Ther. 81 807-816. [17] Lunceford, J. K., Davidian, M. and Tsiatis, A. A. (2002). Estimation of survival distributions of treatment policies in two-stage randomization designs in clinical trials. Biometrics 58 48-57. JSTOR: · Zbl 1209.62307 · doi:10.1111/j.0006-341X.2002.00048.x · links.jstor.org [18] Mammen, E. and Tsybakov, A. B. (1999). Smooth discrimination analysis. Ann. Statist. 27 1808-1829. · Zbl 0961.62058 · doi:10.1214/aos/1017939240 [19] Massart, P. (2005). A non-asymptotic theory for model selection. In European Congress of Mathematics 309-323. Eur. Math. Soc., Zürich. · Zbl 1070.62002 [20] Murphy, S. A. (2003). Optimal dynamic treatment regimes. J. R. Stat. Soc. Ser. B Stat. Methodol. 65 331-366. JSTOR: · Zbl 1065.62006 · doi:10.1111/1467-9868.00389 · links.jstor.org [21] Murphy, S. A. (2005). A generalization error for Q-learning. J. Mach. Learn. Res. 6 1073-1097 (electronic). · Zbl 1222.68271 · www.jmlr.org [22] Murphy, S. A., van der Laan, M. J., Robins, J. M. and (2001). Marginal mean models for dynamic regimes. J. Amer. Statist. Assoc. 96 1410-1423. JSTOR: · Zbl 1051.62114 · doi:10.1198/016214501753382327 · links.jstor.org [23] Piquette-Miller, P. and Grant, D. M. (2007). The art and science of personalized medicine. Clin. Pharmacol. Ther. 81 311-315. [24] Polonik, W. (1995). Measuring mass concentrations and estimating density contour clusters-an excess mass approach. Ann. Statist. 23 855-881. · Zbl 0841.62045 · doi:10.1214/aos/1176324626 [25] Qian, M. and Murphy, S. A. (2011). Supplement to “Performance guarantees for individualized treatment rules.” DOI: . · Zbl 1216.62178 · dx.doi.org [26] Robins, J., Orellana, L. and Rotnitzky, A. (2008). Estimation and extrapolation of optimal treatment and testing strategies. Stat. Med. 27 4678-4721. · doi:10.1002/sim.3301 [27] Robins, J. M. (2004). Optimal-regime structural nested models. In Proceedings of the Second Seattle Symposium on Biostatistics (D. Y. Lin and P. Haegerty eds.). Springer, New York. · Zbl 1279.62024 [28] Stoehlmacher, J., Park, D. J., Zhang, W., Yang, D., Groshen, S., Zahedy, S. and Lenz, H.-J. (2004). A multivariate analysis of genomic polymorphisms: Prediction of clinical outcome to 5-FU/oxaliplatin combination chemotherapy in refractory colorectal cancer. Br. J. Cancer 91 344-354. [29] Thall, P. F., Millikan, R. E. and Sung, H. G. (2000). Evaluating multiple treatment courses in clinical trials. Stat. Med. 19 1011-1028. [30] Thall, P. F., Sung, H.-G. and Estey, E. H. (2002). Selecting therapeutic strategies based on efficacy and death in multicourse clinical trials. J. Amer. Statist. Assoc. 97 29-39. JSTOR: · Zbl 1073.62590 · doi:10.1198/016214502753479202 · links.jstor.org [31] Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Ser. B 58 267-288. JSTOR: · Zbl 0850.62538 · links.jstor.org [32] Tsybakov, A. B. (2004). Optimal aggregation of classifiers in statistical learning. Ann. Statist. 32 135-166. · Zbl 1105.62353 · doi:10.1214/aos/1079120131 · euclid:aos/1079120131 [33] van de Geer, S. A. (2008). High-dimensional generalized linear models and the lasso. Ann. Statist. 36 614-645. · Zbl 1138.62323 · doi:10.1214/009053607000000929 [34] van der Laan, M. J., Petersen, M. L. and Joffe, M. M. (2005). History-adjusted marginal structural models and statically-optimal dynamic treatment regimens. Int. J. Biostat. 1 Art. 4, 41 pp. (electronic). · Zbl 1080.62097 · doi:10.2202/1557-4679.1003 · www.bepress.com [35] Wahed, A. S. and Tsiatis, A. A. (2006). Semiparametric efficient estimation of survival distributions in two-stage randomisation designs in clinical trials with censored data. Biometrika 93 163-177. · Zbl 1152.62397 · doi:10.1093/biomet/93.1.163 [36] Zhang, C.-H. and Huang, J. (2008). The sparsity and bias of the LASSO selection in high-dimensional linear regression. Ann. Statist. 36 1567-1594. · Zbl 1142.62044 · doi:10.1214/07-AOS520 [37] Zou, H. (2006). The adaptive lasso and its oracle properties. J. Amer. Statist. Assoc. 101 1418-1429. · Zbl 1171.62326 · doi:10.1198/016214506000000735
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.