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Collaborative Double Robust Targeted Maximum Likelihood Estimation

Author

Listed:
  • van der Laan Mark J.

    (University of California, Berkeley)

  • Gruber Susan

    (University of California, Berkeley)

Abstract

Collaborative double robust targeted maximum likelihood estimators represent a fundamental further advance over standard targeted maximum likelihood estimators of a pathwise differentiable parameter of a data generating distribution in a semiparametric model, introduced in van der Laan, Rubin (2006). The targeted maximum likelihood approach involves fluctuating an initial estimate of a relevant factor (Q) of the density of the observed data, in order to make a bias/variance tradeoff targeted towards the parameter of interest. The fluctuation involves estimation of a nuisance parameter portion of the likelihood, g. TMLE has been shown to be consistent and asymptotically normally distributed (CAN) under regularity conditions, when either one of these two factors of the likelihood of the data is correctly specified, and it is semiparametric efficient if both are correctly specified.In this article we provide a template for applying collaborative targeted maximum likelihood estimation (C-TMLE) to the estimation of pathwise differentiable parameters in semi-parametric models. The procedure creates a sequence of candidate targeted maximum likelihood estimators based on an initial estimate for Q coupled with a succession of increasingly non-parametric estimates for g. In a departure from current state of the art nuisance parameter estimation, C-TMLE estimates of g are constructed based on a loss function for the targeted maximum likelihood estimator of the relevant factor Q that uses the nuisance parameter to carry out the fluctuation, instead of a loss function for the nuisance parameter itself. Likelihood-based cross-validation is used to select the best estimator among all candidate TMLE estimators of Q0 in this sequence. A penalized-likelihood loss function for Q is suggested when the parameter of interest is borderline-identifiable.We present theoretical results for "collaborative double robustness," demonstrating that the collaborative targeted maximum likelihood estimator is CAN even when Q and g are both mis-specified, providing that g solves a specified score equation implied by the difference between the Q and the true Q0. This marks an improvement over the current definition of double robustness in the estimating equation literature.We also establish an asymptotic linearity theorem for the C-DR-TMLE of the target parameter, showing that the C-DR-TMLE is more adaptive to the truth, and, as a consequence, can even be super efficient if the first stage density estimator does an excellent job itself with respect to the target parameter.This research provides a template for targeted efficient and robust loss-based learning of a particular target feature of the probability distribution of the data within large (infinite dimensional) semi-parametric models, while still providing statistical inference in terms of confidence intervals and p-values. This research also breaks with a taboo (e.g., in the propensity score literature in the field of causal inference) on using the relevant part of likelihood to fine-tune the fitting of the nuisance parameter/censoring mechanism/treatment mechanism.

Suggested Citation

  • van der Laan Mark J. & Gruber Susan, 2010. "Collaborative Double Robust Targeted Maximum Likelihood Estimation," The International Journal of Biostatistics, De Gruyter, vol. 6(1), pages 1-71, May.
  • Handle: RePEc:bpj:ijbist:v:6:y:2010:i:1:n:17
    DOI: 10.2202/1557-4679.1181
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    References listed on IDEAS

    as
    1. Rose Sherri & van der Laan Mark J., 2008. "Simple Optimal Weighting of Cases and Controls in Case-Control Studies," The International Journal of Biostatistics, De Gruyter, vol. 4(1), pages 1-26, September.
    2. van der Laan Mark J. & Petersen Maya L, 2007. "Causal Effect Models for Realistic Individualized Treatment and Intention to Treat Rules," The International Journal of Biostatistics, De Gruyter, vol. 3(1), pages 1-55, March.
    3. van der Laan Mark J. & Dudoit Sandrine & Keles Sunduz, 2004. "Asymptotic Optimality of Likelihood-Based Cross-Validation," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 3(1), pages 1-25, March.
    4. S. A. Murphy, 2003. "Optimal dynamic treatment regimes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 331-355, May.
    5. van der Laan Mark J. & Rubin Daniel, 2006. "Targeted Maximum Likelihood Learning," The International Journal of Biostatistics, De Gruyter, vol. 2(1), pages 1-40, December.
    6. Vaart Aad W. van der & Dudoit Sandrine & Laan Mark J. van der, 2006. "Oracle inequalities for multi-fold cross validation," Statistics & Risk Modeling, De Gruyter, vol. 24(3), pages 351-371, December.
    7. van der Laan Mark J. & Petersen Maya L & Joffe Marshall M, 2005. "History-Adjusted Marginal Structural Models and Statically-Optimal Dynamic Treatment Regimens," The International Journal of Biostatistics, De Gruyter, vol. 1(1), pages 1-41, November.
    8. Rubin Daniel B & van der Laan Mark J., 2008. "Empirical Efficiency Maximization: Improved Locally Efficient Covariate Adjustment in Randomized Experiments and Survival Analysis," The International Journal of Biostatistics, De Gruyter, vol. 4(1), pages 1-42, May.
    9. Rose Sherri & van der Laan Mark J., 2009. "Why Match? Investigating Matched Case-Control Study Designs with Causal Effect Estimation," The International Journal of Biostatistics, De Gruyter, vol. 5(1), pages 1-26, January.
    10. Laan Mark J. van der & Dudoit Sandrine & Vaart Aad W. van der, 2006. "The cross-validated adaptive epsilon-net estimator," Statistics & Risk Modeling, De Gruyter, vol. 24(3), pages 373-395, December.
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