Data-adaptive doubly robust instrumental variable methods for treatment effect heterogeneity

DiazOrdaz, Karla; Daniel, Rhian; Kreif, Noemi

Journal de la société française de statistique, Causality, Tome 161 (2020) no. 1, pp. 135-163.

We consider the estimation of the average treatment effect in the treated as a function of baseline covariates, where there is a valid (conditional) instrument.

We describe two doubly-robust (DR) estimators: a g-estimator and a targeted minimum loss-based estimator (TMLE). These estimators can be viewed as generalisations of the two-stage least squares (TSLS) method to semi-parametric models that make weaker assumptions. We exploit recent theoretical results and use data-adaptive estimation of the nuisance parameters for the g-estimator.

A simulation study is used to compare standard TSLS with the two DR estimators’ finite-sample performance when using (1) parametric or (2) data-adaptive estimation of the nuisance parameters.

Data-adaptive DR estimators have lower bias and improved coverage, when compared to incorrectly specified parametric DR estimators and TSLS. When the parametric model for the treatment effect curve is correctly specified, the g-estimator outperforms all others, but when this model is misspecified, TMLE performs best, while TSLS can result in large biases and zero coverage.

The methods are also applied to the COPERS (COping with persistent Pain, Effectiveness Research in Self-management) trial to make inferences about the causal effect of treatment actually received, and the extent to which this is modified by depression at baseline.

[Abadie, 2003] Abadie, A. (2003). Semiparametric instrumental variable estimation of treatment response models. Journal of econometrics, 113(2):231–263. | MR

[Angrist et al., 1996] Angrist, J. D., Imbens, G. W., and Rubin, D. B. (1996). Identification of causal effects using instrumental variables. Journal of the American Statistical Association, 91(434):pp. 444–455.

[Athey et al., 2018] Athey, S., G. W. Imbens, and S. Wager (2018). Approximate residual balancing: debiased inference of average treatment effects in high dimensions. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 80(4), 597–623.

[Belloni et al., 2012] Belloni, A., Chen, D., Chernozhukov, V., and Hansen, C. (2012). Sparse models and methods for optimal instruments with an application to eminent domain. Econometrica, 80(6):2369–2429. | MR

[Benkeser et al., 2017] Benkeser, D., Carone, M., Laan, M. V. D., and Gilbert, P. (2017). Doubly robust nonparametric inference on the average treatment effect. Biometrika, 104(4):863–880. | MR

[Benkeser and Laan, 2016] Benkeser, D. and Laan, M. V. D. (2016). The highly adaptive lasso estimator. In 2016 IEEE International Conference on Data Science and Advanced Analytics (DSAA), pages 689–696.

[Bickel, 1982] Bickel, P., F. (1982). On adaptive estimation. Annals of Statistics 10, 647–71 | MR

[Bickel et al., 1997] Bickel, P., F. Götze, and W. van Zwet (1997). Resampling fewer than n observations: Gains, losses, and remedies for losses. Statistica Sinica 7, 1–31. | MR

[Chernozhukov et al., 2017] Chernozhukov, V., D. Chetverikov, M. Demirer, E. Duflo, C. Hansen, and W. Newey (2017). Double/debiased/neyman machine learning of treatment effects. American Economic Review 107(5), 261–65.

[Chernozhukov et al., 2018] Chernozhukov, V., D. Chetverikov, M. Demirer, E. Duflo, C. Hansen, W. Newey, and J. Robins (2018). Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal 21(1), C1–C68. | MR

[Dodd et al., 2012] Dodd, S., White, I., and Williamson, P. (2012). Nonadherence to treatment protocol in published randomised controlled trials: a review. Trials, 13(1):84.

[Dunn and Bentall, 2007] Dunn, G. and Bentall, R. (2007). Modelling treatment-effect heterogeneity in randomized controlled trials of complex interventions (psychological treatments). Statistics in Medicine, 26(26):4719–4745. | MR

[Farrell, 2015] Farrell, M. H. (2015). Robust inference on average treatment effects with possibly more covariates than observations. Journal of Econometrics, 189(1):1–23. | MR

[Frölich, 2007] Frölich, M. (2007). Nonparametric IV estimation of local average treatment effects with covariates. Journal of Econometrics, 139(1):35 – 75. Endogeneity, instruments and identification. | MR

[Gruber and van der Laan, 2010] Gruber, S. and van der Laan, M. J. (2010). A targeted maximum likelihood estimator of a causal effect on a bounded continuous outcome. The International Journal of Biostatistics, 6(1). | MR

[Györfi et al., 2006] Györfi, L., Kohler, M., Krzyzak, A., and Walk, H. (2006). A distribution-free theory of nonparametric regression. Springer Science & Business Media.

[Hernán and Robins, 2006] Hernán, M. A. and Robins, J. M. (2006). Instruments for causal inference: an epidemiologist’s dream? Epidemiology, 17(4):360–372.

[Hirano et al., 2000] Hirano, K., Imbens, G. W., Rubin, D. B., and Zhou, X.-H. (2000). Assessing the effect of an influenza vaccine in an encouragement design. Biostatistics, 1(1):69–88.

[Kang et al., 2007] Kang, J. D., Schafer, J. L., et al. (2007). Demystifying double robustness: A comparison of alternative strategies for estimating a population mean from incomplete data. Statistical science, 22(4):523–539. | MR

[Kennedy, 2016] Kennedy, E. H. (2016). Semiparametric Theory and Empirical Processes in Causal Inference, Chapter Statistical Causal Inferences and Their Applications in Public Health Research, pp. 141–167. Cham: Springer International Publishing. | MR

[Little and Yau, 1998] Little, R. J. and Yau, L. H. (1998). Statistical techniques for analyzing data from prevention trials: Treatment of no-shows using Rubin’s causal model. Psychological Methods, 3(2):147.

[Newey, 1990] Newey, W. K. (1990). Semiparametric efficiency bounds. Journal of Applied Econometrics, 5(2):99–135.

[Newey and McFadden, 1994] Newey, W. K. and McFadden, D. (1994) Large sample estimation and hypothesis testing, in Handbook of Econometrics, Elsevier B.V., 2111–2245. | MR

[Ogburn et al., 2015] Ogburn, E. L., Rotnitzky, A., and Robins, J. M. (2015). Doubly robust estimation of the local average treatment effect curve. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 77(2):373–396. | MR

[Okui et al., 2012] Okui, R., Small, D. S., Tan, Z., and Robins, J. M. (2012). Doubly robust instrumental variable regression. Statistica Sinica, 22(1):173–205. | MR

[Petersen et al., 2014] Petersen, M., Schwab, J., Gruber, S., Blaser, N., Schomaker, M., and van der Laan, M. (2014). Targeted maximum likelihood estimation for dynamic and static longitudinal marginal structural working models. Journal of causal inference, 2(2):147–185. | MR

[Pirracchio et al., 2015] Pirracchio, R., Petersen, M. L., and van der Laan, M. (2015). Improving propensity score estimators’ robustness to model misspecification using super learner. American journal of epidemiology, 181(2):108–119.

[Porter et al., 2011] Porter, K. E., Gruber, S., van der Laan, M. J., and Sekhon, J. S. (2011). The relative performance of targeted maximum likelihood estimators. The international journal of biostatistics, 7(1):1–34. | MR

[Robins, 1994] Robins, J. M. (1994). Correcting for non-compliance in randomized trials using structural nested mean models. Communications in Statistics - Theory and Methods, 23(8):2379–2412. | MR

[Robins, 2000] Robins, J. M. (2000). Robust estimation in sequentially ignorable missing data and causal inference models. In Proceedings of the American Statistical Association, Section on Bayesian Statistical Science, 6–10.

[Rubin, 1978] Rubin, D. B. (1978). Bayesian inference for causal effects: The role of randomization. The Annals of statistics, 34–58. | MR

[Swanson et al., 2018] Swanson, S. A., Hernan, M. A., Miller, M., Robins, J. M. and Richardson T. S. (2018) Partial Identification of the Average Treatment Effect Using Instrumental Variables: Review of Methods for Binary Instruments, Treatments, and Outcomes. Journal of the American Statistical Association 113(522):933–947. | MR

[Tan, 2006] Tan, Z. (2006). Regression and weighting methods for causal inference using instrumental variables. Journal of the American Statistical Association, 101(476):1607–1618. | MR

[Tan, 2010] Tan, Z. (2010). Marginal and nested structural models using instrumental variables. Journal of the American Statistical Association, 105(489):157–169. | MR

[Taylor et al., 2016] Taylor, S. J., Carnes, D., and Homer, Kate, e. a. (2016). Improving the self-management of chronic pain: Coping with persistent pain, effectiveness research in self-management (copers). Programme Grants for Applied Research, 4(14).

[Tóth and van der Laan, 2016] Tóth, B. and van der Laan, M. J. (2016). TMLE for marginal structural models based on an instrument. U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 350.

[van der Laan and Rubin, 2006] van der Laan, M. J. and D. Rubin (2006). Targeted maximum likelihood learning. The International Journal of Biostatistics 2(1). | MR

[van der Laan and Rose, 2011] van der Laan, M. and Rose, S. (2011). Targeted Learning: Causal Inference for Observational and Experimental Data. Springer Series in Statistics. | MR

[van der Laan and Rose, 2018] van der Laan, M. and Rose, S. (2018). Targeted Learning in Data Science: Causal Inference for Complex Longitudinal Studies. Springer International Publishing | MR

[van der Laan and Robins, 2003] van der Laan, M. J. and Robins, J.M., (2003). Unified Methods for Censored Longitudinal Data and Causality, Springer Series in Statistics. | MR

[van der Laan and Gruber, 2011] van der Laan, M. J. and Gruber, S. (2011). Targeted minimum loss based estimation of an intervention specific mean outcome. newblock U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 290.

[van der Laan and Gruber, 2012] van der Laan, M. J. and Gruber, S. (2012). Targeted minimum loss based estimation of causal effects of multiple time point interventions. The international journal of biostatistics, 8(1). | MR

[van der Laan and Luedtke, 2015] van der Laan, M. J. and Luedtke, A. R. (2015). Targeted learning of the mean outcome under an optimal dynamic treatment rule. Journal of causal inference, 3(1):61–95. | MR

[van der Laan et al., 2007] van der Laan, M. J., Polley, E. C., and Hubbard, A. E. (2007). Super learner. Statistical applications in genetics and molecular biology, 6(1):1–21. | MR

[van der Vaart, 2014] van der Vaart, A. (2014). Higher Order Tangent Spaces and Influence Functions. Statistical Science 29(4), 679–686. | MR

[VanderWeele, 2009] VanderWeele, T. J. (2009). Concerning the consistency assumption in causal inference. Epidemiology, 20(6):880–883.

[Vansteelandt and Didelez, 2018] Vansteelandt, S. and Didelez, V. (2018). Improving the robustness and efficiency of covariate adjusted linear instrumental variable estimators. to appear in Scandinavian Journal of Statistics. | MR

[Vermeulen and Vansteelandt, 2016] Vermeulen, K. and Vansteelandt, S. (2016). Data-adaptive bias-reduced doubly robust estimation. The international journal of biostatistics, 12(1):253–282. | MR

[Wiles et al., 2014] Wiles, N. J., Fischer, K., Cowen, P., Nutt, D., Peters, T. J., Lewis, G., and White, I. R. (2014). Allowing for non-adherence to treatment in a randomized controlled trial of two antidepressants (citalopram versus reboxetine): an example from the genpod trial. Psychological Medicine, 44(13):2855–2866.

[Wooldridge, 2010] Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data. MIT Press. | MR

[Zhang et al., 2014] Zhang, Z., Peluso, M. J., Gross, C. P., Viscoli, C. M., and Kernan, W. N. (2014). Adherence reporting in randomized controlled trials. Clinical Trials, 11(2):195–204.

[Zheng and van der Laan, 2012] Zheng, W. and van der Laan, M. J. (2012). Targeted maximum likelihood estimation of natural direct effects. The international journal of biostatistics, 8(1):1–40. | MR

[Zheng and van der Laan, 2011] Zheng, W. and van der Laan, M. J. (2011). Cross-validated targeted minimum-loss-based estimation. Targeted Learning. Springer, New York, USA. pp. 459–474. | MR