Efficient Principally Stratified Treatment Effect Estimation in Crossover Studies with Absorbent Binary Endpoints

Luedtke, Alex; Wu, Jiacheng

Journal de la société française de statistique, Causality, Tome 161 (2020) no. 1, pp. 176-200.

We consider the estimation of the effect of a binary treatment on a binary endpoint conditional on a post-randomization quantity in a counterfactual world where all individuals received treatment. It is generally difficult to identify this effect without strong, untestable assumptions. It has been shown that identifiability assumptions become weaker under a crossover design where individuals not receiving treatment are later provided treatment. Under the assumption that the post-treatment biomarker observed in these crossover individuals is the same as would have been observed had they received treatment at the start of the study, the treatment effect can be identified with only mild additional assumptions. This remains true if the endpoint is absorbent, that is, if the post-crossover treatment biomarker is not meaningful if the endpoint has already occurred. Examples of absorbent endpoints include death and HIV infection. We provide identifiability conditions for the principally stratified treatment effect of interest when the data arise from a crossover design and describe situations where these conditions would be falsifiable given a large sample from the observed data distribution. We then introduce a nonparametric estimator for this effect. When the biomarker is discrete, this estimator is efficient among all regular and asymptotically linear estimators.

[1] Anthony, M; Bartlett, P L Neural network learning: theoretical foundations (1999) | DOI | MR | Zbl

[2] Bickel, P J; Klaassen, C A J; Ritov, Y; Wellner, J A Efficient and adaptive estimation for semiparametric models, Johns Hopkins University Press, Baltimore, 1993 | MR

[3] Bareinboim, E; Pearl, J Transportability of Causal Effects: Completeness Results, Twenty-Sixth AAAI Conference on Artificial Intelligence (2012)

[4] Bibaut, A F; van der Laan, M J Data-adaptive smoothing for optimal-rate estimation of possibly non-regular parameters, arXiv preprint arXiv:1706.07408 (2017)

[5] Cole, S R; Hernán, M A Fallibility in estimating direct effects, International journal of epidemiology, Volume 31 (2002) no. 1, pp. 163-165 | DOI

[6] Chiu, S-T Bandwidth selection for kernel density estimation, The Annals of Statistics (1991), pp. 1883-1905 | MR | Zbl

[7] Fan, J; Marron, J S Best possible constant for bandwidth selection, The Annals of Statistics (1992), pp. 2057-2070 | MR | Zbl

[8] Follmann, D Augmented designs to assess immune response in vaccine trials, Biometrics, Volume 62 (2006) no. 4, pp. 1161-1169 | DOI | MR | Zbl

[9] Frangakis, C E; Rubin, D B Principal stratification in causal inference, Biometrics, Volume 58 (2002) no. 1, pp. 21-29 | DOI | MR | Zbl

[10] Gabriel, E E; Follmann, D Augmented trial designs for evaluation of principal surrogates, Biostatistics, Volume 17 (2016) no. 3, pp. 453-467 | DOI | MR

[11] Gabriel, E E; Gilbert, P B Evaluating principal surrogate endpoints with time-to-event data accounting for time-varying treatment efficacy, Biostatistics, Volume 15 (2013) no. 2, pp. 251-265 | DOI

[12] Gilbert, P B; Hudgens, M G Evaluating candidate principal surrogate endpoints, Biometrics, Volume 64 (2008) no. 4, pp. 1146-1154 | DOI | MR | Zbl

[13] Gilbert, P B; Hudgens, M G; Wolfson, J Commentary on" Principal Stratification–a Goal or a Tool?" by Judea Pearl, Int J Biostat, Volume 7 (2011) no. 1, pp. 1-15 | DOI | MR

[14] Huang, Y; Gilbert, P B; Wolfson, J Design and estimation for evaluating principal surrogate markers in vaccine trials, Biometrics, Volume 69 (2013) no. 2, pp. 301-309 | DOI | MR | Zbl

[15] Horowitz, J L Semiparametric and nonparametric methods in econometrics, 12, Springer, 2009 | DOI | MR

[16] Hall, P; Sheather, S J; Jones, M C; Marron, J S On optimal data-based bandwidth selection in kernel density estimation, Biometrika, Volume 78 (1991) no. 2, pp. 263-269 | DOI | MR | Zbl

[17] Hastie, T J; Tibshirani, R J Generalized additive models, Chapman & Hall, London, 1990 | MR

[18] Kennedy, E H; Ma, Z; McHugh, M D; Small, D S Non-parametric methods for doubly robust estimation of continuous treatment effects, J. Royal Stat. Soc Series B (2016) | MR

[19] Marsh, T Efficient inference for an additive gene-treatment interaction from a nested two-phase study (2016)

[20] Nason, M; Follmann, D Design and analysis of crossover trials for absorbing binary endpoints, Biometrics, Volume 66 (2010) no. 3, pp. 958-965 | DOI | MR | Zbl

[21] Pearl, J Direct and indirect effects, Proceedings of the Seventeenth Conference on Uncertainty in artificial intelligence, Morgan Kaufmann, San Francisco (2001), pp. 411-420

[22] Pfanzagl, J Estimation in semiparametric models, Springer, 1990 | DOI | MR

[23] Robins, J M; Greenland, S Identifiability and exchangeability for direct and indirect effects, Epidemiol, Volume 3 (1992), pp. 143-155 | DOI

[24] Rose, S; van der Laan, M J A Targeted Maximum Likelihood Estimator for Two-Stage Designs, Int J Biostat, Volume 7 (2011) no. 17 | MR

[25] Silverman, B W Density Estimation for Statistics and Data analysis, Chapman & Hall, 1986 | MR

[26] VanderWeele, T J Simple relations between principal stratification and direct and indirect effects, Statistics & Probability Letters, Volume 78 (2008) no. 17, pp. 2957-2962 | DOI | MR | Zbl

[27] VanderWeele, T Explanation in causal inference: methods for mediation and interaction, Oxford University Press, 2015

[28] van der Laan, M A Generally Efficient Targeted Minimum Loss Based Estimator based on the Highly Adaptive Lasso, Int J Biostat (2017) | DOI | MR

[29] van der Laan, M J; Hubbard, A E; Kherad, S Statistical inference for data adaptive target Parameters (2013) no. 314 ( Technical report )

[30] van der Laan, M J; Robins, J M Unified methods for censored longitudinal data and causality, Springer, New York Berlin Heidelberg, 2003 | DOI | MR

[31] van der Vaart, A W; Wellner, J A Weak convergence and empirical processes, Springer, Berlin Heidelberg New York, 1996 | DOI | MR

[32] Wasserman, L All of Nonparametric Statistics, Springer-Verlag New York, Inc., Secaucus, NJ, USA, 2006

[33] Wolfson, J; Gilbert, P Statistical identifiability and the surrogate endpoint problem, with application to vaccine trials, Biometrics, Volume 66 (2010) no. 4, pp. 1153-1161 | DOI | MR | Zbl

[34] Wolfson, J; Henn, L Hard, harder, hardest: principal stratification, statistical identifiability, and the inherent difficulty of finding surrogate endpoints, Emerging themes in epidemiology, Volume 11 (2014) no. 1, p. 14 | DOI

[35] Zheng, Wenjing; van der Laan, Mark J Cross-validated targeted minimum-loss-based estimation, Targeted Learning, Springer, 2011, pp. 459-474 | DOI | MR