Conditional Local Risk Ratio (CLRR) is a widely used metric for assessing heterogeneous treatment effects of binary outcomes in randomized clinical trials involving noncompliance. Existing methods, such as moment-based and likelihood-based approaches, often overlook the inherent mixture structure in data, necessitate stringent parametric assumptions, or yield estimates with implausible values. In this paper, we introduce a novel semiparametric likelihood-based (SPL) method for estimating CLRR. Our method requires only three parametric model assumptions, significantly fewer than the six models needed by existing likelihood-based methods, thereby reducing model complexity and enhancing robustness. This simplicity also results in fewer unknown parameters, further boosting computational efficiency. Unlike moment-based methods, our SPL method fully exploits the mixture structure of the observed data and the principal strata framework. Additionally, our method ensures that the final CLRR estimate always fall within a valid range. We establish the asymptotic normality of our estimator and demonstrate its superiority over existing methods through numerical simulations. We further apply our method to analyze the Oregon Health Insurance Experiment dataset, providing valuable insights into the heterogeneous effects of Medicaid on both physical and mental health.

References

• Abadie, A. (2002). Bootstrap tests for distributional treatment effects in instrumental variable models. Journal of the American Statistical Association, 97(457), 284–292. https://doi.org/10.1198/016214502753479419
• Abadie, A. (2003). Semiparametric instrumental variable estimation of treatment response models. Journal of Econometrics, 113(2), 231–263. https://doi.org/10.1016/S0304-4076(02)00201-4
• Anderson, J. E. (1979). A theoretical foundation for the gravity equation. The American Economic Review, 69(1), 106–116.
• Baicker, K., Taubman, S. L., Allen, H. L., Bernstein, M., Gruber, J. H., J. P. Newhouse, Schneider, E. C., Wright, B. J., Zaslavsky, A. M., & A. N. Finkelstein (2013). The Oregon experiment—Effects of Medicaid on clinical outcomes. New England Journal of Medicine, 368(18), 1713–1722. https://doi.org/10.1056/NEJMsa1212321
• Charles, P., Giraudeau, B., Dechartres, A., Baron, G., & Ravaud, P. (2009). Reporting of sample size calculation in randomised controlled trials: Review. British Medical Journal, 338:b1732. https://doi.org/10.1136/bmj.b1732
• Dodd, S., White, I. R., & Williamson, P. (2012). Nonadherence to treatment protocol in published randomised controlled trials: A review. Trials, 13(1), Article 84. https://doi.org/10.1186/1745-6215-13-84
• Finkelstein, A., Taubman, S., Wright, B., Bernstein, M., Gruber, J., Newhouse, J. P., Allen, H., & Baicker, K. & Oregon Health Study Group, t. (2012). The Oregon health insurance experiment: Evidence from the first year. The Quarterly Journal of Economics, 127(3), 1057–1106. https://doi.org/10.1093/qje/qjs020
• Frangakis, C. E., & Rubin, D. B. (2002). Principal stratification in causal inference. Biometrics, 58(1), 21–29. https://doi.org/10.1111/biom.2002.58.issue-1
• Frölich, M. (2007). Nonparametric iv estimation of local average treatment effects with covariates. Journal of Econometrics, 139(1), 35–75. https://doi.org/10.1016/j.jeconom.2006.06.004
• Hattab, Z., Doherty, E., Ryan, A. M., & O'Neill, S. (2024). Heterogeneity within the oregon health insurance experiment: An application of causal forests. PloS one, 19(1), 1–21. https://doi.org/10.1371/journal.pone.0297205
• Have, T. R. T., Joffe, M., & Cary, M. (2003). Causal logistic models for non-compliance under randomized treatment with univariate binary response. Statistics in Medicine, 22(8), 1255–1283. https://doi.org/10.1002/sim.v22:8
• Hirano, K., Imbens, G. W., Rubin, D. B., & Zhou, X.-H. (2000). Assessing the effect of an influenza vaccine in an encouragement design. Biostatistics, 1(1), 69–88. https://doi.org/10.1093/biostatistics/1.1.69
• Imbens, G. W. (2004). Nonparametric estimation of average treatment effects under exogeneity: A review. Review of Economics and Statistics, 86(1), 4–29. https://doi.org/10.1162/003465304323023651
• Imbens, G. W., & Angrist, J. D. (1994). Identification and estimation of local average treatment effects. Econometrica, 62(2), 467–475. https://doi.org/10.2307/2951620
• Imbens, G. W., & Rubin, D. B. (1997). Bayesian inference for causal effects in randomized experiments with noncompliance. The Annals of Statistics, 25(1), 305–327. https://doi.org/10.1214/aos/1034276631
• Jenkinson, C., Coulter, A., & Wright, L. (1993). Short form 36 (SF36) health survey questionnaire: Normative data for adults of working age. British Medical Journal, 306(6890), 1437–1440. https://doi.org/10.1136/bmj.306.6890.1437
• Kennedy, E. H., Balakrishnan, S., & G'Sell, M. (2020). Sharp instruments for classifying compliers and generalizing causal effects. The Annals of Statistics, 48(4), 2008–2030. https://doi.org/10.1214/19-AOS1874
• Kohavi, R., & Thomke, S. (2017). The surprising power of online experiments. Harvard Business Review, 95(5), 74–82.
• Little, R. J., & 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), 147–159. https://doi.org/10.1037/1082-989X.3.2.147
• Lois, N., Burr, J., Norrie, J., Vale, L., Cook, J., & McDonald, A. (2008). Clinical and cost-effectiveness of internal limiting membrane peeling for patients with idiopathic full thickness macular hole. protocol for a randomised controlled trial: Films (full-thickness macular hole and internal limiting membrane peeling study). Trials, 9(1), Article 61. https://doi.org/10.1186/1745-6215-9-61
• Matsouaka, R. A., & Tchetgen Tchetgen, E. J. (2017). Instrumental variable estimation of causal odds ratios using structural nested mean models. Biostatistics, 18(3), 465–476. https://doi.org/10.1093/biostatistics/kxw059
• Ogburn, E. L., Rotnitzky, A., & 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. https://doi.org/10.1111/rssb.12078
• Okui, R., Small, D. S., Tan, Z., & Robins, J. M. (2012). Doubly robust instrumental variable regression. Statistica Sinica, 22(1), 173–205. https://doi.org/10.5705/ss.2011.v22n1a
• Pearl, J. (2009). Causality (2nd ed.). Cambridge University Press.
• Piantadosi, S. (2024). Clinical Trials: A Methodologic Perspective. John Wiley & Sons.
• Qiu, Y., Tao, J., & Zhou, X. (2021). Inference of heterogeneous treatment effects using observational data with high-dimensional covariates. Journal of the Royal Statistical Society Series B: Statistical Methodology, 83(5), 1016–1043. https://doi.org/10.1111/rssb.12469
• Richardson, T. S., & Robins, J. M. (2013). Single world intervention graphs (SWIGs): A unification of the counterfactual and graphical approaches to causality (Working Paper, 128). Center for the Statistics and the Social Sciences, University of Washington.
• Richardson, T. S., Robins, J. M., & Wang, L. (2017). On modeling and estimation for the relative risk and risk difference. Journal of the American Statistical Association, 112(519), 1121–1130. https://doi.org/10.1080/01621459.2016.1192546
• 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. https://doi.org/10.1080/03610929408831393
• Rothman, K. J., Greenland, S., & Lash, T. L. (2008). Modern Epidemiology (3rd ed.). Lippincott Williams & Wilkins.
• Rubin, D. B. (1974). Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of Educational Psychology, 66(5), 688–701. https://doi.org/10.1037/h0037350
• Shadish, W., Cook, T., & Campbell, D. (2002). Experimental and Quasi-Experimental Designs for Generalized Causal Inference. Houghton Mifflin.
• Stamler, J., Wentworth, D., & Neaton, J. D. (1986). Is relationship between serum cholesterol and risk of premature death from coronary heart disease continuous and graded? Findings in 356 222 primary screenees of the multiple risk factor intervention trial (MRFIT). Journal of the American Medical Association, 256(20), 2823–2828. https://doi.org/10.1001/jama.1986.03380200061022
• Strobl, E. V., Zhang, K., & Visweswaran, S. (2019). Approximate kernel-based conditional independence tests for fast non-parametric causal discovery. Journal of Causal Inference, 7(1), 20180017. https://doi.org/10.1515/jci-2018-0017
• Tan, Z. (2006). Regression and weighting methods for causal inference using instrumental variables. Journal of the American Statistical Association, 101(476), 1607–1618. https://doi.org/10.1198/016214505000001366
• Taylor, S. J., Carnes, D., Homer, K., Kahan, B. C., Hounsome, N., Eldridge, S., Spencer, A., Pincus, T., Rahman, A., & Underwood, M. (2016). Novel three-day, community-based, nonpharmacological group intervention for chronic musculoskeletal pain (copers): A randomised clinical trial. PLOS Medicine, 13(6), 1–18. https://doi.org/10.1371/journal.pmed.1002040
• Wang, L., Zhang, Y., Richardson, T. S., & Robins, J. M. (2021). Estimation of local treatment effects under the binary instrumental variable model. Biometrika, 108(4), 881–894. https://doi.org/10.1093/biomet/asab003
• Ware, J. E. J., & Sherbourne, C. D. (1992). The MOS 36-item short-form health survey (SF-36). I. conceptual framework and item selection. Medical Care, 30(6), 473–483. https://doi.org/10.1097/00005650-199206000-00002

To cite this article: Wenli Liu, Jing Qin & Yukun Liu (2025) Leveraging density ratio models in a binary instrumental variable inference with a binary outcome: A retrospective approach, Statistical Theory and Related Fields, 9:4, 331-356, DOI: 10.1080/24754269.2025.2537517

To link to this article: https://doi.org/10.1080/24754269.2025.2537517