Review Articles

Robust analyzes for longitudinal clinical trials with missing and non-normal continuous outcomes

Siyi Liu ,

Department of Statistics, North Carolina State University, Raleigh, NC, USA

Yilong Zhang ,

Merck & Co., Inc., Kenilworth, NJ, USA

Gregory T. Golm ,

Merck & Co., Inc., Kenilworth, NJ, USA

Guanghan(Frank) Liu ,

Merck & Co., Inc., Kenilworth, NJ, USA

Shu Yang

Department of Statistics, North Carolina State University, Raleigh, NC, USA

Pages | Received 11 Oct. 2022, Accepted 16 Sep. 2023, Published online: 26 Sep. 2023,
  • Abstract
  • Full Article
  • References
  • Citations

Missing data are unavoidable in longitudinal clinical trials, and outcomes are not always normally distributed. In the presence of outliers or heavy-tailed distributions, the conventional multiple imputation with the mixed model with repeated measures analysis of the average treatment effect (ATE) based on the multivariate normal assumption may produce bias and power loss. Control-based imputation (CBI) is an approach for evaluating the treatment effect under the assumption that participants in both the test and control groups with missing outcome data have a similar outcome profile as those with an identical history in the control group. We develop a robust framework to handle non-normal outcomes under CBI without imposing any parametric modeling assumptions. Under the proposed framework, sequential weighted robust regressions are applied to protect the constructed imputation model against non-normality in the covariates and the response variables. Accompanied by the subsequent mean imputation and robust model analysis, the resulting ATE estimator has good theoretical properties in terms of consistency and asymptotic normality. Moreover, our proposed method guarantees the analysis model robustness of the ATE estimation in the sense that its asymptotic results remain intact even when the analysis model is misspecified. The superiority of the proposed robust method is demonstrated by comprehensive simulation studies and an AIDS clinical trial data application.


  • Carpenter, J. R., Roger, J. H., & Kenward, M. G. (2013). Analysis of longitudinal trials with protocol deviation: a framework for relevant, accessible assumptions, and inference via multiple imputation. Journal of Biopharmaceutical Statistics23(6), 1352–1371.
  • Carroll, R. J., & Pederson, S. (1993). On robustness in the logistic regression model. Journal of the Royal Statistical Society. Series B, Statistical Methodology55(3), 693–706.  
  • Chang, L., Roberts, S., & Welsh, A. (2018). Robust lasso regression using Tukey's biweight criterion. Technometrics60(1), 36–47.  
  • Cro, S., Morris, T. P., Kenward, M. G., & Carpenter, J. R. (2016). Reference-based sensitivity analysis via multiple imputation for longitudinal trials with protocol deviation. The Stata Journal16(2), 443–463.  
  • Elashoff, R., Li, G., Li, N., & Tseng, C. H. (2012). Robust inference for longitudinal data analysis with non-ignorable and non-monotonic missing values. Statistics and its Interface5(4), 479–490. 
  • Fox, J., & Monette, G. (2002). An R and S-Plus companion to applied regression. Sage.
  • Guan, Q., & Yang, S. (2019). A unified framework for causal inference with multiple imputation using martingale. arXiv preprint arXiv:1911.04663. 
  • Henry, K., Erice, A., Tierney, C., Balfour, H. H., Fischl, M. A., Kmack, A., Liou, S. H., Kenton, A., Hirsch, M. S., Phair, J., & Martinez, A. (1998). A randomized, controlled, double-blind study comparing the survival benefit of four different reverse transcriptase inhibitor therapies (three-drug, two-drug, and alternating drug) for the treatment of advanced AIDS. Journal of Acquired Immune Deficiency Syndromes19(4), 339–349. 
  • Huber, P. J. (1973). Robust regression: Asymptotics, conjectures and Monte Carlo. Annals of Statistics1(5), 799–821. 
  • Huber, P. J. (2004). Robust statistics (Vol. 523). John Wiley & Sons.  
  • ICH (2021). E9(R1) statistical principles for clinical trials: Addendum: Estimands and sensitivity analysis in clinical trials. FDA Guidance Documents. 
  • Jaeckel, L. A. (1972). Estimating regression coefficients by minimizing the dispersion of the residuals. The Annals of Mathematical Statistics, 1449–1458. 
  • Kelly, G. E. (1992). Robust regression estimators-the choice of tuning constans. Journal of the Royal Statistical Society Series D: The Statistician41(3), 303–314. 
  • Leurent, B., Gomes, M., Cro, S., Wiles, N., & Carpenter, J. R. (2020). Reference-based multiple imputation for missing data sensitivity analyses in trial-based cost-effectiveness analysis. Health Economics29(2), 171–184.  
  • Little, R., & Yau, L. (1996). Intent-to-treat analysis for longitudinal studies with drop-outs. Biometrics, 1324–1333. 
  • Little, R. J. (1993). Pattern-mixture models for multivariate incomplete data. Journal of the American Statistical Association88(421), 125–134. 
  • Little, R. J., D'Agostino, R., Cohen, M. L., Dickersin, K., Emerson, S. S., Farrar, J. T., Frangakis, C., Hogan, J. W., Molenberghs, G., Murphy, S. A., & Neaton, J. D. (2012). The prevention and treatment of missing data in clinical trials. N Engl J Med367(14), 1355–1360. 
  • Liu, G. F., & Pang, L. (2016). On analysis of longitudinal clinical trials with missing data using reference-based imputation. Journal of Biopharmaceutical Statistics26(5), 924–936. 
  • Liu, S., Yang, S., Zhang, Y., & Liu, G. F. (2022). Sensitivity analysis in longitudinal clinical trials via distributional imputation. Statistical Methods in Medical Research32(1), 181–194. 
  • Mack, M. J., Leon, M. B., Thourani, V. H., Makkar, R., Kodali, S. K., Russo, M., Kapadia, S. R., Malaisrie, S. C., Cohen, D. J., Pibarot, P., & Leipsic, J. (2019). Transcatheter aortic-valve replacement with a balloon-expandable valve in low-risk patients. N Engl J Med380(18), 1695–1705. 
  • Mallinckrodt, C., Lin, Q., & Molenberghs, M. (2013). A structured framework for assessing sensitivity to missing data assumptions in longitudinal clinical trials. Pharmaceutical Statistics12(1), 1–6. 
  • Mehrotra, D. V., Li, X., Liu, J., & Lu, K. (2012). Analysis of longitudinal clinical trials with missing data using multiple imputation in conjunction with robust regression. Biometrics68(4), 1250–1259.  
  • Meyer, G. P. (2021). An alternative probabilistic interpretation of the huber loss. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition (pp. 5261–5269). IEEE Computer Society.  
  • Miao, W., Gel, Y. R., & Gastwirth, J. L. (2006). A new test of symmetry about an unknown median. In Random walk, sequential analysis and related topics: A festschrift in honor of Yuan-Shih Chow (pp. 199–214). World Scientific. 
  • Mogg, R., & Mehrotra, D. V. (2007). Analysis of antiretroviral immunotherapy trials with potentially non-normal and incomplete longitudinal data. Statistics in Medicine26(3), 484–497. 
  • O'Neil, P. M., Birkenfeld, A. L., McGowan, B., Mosenzon, O., Pedersen, S. D., Wharton, S., Carson, C. G., Jepsen, C. H., Kabisch, M., & Wilding, J. P. (2018). Efficacy and safety of semaglutide compared with liraglutide and placebo for weight loss in patients with obesity: a randomised, double-blind, placebo and active controlled, dose-ranging, phase 2 trial. The Lancet392(10148), 637–649.  
  • Ratitch, B., O'Kelly, M., & Tosiello, R. (2013). Missing data in clinical trials: from clinical assumptions to statistical analysis using pattern mixture models. Pharmaceutical Statistics12(6), 337–347. 
  • Robins, J. M., & Wang, N. (2000). Inference for imputation estimators. Biometrika87(1), 113–124. 
  • Rosenblum, M., & Van Der Laan, M. J. (2009). Using regression models to analyze randomized trials: Asymptotically valid hypothesis tests despite incorrectly specified models. Biometrics65(3), 937–945.  
  • Rubin, D. B. (1976). Inference and missing data. Biometrika63(3), 581–592.  
  • Rubin, D. B. (2004). Multiple imputation for nonresponse in surveys (Vol. 81). John Wiley & Sons.  
  • Sadeghkhani, A., Peng, Y., & Lin, C. D. (2019). A parametric Bayesian approach in density ratio estimation. Stats2(2), 189–201. 
  • Smola, A. J., & Schölkopf, B. (2004). A tutorial on support vector regression. Statistics and Computing14(3), 199–222.  
  • Sun, Q., Zhou, W. X., & Fan, J. (2020). Adaptive huber regression. Journal of the American Statistical Association115(529), 254–265.
  • Takeuchi, I., Bengio, Y., & Kanamori, T. (2002). Robust regression with asymmetric heavy-tail noise distributions. Neural Computation14(10), 2469–2496.  
  • Tan, P. T., Cro, S., Van Vogt, E., Szigeti, M., & Cornelius, V. R. (2021). A review of the use of controlled multiple imputation in randomised controlled trials with missing outcome data. BMC Medical Research Methodology21(1), 1–17.  
  • Tang, Y. (2017). An efficient multiple imputation algorithm for control-based and delta-adjusted pattern mixture models using SAS. Statistics in Biopharmaceutical Research9(1), 116–125. 
  • US Food and Drug Administration (2016). Statistical review and evaluation of tresiba and ryzodeg 70/30. Retrieved May 26, 2016, from
  • Yang, S., & Kim, J. K. (2016). A note on multiple imputation for method of moments estimation. Biometrika103(1), 244–251. 
  • Yang, S., Zhang, Y., Liu, G. F., & Guan, Q. (2021). SMIM: A unified framework of survival sensitivity analysis using multiple imputation and martingale. Biometrics.  
  • Ye, Y., Gao, J., Shao, Y., Li, C., Jin, Y., & Hua, X. (2020). Robust support vector regression with generic quadratic nonconvex ε-insensitive loss. Applied Mathematical Modelling82, 235–251. 

To cite this article: Siyi Liu, Yilong Zhang, Gregory T. Golm, Guanghan (Frank) Liu & Shu Yang (2024) Robust analyzes for longitudinal clinical trials with missing and nonnormal continuous outcomes, Statistical Theory and Related Fields, 8:1, 1-14, DOI: 10.1080/24754269.2023.2261351

To link to this article: