The Annals of Applied Statistics

A penalized Cox proportional hazards model with multiple time-varying exposures

Chenkun Wang, Hai Liu, and Sujuan Gao

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In recent pharmacoepidemiology research, the increasing use of electronic medication dispensing data provides an unprecedented opportunity to examine various health outcomes associated with long-term medication usage. Often, patients may take multiple types of medications intended for the same medical condition and the medication exposure status and intensity may vary over time, posing challenges to the statistical modeling of such data. In this article, we propose a penalized Cox proportional hazards (PH) model with multiple functional covariates and potential interaction effects. We also consider constrained coefficient functions to ensure a diminishing medication effect over time. Hypothesis testing of interaction effect and main effect was discussed under the penalized Cox PH model setting. Our simulation studies demonstrate the adequate performance of the proposed methods for both parameter estimation and hypothesis testing. Application to a primary care depression cohort study was also illustrated to examine the effects of two common types of antidepressants on the risk of coronary artery disease.

Article information

Ann. Appl. Stat., Volume 11, Number 1 (2017), 185-201.

Received: May 2015
Revised: October 2016
First available in Project Euclid: 8 April 2017

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Pharmacoepidemiology time-varying exposure interaction penalized spline


Wang, Chenkun; Liu, Hai; Gao, Sujuan. A penalized Cox proportional hazards model with multiple time-varying exposures. Ann. Appl. Stat. 11 (2017), no. 1, 185--201. doi:10.1214/16-AOAS999.

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  • Abrahamowicz, M., Bartlett, G., Tamblyn, R. and du Berger, R. (2006). Modeling cumulative dose and exposure duration provided insights regarding the associations between benzodiazepines and injuries. J. Clin. Epidemiol. 59 393–403.
  • Bergstrom, R. F., Lemberger, L., Farid, N. A. and Wolen, R. L. (1988). Clinical pharmacology and pharmacokinetics of fluoxetine: A review. Br. J. Psychiatry Suppl. 3 47–50.
  • Berhane, K., Hauptmann, M. and Langholz, B. (2008). Using tensor product splines in modeling exposure-time-response relationships: Application to the Colorado Plateau Uranium Miners cohort. Stat. Med. 27 5484–5496.
  • Bhadra, D., Daniels, M. J., Kim, S., Ghosh, M. and Mukherjee, B. (2012). A Bayesian semiparametric approach for incorporating longitudinal information on exposure history for inference in case–control studies. Biometrics 68 361–370.
  • Breslow, N. E., Lubin, J. H., Marek, P. and Langholz, B. (1983). Multiplicative models and cohort analysis. J. Amer. Statist. Assoc. 78 1–12.
  • Callahan, C. M., Hui, S. L., Nienaber, N. A. and Musick, B. S. (1994). Longitudinal study of depression and health services use among elderly primary care patients. Journal of the American Geriatrics Society 42 833–838.
  • Damush, T. M., Jia, H., Ried, L. D., Qin, H., Cameon, R., Plue, L. and Williams, L. S. (2008). Case-finding algorithm for post-stroke depression in the veterans health administration. International Journal of Geriatric Psychiatry 23 517–522.
  • de la Torre, B. R., Dreher, J., Malevany, I., Bagli, M., Kolbinger, M., Omran, H., Lüderitz, B. and Rao, M. L. (2001). Serum levels and cardiovascular effects of tricyclic antidepressants and selective serotonin reuptake inhibitors in depressed patients. Therapeutic Drug Monitoring 23 435–440.
  • Ferraty, F. and Vieu, P. (2009). Additive prediction and boosting for functional data. Comput. Statist. Data Anal. 53 1400–1413.
  • Fuchs, K., Scheipl, F. and Greven, S. (2015). Penalized scalar-on-functions regression with interaction. Comput. Statist. Data Anal. 81 38–51.
  • Gasparrini, A. (2014). Modeling exposure-lag-response associations with distributed lag non-linear models. Stat. Med. 33 881–899.
  • Glassman, A. H. (1984). Cardiovascular effects of tricyclic antidepressants. Annual Review of Medicine 35 503–511.
  • Goeman, J., Meijer, R. and Chaturvedi, N. (2016). L1 and L2 penalized regression models. Available at
  • Goldsmith, J., Crainiceanu, C. M., Caffo, B. and Reich, D. (2012). Longitudinal penalized functional regression for cognitive outcomes on neuronal tract measurements. J. R. Stat. Soc. Ser. C. Appl. Stat. 61 453–469.
  • Gray, R. J. (1992). Flexible methods for analyzing survival data using splines, with applications to breast cancer prognosis. J. Amer. Statist. Assoc. 87 942–951.
  • Gray, R. J. (1994). Spline-based tests in survival analysis. Biometrics 50 640–652.
  • Gray, S. L., Anderson, M. L., Dublin, S., Hanlon, J. T., Hubbard, R., Walker, R., Yu, O., Crane, P. K. and Larson, E. B. (2015). Cumulative use of strong anticholinergics and incident dementia: A prospective cohort study. JAMA Internal Medicine 175 401–407.
  • Hastie, T. and Tibshirani, R. (1986). Generalized additive models. Statist. Sci. 1 297–318.
  • Hauptmann, M., Wellmann, J., Lubin, J. H., Rosenberg, P. S. and Kreienbrock, L. (2000). Analysis of exposure-time–response relationships using a spline weight function. Biometrics 56 1105–1108.
  • Hurvich, C. M., Simonoff, J. S. and Tsai, C.-L. (1998). Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion. J. R. Stat. Soc. Ser. B. Stat. Methodol. 60 271–293.
  • Jefferson, J. W. (1975). A review of the cardiovascular effects and toxicity of tricyclic antidepressants. Psychosomatic Medicine 37 160–179.
  • Langholz, B., Thomas, D., Xiang, A. and Stram, D. (1999). Latency analysis in epidemiologic studies of occupational exposures: Application to the Colorado Plateau uranium miners cohort. American Journal of Industrial Medicine 35 246–256.
  • Lindsley, C. W. (2012). The top prescription drugs of 2011 in the United States: Antipsychotics and antidepressants once again lead CNS therapeutics. ACS Chemical Neuroscience 3 630–631.
  • McDonald, C. J., Overhage, J. M., Tierney, W. M., Dexter, P. R., Martin, D. K., Suico, J. G., Zafar, A., Schadow, G., Blevins, L., Glazener, T. et al. (1999). The Regenstrief medical record system: A quarter century experience. International Journal of Medical Informatics 54 225–253.
  • Muggeo, V. M. (2008). Modeling temperature effects on mortality: Multiple segmented relationships with common break points. Biostatistics 9 613–620.
  • Obermeier, V., Scheipl, F., Heumann, C., Wassermann, J. and Küchenhoff, H. (2015). Flexible distributed lags for modelling earthquake data. J. R. Stat. Soc. Ser. C. Appl. Stat. 64 395–412.
  • Pacher, P. and Kecskemeti, V. (2004). Cardiovascular side effects of new antidepressants and antipsychotics: New drugs, old concerns? Curr. Pharm. Des. 10 2463–2475.
  • Perperoglou, A. (2014). Cox models with dynamic ridge penalties on time-varying effects of the covariates. Stat. Med. 33 170–180.
  • Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer, New York.
  • Richardson, D. B. (2009). Latency models for analyses of protracted exposures. Epidemiology (Cambridge, Mass.) 20 395.
  • Scheipl, F. and Greven, S. (2015). Identifiability in penalized function-on-function regression models. Preprint. Available at arXiv:1506.03627.
  • Schipper, M., Taylor, J. M. G. and Lin, X. (2008). Generalized monotonic functional mixed models with application to modelling normal tissue complications. J. R. Stat. Soc. Ser. C. Appl. Stat. 57 149–163.
  • Sylvestre, M.-P. and Abrahamowicz, M. (2008). Comparison of algorithms to generate event times conditional on time-dependent covariates. Stat. Med. 27 2618–2634.
  • Sylvestre, M.-P. and Abrahamowicz, M. (2009). Flexible modeling of the cumulative effects of time-dependent exposures on the hazard. Stat. Med. 28 3437–3453.
  • Therneau, T. M., Grambsch, P. M. and Pankratz, V. S. (2003). Penalized survival models and frailty. J. Comput. Graph. Statist. 12 156–175.
  • Thomas, D. C. (1988). Models for exposure-time-response relationships with applications to cancer epidemiology. Annual Review of Public Health 9 451–482.
  • Vacek, P. M. (1997). Assessing the effect of intensity when exposure varies over time. Stat. Med. 16 505–513.
  • Wood, S. N. (2006). Generalized Additive Models: An Introduction with R. Chapman & Hall/CRC, Boca Raton, FL.
  • Wood, S. N. (2011). Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models. J. R. Stat. Soc. Ser. B. Stat. Methodol. 73 3–36.
  • Zhang, D., Lin, X. and Sowers, M. (2007). Two-stage functional mixed models for evaluating the effect of longitudinal covariate profiles on a scalar outcome. Biometrics 63 351–362.