Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-21T15:59:06.797Z Has data issue: false hasContentIssue false
  • English
  • Français

LOESS smoothed density estimates for multivariate survival data subject to censoring and masking

Published online by Cambridge University Press:  22 August 2016

Peter Adamic  and
Jenna Guse
Peter Adamic*
Affiliation:
Laurentian University, Sudbury, Ontario P3E 2C6, Canada
Jenna Guse
Affiliation:
University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
*
*Correspondence to: Peter Adamic, Laurentian University, 935 Ramsey Lake Rd, Sudbury, Ontario P3E 2C6, Canada. Tel: (705)675-1151 x2325; E-mail: padamic@laurentian.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

Actuaries often encounter censored and masked survival data when constructing multiple-decrement tables. In this paper, we propose estimators for the cause-specific failure time density using LOESS smoothing techniques that are employed in the presence of left-censored data, while still allowing for right-censored and exact observations, as well as masked causes of failure. The smoothing mechanism is incorporated as part of an expectation-maximisation algorithm. The proposed models are applied to a bivariate African sleeping sickness data set.

Keywords

LOESS smoothingMultiple decrementLeft censoringRight censoring
Type
Papers
Information
Annals of Actuarial Science , Volume 10 , Issue 2 , September 2016 , pp. 285 - 302
Copyright
© Institute and Faculty of Actuaries 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aalen, O. (1976). Nonparametric inference in connection with multiple decrement models. Scandinavian Journal of Statistics, 3, 15–27.Google Scholar
Adamic, P. (2010). Modeling multiple risks in the presence of double censoring. Scandinavian Actuarial Journal, 1, 68–81.CrossRefGoogle Scholar
Adamic, P. (2012). A model for time-dependent competing risks. Contributed paper presented at the 40th Annual Meeting of the Statistical Society of Canada, June 4, 2012, University of Guelph, Guelph, Ontario, Canada.Google Scholar
Adamic, P., Dixon, S. & Gillis, D. (2010). Multiple decrement modeling in the presence of interval censoring and masking. Scandinavian Actuarial Journal, 4, 312–327.CrossRefGoogle Scholar
Belsley, D.A., Kuh, E. & Welsch, R.E. (1980). Regression Diagnostics. John Wiley, New York.CrossRefGoogle Scholar
Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. & Nesbitt, C.J. (1997). Actuarial Mathematics, 2nd edition. The Society of Actuaries, Schaumburg, IL.Google Scholar
Braun, J., Duchesne, T. & Stafford, J.E. (2005). Local likelihood density estimation for interval censored data. The Canadian Journal of Statistics, 33(1), 39–60.CrossRefGoogle Scholar
Chambers, J.M., Cleveland, W.S., Kleiner, B. & Tukey, P.A. (1983). Graphical Methods for Data Analysis. Wadsworth, Monterey, CA.Google Scholar
Cleveland, W.S. (1979). Robust locally weighted regression and smoothing scatterplots. Journal of the American Statistical Association, 74, 829–836.CrossRefGoogle Scholar
Cleveland, W.S. & Devlin, S.J. (1988). Locally weighted regression: an approach to regression analysis by local fitting. Journal of the American Statistical Association, 83, 596–610.CrossRefGoogle Scholar
Cleveland, W.S. & Loader, C.L. (1996). Smoothing by local regression: principles and methods. In W. Haerdle & M.G. Schimek (Eds.), Statistical Theory and Computational Aspects of Smoothing (pp. 10–49). Springer, New York.CrossRefGoogle Scholar
Cook, R.D. & Weisberg, S. (1982). Influence and Residuals in Regression. Chapman & Hall, New York.Google Scholar
Cox, D.R. (1959). The analysis of exponentially distributed lifetimes with two types of failures. Journal of the Royal Statistical Society, Series B, 21, 411–421.Google Scholar
Craiu, R.V. & Reiser, B. (2006). Inference for the dependent competing risks model with masked causes of failure. Life Time Data Analysis, 12, 21–33.CrossRefGoogle ScholarPubMed
Daniel, C. & Wood, F. (1971). Fitting Equations to Data. John Wiley, New York.Google Scholar
Dempster, A.P., Laird, N.M. & Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39, 1–38.Google Scholar
Dinse, G.E. (1982). Nonparametric estimation for partially-complete time and type of failure data. Biometrics, 38, 417–431.CrossRefGoogle ScholarPubMed
Duchesne, T. & Stafford, J.E. (2001). A kernel density estimate for interval censored data, technical report no. 0106, University of Toronto, Toronto, Ontario, Canada.Google Scholar
Efron, B. (1967). The two-sample problem with censored data. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability: University of California Press, 4, 831–853.Google Scholar
Escarela, G. & Carriere, J.F. (2003). Fitting competing risks with an assumed copula. Statistical Methods in Medical Research, 12(4), 333–349.CrossRefGoogle ScholarPubMed
Finkelstein, D.M. (1986). A proportional hazards model for interval-censored failure time data. Biometrics, 42, 845–854.CrossRefGoogle ScholarPubMed
Flehinger, B.J., Reiser, B. & Yashchin, E. (1998). Survival with competing risks and masked causes of failures. Biometrika, 85, 151–164.CrossRefGoogle Scholar
Gasser, T., Kneip, A. & Köhler, W. (1991). A flexible and fast method for automatic smoothing. Journal of the American Statistical Association, 86, 643–652.CrossRefGoogle Scholar
Gichangi, A. & Vach, W. (2005). The analysis of competing risks data: a guided tour. Statistics in Medicine, 132(4), 1–41.Google Scholar
Goutis, C. (1997). Nonparametric estimation of a mixing density via the kernel method. Journal of the American Statistical Association, 92, 1445–1450.CrossRefGoogle Scholar
Groeneboom, P., Maathuis, M.H. & Wellner, J.A. (2008). Current status data with competing risks: consistency and rates of convergence of the MLE. The Annals of Statistics, 36, 1031–1063.Google Scholar
Hastie, T. & Loader, C.R. (1993). Local regression: automatic kernel carpentry. Statistical Science, 8, 120–129.Google Scholar
Huang, J. (2000). Asymptotic properties of nonparametric estimation based on partly interval-censored data. Statistica Sinica, 9, 501–519.Google Scholar
Hudgens, M.G., Satten, G.A. & Longini, I.M. Jr (2001). Nonparametric maximum likelihood estimation for competing risks survival data subject to interval censoring and truncation. Biometrics, 57, 74–80.CrossRefGoogle ScholarPubMed
Irizarry, R.A. (2001). Chapter 3: local regression. Available online at the address http://www.biostat.jhsph.edu/ ririzarr/Teaching/754/section-03.pdf [accessed Jun-2015].Google Scholar
Jacoby, W.G. (2000). LOESS: a nonparametric, graphical tool for depicting relationships between variables. Electoral Studies, 19, 577–613.CrossRefGoogle Scholar
Jewell, N.P., Van Der Laan, M. & Henneman, T. (2003). Nonparametric estimation from current status data with competing risks. Biometrika, 90, 183–197.CrossRefGoogle Scholar
Kalbfleisch, J.D. & Prentice, R.L. (1980). The Statistical Analysis of Failure Time Data. Wiley, New York.Google Scholar
Klein, J.P. & Moeschberger, M.L. (1997). Survival Analysis: Techniques for Censored and Truncated Data. Springer, New York.CrossRefGoogle Scholar
Lawless, J.F. (2003). Statistical Models and Methods for Lifetime Data, 2nd edition. John Wiley and Sons, New York.Google Scholar
Legros, F. & Ancelle, T. (2006). Trypanosomiase humaine africaine: recensement des cas d’importation observés en France, 1980–2004. Bulletin épidémiologique hebdomadaire, 7, 57–59.Google Scholar
Li, J. & Yu, Q. (2014). A consistent NPMLE of the joint distribution function with competing risks data under the dependent masking and right-censoring model. Life Time Data Analysis, 22, 63–99.CrossRefGoogle ScholarPubMed
Li, L., Watkins, T. & Yu, Q. (1997). An EM algorithm for smoothing the self-consistent estimator of survival functions with interval-censored data. Scandinavian Journal of Statistics, 24, 531–542.CrossRefGoogle Scholar
Loader, C.R. (1999 a). Local Regression and Likelihood. Springer-Verlag, New York.CrossRefGoogle Scholar
Loader, C.R. (1999 b). Bandwidth selection: classical or plug-in? The Annals of Statistics, 27, 415–438.CrossRefGoogle Scholar
Mallows, C.L. (1966). Choosing a subset regression. Presented at the Annual Meeting of the American Statistical Association, Los Angeles, California, August 15–19.Google Scholar
Mallows, C.L. (1973). Some comments on C p . Technometrics, 15, 661–675.Google Scholar
Pan, W. (2000). Smooth estimation of the survival function for interval censored data. Statistics in Medicine, 24, 531–542.Google Scholar
Park, C. & Padgett, W.J. (2006). Analysis of strength distributions of multi-modal failures using the EM algorithm. Journal of Statistical Computation and Simulation, 76, 619–636.CrossRefGoogle Scholar
Ruppert, D., Sheather, S.J. & Wand, M.P. (1995). An effective bandwidth selector for local least squares regression. Journal of the American Statistical Association, 901, 1257–1270.CrossRefGoogle Scholar
Stafford, J. (2005). “Exact” local likelihood estimators. Available online at the address http://www.gla.ac.uk/external/RSS/RSScomp/Stafford.pdf [accessed Jun-2015].Google Scholar
Tsiatis, A. (1975). A nonidentifiability aspect of the problem of competing risks. Proceedings of the National Academy of Sciences USA, 72, 20–22.CrossRefGoogle ScholarPubMed
Turnbull, B.W. (1974). Nonparametric estimation of a survivorship function with doubly censored data. Journal of the American Statistical Association, 69, 169–173.CrossRefGoogle Scholar
Turnbull, B.W. (1976). The empirical distribution function with arbitrarily grouped, censored and truncated data. Journal of the Royal Statistical Society, 38, 290–295.Google Scholar
Wang, J., Yu, Q. & Wong, G.Y.C. (2012). The random partition masking model for interval-censored and masked competing risks data. Journal of Statistical Computation and Simulation, 82, 981–1002.CrossRefGoogle Scholar
Yu, Q. & Li, J. (2012). The NPMLE of the joint distribution function with right-censored and masked competing risks data. Journal of Nonparametric Statistics, 24, 753–764.CrossRefGoogle Scholar
Yu, Q., Li, L. & Wong, G.Y.C. (2000). On consistency of the self-consistent estimator of survival functions with interval-censored data. Scandinavian Journal of Statistics, 27, 35–44.CrossRefGoogle Scholar
Zhou, M. (2004). Nonparametric Bayes estimator of survival functions for doubly/interval censored data. Statistica Sinica, 14, 533–546.Google Scholar