Advertisement

Lifetime Data Analysis

, Volume 19, Issue 4, pp 463–489 | Cite as

A copula model for marked point processes

  • Liqun Diao
  • Richard J. CookEmail author
  • Ker-Ai Lee
Article

Abstract

Many chronic diseases feature recurring clinically important events. In addition, however, there often exists a random variable which is realized upon the occurrence of each event reflecting the severity of the event, a cost associated with it, or possibly a short term response indicating the effect of a therapeutic intervention. We describe a novel model for a marked point process which incorporates a dependence between continuous marks and the event process through the use of a copula function. The copula formulation ensures that event times can be modeled by any intensity function for point processes, and any multivariate model can be specified for the continuous marks. The relative efficiency of joint versus separate analyses of the event times and the marks is examined through simulation under random censoring. An application to data from a recent trial in transfusion medicine is given for illustration.

Keywords

Copula function Joint analysis Marks Recurrent events 

Notes

Acknowledgments

This research was supported by Grants from the Natural Sciences and Engineering Research Council of Canada (RGPIN 155849) and the Canadian Institutes for Health Research (FRN 13887). Richard Cook is a Canada Research Chair in Statistical Methods for Health Research. The authors thank Professor Jerry Lawless and Professor Nancy Heddle for helpful discussions and collaboration and Ray Goodrich for helpful discussion and permission to use the data from the Mirasol Study.

References

  1. Aalen OO, Borgan O, Gjessing HK (2008) Survival and event history analysis: a point process point of view. Springer, New YorkCrossRefGoogle Scholar
  2. Albrecher H, Teugels JL (2006) Exponential behavior in the presence of dependence in risk theory. J Appl Probab 43:257–273MathSciNetCrossRefzbMATHGoogle Scholar
  3. Andersen PK, Gill RD (1982) Cox’s regression model for counting processes: a large sample study. Ann Stat 10:1100–1120MathSciNetCrossRefzbMATHGoogle Scholar
  4. Andersen PK, Borgan O, Gill RD, Keiding N (1993) Statistical models based on counting processes. Springer, New YorkCrossRefzbMATHGoogle Scholar
  5. Cazenave J-P, Folléa G, Bardiaux L, Boiron J-M, Lafeuillade B, Debost M, Lioure B, Harousseau J-L, Tabrizi R, Cahn J-Y, Michallet M, Ambruso D, Schots R, Tissot J-D, Sensebé L, Kondo T, McCullough J, Rebulla P, Escolar G, Mintz P, Heddle NM, Goodrich RP, Bruhwyler J, Le C, Cook RJ, The Mirasol Clinical Evaluation Study Group (2010) A randomized controlled clinical trial evaluating the performance and safety of platelets treated with MIRASOL pathogen reduction technology. Transfusion 50:2362–2375CrossRefGoogle Scholar
  6. Chatterjee N, Kalaylioglu Z, Shih JH, Gail MH (2006) Case–control and case-only designs with genotype and family history data: estimating relative risk, residual familial aggregation, and cumulative risk. Biometrics 62:36–48MathSciNetCrossRefzbMATHGoogle Scholar
  7. Chen F, Zheng Y-S (1997) One-warehouse multiretailer systems with centralized stock information. Operat Res 45:275–287CrossRefzbMATHGoogle Scholar
  8. Cole EH, Cattran DC, Farewell VT, Aprile M, Bear RA, Pei YP, Fenton SS, Tober JA, Cardella CJ (1994) A comparison of rabbit antithymocyte serum and okt3 as prophylaxis against renal allograft rejection. Transplantation 57:60–67CrossRefGoogle Scholar
  9. Cook RJ, Lawless JF, Lee K-A (2003) Cumulative processes related to event histories. SORT 27:13–30MathSciNetzbMATHGoogle Scholar
  10. Cook RJ, Lawless JF (2007) The statistical analysis of recurrent events. Springer, New YorkzbMATHGoogle Scholar
  11. Cox DR, Isham V (1980) Point processes. Chapman and Hall, LondonzbMATHGoogle Scholar
  12. Craiu M, Craiu RV (2008) Choice of parametric families of copulas. Adv Appl Stat 10:25–40MathSciNetzbMATHGoogle Scholar
  13. Daley DJ, Vere-Jones D (2008) An introduction to the theory of point processes: volumn ii: general theory and structure, 2nd edn. Springer, New YorkCrossRefGoogle Scholar
  14. Davis KB, Slichter SJ, Corash L (1999) Corrected count increment and percent platelet recovery as measures of post-transfusion platelet response: problems and a solution. Transfusion 39:586–592CrossRefGoogle Scholar
  15. Descombes X, Zerubia J (2002) Marked point process in image analysis. Sig Process Mag IEEE 19:77–84CrossRefGoogle Scholar
  16. Fok CCT, Ramsay JO, Abrahamowicz M, Fortin P (2012) A functional marked point process model for lupus data. Can J Stat 40:517–529MathSciNetCrossRefzbMATHGoogle Scholar
  17. Genest C, MacKay J (1986) The joy of copulas: bivariate distributions with uniform marginals. Am Stat 40:280–283MathSciNetGoogle Scholar
  18. Goulard M, Säkkä A, Grabarnik P (1996) Parameter estimation for marked Gibbs point processes through the maximum pseudo-likelihood method. Scand J Stat 23:365–379zbMATHGoogle Scholar
  19. Grandell J (1997) Mixed poisson processes. Chapman and Hall, LondonCrossRefzbMATHGoogle Scholar
  20. Guan Y (2006) Tests for independence between marks and points of a marked point process. Biometrics 62:126–134MathSciNetCrossRefzbMATHGoogle Scholar
  21. Holden L, Sannan S, Bungum H (2002) A stochastic marked point process model for earthquakes. Nat Hazards Earth Syst Sci 3:95–101CrossRefGoogle Scholar
  22. Joe H (1997) Multivariate dependence concepts. Chapman and Hall, LondonCrossRefzbMATHGoogle Scholar
  23. Kalbfleisch JD, Prentice RL (2002) The statistical analysis of failure time data, 2nd edn. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  24. Karr AF (1991) Point processes and their statistical inference, 2nd edn. Dekker, New YorkzbMATHGoogle Scholar
  25. Landriault D, Lee WY, Willmot GE, Woo J-K (2013) A note on deficit analysis in dependency models involving Coxian claim amounts. Scand Actuar J. doi: 10.1080/03461238.2012.723044
  26. Lawless JF (1987a) Negative binomial and mixed Poisson regression. Can J Stat 15:209–225Google Scholar
  27. Lawless JF (1987b) Regression methods for Poisson process data. J Am Stat Assoc 82:808–815Google Scholar
  28. Lawless JF (2003) Statistical models and methods for lifetime data, 2nd edn. Wiley, Hoboken, NJzbMATHGoogle Scholar
  29. Lawless JF, Nadeau JC (1995) Nonparametric estimation of cumulative mean functions for recurrent events. Technometrics 37:158–168MathSciNetCrossRefzbMATHGoogle Scholar
  30. Lawless JF, Yilmaz YE (2011a) Comparison of semiparametric maximum likelihood estimation and two-stage semiparametric estimation in copula models. Comput Stat Data Anal 55:2446–2455MathSciNetCrossRefGoogle Scholar
  31. Lawless JF, Yilmaz YE (2011b) Semiparametric estimation in copula models for bivariate sequential survival times. Biometr J 53:779–796MathSciNetCrossRefzbMATHGoogle Scholar
  32. Nelsen RB (2006) An introduction to copulas. Springer, New YorkzbMATHGoogle Scholar
  33. Pascual J, Falk RM, Piessens F, Prusinski A, Docekal P, Robert M, Ferrer P, Luria X, Segarra R, Zayas JM (2000) Consistent efficacy and tolerability of almotriptan in the acute treatment of multiple migraine attacks: results of a large, randomized, double-blind, placebo-controlled study. Cephelegia 20:588–596CrossRefGoogle Scholar
  34. Patton AJ (2006) Modelling asymmetric exchange rate dependence. Int Econ Rev 47:527–556MathSciNetCrossRefGoogle Scholar
  35. Penttinen A, Stoyan D, Henttonen HM (1992) Marked point processes in forest statistics. For Sci 38:806–824Google Scholar
  36. Petri M, Genovese M, Engle E, Hochberg M (1991) Definition, incidence, and clinical description of flare in systemic lupus erythematosus. A prospective cohort study. Arthr Rheum 34:937–944CrossRefGoogle Scholar
  37. Politis DN, Sherman M (2001) Moment estimation for statistics from marked point processes. J Royal Stat Soc Ser B 63:261–275MathSciNetCrossRefzbMATHGoogle Scholar
  38. Prentice RL, Williams BJ, Peterson AV (1981) On the regression analysis of multivariate failure time data. Biometrika 68:373–379MathSciNetCrossRefzbMATHGoogle Scholar
  39. Prigent J-L (2001) Option pricing with a general marked point process. Math Oper Res 26:50–66MathSciNetCrossRefzbMATHGoogle Scholar
  40. Prokhorov A, Schmidt P (2009) Likelihood-based estimation in a panel setting: robustness, redundancy and validity of copulas. J Econ 153:93–104MathSciNetCrossRefGoogle Scholar
  41. Robin S (2002) A compound Poisson model for word occurrences in DNA sequences. J Royal Stat Soc Ser C 51:437–451MathSciNetCrossRefzbMATHGoogle Scholar
  42. Schlather M, Ribeiro PJ Jr, Diggle PJ (2004) Detecting dependence between marks and locations of marked point processes. J Royal Stat Soc Ser B 66:79–93MathSciNetCrossRefzbMATHGoogle Scholar
  43. Schoenberg FP (2004) Testing separability in spatialtemporal marked point processes. Biometrics 60:471–481MathSciNetCrossRefzbMATHGoogle Scholar
  44. Sears MR, Taylor DR, Print CG, Lake DC, Li Q, Flannery M, Yates DM, Lucas MK, Herbison GP (1990) Regular inhaled beta-agonist treatment in bronchial asthma. Lancet 336:1391–1396CrossRefGoogle Scholar
  45. Self SG, Liang K-Y (1987) Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. J Am Stat Assoc 82:605–610MathSciNetCrossRefzbMATHGoogle Scholar
  46. Shih JH, Louis TA (1995) Inferences on the association parameters in copula models for bivariate survival data. Biometrics 51:1384–1399MathSciNetCrossRefzbMATHGoogle Scholar
  47. Snyder DL, Miller MI (1991) Random point processes in time and space. Springer, New YorkCrossRefzbMATHGoogle Scholar
  48. Verona E, Petrov D, Cserhati E, Hofman J, Geppe N, Medley H, Hughes S (2003) Fluticasone propionate in asthma: a long term dose comparison study. Arch Dis Child 88:503–509CrossRefGoogle Scholar
  49. Wei LJ, Lin DY, Weissfeld L (1989) Regression analysis of multivariate incomplete failure time data by modeling marginal distributions. J Am Stat Assoc 84:1065–1073MathSciNetCrossRefGoogle Scholar
  50. Yilmaz YE, Lawless JF (2011) Likelihood ratio procedures and tests of t in parametric and semiparametric copula models with censored data. Lifetime Data Anal 17:386–408MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Statistics and Actuarial ScienceUniversity of WaterlooWaterlooCanada

Personalised recommendations