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Checking unimodality using isotonic regression: an application to breast cancer mortality rates

  • C. Rueda
  • M. D. UgarteEmail author
  • A. F. Militino
Original Paper

Abstract

In some diseases it is well-known that a unimodal mortality pattern exists. A clear example in developed countries is breast cancer, where mortality increased sharply until the nineties and then decreased. This clear unimodal pattern is not necessarily applicable to all regions within a country. In this paper, we develop statistical tools to check if the unimodality pattern persists within regions using order restricted inference. Break points as well as confidence intervals are also provided. In addition, a new test for checking monotonicity against unimodality is derived allowing to discriminate between a simple increasing pattern and an up-then-down response pattern. A comparison with the widely used joinpoint regression technique under unimodality is provided. We show that the joinpoint technique could fail when the underlying function is not piecewise linear. Results will be illustrated using age-specific breast cancer mortality data from Spain in the period 1975–2005.

Keywords

Change point Isotonic regression Order restricted inference Temporal trends Joint point regression Segmented regression 

Notes

Acknowledgments

This work has been supported by the Spanish Ministry of Science and Innovation (project MTM 2011-22664 jointly sponsored with Feder grants, project MTM 2012-37129 and project MTM2014-51992-R). The work has been also partially supported by the Health Department of Navarre Government (Project 113, Res. 2186/2014).

References

  1. Andersson E, Bock D, Frisén M (2004) Detection of truning points in business cycles. J Bus Cycle Meas Anal 1:93–108Google Scholar
  2. Anraku K (1999) An information criterion for parameters under a simple order restriction. Biometrika 86:141–152CrossRefGoogle Scholar
  3. Akaike H (1973) Information theory and the extension of the maximum likelihood principle. In: Petrov BN, Csaki F (eds) Proceedings of the second international symposium on information theory. Akademiai Kiado, Budapest, pp 267–281Google Scholar
  4. Bachettti P (1989) Additive isotonic models. J Am Stat Assoc 84:289–294Google Scholar
  5. Banerjee M, Mukherjee D, Mishra S (2009) Semiparametric binary regression models under shape constraints with an application to Indian schooling data. J Econom 149:101–117CrossRefGoogle Scholar
  6. Bartholomew DJ (1961) A test of homogeneity for means under restricted alternatives. J R Stat Soc Ser B 23:239–281Google Scholar
  7. Basso D, Salmaso L (2011) A permutation test for umbrella alternatives. Stat Comput 21:45–54 (correction: 2001;20:655)CrossRefGoogle Scholar
  8. Berrino F, De Angelis R, Sant M, Rosso S, Lasota MB, Coebergh JW et al (2007) Survival for eight major cancers and all cancers combined for European adults diagnosed in 1995–99: results of the EUROCARE-4 study. Lancet Oncol 8:773–783CrossRefGoogle Scholar
  9. Bock D, Andersson E, Frisén M (2008) Statistical surveillance of epidemics: peak detection of influenza in Swewden. Biom J 1:71–85CrossRefGoogle Scholar
  10. Brunk HD (1970) Estimation of isotonic regression (with discussion). In: Puri ML (ed) Nonparametric techniques in statistical inference. Cambridge University Press, CambridgeGoogle Scholar
  11. De Souza DLB, Curado MP, Bernal MM et al (2013) Mortality trends and prediction of HPV-related cancers in Brazil. Eur J Cancer Prev 22:380–387CrossRefGoogle Scholar
  12. Dykstra R (1983) An algorithm for restricted least squares regression. Ann Stat 3:401–421Google Scholar
  13. Feder PI (1975) On asymptotic distribution theory in segmented regression problems: identified case. Ann Stat 3:76–83Google Scholar
  14. Gunn LH, Dunson DB (2005) A transformation approach for incorporating monotone or unimodal constraints. Biostatistics 6:434–449CrossRefGoogle Scholar
  15. Hastie T, Tibshirani R (1986) Generalized additive models. Stat Sci 3:297–310CrossRefGoogle Scholar
  16. Huang J (2002) A note on estimating a partly linear model under monotonicity constraints. J Stat Plan Inference 107:345–351CrossRefGoogle Scholar
  17. Hudson D (1966) Fitting segmented curves whose join points have to be estimated. J Am Stat Assoc 61:1097–1129CrossRefGoogle Scholar
  18. Hu X, Wright FT (1994) Likelihood ratio tests for a class of non-oblique hypotheses. Ann Inst Stat Math 46(1):137–145CrossRefGoogle Scholar
  19. Hur C, Miller M, Kong CY et al (2013) Trends in esophageal adenocarcinoma incidence and mortality. Cancer 119(6):1149–1158CrossRefGoogle Scholar
  20. Hwang JG, Peddada SD (1994) Confidence interval estimation subject to order restrictions. Ann Stat 22:67–93CrossRefGoogle Scholar
  21. Iverson GJ, Harp SA (1987) A conditional likelihood ratio test for order restrictions in exponential families. Math Soc Sci 14:141–159CrossRefGoogle Scholar
  22. Kato K (2009) On the degrees of freedom in shrinkage estimation. J Multivar Anal 100:1338–1352CrossRefGoogle Scholar
  23. Kim HJ, Fay MP, Feuer EJ, Midthune DN (2000) Permutation tests for joinpoint regression with applications to cancer rates. Stat Med 19:335–351 (correction: 2001, 20,655)CrossRefGoogle Scholar
  24. Köllmann C, Bornkamp B, Ickstadt K (2012) Unimodal regression using Bernstein-Schoenberg-splines and penalties. Technical Report. Universiy of DortmundGoogle Scholar
  25. Lagarde A, Beausoleil C, Belcher SM, Belzunces LP, Emond C, Guerbet M, Rousselle C (2015) Non-monotonic dose-response relationships and endocrine disruptors: a qualitative method of assessment. Environ Health. doi:10.1186/1476-069X-14-13Google Scholar
  26. Lerman PM (1980) Fitting segmented regression models by grid search. Appl Stat 29:77–84CrossRefGoogle Scholar
  27. Lim C, Shen PK, Peddada SD (2013) Robust analysis of high throughput screening (HTS) assay data. Tecnometrics 55:150–160. doi: 10.1080/00401706.2012.749166 CrossRefGoogle Scholar
  28. Liu T, Lin N, Shi N, Zhang B (2009) Information criterion-based clustering with order-restricted candidate profiles in short time-course microarray experiments. BMC Bioinform 10:146. doi: 10.1186/1471-2105-10-146 CrossRefGoogle Scholar
  29. López-Campos JL, Ruiz-Ramos M, Soriano JB (2013) COPD mortality rates in Andalusia, Spain, 1975–2010: a joinpoint regression analysis. Int J Tuberc Lung Dis 17:131–136CrossRefGoogle Scholar
  30. Malvezzi M, Bertuccio P, Levi F, La Vecchia C, Negri E (2012) European cancer mortality predictions for the year 2012. Ann Oncol 23:1044–1052CrossRefGoogle Scholar
  31. Menéndez JA, Salvador B (1991) Anomalies of the likelihood ratio test for testing restricted hypotheses. Ann Stat 19:889–898CrossRefGoogle Scholar
  32. Menéndez JA, Rueda C, Salvador B (1991) Conditional test for testing a face of the tree order cone. Commun Stat Simul Comput 20(2&3):751–762CrossRefGoogle Scholar
  33. Menéndez JA, Rueda C, Salvador B (1992) Dominance of likelihood ratio tests under cone constraints. Ann Stat 20:2087–2099CrossRefGoogle Scholar
  34. Meyer M (2008) Inference using shape-restricted regression splines. Ann Stat 2:1013–1033CrossRefGoogle Scholar
  35. Meyer M, Woodruff M (2000) On the degrees of freedom in shape-restricted regression. Ann Stat 28:1083–1104CrossRefGoogle Scholar
  36. Molodecky NA, Soon IS, Rabi DM et al (2012) Increasing incidence and prevalence of the inflammatory bowel diseases with time, based on systematic review. Gastroenterology 142:46–54CrossRefGoogle Scholar
  37. Morton-Jones T, Diggle P, Parker L, Dickinson HO, Binks K (2000) Additive isotonic regression models in epidemiology. Stat Med 9:849–859CrossRefGoogle Scholar
  38. Muggeo VMR (2003) Estimating regression models with unknown break-points. Stat Med 22:3055–3071CrossRefGoogle Scholar
  39. Muggeo VMR (2008) Segmented: an R package to fit regression models with broken-line relationships. R News 8(1):20–25. URL http://cran.r-project.org/doc/Rnews/
  40. O’Connell RG, Dockree PM, Kelly SP (2012) A supramodal accumulation-to-bound signal that determines perceptual decisions in humans. Nat Neurosci 15:1729–1735CrossRefGoogle Scholar
  41. Pan G (1997) Confidence subset containing the unknown peaks of an umbrella ordering. J Am Stat Assoc 92(437):307–314CrossRefGoogle Scholar
  42. Peddada SD (1997) Confidence interval estimation of population means subject to order restrictions using resampling procedures. Stat Probab Lett 31:255–265CrossRefGoogle Scholar
  43. Robertson T, Wright FT, Dykstra RL (1988) Order restricted statistical inference. Wiley, New yorkGoogle Scholar
  44. Rueda C, Lombardía MJ (2012) Small area semiparametric additive monotone models. Stat Model 12:527–549CrossRefGoogle Scholar
  45. Rueda C (2013) Degrees of freedom and model selection in semiparametric additive monotone regression. J Multivar Anal 117:88–99CrossRefGoogle Scholar
  46. Ruiz-Medina MD, Espejo RM, Ugarte MD, Militino AF (2014) Functional time series analysis of spatio-temporal epidemiological data. Stoch Environ Res Risk Assess 28:943–954. doi: 10.1007/s00477-013-0794-y CrossRefGoogle Scholar
  47. Shapiro A (1988) Towards a unified theory of inequality constraints testing in multivariate analysis. Int Stat Rev 56:49–62CrossRefGoogle Scholar
  48. Shi NZ (1988) A test of homogeneity for umbrella alternatives and tables of the level probabilities. Commun Stat Theory Methods 17:657–670CrossRefGoogle Scholar
  49. Shively TS, Walker SG, Damien P (2011) Nonparametric function estimation subject to monotonicity, convexity, and other shape constraints. J Econom 161:166–181CrossRefGoogle Scholar
  50. Simard EP, Ward EM, Siegel R et al (2012) Cancers with increasing incidence trends in the United States: 1999 through 2000. CA: Cancer J Clin 62:118–128Google Scholar
  51. Sprent P (1961) Some hypotheses concerning two-phase regression lines. Biometrics 17:634–645CrossRefGoogle Scholar
  52. Statistical Research and Applications Branch, National Cancer Institute (2013). Joinpoint Regression Program, Version 4.0.4 - May 2013Google Scholar
  53. Strand M, Zhang Y, Swihart B (2010) Monotone nonparametric regression and confidence intervals. Commun Stat Simul Comput 39(4):828–845CrossRefGoogle Scholar
  54. Susko E (2013) Likelihood ratio tests with boundary constraints using data-dependent degrees of freedom. Biometrika 100:1019–1023CrossRefGoogle Scholar
  55. Thomas SC (2010) Photosynthetic capacity peaks at intermediate size in temperate deciduous trees. Tree Physiol 30:555–573CrossRefGoogle Scholar
  56. Turner R (2013) Iso: functions to perform isotonic regression. R package version 0.0-12. http://CRAN.R-project.org/package=Iso
  57. Turner TR, Wollan PC (1997) Locating a maximum usin isotonic regression. Comput Stat Data Anal 25:305–320CrossRefGoogle Scholar
  58. Ugarte MD, Goicoa T, Etxeberria J, Militino AF, Pollán M (2010) Age-specific spatio-temporal patterns of female breast cancer mortality in Spain (1975–2005). Ann of Epidemiol 20:906–916CrossRefGoogle Scholar
  59. Vandenberg LN, Colborn T, Hayes TB, Heindel JJ, Jacobs DR Jr, Lee DH, Shioda T, Soto AM, vom Saal FS, Welshons WV, Zoeller RT, Myers JP (2012) Hormones and endocrine-disrupting chemicals: low-dose effects and nonmonotonic dose responses. Endocr Rev 33:378–455CrossRefGoogle Scholar
  60. Wolfe DA (2006) Nonparametric distribution-free procedures for order restricted alternatives. In: Ahsanullah M, Raquad MZ (eds) Recent developments in order random variables. Nova Science, New YorkGoogle Scholar
  61. Wollan PC, Dykstra RL (1986) Conditional test with an order restriction as a null hypothesis in advances in order restricted statistical inference. Lect Notes Stat 37:279–295CrossRefGoogle Scholar
  62. Zhao L, Peng L (2002) Model selection under order restrictions. Stat Probab Lett 57:301–306CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Statistics and Operations Research, Facultad de CienciasValladolid UniversityValladolidSpain
  2. 2.Department of Statistics and Operations ResearchPublic University of NavarrePamplonaSpain
  3. 3.INAMATPublic University of NavarrePamplonaSpain

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