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

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


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.


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



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).


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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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