Environmental and Ecological Statistics

, Volume 22, Issue 3, pp 465–491 | Cite as

Modelling grouped survival times in toxicological studies using Generalized Additive Models

  • S. G. Candy
  • B. J. Sfiligoj
  • C. K. King
  • J. A. Mondon


A method for combining a proportional-hazards survival time model with a bioassay model where the log-hazard function is modelled as a linear or smoothing spline function of log-concentration combined with a smoothing spline function of time is described. The combined model is fitted to mortality numbers, resulting from survival times that are grouped due to a common set of observation times, using Generalized Additive Models (GAMs). The GAM fits mortalities as conditional binomials using an approximation to the log of the integral of the hazard function and is implemented using freely-available, general software for fitting GAMs. Extensions of the GAM are described to allow random effects to be fitted and to allow for time-varying concentrations by replacing time with a calibrated cumulative exposure variable with calibration parameter estimated using profile likelihood. The models are demonstrated using data from a studies of a marine and a, previously published, freshwater taxa. The marine study involved two replicate bioassays of the effect of zinc exposure on survival of an Antarctic amphipod, Orchomenella pinguides. The other example modelled survival of the daphnid, Daphnia magna, exposed to potassium dichromate and was fitted by both the GAM and the process-based DEBtox model. The GAM fitted with a cubic regression spline in time gave a 61 % improvement in fit to the daphnid data compared to DEBtox due to a non-monotonic hazard function. A simulation study using each of these hazard functions as operating models demonstrated that the GAM is overall more accurate in recovering lethal concentration values across the range of forms of the underlying hazard function compared to DEBtox and standard multiple endpoint probit analyses.


Dose–response model Generalized Additive Model Grouped survival times Time–response model 



We are grateful to Professor Simon Wood for assistance in understanding the workings of the gam and gamm functions in his mgcv package. The constructive reviews by the Associate Editor and the referees contributed substantially to improvements in this work and its presentation. This research was partly funded by the Australian Antarctic Division through AAS Project 2933 (CI King).

Supplementary material

10651_2014_306_MOESM1_ESM.docx (59 kb)
Supplementary material 1 (docx 59 KB)


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • S. G. Candy
    • 1
    • 2
  • B. J. Sfiligoj
    • 3
    • 4
  • C. K. King
    • 3
  • J. A. Mondon
    • 4
  1. 1.Wildlife Conservation and FisheriesAustralian Antarctic DivisionKingstonAustralia
  2. 2.SCandy Statistical Modelling Pty LtdBlackmans BayAustralia
  3. 3.Terrestrial and Near Shore EcosystemsAustralian Antarctic DivisionKingstonAustralia
  4. 4.Environmental Sustainability Research Group, Faculty of Science and TechnologyDeakin UniversityWarrnamboolAustralia

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