Summary
A method for combining a proportional-hazards survival time model with a bioassay model where the log-hazard function is modelled as a linear function of log-concentration combined with a cubic smoothing spline function of time is described. The combined model is fitted to mortality numbers resulting from interval-censored survival times using a generalized additive model (GAM) with the assumption that mortalities are conditional binomials combined with an approximation to the log of the integral of the hazard function using freely-available, general software for fitting GAM. Extensions of the GAM to allow random effects to be fitted and to allow for time-varying concentrations by replacing time with a calibrated cumulative exposure (CCE) variable with the calibration parameter estimated using profile likelihood are described. Random effects defined by multi-level sampling units such as replicate bioassay or defined by lack-of-fit within bioassay via the interaction of concentration levels with time intervals can be incorporated. Numerical methods are used to obtain lethal concentration values under fixed concentrations over time. The models are demonstrated using data from a study of a marine and, a previously published paper on 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 gave a 30% improvement in fit to the daphnid data compared to DEBtox as measured by the deviance.