Diagnostics to assess toxicokinetic–toxicodynamic models with interval‐censored data

Abstract

In ecotoxicology, toxicokinetic-toxicodynamic models are increasingly being used to relate toxicant concentration to survival. These have a toxicokinetic component that specifies a model for toxicant uptake and elimination and a toxicodynamic component that specifies a model for the relationship between internal body concentration of toxicant and the hazard rate. Often, the toxicodynamic model is a proportional hazards model and the data to evaluate the model is interval censored. We develop diagnostic methods to assess the appropriateness of the toxicokinetic and hazard models for interval-censored data. A log-cumulative hazard plot is used to assess visually the proportional hazard assumption. This can be further evaluated by an approximate test based on a linear regression. Martingale residuals for interval-censored data are poorly behaved, and their use is not recommended. If the proportional hazard assumption is valid, the kinetic model can be assessed by a plot of toxicant concentration against the average of log-cumulative hazard for each concentration. The use of these diagnostic tools is illustrated by using data on Hyalella azteca survival during and after exposure to copper sulfate and sodium pentachlorophenol. The model that was initially proposed did not fit the data well, but the diagnostic tools suggested appropriate modifications. The diagnostic tools can be applied more generally to interval-censored survival data.