A theoretical model is developed for the estimation of the toxicity of a large number of chemicals to a group of species for which the log-toxicity values have strong linear relationships. The model is a multi-dimensional functional relationship (FR) that allows for many missing values. It depends on an unobserved variate (denoted by ξ in the text), with a value for each chemical; for each species, the expectations of the log-toxicity values are each linearly related to ξ. There are worthwhile gains in multi-dimensional over the two-dimensional FRs hitherto used. The model is applied to data on the toxicity of 51 chemicals to 10 species. It allows for species to have unequal variance about their expectations. There is good prediction of log-toxicity from measurements on other species where only 30% of the combinations were measured even when the taxa are not closely related. Among the uses for this technique are (i) to predict toxicity for combinations of species and chemicals that have not been measured, (ii) to suggest a suitable range of concentrations for a new bioassay, (iii) identifying specificity, that is, chemicals that have a different relative toxicity for a particular species. Copyright © 2008 John Wiley & Sons, Ltd.