Risk assessment for quantitative responses using a mixture model

A problem that frequently occurs in biological experiments with laboratory animals is that some subjects are less susceptible to the treatment than ot hers. A mixture model has traditionally been proposed to describe the distr ibution of responses in treatment groups for such experiments. Using a mixt ure dose-response model, Re derive an upper confidence limit on additional risk, defined as the excess risk over the background risk due to an added d ose. Our focus will be on experiments with continuous responses for which r isk is the probability of an adverse effect defined as an event that is ext remely rare in controls. The asymptotic distribution of the likelihood rati o statistic is used to obtain the upper confidence limit on additional risk . The method can also be used to derive a benchmark dose corresponding to a specified level of increased risk. The EM algorithm is utilized to find th e maximum likelihood estimates of model parameters and an extension of the algorithm is proposed to derive the estimates when the model is subject to a specified level of added risk. An example is used to demonstrate the resu lts, and it is shown that by using the mixture model a more accurate measur e of added risk is obtained.