Bias correction of the maximum likelihood estimator for Emax model at the interim analysis

The Emax model is a dose-response model commonly applied in clinical trials, agriculture and environmental experiments. We consider a two-stage adaptive design for collecting \lq\lq optimal" data for estimating the model parameters. At the first stage (interim analysis) a locally D-optimum design is computed to get a sample of independent observations and to produce a first-stage maximum likelihood estimate (MLE). At the second stage, the first-stage MLE is used as initial parameter-value to determine another D-optimum design and then to collect the second-stage observations.\\ %(which depend on the responses observed at the first stage). \\ The first-stage estimate influences the quality of the data gathered at the second stage, where a large number of observations can be collected. In real life problems, instead, the sample size of the interim analysis is usually small; therefore, the first-stage MLE should be precise enough even if based on few data. From this consideration, our guess is that if we improved the behaviour of the first-stage MLE through a bias correction, then the D-optimal design determined at the second stage would produce better experimental points. In this study we provide the analytic expression of the first-order bias correction of the MLE in the Emax model