Nonlinear MLE estimation of the shift parameter for ELISA quality control

The quality control for ELISA dilution procedure should be provided to ensure the accuracy of the concentration estimation. This is potentially important for health studies of allergens and childhood asthma, for which even very low exposures can be dangerous and non-negligible (Gelman and Shnaidman, 2004).;The traditional approach (Linear models) can not estimate the exact shift between sucessive samples in a bi-diluted data setting thus it is not able to ensure linearity among bi-diluted samples. This problem is more severe when the reference and samples are not quite consistent. Hence, we propose a nonlinear procedure based on the maximum likelihood estimation to estimate the shift parameter. We consider three different models: a null model, a constrained model, and a non-constrained model. Simulation studies show a significant advantage of the nonlinear procedure in estimating the shift in various scenarios.