Locally optimal designs for errors-in-variables models

Abstract

This paper considers the construction of optimal designs for nonlinear regression models when there are measurement errors in the predictor. Corresponding (approximate) design theory is developed for maximum likelihood and least squares estimation, where the latter leads to non-concave optimisation problems. For the Michaelis-Menten, EMAX and exponential regression model D-optimal designs can be found explicitly and compared with the corresponding designs derived under the assumption of no measurement error in concrete applications.

Bayesian D‐optimal designs for error‐in‐variables models

“…where p + 1 is the number of model parameters and ξ * θ is the locally D-optimal design for errors-in-variables models with classical errors using the parameter values vector θ, explicitly defined in Konstantinou and Dette [2015]. We begin with an investigation of how the Bayesian D-optimal saturated designs change in the presence of error in the covariates.…”

“…Using a uniform prior with ν = 11 this design is equally supported at points 3.06 and 80 for maximum likelihood estimation and at points 5.82 and 80 when the parameters are estimated via least squares. As for the case of locally D-optimal designs discussed in Konstantinou and Dette [2015], taking the error in the covariate into account results in the non-trivial support point of the Bayesian D-optimal design to move further away from x = 0 compared to its value when ̺ 2 εη is assumed to be zero. As the ̺ 2 εη -value becomes larger, the value of the non-trivial support point increases further.…”

“…In this example the use of error-in-variables models is justified by the possible false measurement of the intensity of an X-ray beam, which is then used to determine the percentage of gas-brine saturation (see Frisillo and Stewart [1980] for more details). Konstantinou and Dette [2015] fit the exponential regression model (4.3) to the data and obtain the parameter estimates Table 1: Left part: Support points of the Bayesian D-optimal two-point designs using a uniform prior with ν = 5 or ν = 11 values for each parameter. Right part: Efficiencies (%) of the Bayesian D-optimal two-point design (ν = 11) for ̺ 2 εη = 0.…”

“…Despite of their importance, the literature on optimal designs for error-in-variables models with classical errors is rather scarce [see Keeler and Reilly [1992] and Dovi et al [1993] for early references]. Recently, Konstantinou and Dette [2015] develop an approximate optimal design theory for local optimality criteria in error-in-variables models with classical errors and provide analytical results on locally D-optimal designs for some commonly used nonlinear models when these are subject to the classical error structure. This paper extends their work and provides the corresponding approximate design theory for Bayesian optimality.…”

“…Finally, in Section 5 we consider the case of a uniform prior on the parameter space. Via several data examples, we establish the superiority of the resulting Bayesian D-optimal designs by comparing them to the corresponding locally D-optimal designs, explicitly defined in Konstantinou and Dette [2015], as well as to other designs frequently used in practice.…”