Design of experiments in the presence of errors in factor levels

Abstract

This paper is concerned with the statistical properties of experimental designs where the factor levels cannot be set precisely. When the errors in setting the factor levels cannot be measured, design robustness is explored. However, when the actual design could be measured at the end of the investigation, its optimality is of interest. D-optimality could be assessed in di erent ways. Several measures are compared. Evaluating them is di cult even in simple cases. Therefore, in general, simulations are used to obtain their values. It is shown that if D-optimality is measured by the expected value of the determinant of the information matrix of the experimental design, as has been suggested in the past, on average the designs appear to improve with the variance of the error in setting the factor levels. However, we argue that the criterion of D-optimality should be based on the inverse of the information matrix. In this case it is shown that the experiment could be better or worse than the planned one. It is also recognized that setting the factor levels with error could lead to an increased risk of losing observations, which on its own could reduce considerably the optimality of the experimental designs. Advice on choosing the design region in such a way that such a risk is controlled to an acceptable level is given.

A weighted prediction‐based selection criterion for response surface designs

“…A concentration on D-optimality is particularly susceptible to such problems because of its tendency to utilize extreme points of the region. In this case, one tool for overcoming the issue might be use of 'guard regions' 19 , but these tends to be conservative and themselves require specification. Probability-based criteria 13 , particularly when compounded with other criteria, might be more useful and, indeed, our own weighted interest criterion could be modified to limit interest in mapping the response at the vaguely specified edges of the region.…”

Robust parameter design based on response surface model under considering measurement errors

“…Under this assumption, the response model derived from the experimental results is unaffected by the measurement errors. Donev [20] is concerned with the statistical properties of experimental designs where the factor levels cannot be set precisely and proposed that the criterion of D-optimality should be based on the inverse of the information matrix. They recognized that errors in variables can affect the mean and the variance of responses.…”

Locally optimal designs for errors-in-variables models

“…More recent results on Berkson error models are given in Pronzato (2002) and Donev (2004). Besides the initial assumption of errors in setting the independent variable values, Pronzato (2002) further assumes that the true values, and therefore the actual design used, is known at the end of the experiment.…”

“…He uses the expected value of the determinant of the information matrix as an extension to the D-optimality criterion. Donev (2004) considers both the cases of unknown and known true values at the end of the study. Under the former scenario he proposes the use of the sum of the variances of the estimated responses as a measure of robustness whereas in the latter case he argues that minimising the expected value of the inverse information matrix is a more appropriate criterion than that proposed by Pronzato (2002).…”