Errors in the Factor Levels and Experimental Design

Draper, Norman R.; Beggs, W.J.

doi:10.1214/aoms/1177693493

Cited by 14 publications

(11 citation statements)

References 2 publications

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“…In this case the information matrix of a complete factorial or fractional factorial design is G = nI p and therefore D = log|G −1 | = −p log n. Deriving explicit expressions for the remaining criteria of optimality even in simple cases is usually not possible. Similar di culties in evaluating criteria of optimality analytically have been reported by Draper and Beggs (1970) and Pronzato (2002). On the other hand such results may not be necessary as the criteria of optimality can be estimated relatively easy with the use of simulations.…”

Section: Two-level Factorial Designsmentioning

confidence: 58%

“…The e ect of errors in the factor levels on the statistical properties of the parameters obtained from a two-level factorial and fractional factorial designs was ÿrst studied by Box (1963). Draper and Beggs (1970) measure the robustness of experimental designs to errors in the factor levels by the sum of squared di erences between the observed and the predicted with the model response values. They assume that the errors are very small and show that zero ÿrst and odd second order moments of the designs are desirable.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2 we discuss how the analysis of the data is a ected if the actual design is not known and introduce a measure for design robustness that is easer to calculate and interpret than those used previously by Draper and Beggs (1970) and Vuchkov and Boyadjieva (1983). We show that when the actual design can be recorded exactly and prior information about the errors in setting the factor levels is available, other generalizations of the D-optimality criterion than that used by Pronzato (1998Pronzato ( , 2002 are more appropriate.…”

Section: Introductionmentioning

confidence: 99%

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Design of experiments in the presence of errors in factor levels

Donev

2004

Journal of Statistical Planning and Inference

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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.

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Section: Two-level Factorial Designsmentioning

confidence: 58%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Design of experiments in the presence of errors in factor levels

Donev

2004

Journal of Statistical Planning and Inference

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“…He considers response surface models with Berkson type errors and explores the robustness of factorial and fractional factorial designs. Following Box (1963), Draper and Beggs (1971) and Vuchkov and Boyadjieva (1983) propose the use of the sum of squared differences of the responses and the maximum element of the information matrix respectively, as a measure of robustness against Berkson type errors. Furthermore, Tang and Bacon-Shone (1992) study the construction of Bayesian optimal designs for the Berkson type probit model.…”

Section: Introductionmentioning

confidence: 99%

Locally optimal designs for errors-in-variables models

Konstantinou

Dette

2015

Biometrika

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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.

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“…E-m ail: tan@uta.® 236 T. Num mi m any growth exp eriments, since the levels of regressors are often controlled by the exp erimenter, so cannot be directly considered as random variables. M easurement error m odels where the levels of regressor variables are controlled by an experim enter have been considered by Berkson (1950), Box (1961) andD raper andBeggs (1971), for exam ple.…”

Section: Introductionmentioning

confidence: 99%