Doe: Response Surface Methods

Origins of the RSM Experimental Arrays (cont.) Allen et al. (2003) proposed “expected integrated mean squared error” (EIMSE) designs as the solution to an optimization problem. To understand their approach, consider that even though experimentation involves uncertainty, much can be predicted before testing begins.

• The formula developed by Allen et al. (2003) suggests that prediction errors are undefined or infinite if the number of runs, n, is less than the number of terms in the fitted model, k.
• This suggests a lower limit on the possible number of runs that can be used. Fortunately, the number of runs is otherwise unconstrained. The formula predicts that as the number of runs increases, the expected prediction errors decrease. This flexibility in the number of runs that can be used may be considered a major advantage of EIMSE designs over CCDs or BBDs. Advanced readers may realize that BBDs are a subset of the EIMSE designs in the sense that, for specific assumption choices, EIMSE designs also minimize the expected bias.

Decision Support Information (Optional):

• It is relevant to decisions about which experimental array should be used to achieve the desired prediction accuracy. Response surface methods (RSM) generate prediction models, yest(x) intended to predict accurately the prototype system’s input-output relationships.
• The phrase “prediction point” refers to a combination of settings, x, at which prediction is of potential interest. The phrase “region of interest” refers to a set of prediction points, R. This name derives from the fact that possible settings define a vector space and the settings of interest define a region in that space.
• The prediction model, yest(x) with the extra subscript is called an “empirical model” since it is derived from data. If there is only one response, then thesubscript is omitted.
• The empirical model, yest(x), is intended to predict the average prototype system response at the prediction point x.
• Ideally, it can predict the engineered system response at x. Through the logical construct of a thought experiment, it is possible to develop an expectation of the prediction errors that will result from performing experiments, fitting a model, and using that model to make a prediction.

Many authors have explored the implications of specific assumptions about yest(x) including Box and Draper (1987) and Myers and Montgomery (2001). The assumptions explored in this section are that the true model is a third order polynomial. Third order polynomials contain all the terms in second order RSM models with the addition of third order terms involving, e.g., x2

3 and x₁ ²x₂. Further,

it assumes that coefficients are random with standard deviation γ.