TABLE OF CONTENTS Once data has been obtained from a designed experiment by measuring the response of interest at various combinations of input factor levels dictated by the design, the results may be summarized in the form of a response surface. A response surface is simply a polynomial fit to the measured data. The proper fit is obtained using statistical regression techniques such as the method of least squares, also known as regression analysis.
The goal of regression analysis is to develop a quantitative model, usually in the form, of a polynomial, which predicts a relationship between input factors and a given response. An accurate model should minimize the difference between the observed values of the response and its own predictions.
The simplest polynomial response surface is merely a straight line. Models fit to a straight line are derived using linear regression. Consider fitting experimental data to a straight line which passes through the origin. Although rather elementary, this example illustrates many of the basic principles of least squares.
Suppose the electroplating of copper is being studied and n = 9 observations of the data shown in Table 20.11 are collected, where x is the time in minutes and y is the thickness of the copper film. Physical considerations indicate that a simple proportional relationship between x and y is reasonable. That is, the relationship between x and y should be described by a straight line through the origin, or:
