ABSTRACT
In previous chapters, optimal settings were obtained through the prediction (regression) equation, which was assumed to provide a good fit. In this chapter, we present simple graphic methods and formal statistical tests for studying the aptness of the prediction equation. The graphic methods include the plot of the residuals against the fitted values, the normal probability plot of the residuals, and the histogram of the residuals. The formal statistical methods include the adjusted coefficient of determination and the test for lack of fit.-
8.2 GRAPHIC ANALYSIS
A residual e is defined as the difference between the observed value and the fitted value:
e Y Y . ~ c V
There are four important types of departures from an appropriate regression equation: 1. The regression equation is not linear. 2. The error terms do not have a constant variance. 3. The error terms are not independent. 4. The error terms are not normally distributed. The first three important types of departure can be verified through a simple plot, that is, a plot of the residuals versus the fitted values. There are four prototypes of residual plots. Figure 8.1 shows that the residuals tend to fall within a horizontal band centered around zero. This fact alone shows that the predicted values are close to the observed values and, since the residuals fall between two parallel lines, the error terms have a constant variance, and finally, since the residuals seem to randomly fluctuate around zero, the error terms are independent.
