Experimental Design - Critical Values for Central Composite Designs

Both the Quick tab and the Prediction & profiling tab of the Analysis of a Central Composite (Response Surface) Experiment dialog box contain the Critical values (min, max, saddle) button, available whenever a quadratic response surface model is used to predict the dependent variable. Click the Critical values (min, max, saddle) button to produce three spreadsheets that provide results of the analysis of the quadratic response surface:

Response surface

The Response surface spreadsheet displays the second-order effects of the predictor variables on the response (i.e., the linear by linear interactions and the quadratic effects), and the first-order effects of the predictor variables on the response (i.e., the main effects). Note that these effects correspond to those in the spreadsheet displayed when you click the Regression coefficients button on the ANOVA/Effects tab.

Eigenvalues and eigenvectors

The Eigenvalues and eigenvectors spreadsheet is useful in identifying the shape and the orientation of the quadratic response surface. The determinant of the matrix of second-order effects of the predictor variables on the response is displayed in the information box at the top of the spreadsheet. If the determinant is close to zero, the response surface is nearly flat in at least one direction. The eigenvalues of the second-order effects represent the curvature of the quadratic response surface. The eigenvalues are positive if the response surface curves upward from a minimum, and are negative if the surface curves downward from a maximum. Mixed eigenvalues indicate that the surface is shaped like a saddle, curving upward in one direction and downward in another. The eigenvectors show the orientations of the principal axes of the quadratic response surface relative to the axes of the original predictor variables. A high "loading" of a predictor variable on a principal axis indicates that the axis of the response surface is oriented in the same direction as the axis of the predictor variable. Inspecting the eigenvectors and their corresponding eigenvalues provides useful information about the curvature (upward or downward), or lack thereof (flatness) of the quadratic response surface in each direction defined by the axes of the original predictor variables.

Critical values

The Critical values spreadsheet displays information that identifies the point on the quadratic response surface that defines the curvature of the surface. The critical values for the predictor variables are the coordinates (on the axes of the predictor variables) of the origin of the quadratic response surface. The information box at the top of the spreadsheet displays whether this point represents a minimum, a maximum, or a saddle point on the response surface. The predicted value of the dependent variable at the critical values for each of the predictor variables is also displayed in the information box. The three columns of the Critical values spreadsheet list the Observed minimum values of the predictor variables, the Critical values of the predictor variables, and the Observed maximum values of the predictor variables. Rows in the Critical values spreadsheet in which the critical value lies outside the observed range of the predictor variable are highlighted. This draws attention to an origin for the response surface that lies outside the experimental region, and to the predictor variable (or variables) for which the origin is beyond the observed range.

See also the note on Main Effects and Interactions.