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The RSREG Procedure |

Suppose that a response variable *y* is measured at combinations
of values of two factor variables, *x _{1}* and

- model fitting and analysis of variance to estimate parameters
- canonical analysis to investigate the shape of the predicted response surface
- ridge analysis to search for the region of optimum response

- What is the contribution of each type of effect -linear, quadratic, and crossproduct -to the statistical fit? The ANOVA table with sources labeled "Regression" addresses this question.
- What part of the residual error is due to lack of fit? Does the quadratic response model adequately represent the true response surface? If you specify the LACKFIT option in the MODEL statement, then the ANOVA table with sources labeled "Residual" addresses this question.
- What is the contribution of each factor variable to the statistical fit? Can the response be predicted as well if the variable is removed? The ANOVA table with sources labeled "Factor" addresses this question.
- What are the predicted responses for a grid of factor values? (See the section "Plotting the Surface" and the "Searching for Multiple Response Conditions" section.)

If some observations in your design are replicated, you can test for lack of fit by specifying the LACKFIT option in the MODEL statement. Note that, since all other tests use total error rather than pure error, you may want to hand-calculate the tests with respect to pure error if the lack-of-fit is significant. On the other hand, significant lack-of-fit indicates the quadratic model is inadequate, so if this is a problem you can also try to refine the model, possibly using PROC GLM for general polynomial modeling; refer to Chapter 30, "The GLM Procedure," for more information. Example 56.1 illustrates the use of the LACKFIT option.

- Is the surface shaped like a hill, a valley, a saddle surface, or a flat surface?
- If there is a unique optimum combination of factor values, where is it?
- To which factor or factors are the predicted responses most sensitive?

The eigenvalues and eigenvectors in the matrix of second-order parameters characterize the shape of the response surface. The eigenvectors point in the directions of principle orientation for the surface, and the signs and magnitudes of the associated eigenvalues give the shape of the surface in these directions. Positive eigenvalues indicate directions of upward curvature, and negative eigenvalues indicate directions of downward curvature. The larger an eigenvalue is in absolute value, the more pronounced is the curvature of the response surface in the associated direction. Often, all of the coefficients of an eigenvector except for one are relatively small, indicating that the vector points roughly along the axis associated with the factor corresponding to the single large coefficient. In this case, the canonical analysis can be used to determine the relative sensitivity of the predicted response surface to variations in that factor. (See the "Getting Started" section for an example.)

- If there is not a unique optimum of the response surface within the range of experimentation, in which direction should further searching be done in order to locate the optimum?

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