Least squares estimation

As with other normal models, the parameters are usually estimated by least squares.

Displaying the fitted model in two dimensions

The fitted quadratic model corresponds to a surface in 3 dimensions. This surface was shown on a rotatable 3-dimensional display in earlier pages in this section.

To display the response surface in 2 dimensions (e.g. on a printed page), the surface is represented on axes for the two factors, X and Z. The fitted mean response (i.e. the surface height) is either represented by:

These displays are easier to explain with an example than in words.

Chemical process yield

The diagram below shows experimental results from a single replicates of a 22 factorial experiment for the yield of a chemical process with factors Time and Temperature. The experiment was augmented with a single replicate of the four star points and four replicates of the centre point of the design.

The surface above is the least squares quadratic response surface fitted to the data. Rotate the diagram (by dragging with the mouse or using the rotation buttons) to get a better feel for the shape of the surface.

Click the x-z rotation button. The resulting 2-dimensional display uses colour to indicate the mean response. The key on the right shows how the colours relate to the mean value of the response, Yield.

Now click the y-x-z rotation button and drag the black arrow on the lefthand key. The black curve on the surface shows Time-Temperature combinations with equal quality — i.e. contours in the diagram.

Click the x-z rotation button and select Selected contours from the pop-up menu under the display. Continue dragging the black arrow on the key and observe that the diagram shows the contours for mean Yield equal to 72, 74, 76, ..., 88.

This display of selected contours is the most commonly used representation of a response surface.