5.6. Case Studies
5.6.1. Eddy Current Probe Sensitivity Case Study
|Parameter Estimates Don't Change as Additional Terms Added||In most cases of least-squares fitting, the model coefficient estimates for previously added terms change depending on what was successively added. For example, the estimate for the X1 coefficient might change depending on whether or not an X2 term was included in the model. This is not the case when the design is orthogonal, as is this 23 full factorial design. In such a case, the estimates for the previously included terms do not change as additional terms are added. This means the list of effect estimates in section 220.127.116.11 serves as the least-squares coefficient estimates for progressively more complicated models.|
|Default Model: Grand Mean||
If none of the factors are important, the prediction equation
defaults to the mean of all the response values (the overall
or grand mean). That is,
For our example, the default model has a grand mean of 2.65875 with a residual standard deviation (a measure of goodness of fit) of 1.74106 ohms.
|Possible Prediction Equations||
We add effects to the default model in decreasing order of
absolute magnitude and compute the residual standard deviation
after adding each effect. The prediction equations and their
residual standard deviations are shown below.
Residual Model Terms Std. Dev. ----------------------------------------------------- --------- Mean + X1 0.57272 Mean + X1 + X2 0.30429 Mean + X1 + X2 + X2*X3 0.26737 Mean + X1 + X2 + X2*X3 + X1*X3 0.23341 Mean + X1 + X2 + X2*X3 + X1*X3 + X3 0.19121 Mean + X1 + X2 + X2*X3 + X1*X3 + X3 + X1*X2*X3 0.18031 Mean + X1 + X2 + X2*X3 + X1*X3 + X3 + X1*X2*X3 + X1*X2 NANote that the full model is a perfect fit to the data.