5. Process Improvement
5.6. Case Studies
5.6.1. Eddy Current Probe Sensitivity Case Study

## Modeling and Prediction Equations

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 5.6.1.5 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,
$$\hat{Y} = 2.65875$$

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            NA

Note that the full model is a perfect fit to the data.