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

5.6.1.6.

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 ranked list of effect estimates in the Yates table simultaneously serves as the least squares coefficient estimates for progressively more complicated models.

The last column of the Yates table gave the residual standard deviation for 8 possible models, each one progressively more complicated.

Default Model: Grand Mean

At the top of the Yates table, 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,

From the last column of the Yates table, it can be seen that this simplest of all models has a residual standard deviation (a measure of goodness of fit) of 1.74106 ohms. Finding a good-fitting model was not one of the stated goals of this experiment, but the determination of a good-fitting model is "free" along with the rest of the analysis, and so it is included.

Conclusions

From the last column of the Yates table, we can summarize the following prediction equations:

• 

  has a residual standard deviation of 1.74106 ohms.

• 

  has a residual standard deviation of 0.57272 ohms.

• 

  has a residual standard deviation of 0.30429 ohms.

• 

  has a residual standard deviation of 0.29750 ohms.

• The remaining models can be listed in a similar fashion. Note that the full model provides a perfect fit to the data.