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

Validate the Fitted Model

Model Validation In the Important Factors and Parsimonious Prediction section, we selected the following model
    \( \hat{Y} = 2.65875 + 1.55125 X_{1} - 0.43375 X_{2} \)
The residual standard deviation for this model is 0.30429.

The next step is to validate the model. The primary method of model validation is graphical residual analysis; that is, through an assortment of plots of the differences between the observed data Y and the predicted value \( \scriptsize \hat{Y} \) from the model. For example, the design point (-1, -1, -1) has an observed data point (from the Background and data section) of Y = 1.70, while the predicted value from the above fitted model for this design point is

    \( \hat{Y} = 2.65875 + 1.55125 (-1) - 0.43375 (-1) \)
which leads to the residual 0.15875.
Table of Residuals If the model fits well, \( \scriptsize \hat{Y} \) should be near Y for all eight design points. Hence the eight residuals should all be near zero. The eight predicted values and residuals for the model with these data are:
   X1   X2   X3  Observed Predicted  Residual
   -1   -1   -1    1.70    1.54125    0.15875
   +1   -1   -1    4.57    4.64375   -0.07375
   -1   +1   -1    0.55    0.67375   -0.12375
   +1   +1   -1    3.39    3.77625   -0.38625
   -1   -1   +1    1.51    1.54125   -0.03125
   +1   -1   +1    4.59    4.64375   -0.05375
   -1   +1   +1    0.67    0.67375   -0.00375
   +1   +1   +1    4.29    3.77625    0.51375
Residual Standard Deviation What is the magnitude of the typical residual? There are several ways to compute this, but the statistically optimal measure is the residual standard deviation:
    \( s_{res} = \sqrt{ \frac{\sum_{i=1}^{N}{r_{i}^{2}}} {N-P}} \)
with ri denoting the ith residual, N = 8 is the number of observations, and P = 3 is the number of fitted parameters. From the table of prediction equations, the residual standard deviation is 0.30429.
How Should Residuals Behave? If the prediction equation is adequate, the residuals from that equation should behave like random drawings (typically from an approximately normal distribution), and should, since presumably random, have no structural relationship with any factor. This includes any and all potential terms (X1, X2, X3, X1*X2, X1*X3, X2*X3, X1*X2*X3).

Further, if the model is adequate and complete, the residuals should have no structural relationship with any other variables that may have been recorded. In particular, this includes the run sequence (time), which is really serving as a surrogate for any physical or environmental variable correlated with time. Ideally, all such residual scatter plots should appear structureless. Any scatter plot that exhibits structure suggests that the factor should have been formally included as part of the prediction equation.

Validating the prediction equation thus means that we do a final check as to whether any other variables may have been inadvertently left out of the prediction equation, including variables drifting with time.

The graphical residual analysis thus consists of scatter plots of the residuals versus all three factors and four interactions (all such plots should be structureless), a scatter plot of the residuals versus run sequence (which also should be structureless), and a normal probability plot of the residuals (which should be near linear). We present such plots below.

Residual Plots

various residual plots do not indicate any serious problems with the model

The first plot is a normal probability plot of the residuals. The second plot is a run sequence plot of the residuals. The remaining plots show the residuals plotted against each of the factors and each of the interaction terms.

Conclusions We make the following conclusions based on the above plots.
  1. Main Effects and Interactions: The X1 and X2 scatter plots are "flat" (as they must be since X1 and X2 were explicitly included in the model). The X3 plot shows some structure as does the X1*X3, the X2*X3, and the X1*X2*X3 plots. The X1*X2 plot shows little structure. The net effect is that the relative ordering of these scatter plots is very much in agreement (again, as it must be) with the relative ordering of the "unimportant" factors. From the table of effects and the X2*X3 residual plot, the third most influential term to be added to the model would be X2*X3. In effect, these plots offer a higher-resolution confirmation of the ordering of effects. On the other hand, none of these other factors "passed" the criteria given in the previous section, and so these factors, suggestively influential as they might be, are still not influential enough to be added to the model.

  2. Time Drift: The run sequence scatter plot is random. Hence there does not appear to be a drift either from time, or from any factor (e.g., temperature, humidity, pressure, etc.) possibly correlated with time.

  3. Normality: The normal probability plot of the eight residuals has some trend, which suggests that additional terms might be added. On the other hand, the correlation coefficient of the 8 ordered residuals and the eight theoretical normal N(0,1) order statistic medians (which define the two axes of the plot) has the value 0.934, which is well within acceptable (5 %) limits of the normal probability plot correlation coefficient test for normality. Thus, the plot is not so non-linear as to reject normality.
In summary, therefore, we accept the fitted model
    \( \hat{Y} = 2.65875 + 1.55125 X_{1} - 0.43375 X_{2} \)
as a parsimonious, but good, representation of the sensitivity phenomenon under study.
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