5. Process Improvement
5.6. Case Studies
5.6.2. Sonoluminescent Light Intensity Case Study

## Cumulative Residual Standard Deviation Plot

Purpose The cumulative residual standard deviation plot is used to identify the best (parsimonious) model.
Sample Cumulative Residual Standard Deviation Plot
Conclusions from the Cumulative Residual SD Plot We can make the following conclusions from the cumulative residual standard deviation plot.
1. The baseline model consisting only of the average ($$\scriptsize \hat{Y}$$ = 110.6063) has a high residual standard deviation (95).

2. The cumulative residual standard deviation shows a significant and steady decrease as the following terms are added to the average: X2, X7, X1*X3, X1, X3, X2*X3, and X1*X2. Including these terms reduces the cumulative residual standard deviation from approximately 95 to approximately 17.

3. Exclude from the model any term after X1*X2 as the decrease in the residual standard deviation becomes relatively small.

4. From the |effects| plot, we see that the average is 110.6063, the estimated X2 effect is -78.6126, and so on. (The model coefficients are one half of the effect estimates.) We use this to from the following prediction equation:

$$\begin{eqnarray*} \hat{Y} & = & 110.6063 - 39.3063 X_2 - 39.0563 X_7 + \\ & & 35.00625 X_1 X_3 + 33.106245 X_1 + 31.90625 X_3 - \\ & & 31.7313 X_1 X_5 - 29.781 X_1 X_2 \end{eqnarray*}$$

Note that X1*X3 is confounded with X2*X7 and X4*X6, X1*X5 is confounded with X2*X6 and X4*X7, and X1*X2 is confounded with X3*X7 and X5*X6.

From the above graph, we see that the residual standard deviation for this model is approximately 17.