# Tutorial #3: Regression Analysis

To analyze our design using Multiple Regression we will select "DOE PRO" - "Analyze Design" - "Multiple Response Regression" from the Excel menu bar. DOE PRO will read our design sheets, analyze our designs using least squares regression, and then produce a regression table for both the Y-hat and S-hat models.

The first regression analysis (shown below) will include all the main effects as well as 2-way, 3-way, and quadratic interactions. The columns entitled "Factor" and "Name" indicate which row is used for which factor/interaction. For each response (output) we also have columns for "Coef" (coefficient), "P(2 Tail)", "Tol" (Tolerance), and "Active". The last three factors in the table are "AA", "BB", and "CC". These are the "A Squared", "B Squared", and "C Squared" quadratics respectively.

##### Y-hat Regression Analysis

In addition to the Y-hat model, if you scroll right you will see the S-hat model also.

##### S-hat Regression Analysis

Now that we have our initial models, we are going to remove insignificant effects and recompute the regression. To do this we will remove the "X" in the Active column next to the insignificant factors. Note that in the models shown above, the "Xs" have already been removed. After the "Xs" have been removed, select "DOE PRO" - "Analyze Design" - "Multiple Response Regression" from the menu bar **with the regression table still active**. The factors that did not have an "X" in the Active column will be removed, and the regression recomputed without them. The reduced models are shown below.

##### Y-hat Model with insignificant terms removed

##### S-hat Model with insignificant terms removed

Note that between the Y-hat models and the S-hat models there is a "Prediction" area. This area uses the models to predict a "Y-hat" and "S-hat" value for the output. In the column "Exper" (Experimental) type the following values:

Pull Back Angle 170

Stop Angle 3

Cup Angle 30

The area entitled "Multiple Response Prediction" shows a predicted Red Ball Y-hat of 94.63 and a predicted Green Ball Y-hat of 76.17. This tells us that if we set the settings as we did above, we would expect the Red ball to travel 94.63 inches and the Green ball to travel 76.17 inches.

However, what do we do if we don’t want the Red Ball to go 94.63 inches and the Green Ball to go 76.17 inches? Perhaps we want the Red ball to go 70 inches and the Green ball to go 60 inches. How do we find combinations of Pull Back Angle, Stop Angle, and Cup Angle that will result in our desired distances? The next topic of plotting and optimization discusses methods to hit our desired targets.