1. Analysis of Residuals ©2005 Dr. B. C. Paul

2. Examining Residuals of Regression (From our Previous Example) Set up your linear regression in the Usual manner.

3. Selecting Plots After setting your dependent and Independent variables and before Clicking ok, click plots instead.

4. Picking Residual Plots Plot the residual on the Y axis Against the predicted value on The X axis. Ask for Histograms and normal Probability plots.

5. More Plots Use the next button to allow you To select another plot. Then enter the residual on the Y axis against the dependent Variable. Finally tell the computer to Continue.

6. You Will Still Get the Normal Tables we Saw Before Scroll down To see what Is new.

7. Some Abnormality in the Histogram A Histogram is a bar chart Showing the number of Results in different numeric Intervals. In this case we can see there May be two families of Unexplained events and One of them is causing the Model to over-predict (note the negative tail).

8. We Have a Cumulative Probability Plot Cumulative probability Counts all the samples That should have come Up by a certain point (it is an integration of the Probability distribution). Normal would plot on a Straight line. This is Somewhat straight but The slope at the center is Wrong and the tails Drift off. (More commentary On reading cumulative Probability plots later).

9. Look for Trends that have been systematically missed This plot shows The residual (amount we Missed by) against The predicted Value. If there is a trend In the points it May tell us What we missed. In this case it is Pretty scattered.

10. Missing Trends We are still missing Something because There is a definite Trend in the residuals Relative to the actual MPG. We are missing a Variable or factor. (it might be linear).

11. Consider Another Data Set We have an Independent and Dependent Variable. (The data set could represent Any problem we wished to Model).

12. Tell it to do a Regression of the Dependent against the Independent Variable. Be sure we also ask for our Residual plots.

13. Go to Results The R^2 value is 0.996 – darn One is a straight line. How much Closer do you want to be. This regression looks like it Fits like a glove – The Mean Square for regression Is 5 orders of magnitude Greater than the MS for error. The F statistic blows the null Hypothesis off the map.

14. No Chance the Slope or Constant are Zero

15. There is some evidence the distribution of residuals is a little skewed.

16. The residual distribution is definitely skewed off to one side

17. Oh Boy – Can You See the Trend we missed here? Here the residuals Follow a clear and Unmistakable shape of An effect we missed.

18. This Thing Has a Second Order or Curved Effect

19. OK – Now What Do I Do? Linear Regression Rapidly and Quantitatively Fits a simple linear function of one variable to another. We noted that there had to be other effects present on the gas mileage but linear regression only handles one independent variable. We also noted that sometimes there our second or higher order effects of a variable present – a straight line just doesn’t fit that We may want to have some more powerful tools to fall back on (we just try the easy stuff first).