Two-Variable Relationships

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These activities are designed to help you understand technical aspects of regression. Feel free to explore. The instructions we give are meant to guide your explorations, but of course you might make note of how you would use these in your own class.

1. This applet helps in understanding two (at least) fundamental aspects of regression:
a) The regression line can be strongly influenced by "unusual" observations.
b) Visualizing the residuals plot.

Do this:
I. Place 4 or five points in a mostly linear fashion. Notice how the residual plot (to the right) is clustered about the line y = 0 and that the residual plot displays the vertical distance that each point falls from the regression line.
II. Now place a point off to the extreme right or left corner (upper or lower). How does the regression line react? What does the residual plot show?
III. Now place a point near the top or bottom of the plot, but towards the middle of the points. Does this affect the line as much?
IV. Clear the points and start all over. This time, place your points to follow a quadratic relation or some other curve. Note that regression line still exists (and points to the general trend of the data), and pay attention to how the residuals look.

2. This applet shows the correlation coefficient and also helps you compare your intuition about "best fit" lines with the actual best-fit. You can also explore the role of influential points. Click on the link below and then select: Correlation and Regression.

Do this:
I. Add points so that the correlation coefficient is as close to 1 as you can get it.
II. Start again; add points so that the correlation is close to 0.
III. Add points to roughly follow a linear trend so that the correlation is somewhere around 0.5. Display the least-squares line. Choose one point near the extremes of the plot and move it up and down. How does the line react? Compare this to what happens when you move a point more in the middle of the plot.
IV. Start again: create an approximately linear relationship with about 10 points. Now add another point far away from the others but still following the linear trend. What happens as you move this distant point up and down?