Correlation and Regression

1. Correlation and Regression Statistics 2126

2. Introduction • Means etc are of course useful • We might also wonder, “how do variables go together?” • IQ is a great example • It goes together with so much stuff

4. A scatterplot • You tend to put the predictor on the x axis and the predicted on the y, though this is not a hard and fast rule • A scatterplot is a pretty good EDA tool too eh • Pick an appropriate scale for you axes • Plot the (x,y) pairs

5. So what does it mean • If, as one variable increases, the other variable increases we have a positive association • If, as one goes up, the other goes down, we have a negative association • There could be no association at all

6. Linear relationships • BTW, I am only talking about straight line relationships • Not curvilinear • Say like the Yerkes Dotson Law, as far as a the stuff we will talk about, there is no relationship, yet we know there is

7. The strength is important too • The more the points cluster around a line, the stronger the relationship is • Height and weight vs height in cm vs height in inches • We need something that ignores the units though, so if I did IQ and your income in real money or IQ and your income in that worthless stuff they use across the river, the numbers would be the same

8. The Pearson Product Moment Correlation Coefficient

9. Properties of r • -1.00 <= r <= +1.00 • The sign indicates ONLY the direction (think of it as going uphill or downhill) • |r| indicates the strength • So, r = -.77 is a stronger correlation than r = .40

10. Some examples

11. EDA is KEY

12. Check these out.. • All of these have have the same correlation • R = .7 in each case • Note the problem of outliers • Note the problem of two subpopulations

13. Remember this • Correlation is not causation • I said, correlation is not causation • Let me say it again, correlation is not causation • Birth control and the toaster method

14. Wouldn’t it be nice • If we could predict y from x • You know, like an equation • Remember that in school, you would get an equation, plug in the x and get the y • Well surprise surprise, there is a method like this in statistics

15. If we are going to predict with a line • Well, we will make mistakes • We will want to minimize those mistakes

16. There is a problem, a common problem • Those prediction errors or residuals (e) sum to 0 • Damn • Though guess what we could do… • Why square them of course • So we get a line that minimizes squared residuals

17. The line will look like this

18. In general the equation of the line is….. Y intercept slope Y hat (predicted y)

19. This might help

20. So…. • With a regression line you can predict y from x • Just because it says that some value = a linear combination of numbers it does not mean that there is necessarily a causal link • Don’t go outside the range • Linear only