1. Monday, October 8 Wednesday, October 10
Correlation and Linear Regression

2. When X and Y are perfectly correlated
zy = zx
When X and Y are perfectly correlated

3. We can say that zx perfectly predicts zy
zy’ = zx Or
zy = zx ^

4. When they are imperfectly correlated, i.e., rxy ≠ 1 or -1
zy’ = rxyzx

5. Example from hands…
Let’s double-check our understanding of what a correlation coefficient is with respect to z-scores on X and Y variables.

8. When we want to express the prediction in terms of raw units:
zy’ = rxyzx
Y’ = bYXX + aYX
bYX = rYX (σy / σx)
aYX = Y - bYXX _ _

9. Explained and unexplained variance
SStotal = SSexplained + SSunexplained
SStotal = SSexplained + SSunexplained N

10. Explained and unexplained variance
r2XY = 1 -
σ2Y’ [ =unexplained]
σ2Y [ =total]
σ2Y - σ2Y’ =
σ2Y
r2 is the proportion explained variance to the total variance.