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

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

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

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

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.

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

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

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