Monday, November 9

Correlation and Linear Regression

You will not leave the room until…

• you have understood that a correlation is a systematic quantitative expression of the proportion of explained and unexplained co-variation of two variables.

You will not leave the room until…

• you have understood that a correlation is a systematic quantitative expression of the proportion of explained and unexplained co-variation of two variables … and you love knowing this fact!

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…

When they are imperfectly correlated, i.e., rxy ≠ 1 or -1

zy’ = rxyzx

Y’ = bYXX + aYX

bYX = rYX (sy / sx)

aYX = Y - bYXX

_ _

When they are imperfectly correlated, i.e., rxy ≠ 1 or -1

zy’ = rxyzx

Y’ = bYXX + aYX

bYX = rYX (sy / sx)

aYX = Y - bYXX

_ _

Assumptions

• Linearity

• Homoscedasticity

Explained and unexplained variance

SStotal = SSexplained + SSunexplained

SStotal = SSexplained + SSunexplained

N N N

Explained and unexplained variance

r2XY = 1 -

σ2Y’ [ =unexplained]

σ2Y [ =total]

=

σ2Y - σ2

Y’

σ2Y

r2 is the proportion explained variance to the total variance.

Point-biserial correlation rpb

• A correlation coefficient r that is calculated when one of the variables being correlated has only two levels, which are assigned arbitrary values (e.g., 0, 1).

• This coefficient is useful in expressing the effect size of an independent samples t-test, as the proportion of the variance in the dependent variable that is explained by the independent variable.