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!

z y = z x When X and Y are perfectly correlated

We can say that z x perfectly predicts z y z y ’ = z x Or z y = z x ^

When they are imperfectly correlated, i.e., r xy ≠ 1 or -1 z y ’ = r xy z x

Example from hands…

When they are imperfectly correlated, i.e., r xy ≠ 1 or -1 z y ’ = r xy z x Y’ = b YX X + a YX b YX = r YX (s y / s x ) a YX = Y - b YX X __

When they are imperfectly correlated, i.e., r xy ≠ 1 or -1 z y ’ = r xy z x Y’ = b YX X + a YX b YX = r YX (s y / s x ) a YX = Y - b YX X __

Assumptions Linearity Homoscedasticity

Explained and unexplained variance SS total = SS explained + SS unexplained N NN

Explained and unexplained variance r 2 XY = 1 - σ 2 Y’ [ =unexplained] σ 2 Y [ =total] = σ 2 Y - σ 2 Y’ σ2Yσ2Y r 2 is the proportion explained variance to the total variance.

Point-biserial correlation r pb 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.