CK-12 Probability and Statistics - Advanced

http://www.ck12.org Chapter 9. Regression and Correlation

When predicting values using multiple regression, we can also use the standard score form of the formula:

zYˆ=β 1 z 1 β 2 z 2 +etc...

where:

zYˆ=the predicted or criterion variable

β=the regression coefficient

z=the predictor variable

To solve for the regression and constant coefficients, we first need to determine the multiple correlation coefficient
(r)andcoefficient of determination, also known as the proportion of shared variance(R^2 ). In a linear regression
model, we measuredR^2 by adding the sum of the distances from the actual to the points predicted by the regression
line. So what doesR^2 look like in a multiple regression model? Let’s take a look at the figure above. Essentially, like
the linear regression model, the theory behind the computation of the multiple regression equation is to minimize
the sum of the squared deviations from the observation to the regression plane.

In most situations, we use thecomputer to calculate the multiple regression equationand determine the coeffi-
cients in this equation. We can also do multiple regression on a TI83/84 calculator (this program can be downloaded
from http://www.wku.edu/ david.neal/manual/ti83.html). However, it is helpful to explain the calculations that go
into the multiple regression equation so we can get a better understanding of how this formula works.

After we find the correlation values(r)between the variables, we can use the following formulas to determine the
regression coefficients for each of the predictor(X)variables:

β 1 =

rY 1 −(rY 2 )(r 12 )

1 −r 122

β 2 =

rY 2 −(rY 1 )(r 12 )

1 −r 122

where:

β 1 =the correlation coefficient

rY 1 =correlation between the criterion variables(Y)and the first predictor variable(X 1 )