W7 multiple regression analysis

Question 1

Ordinary least squares regression

Answer 1

When we have > 1 predictor (X) variable, we simply need a way to combine them
The regression procedure expands seamlessly to do this…

Essentially, we extend the least squares procedure to estimate b0 and b1 and b2… bk to give us the best possible prediction of Y from all the variables jointly

This type of regression might be referred to in some texts as OLS, Ordinary Least Squares Regression

Question 2

Equation for multiple regression

Answer 2

Y’ = b0 + b1X1 + b2X2 + e

Question 3

Testing the significance of R

Answer 3

fancy p = rho
-> population correlation coefficient
H0 : fancy p = 0
H1 : fancy p not equal to 0

H0 : there is no linear relationship between X and Y in the population
H1: there is a linear relationship between X and Y in the population

Question 4

R squared and variance

Answer 4

In correlation, r squared = proportion of variance shared by X and Y, i.e., the two correlates—overlap

Question 5

Multiple R squared and variance

Answer 5

• Multiple R2 is variance in Y accounted for by Xk
  usually referred to simply as R2
• Not specific to any one single X, it is for the set of them.
• The regression equation is also referred to as the Model
  we check the value of R2 in the Model Summary box in the SPSS output

Question 6

R squared equation

Answer 6

R squared = SSreg / SStot

• these can be manually calculated (i guess)
• found in ANOVA table

Question 7

SStot

Answer 7

Y - Y bar

Question 8

SSreg

Answer 8

Y’ - Y bar

Question 9

SSres

Answer 9

Y - Y’

Question 10

Extending our interpretation of regression predictors

Answer 10

Semipartial correlations—sr and sr2

• > Relationship between Xk and Y when all other predictors partialled out of only Xk.
• > Good indicator of the unique relationship between Xk and Y
• > Always squared (i.e., sr2) to give proportion of unique variance in Y accounted for by Xk

Question 11

Calculating the unique variance

Answer 11

• unique variance of X1 on Y is equal to X1’s “part” squared, found in correlations coefficients table when using zpp command
• unique variance of X2 on Y is equal to X2’s “part” squared from correlations table
• the “part” has removed the variance accounted for from the other variable and gives the variance (when squared) of that variable alone

Question 12

Shared variance

Answer 12

• the difference between the combined ‘unique variance’ (sr squared X1 + sr squared X2) and ‘total variance explained’ (R squared)

Shared variance = R squared - (sr squared X1 + sr squared X2)