Multiple Regression

Multiple Regression

• Multiple regression extends linear regression to allow for 2 or more independent variables.

• There is still only one dependent (criterion) variable.• We can think of the independent variables as ‘predictors’

of the dependent variable.• The main complication in multiple regression arises

when the predictors are not statistically independent.

Example 1: Predicting Income

Age

Hours Worked

MultipleRegression Income

Example 2: Predicting Final Exam Grades

Assignments

Midterm

MultipleRegression Final

Coefficient of Multiple Determination

• The proportion of variance explained by all of the independent variables together is called the coefficient of multiple determination (R2).

• R is called the multiple correlation coefficient.• R measures the correlation between the predictions and

the actual values of the dependent variable.

• The correlation riY of predictor i with the criterion (dependent variable) Y is called the validity of predictor i.

Uncorrelated Predictors

21 Yr 2

2Yr

Total variance

Variance explained by assignments Variance explained by midterm

2 2 2 2 21 2=Total proportion of variance explained = Y Y Y YR r r

Uncorrelated Predictors• Recall the regression formula for a single predictor:

• If the predictors were not correlated, we could easily generalize this formula:

Y Xz rz

1 1 2 2Y Y Yz r z r z

Example 1. Predicting Income

Correlations

1 .040* .229**.012 .000

3975 3975 3975.040* 1 .187**

.012 .000

3975 3975 3975

.229** .187** 1

.000 .0003975 3975 3975

Pearson CorrelationSig. (2-tailed)NPearson CorrelationSig. (2-tailed)

N

Pearson CorrelationSig. (2-tailed)N

AGE

HOURS WORKEDFOR PAY OR INSELF-EMPLOYMENT- in Reference Week

TOTAL INCOME

AGE

HOURSWORKEDFOR PAY

OR INSELF-

EMPLOYMENT - inReference Week

TOTALINCOME

Correlation is significant at the 0.05 level (2-tailed).*.

Correlation is significant at the 0.01 level (2-tailed).**.

Correlated Predictors

21 Yr 2

2Yr

Total variance

Variance explained by assignments Variance explained by midterm

2 2 21 2=Total proportion of variance explained < Y YR r r

Correlated Predictors

• Due to the correlation in the predictors, the optimal regression weights must be reduced:

1 1 2 2Yz z z

1 2 12 2 1 121 22 2

12 12

where

and 1 1

Y Y Y Yr r r r r rr r

1 2 beta weights (standardized partial re

andgres

are callesion coeffi

d thc

s)

eient

2 22 1 2 1 2 12

1 1 2 2 212

21

Y Y Y YY Y

r r r r rR r rr

Raw-Score Formulas

0 1 1 2 2Y B B X B X

1 2

1 1 2 2

0 1 1 2 2

where

and

and

Y Y

X X

s sB Bs s

B Y B X B X

Example 1. Predicting Income

Correlations

1 .040* .229**.012 .000

3975 3975 3975.040* 1 .187**

.012 .000

3975 3975 3975

.229** .187** 1

.000 .0003975 3975 3975

Pearson CorrelationSig. (2-tailed)NPearson CorrelationSig. (2-tailed)

N

Pearson CorrelationSig. (2-tailed)N

AGE

HOURS WORKEDFOR PAY OR INSELF-EMPLOYMENT- in Reference Week

TOTAL INCOME

AGE

HOURSWORKEDFOR PAY

OR INSELF-

EMPLOYMENT - inReference Week

TOTALINCOME

Correlation is significant at the 0.05 level (2-tailed).*.

Correlation is significant at the 0.01 level (2-tailed).**.

1 1 2 2Yz z z

1 2 12 2 1 121 22 2

12 12

where

and 1 1

Y Y Y Yr r r r r rr r

2 22 1 2 1 2 12

1 1 2 2 212

21

Y Y Y YY Y

r r r r rR r rr

Example 1. Predicting Income

020

4060

80

0

20

40

60

800

1

2

3

4

5

6

7

x 104

Age (years)Hours worked per week (hours)

Ann

ual I

ncom

e (C

AD

)

Degrees of freedom

1 wheresample sizenumber of predictors

df n knk

Semipartial (Part) Correlations

• The semipartial correlations measure the correlation between each predictor and the criterion when all other predictors are held fixed.

• In this way, the effects of correlations between predictors are eliminated.

• In general, the semipartial correlations are smaller than the validities.

Calculating Semipartial Correlations

• One way to calculate the semipartial correlation for a predictor (say Predictor 1) is to partial out the effects of all other predictors on Predictor 1and then calculate the correlation between the residual of Predictor 1 and the criterion.

• For example, we could partial out the effects of age on hours worked, and then measure the correlation between income and the residual hours worked.

Calculating Semipartial Correlations

• A more straightforward method:

1 2 12(1.2) 2

121Y Y

Yr r rr

r

(1.2)where is the semipartial correlation between Predictor 1 and Yr Y

i.e., the correlation between and Predictor 1 after partialling out the effects of Predictor 2 on Predictor 1.

Y

Example 2: Predicting Final Exam Grades

Assignments

Midterm

MultipleRegression Final

Example 2. Predicting Final Exam Grades (PSYC 6130A, 2005-2006)

Correlations

1 .356 .127.233 .680

13 13 13.356 1 .615*.233 .025

13 13 13.127 .615* 1.680 .025

13 13 13

Pearson CorrelationSig. (2-tailed)NPearson CorrelationSig. (2-tailed)NPearson CorrelationSig. (2-tailed)N

Assignments

Midterm

Final

Assignments Midterm Final

Correlation is significant at the 0.05 level (2-tailed).*.

212 120.356 0.127r r 2

1 120.127 0.016Yr r 22 20.615 0.378Y Yr r

Example 2. Predicting Final Exam Grades (PSYC 6130A, 2005-2006)

212 120.356 0.127r r 2

1 120.127 0.016Yr r 22 20.615 0.378Y Yr r

1 1 2 2Yz z z

1 2 12 2 1 121 22 2

12 12

where

and 1 1

Y Y Y Yr r r r r rr r

2 22 1 2 1 2 12

1 1 2 2 212

21

Y Y Y YY Y

r r r r rR r rr

Example 2. Predicting Final Exam Grades

0 1 1 2 2Y B B X B X

1 2

1 1 2 2

0 1 1 2 2

where

and

and

Y Y

X X

s sB Bs s

B Y B X B X

Example 2. Predicting Final Exam Grades

7080

90100

2040

6080

0

50

100

150

Assignment grade (%)Midterm grade (%)

Fina

l gra

de (%

)

SPSS Output

Example 3. 2006-07 6130 Grades

• Try doing the calculations on this dataset for practice.