Multiple Regression

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# Multiple Regression - PowerPoint PPT Presentation

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.

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### 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.

Statistics

Example 1: Predicting Income

Age

Multiple

Regression

Income

Hours Worked

Statistics

Example 2: Predicting Final Exam Grades

Assignments

Multiple

Regression

Final

Midterm

Statistics

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

Variance explained by assignments

Variance explained by midterm

Statistics

Uncorrelated Predictors
• Recall the regression formula for a single predictor:
• If the predictors were not correlated, we could easily generalize this formula:

Statistics

Example 1. Predicting Income

Correlations

HOURS

WORKED

FOR PAY

OR IN

SELF-

EMPLOY

MENT - in

Referenc

TOTAL

AGE

e Week

INCOME

AGE

Pearson Correlation

1

.040

*

.229

**

Sig. (2-tailed)

.012

.000

N

3975

3975

3975

HOURS WORKED

Pearson Correlation

.040

*

1

.187

**

FOR PAY OR IN

Sig. (2-tailed)

.012

.000

SELF-EMPLOYMENT

- in Reference Week

N

3975

3975

3975

TOTAL INCOME

Pearson Correlation

.229

**

.187

**

1

Sig. (2-tailed)

.000

.000

N

3975

3975

3975

*.

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

**.

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

Statistics

Correlated Predictors

Variance explained by assignments

Variance explained by midterm

Statistics

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

Statistics

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.

Statistics

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.

Statistics

Calculating Semipartial Correlations
• A more straightforward method:

Statistics

Example 2: Predicting Final Exam Grades

Assignments

Multiple

Regression

Final

Midterm

Statistics

Statistics

Statistics

SPSS Output

Statistics