Multiple Regression

PSYC 6130, PROF. J. ELDER 2 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.

PSYC 6130, PROF. J. ELDER 3 Example 1: Predicting Income Age Hours Worked Multiple Regression Income

PSYC 6130, PROF. J. ELDER 4 Example 2: Predicting Final Exam Grades Assignments Midterm Multiple Regression Final

PSYC 6130, PROF. J. ELDER 5 Coefficient of Multiple Determination The proportion of variance explained by all of the independent variables together is called the coefficient of multiple determination (R 2 ). R is called the multiple correlation coefficient. R measures the correlation between the predictions and the actual values of the dependent variable. The correlation r iY of predictor i with the criterion (dependent variable) Y is called the validity of predictor i.

PSYC 6130, PROF. J. ELDER 6 Uncorrelated Predictors Variance explained by assignmentsVariance explained by midterm

PSYC 6130, PROF. J. ELDER 7 Uncorrelated Predictors Recall the regression formula for a single predictor: If the predictors were not correlated, we could easily generalize this formula:

PSYC 6130, PROF. J. ELDER 8 Example 1. Predicting Income Correlations 1.040*.229** *1.187** **.187** Pearson Correlation Sig. (2-tailed) N Pearson Correlation Sig. (2-tailed) N Pearson Correlation Sig. (2-tailed) N AGE HOURS WORKED FOR PAY OR IN SELF-EMPLOYMENT - in Reference Week TOTAL INCOME AGE HOURS WORKED FOR PAY OR IN SELF- EMPLOY MENT - in Referenc e Week TOTAL INCOME Correlation is significant at the 0.05 level (2-tailed). *. Correlation is significant at the 0.01 level (2-tailed). **.

PSYC 6130, PROF. J. ELDER 9 Correlated Predictors Variance explained by assignmentsVariance explained by midterm

PSYC 6130, PROF. J. ELDER 10 Correlated Predictors Due to the correlation in the predictors, the optimal regression weights must be reduced:

PSYC 6130, PROF. J. ELDER 11 Raw-Score Formulas

PSYC 6130, PROF. J. ELDER 12 Example 1. Predicting Income

PSYC 6130, PROF. J. ELDER 13 Example 1. Predicting Income

PSYC 6130, PROF. J. ELDER 14 Degrees of freedom

PSYC 6130, PROF. J. ELDER 15 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.

PSYC 6130, PROF. J. ELDER 16 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.

PSYC 6130, PROF. J. ELDER 17 Calculating Semipartial Correlations A more straightforward method:

PSYC 6130, PROF. J. ELDER 18 Example 2: Predicting Final Exam Grades Assignments Midterm Multiple Regression Final

PSYC 6130, PROF. J. ELDER 19 Example 2. Predicting Final Exam Grades (PSYC 6130A, )

PSYC 6130, PROF. J. ELDER 20 Example 2. Predicting Final Exam Grades (PSYC 6130A, )

PSYC 6130, PROF. J. ELDER 21 Example 2. Predicting Final Exam Grades

PSYC 6130, PROF. J. ELDER 22 Example 2. Predicting Final Exam Grades

PSYC 6130, PROF. J. ELDER 23 SPSS Output

PSYC 6130, PROF. J. ELDER 24 Example Grades Try doing the calculations on this dataset for practice.