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Multiple Regression

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

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  1. Multiple Regression

  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 Multiple Regression Income Hours Worked PSYC 6130, PROF. J. ELDER

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

  6. Uncorrelated Predictors Variance explained by assignments Variance 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 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). PSYC 6130, PROF. J. ELDER

  9. Correlated Predictors Variance explained by assignments Variance 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 Multiple Regression Final Midterm PSYC 6130, PROF. J. ELDER

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

  20. Example 2. Predicting Final Exam Grades (PSYC 6130A, 2005-2006) 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 3. 2006-07 6130 Grades • Try doing the calculations on this dataset for practice. PSYC 6130, PROF. J. ELDER

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