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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. Statistics

  3. Example 1: Predicting Income Age Multiple Regression Income Hours Worked Statistics

  4. Example 2: Predicting Final Exam Grades Assignments Multiple Regression Final Midterm Statistics

  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.

  6. Uncorrelated Predictors Variance explained by assignments Variance explained by midterm Statistics

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

  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). Statistics

  9. Correlated Predictors Variance explained by assignments Variance explained by midterm Statistics

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

  11. Raw-Score Formulas Statistics

  12. Example 1. Predicting Income Statistics

  13. Example 1. Predicting Income Statistics

  14. Degrees of freedom Statistics

  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. Statistics

  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. Statistics

  17. Calculating Semipartial Correlations • A more straightforward method: Statistics

  18. Example 2: Predicting Final Exam Grades Assignments Multiple Regression Final Midterm Statistics

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

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

  21. Example 2. Predicting Final Exam Grades Statistics

  22. Example 2. Predicting Final Exam Grades Statistics

  23. SPSS Output Statistics

  24. Example 3. 2006-07 6130 Grades • Try doing the calculations on this dataset for practice. Statistics

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