1. PS 225 Lecture 20 Linear Regression Equation and Prediction

2. Adding Regression Line

3. Dependence  What if two variables are correlated?  What if the mean of a variable is dependent on the value of another variable? Is it dependent? How much is it dependent? How can we express the dependence algebraically?

4. Examples of Dependence  The distance traveled at a given speed = x  The cost of a bag of bulk mixed nuts with a given price per pound = x DistanceSpeedTime Cost Weight Price Linear Relationships

5. Types of Relationships  Deterministic Relationship One variable totally determines the value of another variable with perfect accuracy Algebraic linear relationship Previous examples  Variable One variable affects the value of another variable with some element of variability Example: Height and weight

6. Using SPSS to Determine a Linear Relationship  Is there a relationship?

7. Linear Regression Form of a Line  Algebraic Form of Line: A is the y-intercept B is the slope  Linear Regression Meaning of the Line A is the ‘constant’ B is a ‘coefficient’

8. SPSS Output for A Regression Line Y = -18331.2 + 3909.907*x X = Education Level Y = Current Salary

9. Interpreting the Constant Only has meaning if: Data present to validate Can naturally occur

10. Interpreting the Coefficient Change in dependent variable for each unit change in the independent variable

11. 2-Step Hypothesis Process  Test Overall Linear Relationship  Test Contribution of Each Component Similar to 2-Way ANOVA

12. Step 1: Overall Test  Is there a linear relationship?  Ho: Means are the same at all values of x (No relationship)  Ha: There is a linear relationship between x and y  If significance<.05 conclude relationship  Otherwise, stop analysis

13. Step 2: Component Tests  Is the component significant? Intercept Coefficient  Ho: Not Significant  Ha: Significant  If significance<.05 conclude significant  Otherwise, eliminate from analysis and recreate model

14. Line of Best Fit  Regression line that minimizes the distance to data points  SPSS calculations

15. Sum of Squares  Sum of squared differences for each data point  Regression- Difference between overall mean and regression line  Residual- Difference Between the regression line and data points  Regression lines minimize the residual sum of squares

16. Deviations

17. Sum of Squares

18. Predicting Values from a Linear Regression  Write equation for the regression line  ‘Plug in’ independent variable  Gain a prediction for the dependent variable  The relationship between the values of the independent variable and the prediction are deterministic

19. Accuracy of Predictions  The BEST guess  Probably not exact due to variability  Correct on average

20. Quality of Prediction  Predicted values must be within the range of the data  Relationship must be linear over the entire range of the data  Line must not depend too strongly on one point

21. SPSS Assignment Last class we answered the following questions: Does the number of years of education an individual has affect the hours of television a person watches? Does age affect the hours of television a person watches? This class: Use SPSS to find the regression equation that best represents each relationship. Write the full regression equation. Make a prediction for yourself with each regression equation How different is each prediction from the number of hours you watch? If the equation under predicts, report your answer as a negative number. If it over predicts report your answer as a positive number. Add your prediction error to the class data.