1. Regression Analysis Simple Regression

2. y = mx + b y = a + bx

3. where: y dependent variable (value depends on x) a y-intercept (value of y when x = 0) b slope (rate of change in ratio of delta y divided by delta x) x independent variable

4. Assumptions Linearity Independence of Error Homoscedasticity Normality

5. Linearity The most fundamental assumption is that the model fits the situation [i.e.: the Y variable is linearly related to the value of the X variable].

6. Independence of Error The error (residual) is independent for each value of X. [Residual = observed - predicted]

7. Homoscedasticity The variation around the line of regression constant for all values of X.

8. Normality T he values of Y be normally distributed at each value of X.

9. Diagnostic Checking u Linearity u Independence u Examine scatter plot of residuals versus fitted [Y hat ] for evidence of nonlinearity u Plot residuals in time order and look for patterns

10. Diagnostic Checking u Homoscedasticity u Normality u Examine scatter plots of residuals versus fitted [Y hat ] and residuals vs time order and look for changing scatter. u Examine histogram of residuals. Look for departures from normal curve.

11. Goal Develop a statistical model that can predict the values of a dependent (response) variable based upon the values of the independent (explanatory) variable(s).

12. Goal

13. Simple Regression quantitative quantitative A statistical model that utilizes one quantitative independent variable “X” to predict the quantitative dependent variable “Y.”

14. Mini-Case Since a new housing complex is being developed in Carmichael, management is under pressure to open a new pie restaurant. Assuming that population and annual sales are related, a study was conducted to predict expected sales.

15. Mini-Case (Descartes Pie Restaurants)

16. Mini-Case u What preliminary conclusions can management draw from the data? u What could management expect sales to be if population of the new complex is approximately 18,000 people?

17. Scatter Diagrams u The values are plotted on a two- dimensional graph called a “scatter diagram.” u Each value is plotted at its X and Y coordinates.

18. Scatter Plot of Pieshop

19. Types of Models No relationship between X and Y Positive linear relationship Negative linear relationship

20. Method of Least Squares u The straight line that best fits the data. u Determine the straight line for which the differences between the actual values (Y) and the values that would be predicted from the fitted line of regression (Y-hat) are as small as possible.

21. Measures of Variation Explained Unexplained Total

22. Explained Variation Sum of Squares  (Y hat - Y bar ) 2 due to Regression [SSR]

23. Unexplained Variation Sum of Squares  (Y obs - Y hat ) 2 Error [SSE]

24. Total Variation Sum of Squares  (Y obs - Y bar ) 2 Total [SST]

25. H0:H0: There is no linear relationship between the dependent variable and the explanatory variable

26. Hypotheses  H 0 :  = 0  H 1 :   0 or H 0 : No relationship exists H 1 : A relationship exists

27. Analysis of Variance for Regression

28. Standard Error of the Estimate s y.x - the measure of variability around the line of regression

29. Relationship When null hypothesis is rejected, a relationship between Y and X variables exists.

30. Coefficient of Determination R 2 measures the proportion of variation that is explained by the independent variable in the regression model. R 2 = SSR / SST

31. Confidence interval estimates »True mean  YX »Individual Y-hat

32. Pieshop Forecasting

33. Coefficient of Sanity

34. Diagnostic Checking u H 0 retain or reject {Reject if p-value  0.05} u R 2 (larger is “better”) u s y.x (smaller is “better”)

35. Analysis of Variance for Regression for Pieshop

36. Coefficient of Determination R 2 = SSR / SST = 90.27 % thus, 90.27 percent of the variation in annual sales is explained by the population.

37. Standard Error of the Estimate s y.x = 13.8293 with SSE = 1,530.0

38. Regression Analysis [Simple Regression] *** End of Presentation *** Questions?