1 / 46

Slides Prepared by JOHN S. LOUCKS St. Edward’s University

Slides Prepared by JOHN S. LOUCKS St. Edward’s University. Chapter 15 Multiple Regression. Multiple Regression Model Least Squares Method Multiple Coefficient of Determination Model Assumptions Testing for Significance Using the Estimated Regression Equation

takara
Download Presentation

Slides Prepared by JOHN S. LOUCKS St. Edward’s University

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Slides Prepared by JOHN S. LOUCKS St. Edward’s University

  2. Chapter 15 Multiple Regression • Multiple Regression Model • Least Squares Method • Multiple Coefficient of Determination • Model Assumptions • Testing for Significance • Using the Estimated Regression Equation for Estimation and Prediction • Qualitative Independent Variables • Residual Analysis

  3. The Multiple Regression Model • The Multiple Regression Model y = 0 + 1x1 + 2x2 + . . . + pxp+  • The Multiple Regression Equation E(y) = 0 + 1x1 + 2x2 + . . . + pxp • The Estimated Multiple Regression Equation y = b0 + b1x1 + b2x2 + . . . + bpxp ^

  4. The Least Squares Method • Least Squares Criterion • Computation of Coefficients Values The formulas for the regression coefficients b0, b1, b2, . . . bp involve the use of matrix algebra. We will rely on computer software packages to perform the calculations. • A Note on Interpretation of Coefficients bi represents an estimate of the change in y corresponding to a one-unit change in xi when all other independent variables are held constant. ^

  5. The Multiple Coefficient of Determination • Relationship Among SST, SSR, SSE SST = SSR + SSE • Multiple Coefficient of Determination R 2 = SSR/SST • Adjusted Multiple Coefficient of Determination ^ ^

  6. Model Assumptions • Assumptions About the Error Term  • The error  is a random variable with mean of zero. • The variance of  , denoted by 2, is the same for all values of the independent variables. • The values of  are independent. • The error  is a normally distributed random variable reflecting the deviation between the y value and the expected value of y given by 0 + 1x1 + 2x2 + . . . + pxp

  7. Testing for Significance: F Test • Hypotheses H0: 1 = 2 = . . . = p = 0 Ha: One or more of the parameters is not equal to zero. • Test Statistic F = MSR/MSE

  8. Testing for Significance: F Test • Rejection Rule Using test statistic: Reject H0 if F > F Using p-value:Reject H0 if p-value < a where F is based on an F distribution with p d.f. in the numerator and n - p - 1 d.f. in the denominator

  9. Testing for Significance: F Test • ANOVA Table (assuming p independent variables) Source of Sum of Degrees of Mean Variation Squares Freedom Squares F RegressionSSR p Error SSE n- p - 1 TotalSST n - 1

  10. Testing for Significance: t Test • Hypotheses H0: i = 0 Ha: i = 0 • Test Statistic

  11. Testing for Significance: t Test • Rejection Rule Using test statistic: Reject H0 if t < -tor t > t Using p-value:Reject H0 if p-value < a where tis based on a t distribution with n - p - 1 degrees of freedom

  12. Testing for Significance: Multicollinearity • The term multicollinearity refers to the correlation among the independent variables. • When the independent variables are highly correlated (say, |r | > .7), it is not possible to determine the separate effect of any particular independent variable on the dependent variable. • If the estimated regression equation is to be used only for predictive purposes, multicollinearity is usually not a serious problem. • Every attempt should be made to avoid including independent variables that are highly correlated.

  13. Using the Estimated Regression Equationfor Estimation and Prediction • The procedures for estimating the mean value of y and predicting an individual value of y in multiple regression are similar to those in simple regression. • We substitute the given values of x1, x2, . . . , xp into the estimated regression equation and use the corresponding value of y as the point estimate. • The formulas required to develop interval estimates for the mean value of y and for an individual value of y are beyond the scope of the text. • Software packages for multiple regression will often provide these interval estimates. ^

  14. Example: Programmer Salary Survey A software firm collected data for a sample of 20 computer programmers. A suggestion was made that regression analysis could be used to determine if salary was related to the years of experience and the score on the firm’s programmer aptitude test. The years of experience, score on the aptitude test, and corresponding annual salary ($1000s) for a sample of 20 programmers is shown on the next slide.

  15. Example: Programmer Salary Survey Exper.ScoreSalaryExper.ScoreSalary 4 78 24 9 88 38 7 100 43 2 73 26.6 1 86 23.7 10 75 36.2 5 82 34.3 5 81 31.6 8 86 35.8 6 74 29 10 84 38 8 87 34 0 75 22.2 4 79 30.1 1 80 23.1 6 94 33.9 6 83 30 3 70 28.2 6 91 33 3 89 30

  16. Example: Programmer Salary Survey • Multiple Regression Model Suppose we believe that salary (y) is related to the years of experience (x1) and the score on the programmer aptitude test (x2) by the following regression model: y = 0 + 1x1 + 2x2 +  where y = annual salary ($000) x1 = years of experience x2 = score on programmer aptitude test

  17. Example: Programmer Salary Survey • Multiple Regression Equation Using the assumption E ( ) = 0, we obtain E(y ) = 0 + 1x1 + 2x2 • Estimated Regression Equation b0, b1, b2 are the least squares estimates of 0, 1, 2 Thus y = b0 + b1x1 + b2x2 ^

  18. Example: Programmer Salary Survey • Solving for the Estimates of 0, 1, 2 Least Squares Output Input Data x1x2y 4 78 24 7 100 43 . . . . . . 3 89 30 Computer Package for Solving Multiple Regression Problems b0 = b1 = b2 = R2 = etc.

  19. Using Excel’s Regression Tool to Developthe Estimated Multiple Regression Equation • Formula Worksheet (showing data entered) Note: Rows 10-21 are not shown.

  20. Using Excel’s Regression Tool to Developthe Estimated Multiple Regression Equation • Performing the Multiple Regression Analysis Step 1 Select the Tools pull-down menu Step 2 Choose the Data Analysis option Step 3 Choose Regression from the list of Analysis Tools … continued

  21. Using Excel’s Regression Tool to Developthe Estimated Multiple Regression Equation • Performing the Multiple Regression Analysis Step 4 When the Regression dialog box appears: Enter D1:D21 in the Input Y Range box Enter B1:C21 in the Input X Range box Select Labels Select Confidence Level Enter 95 in the Confidence Level box Select Output Range and enter A24 in the Output Range box Click OK

  22. Using Excel’s Regression Tool to Developthe Estimated Multiple Regression Equation • Value Worksheet (Regression Statistics)

  23. Using Excel’s Regression Tool to Developthe Estimated Multiple Regression Equation • Value Worksheet (ANOVA Output) The Significance F value in cell F35 is the p-value used to test for overall significance.

  24. Using Excel’s Regression Tool to Developthe Estimated Multiple Regression Equation • Value Worksheet (Regression Equation Output) Note: Columns F-I are not shown. The P-value in cell E41 is used to test for the individual significance of Experience.

  25. Using Excel’s Regression Tool to Developthe Estimated Multiple Regression Equation • Value Worksheet (Regression Equation Output) Note: Columns F-I are not shown. The P-value in cell E42 is used to test for the individual significance of Test Score.

  26. Using Excel’s Regression Tool to Developthe Estimated Multiple Regression Equation • Estimated Regression Equation SALARY = 3.174 + 1.404(EXPER) + 0.2509(SCORE) Note: Predicted salary will be in thousands of dollars

  27. Using Excel’s Regression Tool to Developthe Estimated Multiple Regression Equation • Value Worksheet (Regression Equation Output) Note: Columns C-E are hidden.

  28. Example: Programmer Salary Survey • F Test • Hypotheses H0: 1 = 2 = 0 Ha: One or both of the parameters is not equal to zero. • Rejection Rule For  = .05 and d.f. = 2, 17: F.05 = 3.59 Reject H0 if F > 3.59. • Test Statistic F = MSR/MSE = 250.16/5.85 = 42.76 • Conclusion We can reject H0.

  29. Example: Programmer Salary Survey • t Test for Significance of Individual Parameters • Hypotheses H0: i = 0 Ha: i = 0 • Rejection Rule For  = .05 and d.f. = 17, t.025 = 2.11 Reject H0 if t > 2.11 • Test Statistics • Conclusions Reject H0: 1 = 0 Reject H0: 2 = 0

  30. Qualitative Independent Variables • In many situations we must work with qualitative independent variablessuch as gender (male, female), method of payment (cash, check, credit card), etc. • For example, x2 might represent gender where x2 = 0 indicates male and x2 = 1 indicates female. • In this case, x2 is called a dummy or indicator variable. • If a qualitative variable has k levels, k - 1 dummy variables are required, with each dummy variable being coded as 0 or 1. • For example, a variable with levels A, B, and C would be represented by x1 and x2 values of (0, 0), (1, 0), and (0,1), respectively.

  31. Example: Programmer Salary Survey (B) As an extension of the problem involving the computer programmer salary survey, suppose that management also believes that the annual salary is related to whether or not the individual has a graduate degree in computer science or information systems. The years of experience, the score on the programmer aptitude test, whether or not the individual has a relevant graduate degree, and the annual salary ($000) for each of the sampled 20 programmers are shown on the next slide.

  32. Example: Programmer Salary Survey (B) Exp.ScoreDegr.SalaryExp.ScoreDegr.Salary 4 78 No 24 9 88 Yes 38 7 100 Yes 43 2 73 No 26.6 1 86 No 23.7 10 75 Yes 36.2 5 82 Yes 34.3 5 81 No 31.6 8 86 Yes 35.8 6 74 No 29 10 84 Yes 38 8 87 Yes 34 0 75 No 22.2 4 79 No 30.1 1 80 No 23.1 6 94 Yes 33.9 6 83 No 30 3 70 No 28.2 6 91 Yes 33 3 89 No 30

  33. Example: Programmer Salary Survey (B) • Multiple Regression Equation E(y ) = 0 + 1x1 + 2x2 + 3x3 • Estimated Regression Equation y = b0 + b1x1 +b2x2 + b3x3 where y = annual salary ($000) x1 = years of experience x2 = score on programmer aptitude test x3 = 0 if individual does not have a grad. degree 1 if individual does have a grad. degree Note: x3 is referred to as a dummy variable. ^

  34. Using Excel’s Regression Tool to Developthe Estimated Multiple Regression Equation • Formula Worksheet (showing data) Note: Rows 9-21 are not shown.

  35. Using Excel’s Regression Tool to Developthe Estimated Multiple Regression Equation • Value Worksheet (Regression Statistics)

  36. Using Excel’s Regression Tool to Developthe Estimated Multiple Regression Equation • Value Worksheet (ANOVA Output)

  37. Using Excel’s Regression Tool to Developthe Estimated Multiple Regression Equation • Value Worksheet (Regression Equation Output) Note: Columns F-I are not shown.

  38. Using Excel’s Regression Tool to Developthe Estimated Multiple Regression Equation • Value Worksheet (Regression Equation Output) Note: Columns C-E are hidden.

  39. Example: Programmer Salary Survey (B) • Interpreting the Parameters • b1 = 1.15 Salary is expected to increase by $1,150 for each additional year of experience (when all other independent variables are held constant)

  40. Example: Programmer Salary Survey (B) • Interpreting the Parameters • b2 = 0.197 Salary is expected to increase by $197 for each additional point scored on the programmer aptitude test (when all other independent variables are held constant)

  41. Example: Programmer Salary Survey (B) • Interpreting the Parameters • b3 = 2.28 Salary is expected to be $2,280 higher for an individual with a graduate degree than one without a graduate degree (when all other independent variables are held constant)

  42. Residual Analysis • For simple linear regression the residual plot against and the residual plot against x provide the same information. • In multiple regression analysis it is preferable to use the residual plot against to determine if the model assumptions are satisfied.

  43. Residual Analysis • Standardized residuals are frequently used in residual plots for purposes of: • Identifying outliers (typically, standardized residuals < -2 or > +2) • Providing insight about the assumption that the error term e has a normal distribution • The computation of the standardized residuals in multiple regression analysis is too complex to be done by hand • Excel’s Regression tool can be used

  44. Using Excel to Construct a Standardized Residual Plot • Value Worksheet (Residual Output) Note: Rows 37-51 are not shown.

  45. Using Excel to Construct a Standardized Residual Plot Outlier

  46. End of Chapter 15

More Related