1. 1 1 Slide 統計學 Spring 2004 授課教師：統計系余清祥 日期： 2004 年 5 月 4 日 第十二週：複迴歸

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

3. 3 3 Slide The Multiple Regression Model n The Multiple Regression Model y =  0 +  1 x 1 +  2 x 2 +... +  p x p +  n The Multiple Regression Equation E( y ) =  0 +  1 x 1 +  2 x 2 +... +  p x p n The Estimated Multiple Regression Equation y = b 0 + b 1 x 1 + b 2 x 2 +... + b p x p ^

4. 4 4 Slide The Least Squares Method n Least Squares Criterion n Computation of Coefficients’ Values The formulas for the regression coefficients b 0, b 1, b 2,... b p involve the use of matrix algebra. We will rely on computer software packages to perform the calculations. n A Note on Interpretation of Coefficients b i represents an estimate of the change in y corresponding to a one-unit change in x i when all other independent variables are held constant. ^

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

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

7. 7 7 Slide Testing for Significance: F Test n Hypotheses H 0 :  1 =  2 =... =  p = 0 H 0 :  1 =  2 =... =  p = 0 H a : One or more of the parameters H a : One or more of the parameters is not equal to zero. is not equal to zero. n Test Statistic F = MSR/MSE n Rejection Rule Reject H 0 if F > F  where F  is based on an F distribution with p d.f. in the numerator and n - p - 1 d.f. in the denominator.

8. 8 8 Slide Testing for Significance: t Test n Hypotheses H 0 :  i = 0 H 0 :  i = 0 H a :  i = 0 H a :  i = 0 n Test Statistic n Rejection Rule Reject H 0 if t t  where t  is based on a t distribution with where t  is based on a t distribution with n - p - 1 degrees of freedom. n - p - 1 degrees of freedom.

9. 9 9 Slide Testing for Significance: Multicollinearity n The term multicollinearity refers to the correlation among the independent variables. n 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. n If the estimated regression equation is to be used only for predictive purposes, multicollinearity is usually not a serious problem. n Every attempt should be made to avoid including independent variables that are highly correlated.

10. 10 Slide Using the Estimated Regression Equation for Estimation and Prediction n 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. n We substitute the given values of x 1, x 2,..., x p into the estimated regression equation and use the corresponding value of y as the point estimate. n 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. n Software packages for multiple regression will often provide these interval estimates. ^

11. 11 Slide 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.

12. 12 Slide Example: Programmer Salary Survey Exper. Score Salary Exper. Score Salary Exper. Score Salary Exper. Score Salary 4782498838 71004327326.6 18623.7107536.2 58234.358131.6 88635.867429 10843888734 07522.247930.1 18023.169433.9 6833037028.2 6913338930

13. 13 Slide Example: Programmer Salary Survey n Multiple Regression Model Suppose we believe that salary ( y ) is related to the years of experience ( x 1 ) and the score on the programmer aptitude test ( x 2 ) by the following regression model: y =  0 +  1 x 1 +  2 x 2 +  where y = annual salary ($000) y = annual salary ($000) x 1 = years of experience x 1 = years of experience x 2 = score on programmer aptitude test x 2 = score on programmer aptitude test

14. 14 Slide Example: Programmer Salary Survey n Multiple Regression Equation Using the assumption E (  ) = 0, we obtain E( y ) =  0 +  1 x 1 +  2 x 2 n Estimated Regression Equation b 0, b 1, b 2 are the least squares estimates of  0,  1,  2 b 0, b 1, b 2 are the least squares estimates of  0,  1,  2Thus y = b 0 + b 1 x 1 + b 2 x 2 ^

15. 15 Slide Example: Programmer Salary Survey Solving for the Estimates of  0,  1,  2 Solving for the Estimates of  0,  1,  2 ComputerPackage for Solving MultipleRegressionProblemsComputerPackage MultipleRegressionProblems b 0 = b 1 = b 1 = b 2 = b 2 = R 2 = etc. b 0 = b 1 = b 1 = b 2 = b 2 = R 2 = etc. Input Data Least Squares Output x 1 x 2 y 4 78 24 4 78 24 7 100 43 7 100 43...... 3 89 30 3 89 30 x 1 x 2 y 4 78 24 4 78 24 7 100 43 7 100 43...... 3 89 30 3 89 30

16. 16 Slide Example: Programmer Salary Survey n Minitab Computer Output The regression is Salary = 3.17 + 1.40 Exper + 0.251 Score Predictor Coef Stdev t-ratio p Constant3.1746.156.52.613 Exper1.4039.19867.07.000 Score.25089.077353.24.005 s = 2.419 R-sq = 83.4% R-sq(adj) = 81.5%

17. 17 Slide Example: Programmer Salary Survey n Minitab Computer Output (continued) Analysis of Variance SOURCE DF SS MS F P Regression2500.33250.1642.760.000 Error1799.465.85 Total19599.79

18. 18 Slide Example: Programmer Salary Survey n F Test Hypotheses H 0 :  1 =  2 = 0 Hypotheses H 0 :  1 =  2 = 0 H a : One or both of the parameters H a : One or both of the parameters is not equal to zero. is not equal to zero. Rejection Rule Rejection Rule For  =.05 and d.f. = 2, 17: F.05 = 3.59 For  =.05 and d.f. = 2, 17: F.05 = 3.59 Reject H 0 if F > 3.59. Test Statistic Test Statistic F = MSR/MSE = 250.16/5.85 = 42.76 Conclusion Conclusion We can reject H 0. We can reject H 0.

19. 19 Slide Example: Programmer Salary Survey n t Test for Significance of Individual Parameters Hypotheses H 0 :  i = 0 Hypotheses H 0 :  i = 0 H a :  i = 0 H a :  i = 0 Rejection Rule Rejection Rule For  =.05 and d.f. = 17, t.025 = 2.11 For  =.05 and d.f. = 17, t.025 = 2.11 Reject H 0 if t > 2.11 Reject H 0 if t > 2.11 Test Statistics Test Statistics Conclusions Conclusions Reject H 0 :  1 = 0 Reject H 0 :  2 = 0 Reject H 0 :  1 = 0 Reject H 0 :  2 = 0

20. 20 Slide Qualitative Independent Variables n In many situations we must work with qualitative independent variables such as gender (male, female), method of payment (cash, check, credit card), etc. n For example, x 2 might represent gender where x 2 = 0 indicates male and x 2 = 1 indicates female. n In this case, x 2 is called a dummy or indicator variable. n If a qualitative variable has k levels, k - 1 dummy variables are required, with each dummy variable being coded as 0 or 1. n For example, a variable with levels A, B, and C would be represented by x 1 and x 2 values of (0, 0), (1, 0), and (0,1), respectively.

21. 21 Slide 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.

22. 22 Slide Example: Programmer Salary Survey (B) Exp. Score Degr. Salary Exp. Score Degr. Salary Exp. Score Degr. Salary Exp. Score Degr. Salary 478No24988Yes38 7100Yes43273No26.6 186No23.71075Yes36.2 582Yes34.3581No31.6 886Yes35.8674No29 1084Yes38887Yes34 075No22.2479No30.1 180 No 23.1694Yes33.9 683No30370No28.2 691Yes33389No30

23. 23 Slide Example: Programmer Salary Survey (B) n Multiple Regression Equation E( y ) =  0 +  1 x 1 +  2 x 2 +  3 x 3 n Estimated Regression Equation y = b 0 + b 1 x 1 + b 2 x 2 + b 3 x 3 y = b 0 + b 1 x 1 + b 2 x 2 + b 3 x 3where y = annual salary ($000) y = annual salary ($000) x 1 = years of experience x 1 = years of experience x 2 = score on programmer aptitude test x 3 = 0 if individual does not have a grad. degree 1 if individual does have a grad. degree 1 if individual does have a grad. degree Note: x 3 is referred to as a dummy variable. ^

24. 24 Slide Example: Programmer Salary Survey (B) n Minitab Computer Output The regression is Salary = 7.95 + 1.15 Exp + 0.197 Score + 2.28 Deg Predictor Coef Stdev t-ratio p Constant7.9457.3811.08.298 Exp1.1476.29763.86.001 Score.19694.08992.19.044 Deg2.2801.9871.15.268 s = 2.396 R-sq = 84.7% R-sq(adj) = 81.8% s = 2.396 R-sq = 84.7% R-sq(adj) = 81.8%

25. 25 Slide Example: Programmer Salary Survey (B) n Minitab Computer Output (continued) Analysis of Variance SOURCE DF SS MS F P Regression3507.90169.3029.480.000 Error1691.895.74 Total19599.79

26. 26 Slide Residual Analysis n Residual for Observation i y i - y i n Standardized Residual for Observation i where The standardized residual for observation i in multiple regression analysis is too complex to be done by hand. However, this is part of the output of most statistical software packages. ^ ^ ^ ^

27. 27 Slide n Detecting Outliers An outlier is an observation that is unusual in comparison with the other data. An outlier is an observation that is unusual in comparison with the other data. Minitab classifies an observation as an outlier if its standardized residual value is +2. Minitab classifies an observation as an outlier if its standardized residual value is +2. This standardized residual rule sometimes fails to identify an unusually large observation as being an outlier. This standardized residual rule sometimes fails to identify an unusually large observation as being an outlier. This rule’s shortcoming can be circumvented by using studentized deleted residuals. This rule’s shortcoming can be circumvented by using studentized deleted residuals. The | i th studentized deleted residual| will be larger than the | i th standardized residual|. The | i th studentized deleted residual| will be larger than the | i th standardized residual|. Residual Analysis

28. 28 Slide End of Chapter 15