1. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-1 Business Statistics: A Decision-Making Approach 8 th Edition Chapter 15 Multiple Regression Analysis and Model Building

2. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-2 Chapter Goals After completing this chapter, you should be able to: Explain model building using multiple regression analysis Apply multiple regression analysis to business decision-making situations Analyze and interpret the computer output for a multiple regression model Test the significance of the independent variables in a multiple regression model

3. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-3 Chapter Goals After completing this chapter, you should be able to: Recognize potential problems in multiple regression analysis and take steps to correct the problems Incorporate qualitative variables into the regression model by using dummy variables Use variable transformations to model nonlinear relationships Test if the coefficients of a regression model are useful (continued)

4. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-4 The Multiple Regression Model Idea: Examine the linear relationship between 1 dependent (y) & 2 or more independent variables (x i ) Population model: Y-interceptPopulation slopesRandom Error Estimated (or predicted) value of y Estimated slope coefficients Estimated multiple regression model: Estimated intercept

5. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-5 Multiple Regression Model Two variable model y x1x1 x2x2 Slope for variable x 1 Slope for variable x 2 Called the regression hyperplane

6. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-6 Multiple Regression Model Two variable model y x1x1 x2x2 yiyi y i < e = (y – y) < x 2i x 1i The best fit equation, y, is found by minimizing the sum of squared errors,  e 2 < Sample observation

7. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-7 Multiple Regression Assumptions The model errors are statistically independent and represent a random sample from the population of all possible errors For a given x, there can exist many values of y; thus many possible values for  The errors are normally distributed The mean of the errors is zero Errors have a constant variance e = (y – y) < Errors (residuals) from the regression model:

8. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-8 Basic Model-Building Concepts Models are used to test changes without actually implementing the changes Can be used to predict outputs based on specified inputs Consists of 3 components: Model specification Model fitting Model diagnosis

9. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-9 Model Specification Sometimes referred to as model identification Is a process for establishing the framework for the model Decide what you want to do and select the dependent variable (y) Determine the potential independent variables (x) for your model Gather sample data (observations) for all variables

10. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-10 Model Building Process of actually constructing the equation for the data May include some or all of the independent variables (x) The goal is to explain the variation in the dependent variable (y) with the selected independent variables (x)

11. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-11 Model Diagnosis Analyzing the quality of the model (perform diagnostic checks) Assess the extent to which the assumptions appear to be satisfied If unacceptable, begin the model-building process again Should use the simplest model available to meet needs The goal is to help you make better decisions

12. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-12 Example A distributor of frozen desert pies wants to evaluate factors thought to influence demand Dependent variable: Pie sales (units per week) Independent variables: Price (in $) Advertising ($100’s) Data are collected for 15 weeks

13. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-13 Formulate the Model Sales = b 0 + b 1 (Price) + b 2 (Advertising) Week Pie Sales Price ($) Advertising ($100s) 13505.503.3 24607.503.3 33508.003.0 44308.004.5 53506.803.0 63807.504.0 74304.503.0 84706.403.7 94507.003.5 104905.004.0 113407.203.5 123007.903.2 134405.904.0 144505.003.5 153007.002.7 Pie SalesPriceAdvertising Pie Sales1 Price-0.443271 Advertising0.556320.030441 Correlation matrix: Multiple regression model:

14. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-14 Interpretation of Estimated Coefficients Slope (b i ) Estimates that the average value of y changes by b i units for each 1 unit increase in X i holding all other variables constant Example: if b 1 = -20, then sales (y) is expected to decrease by an estimated 20 pies per week for each $1 increase in selling price (x 1 ), net of the effects of changes due to advertising (x 2 ) y-intercept (b 0 ) The estimated average value of y when all x i = 0 (assuming all x i = 0 is within the range of observed values)

15. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-15 Scatter Diagrams Sales Price Advertising

16. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-16 The Correlation Matrix Correlation between the dependent variable and selected independent variables can be found using Excel: Formula Tab: Data Analysis / Correlation Can check for statistical significance of correlation with a t test

17. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-17 Pie Sales Correlation Matrix Price vs. Sales : r = -0.44327 There is a negative association between price and sales Advertising vs. Sales : r = 0.55632 There is a positive association between advertising and sales Pie SalesPriceAdvertising Pie Sales1 Price-0.443271 Advertising0.556320.030441

18. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-18 Estimating a Multiple Linear Regression Equation Computer software is generally used to generate the coefficients and measures of goodness of fit for multiple regression Excel: Data / Data Analysis / Regression PHStat: Add-Ins / PHStat / Regression / Multiple Regression…

19. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-19 Estimating a Multiple Linear Regression Equation Excel: Data / Data Analysis / Regression

20. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-20 Estimating a Multiple Linear Regression Equation PHStat: Add-Ins / PHStat / Regression / Multiple Regression…

21. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-21 Multiple Regression Output Regression Statistics Multiple R0.72213 R Square0.52148 Adjusted R Square0.44172 Standard Error47.46341 Observations15 ANOVA dfSSMSFSignificance F Regression229460.02714730.0136.538610.01201 Residual1227033.3062252.776 Total1456493.333 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept306.52619114.253892.682850.0199357.58835555.46404 Price-24.9750910.83213-2.305650.03979-48.57626-1.37392 Advertising74.1309625.967322.854780.0144917.55303130.70888

22. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-22 The Multiple Regression Equation b 1 = -24.975: sales will decrease, on average, by 24.975 pies per week for each $1 increase in selling price, net of the effects of changes due to advertising b 2 = 74.131: sales will increase, on average, by 74.131 pies per week for each $100 increase in advertising, net of the effects of changes due to price where Sales is in number of pies per week Price is in $ Advertising is in $100’s.

23. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-23 Using The Model to Make Predictions Predict sales for a week in which the selling price is $5.50 and advertising is $350: Predicted sales is 428.62 pies Note that Advertising is in $100’s, so $350 means that x 2 = 3.5

24. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-24 Predictions in PHStat PHStat | regression | multiple regression … Check the “confidence and prediction interval estimates” box

25. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-25 Input values Predictions in PHStat (continued) Predicted y value < Confidence interval for the mean y value, given these x’s < Prediction interval for an individual y value, given these x’s <

26. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-26 Multiple Coefficient of Determination (R 2 ) Reports the proportion of total variation in y explained by all x variables taken together

27. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-27 Regression Statistics Multiple R0.72213 R Square0.52148 Adjusted R Square0.44172 Standard Error47.46341 Observations15 ANOVA dfSSMSFSignificance F Regression229460.02714730.0136.538610.01201 Residual1227033.3062252.776 Total1456493.333 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept306.52619114.253892.682850.0199357.58835555.46404 Price-24.9750910.83213-2.305650.03979-48.57626-1.37392 Advertising74.1309625.967322.854780.0144917.55303130.70888 52.1% of the variation in pie sales is explained by the variation in price and advertising Multiple Coefficient of Determination (continued)

28. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-28 Adjusted R 2 R 2 never decreases when a new x variable is added to the model This can be a disadvantage when comparing models What is the net effect of adding a new variable? We lose a degree of freedom when a new x variable is added Did the new x variable add enough explanatory power to offset the loss of one degree of freedom?

29. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-29 Shows the proportion of variation in y explained by all x variables adjusted for the number of x variables used (where n = sample size, k = number of independent variables) Penalize excessive use of unimportant independent variables Smaller than R 2 Useful in comparing among models Adjusted R 2 (continued)

30. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-30 Regression Statistics Multiple R0.72213 R Square0.52148 Adjusted R Square0.44172 Standard Error47.46341 Observations15 ANOVA dfSSMSFSignificance F Regression229460.02714730.0136.538610.01201 Residual1227033.3062252.776 Total1456493.333 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept306.52619114.253892.682850.0199357.58835555.46404 Price-24.9750910.83213-2.305650.03979-48.57626-1.37392 Advertising74.1309625.967322.854780.0144917.55303130.70888 44.2% of the variation in pie sales is explained by the variation in price and advertising, taking into account the sample size and number of independent variables Multiple Coefficient of Determination (continued)

31. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-31 Model Diagnosis: F-Test F-Test for Overall Significance of the Model Shows if there is a linear relationship between all of the x variables considered together and y Use F test statistic Hypotheses: H 0 : β 1 = β 2 = … = β k = 0 (no linear relationship) H A : at least one β i ≠ 0 (at least one independent variable affects y)

32. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-32 F-Test for Overall Significance Test statistic: where F has (numerator) D 1 = k and (denominator) D 2 = (n – k – 1) degrees of freedom (continued)

33. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-33 Regression Statistics Multiple R0.72213 R Square0.52148 Adjusted R Square0.44172 Standard Error47.46341 Observations15 ANOVA dfSSMSFSignificance F Regression229460.02714730.0136.538610.01201 Residual1227033.3062252.776 Total1456493.333 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept306.52619114.253892.682850.0199357.58835555.46404 Price-24.9750910.83213-2.305650.03979-48.57626-1.37392 Advertising74.1309625.967322.854780.0144917.55303130.70888 (continued) F-Test for Overall Significance With 2 and 12 degrees of freedom P-value for the F-Test

34. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-34 H 0 : β 1 = β 2 = 0 H A : β 1 and β 2 not both zero  = 0.05 df 1 = 2 df 2 = 12 Test Statistic: Decision: Conclusion: Reject H 0 at  = 0.05 The regression model does explain a significant portion of the variation in pie sales (There is evidence that at least one independent variable affects y ) 0  = 0.05 F 0.05 = 3.885 Reject H 0 Do not reject H 0 Critical Value: F  = 3.885 F-Test for Overall Significance (continued) F

35. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-35 Model Diagnosis: Are Individual Variables Significant? Use t-tests of individual variable slopes Shows if there is a linear relationship between the variable x i and y Hypotheses: H 0 : β i = 0 (no linear relationship) H A : β i ≠ 0 (linear relationship does exist between x i and y)

36. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-36 Are Individual Variables Significant? H 0 : β i = 0 (no linear relationship) H A : β i ≠ 0 (linear relationship does exist between x i and y ) Test Statistic: ( df = n – k – 1) (continued)

37. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-37 Regression Statistics Multiple R0.72213 R Square0.52148 Adjusted R Square0.44172 Standard Error47.46341 Observations15 ANOVA dfSSMSFSignificance F Regression229460.02714730.0136.538610.01201 Residual1227033.3062252.776 Total1456493.333 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept306.52619114.253892.682850.0199357.58835555.46404 Price-24.9750910.83213-2.305650.03979-48.57626-1.37392 Advertising74.1309625.967322.854780.0144917.55303130.70888 t-value for Price is t = -2.306, with p-value 0.0398 t-value for Advertising is t = 2.855, with p-value 0.0145 (continued) Are Individual Variables Significant?

38. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-38 d.f. = 15-2-1 = 12  = 0.05 t  /2 = 2.1788 Inferences about the Slope: t Test Example H 0 : β i = 0 H A : β i  0 The test statistic for each variable falls in the rejection region (p-values < 0.05) There is evidence that both Price and Advertising affect pie sales at  = 0.05 From Excel output: Reject H 0 for each variable CoefficientsStandard Errort StatP-value Price-24.9750910.83213-2.305650.03979 Advertising74.1309625.967322.854780.01449 Decision: Conclusion: Reject H 0  /2=0.025 -t α/2 Do not reject H 0 0 t α/2  /2=0.025 -2.17882.1788

39. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-39 Standard Deviation of the Regression Model The estimate of the standard deviation of the regression model is: Is this value large or small? Must compare to the mean size of y for comparison

40. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-40 Regression Statistics Multiple R0.72213 R Square0.52148 Adjusted R Square0.44172 Standard Error47.46341 Observations15 ANOVA dfSSMSFSignificance F Regression229460.02714730.0136.538610.01201 Residual1227033.3062252.776 Total1456493.333 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept306.52619114.253892.682850.0199357.58835555.46404 Price-24.9750910.83213-2.305650.03979-48.57626-1.37392 Advertising74.1309625.967322.854780.0144917.55303130.70888 The standard deviation of the regression model is 47.46 (continued) Standard Deviation of the Regression Model

41. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-41 The standard deviation of the regression model is 47.46 A rough prediction range for pie sales in a given week is Pie sales in the sample were in the 300 to 500 per week range, so this range is probably too large to be acceptable. The analyst may want to look for additional variables that can explain more of the variation in weekly sales (continued) Standard Deviation of the Regression Model

42. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-42 Model Diagnosis: Multicollinearity Multicollinearity: High correlation exists between two independent variables and therefore the variables overlap This means the two variables contribute redundant information to the multiple regression model

43. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-43 Multicollinearity Including two highly correlated independent variables can adversely affect the regression results No new information provided Can lead to unstable coefficients (large standard error and low t-values) Coefficient signs may not match prior expectations (continued)

44. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-44 Some Indications of Severe Multicollinearity Incorrect signs on the coefficients Large change in the value of a previous coefficient when a new variable is added to the model A previously significant variable becomes insignificant when a new independent variable is added The estimate of the standard deviation of the model increases when a variable is added to the model

45. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-45 Detect Collinearity (Variance Inflationary Factor) VIF j is used to measure collinearity: If VIF j ≥ 5, x j is highly correlated with the other explanatory variables R 2 j is the coefficient of determination when the j th independent variable is regressed against the remaining k – 1 independent variables

46. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-46 Detect Collinearity in PHStat Output for the pie sales example: Since there are only two explanatory variables, only one VIF is reported VIF is < 5 There is no evidence of collinearity between Price and Advertising Regression Analysis Price and all other X Regression Statistics Multiple R0.030437581 R Square0.000926446 Adjusted R Square-0.075925366 Standard Error1.21527235 Observations15 VIF1.000927305 Add-Ins/ PHStat / Regression / Multiple Regression … Check the “variance inflationary factor (VIF)” box

47. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-47 Confidence Interval Estimate for the Slope Confidence interval for the population slope β 1 (the effect of changes in price on pie sales): Example: Weekly sales are estimated to be reduced by between 1.37 to 48.58 pies for each increase of $1 in the selling price CoefficientsStandard Error…Lower 95%Upper 95% Intercept306.52619114.25389…57.58835555.46404 Price-24.9750910.83213…-48.57626-1.37392 Advertising74.1309625.96732…17.55303130.70888 where t has (n – k – 1) d.f.

48. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-48 Qualitative (Dummy) Variables Categorical explanatory variable (dummy variable) with two or more levels: yes or no, on or off, male or female freshman, sophomore, etc., class standing Sometimes called indicator variables Regression intercepts are different if the variable is significant Assumes equal slopes for other variables The number of dummy variables needed is (number of levels – 1)

49. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-49 Dummy-Variable Model Example (with 2 Levels) Let: y = pie sales x 1 = price x 2 = holiday (X 2 = 1 if a holiday occurred during the week) (X 2 = 0 if there was no holiday that week)

50. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-50 Same slope Dummy-Variable Model Example (with 2 Levels) (continued) x 1 (Price) y (sales) b 0 + b 2 b0b0 Holiday No Holiday Different intercept Holiday No Holiday If H 0 : β 2 = 0 is rejected, then “Holiday” has a significant effect on pie sales

51. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-51 Sales: number of pies sold per week Price: pie price in $ Holiday: Interpreting the Dummy Variable Coefficient (with 2 Levels) Example: 1 If a holiday occurred during the week 0 If no holiday occurred b 2 = 15: on average, sales were 15 pies greater in weeks with a holiday than in weeks without a holiday, given the same price

52. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-52 Dummy-Variable Models (more than 2 Levels) The number of dummy variables is one less than the number of levels Example: y = house price ; x 1 = square feet The style of the house is also thought to matter: Style = ranch, split level, condo Three levels, so two dummy variables are needed

53. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-53 Dummy-Variable Models (more than 2 Levels) b 2 shows the impact on price if the house is a ranch style, compared to a condo b 3 shows the impact on price if the house is a split level style, compared to a condo (continued) Let the default category be “condo”

54. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-54 Interpreting the Dummy Variable Coefficients (with 3 Levels) With the same square feet, a ranch will have an estimated average price of 23.53 thousand dollars more than a condo With the same square feet, a ranch will have an estimated average price of 18.84 thousand dollars more than a condo. Suppose the estimated equation is For a condo: x 2 = x 3 = 0 For a ranch: x 3 = 0 For a split level: x 2 = 0

55. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-55 The relationship between the dependent variable and an independent variable may not be linear Useful when scatter diagram indicates non- linear relationship Example: Quadratic model (a type of polynomial model) The second independent variable is the square of the first variable Nonlinear Relationships

56. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-56 Polynomial Regression Model where: β 0 = Population regression constant β i = Population regression coefficient for variable x j : j = 1, 2, …k p = Order of the polynomial  i = Model error If p = 2 the model is a quadratic model: General form:

57. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-57 Linear vs. Nonlinear Fit y x y x

58. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-58 Quadratic Regression Model Quadratic models may be considered when scatter diagram takes on the following shapes: x1x1 y x1x1 x1x1 yyy β 1 < 0β 1 > 0β 1 < 0β 1 > 0 β 1 = the coefficient of the linear term β 2 = the coefficient of the squared term x1x1 β 2 > 0 β 2 < 0

59. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-59 Nonlinear Model-Building Model specification Determine dependent and potential independent variables Formulate the model Perform diagnostic checks Model the curvilinear relationship Perform diagnostic checks on the revised curvilinear model

60. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-60 Model Diagnosis: Quadratic Model Test for Overall Relationship F test statistic = Testing the Quadratic Effect Compare quadratic model with the linear model Hypotheses (No 2 nd order polynomial term) (2 nd order polynomial term is needed) H 0 : β 2 = 0 H A : β 2  0

61. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-61 Higher Order Models y x If p = 3 the model is a cubic form:

62. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-62 Interaction Effects Hypothesizes interaction between pairs of x variables Response to one x variable varies at different levels of another x variable Contains two-way cross product terms Basic Terms Interactive Terms

63. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-63 Effect of Interaction Given: Without interaction term, effect of x 1 on y is measured by β 1 With interaction term, effect of x 1 on y is measured by β 1 + β 3 x 2 Effect changes as x 2 increases

64. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-64 x 2 = 1 x 2 = 0 y = 1 + 2x 1 + 3(1) + 4x 1 (1) = 4 + 6x 1 y = 1 + 2x 1 + 3(0) + 4x 1 (0) = 1 + 2x 1 Interaction Example Effect (slope) of x 1 on y does depend on x 2 value x1x1 4 8 12 0 010.51.5 y y = 1 + 2x 1 + 3x 2 + 4x 1 x 2 where x 2 = 0 or 1 (dummy variable)

65. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-65 Interaction Regression Model Worksheet multiply x 1 by x 2 to get x 1 x 2, then run regression with y, x 1, x 2, x 1 x 2

66. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-66 Hypothesize interaction between pairs of independent variables Hypotheses: H 0 : β 3 = 0 (no interaction between x 1 and x 2 ) H A : β 3 ≠ 0 (x 1 interacts with x 2 ) Evaluating Presence of Interaction

67. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-67 Partial-F Test Method to test more than one but fewer than all of the regression coefficients If choosing between two models, one model is a better fit if its SSE is significantly smaller than the other Need the SSE with interaction (complete model) and without interaction (reduced model)

68. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-68 Partial-F Test Statistic where: MSE C = mean square error for the complete model = SSEC/(n-c) r = the number of coefficients in the reduced model c = the number of coefficients in the complete model

69. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-69 Model Building Goal is to develop a model with the best set of independent variables Easier to interpret if unimportant variables are removed Lower probability of collinearity Stepwise regression procedure Provide evaluation of alternative models as variables are added Best-subset approach Try all combinations and select the best using the highest adjusted R 2 and lowest s ε

70. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-70 Idea: develop the least squares regression equation in steps, either through forward selection, backward elimination, or through standard stepwise regression The Coefficient of Partial Determination is the measure of the marginal contribution of each independent variable, given that other independent variables are in the model Stepwise Regression

71. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-71 Best Subsets Regression Idea: estimate all possible regression equations using all possible combinations of independent variables Select subsets from the chose possible independent variables to form models Choose the best fit by looking for the highest adjusted R 2 and lowest standard error s ε Stepwise regression and best subsets regression can be performed using PHStat, Minitab, or other statistical software packages

72. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-72 Model Diagnosis: Aptness of the Model Diagnostic checks on the model include verifying the assumptions of multiple regression: Individual model errors are independent and random Error are normally distributed Errors have constant variance Each x i is linearly related to y Errors (or Residuals) are given by

73. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-73 Analysis of residuals can reveal: A nonlinear regression function Residuals that do not have a constant variance Residuals that are non independent Residual terms that are not normally distributed Residual Analysis

74. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-74 Residual Analysis Non-constant variance Constant variance xx residuals Not IndependentIndependent x residuals x (continued)

75. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-75 The Normality Assumption Errors are assumed to be normally distributed Standardized residuals can be calculated by computer Examine a histogram or a normal probability plot of the standardized residuals to check for normality

76. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-76 Standardized Residuals where: e i = ith residual value s  = Standard error of the estimate x i = Value of x used to generated predicted y-value for the ith observation

77. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-77 Corrective Actions 3 approaches if the model is not appropriate: Transform some of the independent variables (x) Raising x to a power Taking the square root of x Taking the log of x If the normality assumption is not met, transforming the dependent variable (y) may help also Remove some variables from the model Start over

78. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-78 Chapter Summary Developed the multiple regression model Tested the significance of the multiple regression model Developed adjusted R 2 Tested individual regression coefficients Used dummy variables Examined interaction in a multiple regression model

79. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-79 Chapter Summary Described nonlinear regression models Described multicollinearity Discussed model building Stepwise regression Best subsets regression Examined residual plots to check model assumptions (continued)

80. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 15-80 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America.