1. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice- Hall, Inc. Chap 14-1 Business Statistics: A Decision-Making Approach 6 th Edition Multiple Regression Analysis and Model Building

2. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-2 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

3. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-3 Model Specification Decide what you want to do and select the dependent variable Determine the potential independent variables for your model Gather sample data (observations) for all variables

4. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-4 The Correlation Matrix Correlation between the dependent variable and selected independent variables can be found using Excel: – Tools / Data Analysis… / Correlation Can check for statistical significance of correlation with a t test

5. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-5 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 is collected for 15 weeks

6. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-6 Pie Sales 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 SalesPrice Advertisin g Pie Sales1 Price-0.443271 Advertising0.556320.030441 Correlation matrix: Multiple regression model:

7. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-7 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 SalesPrice Advertisin g Pie Sales1 Price-0.443271 Advertising0.556320.030441

8. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-8 Multiple Regression Output Regression Statistics Multiple R0.72213 R Square0.52148 Adjusted R Square0.44172 Standard Error47.46341 Observations15 ANOVA dfSSMSF Significance F Regression229460.027 14730.01 36.538610.01201 Residual1227033.3062252.776 Total1456493.333 Coefficient s Standard 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

9. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-9 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.

10. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-10 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

11. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-11 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 <

12. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-12 Multiple Coefficient of Determination Reports the proportion of total variation in y explained by all x variables taken together

13. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-13 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 Coefficient s Standard 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)

14. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-14 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?

15. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-15 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)

16. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-16 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 Coefficient s Standard 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)

17. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-17 Is the Model Significant? 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)

18. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-18 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)

19. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-19 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 Coefficient s Standard 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

20. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-20 H 0 : β 1 = β 2 = 0 H A : β 1 and β 2 not both zero  =.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  =.05 F.05 = 3.885 Reject H 0 Do not reject H 0 Critical Value: F  = 3.885 F-Test for Overall Significance (continued) F

21. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-21 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)

22. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-22 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)

23. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-23 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 Coefficient s Standard 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.0398 t-value for Advertising is t = 2.855, with p-value.0145 (continued) Are Individual Variables Significant?

24. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-24 d.f. = 15-2-1 = 12  =.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 <.05) There is evidence that both Price and Advertising affect pie sales at  =.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=.025 -t α/2 Do not reject H 0 0 t α/2  /2=.025 -2.17882.1788

25. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-25 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 Coefficient s Standard 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.

26. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-26 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

27. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-27 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 Coefficient s Standard 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

28. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-28 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

29. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-29 Multicollinearity Multicollinearity: High correlation exists between two independent variables This means the two variables contribute redundant information to the multiple regression model

30. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-30 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)

31. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-31 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

32. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-32 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

33. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-33 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 PHStat / regression / multiple regression … Check the “variance inflationary factor (VIF)” box

34. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-34 Qualitative (Dummy) Variables Categorical explanatory variable (dummy variable) with two or more levels: – yes or no, on or off, male or female – coded as 0 or 1 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)

35. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-35 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)

36. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-36 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

37. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-37 Sales: number of pies sold per week Price: pie price in $ Holiday: Interpretation of 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

38. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-38 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

39. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-39 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”

40. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-40 Interpreting the Dummy Variable Coefficients (with 3 Levels) With the same square feet, a split-level will have an estimated average price of 18.84 thousand dollars more than a condo With the same square feet, a ranch will have an estimated average price of 23.53 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

41. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-41 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 – – The second independent variable is the square of the first variable Nonlinear Relationships

42. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-42 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:

43. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-43 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

44. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-44 Testing for Significance: 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

45. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-45 Higher Order Models y x If p = 3 the model is a cubic form:

46. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-46 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

47. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-47 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

48. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-48 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)

49. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-49 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

50. Business Statistics: A Decision- Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 14-50 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