1. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-1 Statistics for Managers Using Microsoft® Excel 5th Edition Chapter 14 Introduction to Multiple Regression

2. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-2 Learning Objectives In this chapter, you learn:  How to develop a multiple regression model  How to interpret the regression coefficients  How to determine which independent variables to include in the regression model  How to determine which independent variables are most important in predicting a dependent variable  How to use categorical variables in a regression model

3. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-3 Multiple Regression Equation The coefficients of the multiple regression model are estimated using sample data Estimated (or predicted) value of Y Estimated slope coefficients Multiple regression equation with k independent variables: Estimated intercept In this chapter we will always use Excel to obtain the regression slope coefficients and other regression summary measures.

4. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-4 Multiple Regression Model Two variable model Y X1X1 X2X2 Slope for variable X 1 Slope for variable X 2

5. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-5 Multiple Regression Equation 2 Variable Example  A distributor of frozen dessert 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

6. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-6 Multiple Regression Equation 2 Variable Example  Sales = b 0 + b 1 (Price) + b 2 (Advertising)  Sales = b 0 +b 1 X 1 + b 2 X 2 Where X 1 = Price X 2 = Advertising WeekPie 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 Multiple regression equation:

7. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-7 Estimating a Multiple Linear Regression Equation  Excel will be used to generate the coefficients and measures of goodness of fit for multiple regression  Excel:  Tools / Data Analysis... / Regression  PHStat:  PHStat / Regression / Multiple Regression…

8. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-8 Multiple Regression Equation 2 Variable Example, Excel 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

9. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-9 Multiple Regression Equation 2 Variable Example 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. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-10 Multiple Regression Equation 2 Variable Example 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. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-11 Predictions in PHStat  PHStat | regression | multiple regression … Check the “confidence and prediction interval estimates” box

12. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-12 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 <

13. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-13 Coefficient of Multiple Determination  Reports the proportion of total variation in Y explained by all X variables taken together

14. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-14 Coefficient of Multiple Determination (Excel) 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

15. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-15 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 independent power to offset the loss of one degree of freedom?

16. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-16 Adjusted r 2  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)  Penalizes excessive use of unimportant independent variables  Smaller than r 2  Useful in comparing models

17. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-17 Adjusted r 2 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

18. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-18 Residual Plots Used in Multiple Regression  These residual plots are used in multiple regression:  Residuals vs. Y i  Residuals vs. X 1i  Residuals vs. X 2i etc. <

19. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-19 Residual Plots Pie Example Use Tools | Data Analysis | Regression

20. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-20 Residual Plots Pie Example Use Tools | Data Analysis | Regression

21. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-21 F-Test for Overall Significance  F-Test for Overall Significance of the Model  Shows if there is a linear relationship between any of the X variables and Y  Use F test statistic  Hypotheses: H 0 : β 1 = β 2 = … = β k = 0 (no linear relationship) H 1 : at least one β i ≠ 0 (at least one independent variable affects Y)

22. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-22 F-Test for Overall Significance 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 P-value for the F-Test

23. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-23 F-Test for Overall Significance  H 0 : β 1 = β 2 = 0  H 1 : β 1 and β 2 not both zero   =.05  df 1 = 2 df 2 = 12 Test Statistic: Decision: Conclusion: There is evidence that at least one independent variable affects Y 0  =.05 F.05 = 3.89 Reject H 0 Do not reject H 0 Critical Value: F  = 3.885 F p-value=.01201 Reject H 0 at  = 0.05

24. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-24 Individual Variables Tests of Hypothesis  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 1 : β i ≠ 0 (linear relationship does exist between X i and Y)

25. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-25 Individual Variables Tests of Hypothesis H 0 : β j = 0 (no linear relationship) H 1 : β j ≠ 0 (linear relationship does exist between X i and Y)  Test Statistic: (df = n – k – 1)

26. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-26 Individual Variables Tests of Hypothesis 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.0398 t-value for Advertising is t = 2.855, with p-value.0145

27. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-27 d.f. = 15-2-1 = 12  =.05 Inferences about the Slope: t Test Example H 0 : β 1 = 0 H 1 : β 1  0 The test statistic for each variable falls in the rejection region (p-values <.05) There is sufficient 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: H 0 : β 2 = 0 H 1 : β 2  0

28. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-28 Confidence Interval Estimate for the Slope Confidence interval for the population slope β i Example: Excel output also reports these interval endpoints: 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. (15 – 2 – 1) = 12 d.f.

29. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-29 Coefficient of Partial Determination for k Variable Model Measures the proportion of variation in the dependent variable that is explained by X j while controlling for (holding constant) the other independent variables

30. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-30 Coefficient of Partial Determination in Excel  Coefficients of Partial Determination can be found using Excel:  PHStat | regression | multiple regression …  Check the “coefficient of partial determination” box

31. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-31 Using Dummy Variables  A dummy variable is a categorical independent variable with two levels:  yes or no, on or off, male or female  coded as 0 or 1  Assumes equal slopes for other variables  If more than two levels, the number of dummy variables needed is (number of levels - 1)

32. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-32 Dummy Variable Example 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)

33. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-33 Dummy Variable Example Same slope X 1 (Price) Y (sales) b 0 + b 2 b0b0 Holiday No Holiday Different intercept Holiday (X 2 = 1) No Holiday (X 2 = 0) If H 0 : β 2 = 0 is rejected, then “Holiday” has a significant effect on pie sales

34. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-34 Dummy Variable Example Sales: number of pies sold per week Price: pie price in $ Holiday: 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

35. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-35 Dummy Variable Model More Than Two Levels  The number of dummy variables is one less than the number of levels  Example: Y = house price ; X 1 = square feet  If style of the house is also thought to matter Style = ranch, split level, colonial  Three levels, so two dummy variables are needed

36. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-36 Interaction Between Independent Variables  Hypothesizes interaction between pairs of X variables  Response to one X variable may vary at different levels of another X variable  Contains a two-way cross product term 

37. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-37 Interaction - Regression Model Worksheet Multiply X 1 by X 2 to get X 1 *X 2. Run regression with Y, X 1, X 2, X 1 X 2 Item, iYiYi X 1i X 2i X 1i *X 2i 11133 248540 31326 435630 ……………

38. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-38 Chapter Summary  Developed the multiple regression model  Tested the significance of the multiple regression model  Discussed adjusted r 2  Discussed using residual plots to check model assumptions In this chapter, we have

39. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-39 Chapter Summary  Tested individual regression coefficients  Tested portions of the regression model  Used dummy variables  Evaluated interaction effects In this chapter, we have