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- 1. Learning Objectives Statistics for Managers In this chapter, you learn: Using Microsoft® Excel How to develop a multiple regression model How to interpret the regression coefficients 5th Edition 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 Chapter 14 How to use categorical variables in a regression model Introduction to Multiple Regression Some revisions made by Jennifer J. Edmonds, PhD @ Wilkes University Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 14-1 Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 14-2 The Estimated The Multiple Regression Model Multiple Regression Equation The coefficients of the multiple regression model are estimated using sample data Examine the linear relationship between 1 dependent (Y) & 2 or more independent variables (Xi). Multiple regression equation with k independent variables: Multiple Regression Model with k Independent Variables: Estimated Estimated (or predicted) Estimated slope coefficients intercept value of Y Y-intercept Population slopes Random Error ˆ Yi b0 b1X1i b2 X2i bk Xki Yi β 0 β1X1i β 2 X 2i β k X ki εi Still only For every independent variable (xi), there is a For every independent variable (xi), we estimate a one corresponding slope (bi) Some revisions made by Jennifer J. Edmonds, PhD @ Wilkes University corresponding slope (bi) Some revisions made by Jennifer J. Edmonds, PhD @ Wilkes University intercept Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 14-3 Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 14-4 1
- 2. Coefficient of Regression Analysis Multiple Determination When performing a regression analysis, use this list as a guide… Reports the proportion of total variation in Y Define the variables Describe the relationship explained by all X variables taken together Calculate the regression coefficients How good is the model? SSR regressionsum of squares Is the model significant? r2 Is each x variable significant? SST total sum of squares Some revisions made by Jennifer J. Edmonds, PhD @ Wilkes University Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 14-6 Test Adjusted r2 for Overall Significance r2 never decreases when a new X variable is Is there a linear relationship between all of the X added to the model variables considered together and Y? This can be a disadvantage when comparing models… Examines the effect of using regression analysis to What is the net effect of adding a new variable? describe the relationship between X(‘s) and Y We lose a degree of freedom when a new X variable is To examine this effect, we compare the variability (in Y) added explained by the regressionFanalysis to the variability due to 0 Did the new X variable add enough independent power randomness (noise) to offset the loss of one degree of freedom? Variability comparisons require a different test than mean comparisons Some revisions made by Jennifer J. Edmonds, PhD @ Wilkes University Some revisions made by Jennifer J. Edmonds, PhD @ Wilkes University Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 14-7 Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 14-8 2
- 3. F-Test F-Test for Overall Significance for Overall Significance H0: β1 = β2 = 0 Test Statistic: Is there a linear relationship between all of the X H1: β1 and β2 not both zero variables considered together and Y? MSR = .05 F Use F test statistic (to test the variance) MSE df1= 2 df2 = 12 Decision: NULL H0: β1 = β2 = … = βk = 0 no linear relationship with any of Since F test statistic is in the rejection the variables region (p-value < .05), reject H0 Critical Value ALT H1: at least one βi ≠ 0 at least one independent variable = .05 has a relationship with Y Conclusion: 0 F There is evidence that at least one Do not Reject H0 Some revisions made by Jennifer J. Edmonds, PhD @ Wilkes University reject H0 F.05 = 3.89 independent variable affects Y Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 14-9 Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 14-10 Individual Variables Multiple Regression Assumptions Tests of Hypothesis Is there a linear relationship between the variable Xi Errors (residuals) from the regression model: and Y? < Use t-tests of individual variable slopes ei = (Yi – Yi) Hypotheses: NULL H0: βi = 0 no linear relationship Assumptions: ALT H1: βi ≠ 0 linear relationship does exist The errors are independent between Xi and Y The errors are normally distributed Errors have an equal variance Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 14-11 Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 14-13 3
- 4. Multiple Regression Assumptions Using Dummy Variables These residual plots are used in multiple regression: A dummy variable is a categorical < Residuals vs. Yi independent variable with two levels: Residuals vs. X1i yes or no, on or off, male or female Residuals vs. X2i coded as 0 or 1 Residuals vs. time (if time series data) Assumes equal slopes for other variables If more than two levels, the number of dummy Use the residual plots to check for variables needed is violations of regression assumptions (number of levels) - 1 Some revisions made by Jennifer J. Edmonds, PhD @ Wilkes University Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 14-14 Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 14-15 Dummy Variable Example Dummy Variable Example ˆ Y b0 b1X1 b2 (1) (b0 b2 ) b1X1 Holiday ˆ Y b0 b1X1 b 2 X 2 ˆ Y b0 b1X1 b2 (0) b0 b1X1 No Holiday Let: Y (sales) Different Same intercept slope Y = pie sales b0 + b2 If H0: β2 = 0 is X1 = price b0 rejected, then “Holiday” has a X2 = holiday (X2 = 1 if a holiday occurred during the week) significant effect (X2 = 0 if there was no holiday that week) on pie sales Some revisions made by Jennifer J. Edmonds, PhD @ Wilkes University X1 (Price) Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 14-16 Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 14-17 4
- 5. Chapter Summary In this chapter, we have Tested individual regression coefficients Tested portions of the regression model Used dummy variables Some revisions made by Jennifer J. Edmonds, PhD @ Wilkes University Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 14-18 5

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