Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 14-1 Chapter 14 Introduction to Multiple Regression Statistics for Managers using Microsoft Excel 6 th Edition

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 more important in predicting a dependent variable How to use categorical independent variables in a regression model

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 14-3 The Multiple Regression Model Idea: Examine the linear relationship between 1 dependent (Y) & 2 or more independent variables (X i ) Multiple Regression Model with k Independent Variables: Y-intercept Population slopesRandom Error DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 14-4 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 use Excel or Minitab to obtain the regression slope coefficients and other regression summary measures. DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 14-5 Two variable model Y X1X1 X2X2 Slope for variable X 1 Slope for variable X 2 Multiple Regression Equation (continued) DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 14-6 Example: 2 Independent Variables 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 DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 14-7 Pie Sales Example Sales = b 0 + b 1 (Price) + b 2 (Advertising) Week Pie Sales Price ($) Advertising ($100s) Multiple regression equation: DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 14-8 Excel Multiple Regression Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations15 ANOVA dfSSMSFSignificance F Regression Residual Total CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept Price Advertising DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 14-9 The Multiple Regression Equation b 1 = : sales will decrease, on average, by pies per week for each $1 increase in selling price, net of the effects of changes due to advertising b 2 = : sales will increase, on average, by 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. DCOVA

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

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

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Input values Predictions in PHStat (continued) Predicted Y value < Confidence interval for the mean value of Y, given these X values Prediction interval for an individual Y value, given these X values DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Coefficient of Multiple Determination Reports the proportion of total variation in Y explained by all X variables taken together DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations15 ANOVA dfSSMSFSignificance F Regression Residual Total CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept Price Advertising % of the variation in pie sales is explained by the variation in price and advertising Multiple Coefficient of Determination In Excel DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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? DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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) DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations15 ANOVA dfSSMSFSignificance F Regression Residual Total CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept Price Advertising % 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 Adjusted r 2 in Excel DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 1 : at least one β i ≠ 0 (at least one independent variable affects Y) DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall F Test for Overall Significance Test statistic: where F STAT has numerator d.f. = k and denominator d.f. = (n – k - 1) DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations15 ANOVA dfSSMSFSignificance F Regression Residual Total CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept Price Advertising (continued) F Test for Overall Significance In Excel With 2 and 12 degrees of freedom P-value for the F Test DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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: Since F STAT test statistic is in the rejection region (p- value <.05), reject H 0 There is evidence that at least one independent variable affects Y 0  =.05 F 0.05 = Reject H 0 Do not reject H 0 Critical Value: F 0.05 = F Test for Overall Significance (continued) F DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Two variable model Y X1X1 X2X2 YiYi Y i < x 2i x 1i The best fit equation is found by minimizing the sum of squared errors,  e 2 Sample observation Residuals in Multiple Regression Residual = e i = (Y i – Y i ) < DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Multiple Regression Assumptions Assumptions: The errors are normally distributed Errors have a constant variance The model errors are independent e i = (Y i – Y i ) < Errors ( residuals ) from the regression model: DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 Residuals vs. time (if time series data) < Use the residual plots to check for violations of regression assumptions DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Are Individual Variables Significant? Use t tests of individual variable slopes Shows if there is a linear relationship between the variable X j and Y holding constant the effects of other X variables Hypotheses: H 0 : β j = 0 (no linear relationship) H 1 : β j ≠ 0 (linear relationship does exist between X j and Y) DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Are Individual Variables Significant? H 0 : β j = 0 (no linear relationship) H 1 : β j ≠ 0 (linear relationship does exist between X j and Y) Test Statistic: ( df = n – k – 1) (continued) DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations15 ANOVA dfSSMSFSignificance F Regression Residual Total CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept Price Advertising t Stat for Price is t STAT = , with p-value.0398 t Stat for Advertising is t STAT = 2.855, with p-value.0145 (continued) Are Individual Variables Significant? Excel Output DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall d.f. = = 12  =.05 t  /2 = Inferences about the Slope: t Test Example H 0 : β j = 0 H 1 : β j  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 the Excel output: Reject H 0 for each variable Decision: Conclusion: Reject H 0  /2=.025 -t α/2 Do not reject H 0 0 t α/2  /2= For Price t STAT = , with p-value.0398 For Advertising t STAT = 2.855, with p-value.0145 DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Confidence Interval Estimate for the Slope Confidence interval for the population slope β j Example: Form a 95% confidence interval for the effect of changes in price (X 1 ) on pie sales: ± (2.1788)(10.832) So the interval is ( , ) (This interval does not contain zero, so price has a significant effect on sales holding constant the effect of advertising) CoefficientsStandard Error Intercept Price Advertising where t has (n – k – 1) d.f. Here, t has (15 – 2 – 1) = 12 d.f. DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Confidence Interval Estimate for the Slope Confidence interval for the population slope β j Example: Excel output also reports these interval endpoints: Weekly sales are estimated to be reduced by between 1.37 to pies for each increase of $1 in the selling price, holding the effect of advertising constant CoefficientsStandard Error…Lower 95%Upper 95% Intercept … Price … Advertising … (continued) DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Contribution of a Single Independent Variable X j SSR(X j | all variables except X j ) = SSR (all variables) – SSR(all variables except X j ) Measures the contribution of X j in explaining the total variation in Y (SST) Testing Portions of the Multiple Regression Model DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Measures the contribution of X 1 in explaining SST From ANOVA section of regression for Testing Portions of the Multiple Regression Model Contribution of a Single Independent Variable X j, assuming all other variables are already included (consider here a 2-variable model): SSR(X 1 | X 2 ) = SSR (all variables) – SSR(X 2 ) (continued) DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall The Partial F-Test Statistic Consider the hypothesis test: H 0 : variable X j does not significantly improve the model after all other variables are included H 1 : variable X j significantly improves the model after all other variables are included Test using the F-test statistic: (with 1 and n-k-1 d.f.) DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Testing Portions of Model: Example Test at the  =.05 level to determine whether the price variable significantly improves the model given that advertising is included Example: Frozen dessert pies DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Testing Portions of Model: Example H 0 : X 1 (price) does not improve the model with X 2 (advertising) included H 1 : X 1 does improve model  =.05, df = 1 and 12 F 0.05 = 4.75 (For X 1 and X 2 )(For X 2 only) ANOVA dfSSMS Regression Residual Total ANOVA dfSS Regression Residual Total (continued) DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Testing Portions of Model: Example Conclusion: Since F STAT = > F 0.05 = 4.75 Reject H 0 ; Adding X 1 does improve model (continued) (For X 1 and X 2 )(For X 2 only) ANOVA dfSSMS Regression Residual Total ANOVA dfSS Regression Residual Total DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Relationship Between Test Statistics The partial F test statistic developed in this section and the t test statistic are both used to determine the contribution of an independent variable to a multiple regression model. The hypothesis tests associated with these two statistics always result in the same decision (that is, the p-values are identical). Where a = degrees of freedom DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 the slopes associated with numerical independent variables do not change with the value for the categorical variable If more than two levels, the number of dummy variables needed is (number of levels - 1) DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Dummy-Variable 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) DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Same slope Dummy-Variable Example (with 2 Levels) (continued) 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 DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 If style of the house is also thought to matter: Style = ranch, split level, colonial Three levels, so two dummy variables are needed DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Dummy-Variable Models (more than 2 Levels) Example: Let “colonial” be the default category, and let X 2 and X 3 be used for the other two categories: Y = house price X 1 = square feet X 2 = 1 if ranch, 0 otherwise X 3 = 1 if split level, 0 otherwise The multiple regression equation is: (continued) DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Interpreting the Dummy Variable Coefficients (with 3 Levels) With the same square feet, a ranch will have an estimated average price of thousand dollars more than a colonial. With the same square feet, a split-level will have an estimated average price of thousand dollars more than a colonial. Consider the regression equation: For a colonial: X 2 = X 3 = 0 For a ranch: X 2 = 1; X 3 = 0 For a split level: X 2 = 0; X 3 = 1 DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 two-way cross product terms DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 changes DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall X 2 = 1: Y = 1 + 2X 1 + 3(1) + 4X 1 (1) = 4 + 6X 1 X 2 = 0: Y = 1 + 2X 1 + 3(0) + 4X 1 (0) = 1 + 2X 1 Interaction Example Slopes are different if the effect of X 1 on Y depends on X 2 value X1X Y = 1 + 2X 1 + 3X 2 + 4X 1 X 2 Suppose X 2 is a dummy variable and the estimated regression equation is DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Significance of Interaction Term Can perform a partial F test for the contribution of a variable to see if the addition of an interaction term improves the model Multiple interaction terms can be included Use a partial F test for the simultaneous contribution of multiple variables to the model DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Simultaneous Contribution of Independent Variables Use partial F test for the simultaneous contribution of multiple variables to the model Let m variables be an additional set of variables added simultaneously To test the hypothesis that the set of m variables improves the model: (where F STAT has m and n-k-1 d.f.) DCOVA

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 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 Tested individual regression coefficients Tested portions of the regression model Used dummy variables Evaluated interaction effects

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