Basic Business Statistics, 10/e

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1 Chapter 4 4- Basic Business Statistics th Edition Chapter 4 Introduction to Multiple Regression Basic Business Statistics, e 9 Prentice-Hall, Inc. Chap 4- 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 variables in a regression model How to predict a categorical dependent variable using logistic regression Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-

2 Chapter 4 4- The Multiple Regression Model Idea: Examine the linear relationship between dependent (Y) & or more independent variables ( i ) Multiple Regression Model with k Independent Variables: Y-intercept Population slopes Random Error Y i β β i β i β k ki ε i Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-3 Multiple Regression Equation The coefficients of the multiple regression model are estimated using sample data Multiple regression equation with k independent variables: Estimated (or predicted) value of Y Yˆ i b Estimated intercept b i Estimated slope coefficients b i b k ki In this chapter we will use Excel or Minitab to obtain the regression slope coefficients and other regression summary measures. Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-4

3 Chapter Multiple Regression Equation Two variable model Y Ŷ b (continued) b b Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-5 Example: 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 $) Data are collected for 5 weeks Advertising ($ s) Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-6

4 Chapter Pie Sales Example Week Pie Sales Price ($) Advertising ($s) Multiple regression equation: Sales = b + b (Price) + b (Advertising) Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-7 Excel Multiple Regression Output Regression Statistics Multiple R.73 R Square.548 Adjusted R Square.447 Standard Error Observations 5 Sales (Pri ce) 74.3(Advertising) ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Price Advertising Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-8

5 Chapter Minitab Multiple Regression Output Sales (Pri ce) 74.3(Advertising) The regression equation is Sales = Price Advertising Predictor Coef SE Coef T P Constant Price Advertising S = R-Sq = 5.% R-Sq(adj) = 44.% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-9 The Multiple Regression Equation Sales (Price) 74.3(Advertising) where Sales is in number of pies per week Price is in $ Advertising is in $ s. b = : sales will decrease, on average, by pies per week for each $ increase in selling price, net of the effects of changes due to advertising b = 74.3: sales will increase, on average, by 74.3 pies per week for each $ increase in advertising, net of the effects of changes due to price Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-

6 Chapter Using The Equation to Make Predictions Predict sales for a week in which the selling price is $5.5 and advertising is $35: Sales (Price) 74.3(Advertising) (5.5) 74.3(3.5) 48.6 Predicted sales is 48.6 pies Note that Advertising is in $ s, so $35 means that = 3.5 Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4- Predictions in Excel using PHStat PHStat regression multiple regression Check the confidence and prediction interval estimates box Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-

7 Chapter Predictions in PHStat (continued) Input values Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-3 < Predicted Y value Confidence interval for the mean value of Y, given these values Prediction interval for an individual Y value, given these values Predictions in Minitab Predicted Values for New Observations Confidence interval for the mean value of Y, given these values Predictedˆ Y value New Obs Fit SE Fit 95% CI 95% PI (39., 466.) (38.6, 538.6) Values of Predictors for New Observations New Obs Price Advertising Input values Prediction interval for an individual Y value, given these values Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-4

8 Chapter Coefficient of Multiple Determination Reports the proportion of total variation in Y explained by all variables taken together r SSR SST regressionsumof squares total sumof squares Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-5 Multiple Coefficient of Determination In Excel Regression Statistics Multiple R.73 R Square.548 Adjusted R Square.447 Standard Error Observations 5 r SSR SST % of the variation in pie sales is explained by the variation in price and advertising ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Price Advertising Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-6

9 Chapter Multiple Coefficient of Determination In Minitab The regression equation is Sales = Price Advertising Predictor Coef SE Coef T P Constant Price Advertising r SSR SST S = R-Sq = 5.% R-Sq(adj) = 44.% Analysis of Variance Source DF SS MS F P Regression Residual Error Total % of the variation in pie sales is explained by the variation in price and advertising Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-7 Adjusted r r never decreases when a new 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 variable is added Did the new variable add enough explanatory power to offset the loss of one degree of freedom? Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-8

10 Chapter 4 4- Adjusted r Shows the proportion of variation in Y explained by all variables adjusted for the number of variables used r adj ( r n ) n k (where n = sample size, k = number of independent variables) Penalize excessive use of unimportant independent variables Smaller than r Useful in comparing among models (continued) Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-9 Adjusted r in Excel Regression Statistics Multiple R.73 R Square.548 Adjusted R Square.447 Standard Error Observations 5 r adj % 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 ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Price Advertising Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-

11 Chapter 4 4- Adjusted r in Minitab The regression equation is Sales = Price Advertising Predictor Coef SE Coef T P Constant Price Advertising S = R-Sq = 5.% R-Sq(adj) = 44.% Analysis of Variance Source DF SS MS F P Regression Residual Error Total r adj % 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 Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4- Is the Model Significant? F Test for Overall Significance of the Model Shows if there is a linear relationship between all of the variables considered together and Y Use F-test statistic Hypotheses: H : β = β = = β k = (no linear relationship) H : at least one β i (at least one independent variable affects Y) Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-

12 Chapter 4 4- F Test for Overall Significance Test statistic: F STAT MSR MSE SSR k SSE n k where F STAT has numerator d.f. = k and denominator d.f. = (n k - ) Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-3 F Test for Overall Significance In Excel Regression Statistics (continued) Multiple R.73 R Square.548 Adjusted R Square.447 Standard Error Observations 5 F STAT MSR MSE With and degrees of freedom P-value for the F Test ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Price Advertising Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-4

13 Chapter F Test for Overall Significance In Minitab The regression equation is Sales = Price Advertising Predictor Coef SE Coef T P Constant Price Advertising S = R-Sq = 5.% R-Sq(adj) = 44.% F STAT MSR MSE Analysis of Variance Source DF SS MS F P Regression Residual Error Total With and degrees of freedom P-value for the F Test Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-5 F Test for Overall Significance (continued) H : β = β = H : β and β not both zero =.5 df = df = Critical Value: F.5 = Test Statistic: MSR F STAT MSE Decision: Since F STAT test statistic is in the rejection region (pvalue <.5), reject H =.5 Conclusion: Do not reject H Reject H F.5 = F There is evidence that at least one independent variable affects Y Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-6

14 Chapter Residuals in Multiple Regression Two variable model Residual = e i = (Y i Y i ) < Y i Y < Y i Sample observation Ŷ b b b x i x i The best fit equation is found by minimizing the sum of squared errors, e Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-7 Multiple Regression Assumptions Errors (residuals) from the regression model: < e i = (Y i Y i ) Assumptions: The errors are normally distributed Errors have a constant variance The model errors are independent Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-8

15 Chapter Residual Plots Used in Multiple Regression These residual plots are used in multiple regression: < Residuals vs. Y i Residuals vs. i Residuals vs. i Residuals vs. time (if time series data) Use the residual plots to check for violations of regression assumptions Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-9 Are Individual Variables Significant? Use t tests of individual variable slopes Shows if there is a linear relationship between the variable j and Y holding constant the effects of other variables Hypotheses: H : β j = (no linear relationship) H : β j (linear relationship does exist between j and Y) Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-3

16 Chapter H : β j Are Individual Variables Significant? = (no linear relationship) H : β j (linear relationship does exist between j and Y) (continued) Test Statistic: t STAT b j S b j (df = n k ) Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-3 Are Individual Variables Significant? Excel Output (continued) Regression Statistics Multiple R.73 R Square.548 Adjusted R Square.447 Standard Error Observations 5 t Stat for Price is t STAT = -.36, with p-value.398 t Stat for Advertising is t STAT =.855, with p-value.45 ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Price Advertising Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-3

17 Chapter Are Individual Variables Significant? Minitab Output The regression equation is Sales = Price Advertising Predictor Coef SE Coef T P Constant Price Advertising S = R-Sq = 5.% R-Sq(adj) = 44.% Analysis of Variance Source DF SS MS F P Regression Residual Error Total t Stat for Price is t STAT = -.36, with p-value.398 t Stat for Advertising is t STAT =.855, with p-value.45 Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-33 H : β j = H : β j d.f. = 5-- = =.5 t / =.788 /=.5 Reject H Inferences about the Slope: t Test Example Do not reject H -t α/ t α/ From the Excel and Minitab output: For Price t STAT = -.36, with p-value.398 For Advertising t STAT =.855, with p-value.45 The test statistic for each variable falls in the rejection region (p-values <.5) /=.5 Reject H Decision: Reject H for each variable Conclusion: There is evidence that both Price and Advertising affect pie sales at =.5 Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-34

18 Chapter Confidence Interval Estimate for the Slope Confidence interval for the population slope β j b j tα / S bj where t has (n k ) d.f. Coefficients Standard Error Intercept Price Advertising Here, t has (5 ) = d.f. Example: Form a 95% confidence interval for the effect of changes in price ( ) on pie sales: ± (.788)(.83) So the interval is ( , -.374) (This interval does not contain zero, so price has a significant effect on sales) Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-35 Confidence Interval Estimate for the Slope Confidence interval for the population slope β j (continued) Coefficients Standard Error Lower 95% Upper 95% Intercept Price Advertising Example: Excel output also reports these interval endpoints: Weekly sales are estimated to be reduced by between.37 to pies for each increase of $ in the selling price, holding the effect of price constant Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-36

19 Chapter Testing Portions of the Multiple Regression Model Contribution of a Single Independent Variable j SSR( j all variables except j ) = SSR (all variables) SSR(all variables except j ) Measures the contribution of j in explaining the total variation in Y (SST) Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-37 Testing Portions of the Multiple Regression Model Contribution of a Single Independent Variable j, assuming all other variables are already included (consider here a -variable model): SSR( ) = SSR (all variables) SSR( ) (continued) From ANOVA section of regression for Yˆ b b b From ANOVA section of regression for Yˆ b b Measures the contribution of in explaining SST Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-38

20 Chapter 4 4- The Partial F-Test Statistic Consider the hypothesis test: H : variable j does not significantly improve the model after all other variables are included H : variable j significantly improves the model after all other variables are included Test using the F-test statistic: (with and n-k- d.f.) F STAT SSR ( j all variablesexcept j) MSE Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-39 Testing Portions of Model: Example Example: Frozen dessert pies Test at the =.5 level to determine whether the price variable significantly improves the model given that advertising is included Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-4

21 Chapter 4 4- Testing Portions of Model: Example H : (price) does not improve the model with (advertising) included H : does improve model =.5, df = and F.5 = 4.75 (continued) (For and ) ANOVA df SS MS Regression Residual Total (For only) ANOVA df SS Regression Residual Total Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-4 Testing Portions of Model: Example (For and ) (For only) (continued) ANOVA ANOVA df SS MS df SS Regression Regression Residual Residual Total Total F STAT SSR ( ) MSE(all) 9,46.37, Conclusion: Since F STAT = 5.36 > F.5 = 4.75 Reject H ; Adding does improve model Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-4

22 Chapter 4 4- 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). t a F, a Where a = degrees of freedom Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-43 Coefficient of Partial Determination for k variable model r Yj.(allvariables except j) SSR (j SST SSR(all variables) SSR( all variables except j j) all variables except j) Measures the proportion of variation in the dependent variable that is explained by j while controlling for (holding constant) the other independent variables Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-44

23 Chapter 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 Regression Analysis Coefficients of Partial Determination Intermediate Calculations SSR(,) SST SSR() SSR( ) SSR().4383 SSR( ) Coefficients r Y r Y Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-45 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 or 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 - ) Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-46

24 Chapter Dummy-Variable Example (with Levels) Let: Y = pie sales = price Ŷ b b b = holiday ( = if a holiday occurred during the week) ( = if there was no holiday that week) Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-47 Ŷ b Ŷ b Y (sales) b + b b Dummy-Variable Example (with Levels) (continued) b b b () (b b () b b Different intercept ) b b Same slope Holiday No Holiday If H : β = is rejected, then Holiday has a significant effect on pie sales (Price) Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-48

25 Chapter Interpreting the Dummy Variable Coefficient (with Levels) Example: Sales 3-3(Price) 5(Holiday) Sales: number of pies sold per week Price: pie price in $ Holiday: If a holiday occurred during the week If no holiday occurred b = 5: on average, sales were 5 pies greater in weeks with a holiday than in weeks without a holiday, given the same price Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-49 Dummy-Variable Models (more than Levels) The number of dummy variables is one less than the number of levels Example: Y = house price ; = 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 Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-5

26 Chapter Dummy-Variable Models (more than Levels) (continued) Example: Let colonial be the default category, and let and 3 be used for the other two categories: Y = house price = square feet = if ranch, otherwise 3 = if split level, otherwise The multiple regression equation is: Ŷ b b b b33 Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-5 Interpreting the Dummy Variable Coefficients (with 3 Levels) Consider the regression equation: Ŷ For a colonial: = 3 = Ŷ For a ranch: = ; 3 = Ŷ Ŷ For a split level: = ; 3 = 8.84 With the same square feet, a ranch will have an estimated average price of 3.53 thousand dollars more than a colonial. With the same square feet, a split-level will have an estimated average price of 8.84 thousand dollars more than a colonial. Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-5

27 Chapter Interaction Between Independent Variables Hypothesizes interaction between pairs of variables Response to one variable may vary at different levels of another variable Contains two-way cross product terms Ŷ b b b b 3 3 b b b b 3 ( ) Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-53 Effect of Interaction Given: Y β β β β3 ε Without interaction term, effect of on Y is measured by β With interaction term, effect of on Y is measured by β + β 3 Effect changes as changes Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-54

28 Chapter Interaction Example Suppose is a dummy variable and the estimated regression equation is Ŷ = Y 8 = : Y = + + 3() + 4 () = = : Y = + + 3() + 4 () = + Slopes are different if the effect of on Y depends on value Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-55 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 Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-56

29 Chapter 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: F STAT [SSR(all) SSR (allexcept newset of m variables)]/ m MSE(all) (where F STAT has m and n-k- d.f.) Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-57 Logistic Regression Used when the dependent variable Y is binary (i.e., Y takes on only two values) Examples Customer prefers Brand A or Brand B Employee chooses to work full-time or part-time Loan is delinquent or is not delinquent Person voted in last election or did not Logistic regression allows you to predict the probability of a particular categorical response Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-58

30 Chapter Logistic Regression (continued) Logistic regression is based on the odds ratio, which represents the probability of a success compared with the probability of failure probability of success Odds ratio probability of success The logistic regression model is based on the natural log of this odds ratio Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-59 Logistic Regression (continued) Logistic Regression Model: ln(odds ratio) β β i β i β k ki ε i Where k = number of independent variables in the model ε i = random error in observation i Logistic Regression Equation: ln(estimated odds ratio) b b i b i b k ki Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-6

31 Chapter Estimated Odds Ratio and Probability of Success Once you have the logistic regression equation, compute the estimated odds ratio: Estimated oddsratio ln(estimat ed odds ratio) e The estimated probability of success is Estimatedprobability of estimatedoddsratio success estimatedoddsratio Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-6 Chapter Summary Developed the multiple regression model Tested the significance of the multiple regression model Discussed adjusted r Discussed using residual plots to check model assumptions Tested individual regression coefficients Tested portions of the regression model Used dummy variables Evaluated interaction effects Discussed logistic regression Basic Business Statistics, e 9 Prentice-Hall, Inc.. Chap 4-6

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