Chapter 13 In this chapter, you learn: Multiple Regression Model with k Independent Variables:
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1 Chapter 4 4- Business Statistics: A First Course Fifth Edition Chapter 3 Multiple Regression Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc. Chap 3- 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 Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3- The Multiple Regression Model Idea: Examine the linear relationship between dependent (Y) & or more independent variables (X i ) Multiple Regression Model with k Independent Variables: Y-intercept Population slopes Random Error Y i β β X i β X i β k X ki ε i Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.
2 Chapter 4 4- 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 b i Estimated intercept b X i Estimated slope coefficients b Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-4 X i In this chapter we will use Excel or Minitab to obtain the regression slope coefficients and other regression summary measures. b k X ki Multiple Regression Equation Two variable model Y Ŷ b bx bx X X Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-5 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 $) Advertising ($ s) Data are collected for 5 weeks Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.
3 Chapter Week Pie Sales Price ($) Advertising ($s) Pie Sales Example Multiple regression equation: Sales = b + b (Price) + b (Advertising) Business Statistics: A First Course, 5e Prentice-Hall, 9 Inc.. Chap 3-7 Excel Multiple Regression Output Regression Statistics Multiple R.73 R Square.548 Adjusted R Square.447 Standard Error Observations 5 Sales (Price) 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 Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-8 Minitab Multiple Regression Output Sales (Price) 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 Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.
4 Chapter 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 Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3- 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 X = 3.5 Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3- Predictions in Excel using PHStat PHStat regression multiple regression Check the confidence and prediction interval estimates box Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-6 Prentice Hall, Inc.
5 Chapter Predictions in PHStat Input values Predicted Y value Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-3 < Confidence interval for the mean value of Y, given these X values Prediction interval for an individual Y value, given these X values Predictions in Minitab Predicted Values for New Observations Confidence interval for the mean value of Y, given these X values Predicted Y ˆ 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 X values Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-4 Coefficient of Multiple Determination Reports the proportion of total variation in Y explained by all X variables taken together r SSR regression sum of squares SST total sum of squares Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.
6 Chapter 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 Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-6 Multiple Coefficient of Determination In Minitab The regression equation is Sales = Price Advertising Predictor Coef SE Coef T P Constant SSR 946. r.548 Price SST Advertising 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 Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-7 Adjusted r r 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? Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.
7 Chapter Adjusted r Shows the proportion of variation in Y explained by all X variables adjusted for the number of X variables used and sample size n r adj ( r ) 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 Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-9 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 d 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 Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3- 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 Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-6 Prentice Hall, Inc.
8 Chapter 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 : β = β = = β k = (no linear relationship) H : at least one β i (at least one independent variable affects Y) Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3- F Test for Overall Significance Test statistic: SSR MSR F k STAT MSE SSE n k where F STAT has numerator d.f. = k and denominator d.f. = (n k - ) Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-3 F Test for Overall Significance In Excel Regression Statistics Multiple R.73 R Square.548 Adjusted R Square.447 Standard Error Observations 5 MSR 473. F STAT MSE 5.8 With and degrees P-value for of freedom 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 Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.
9 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.% MSR 473. F STAT MSE 5.8 Analysis of Variance Source DF SS MS F P Regression Residual Error Total With and degrees of freedom P-value for the F Test Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-5 F Test for Overall Significance H : β = β = H : β and β not both zero =.5 df = df = Do not reject H Critical Value: F.5 = =.5 Reject H F.5 = F Test Statistic: MSR F STAT MSE Decision: Since F STAT test statistic is in the rejection region (pvalue <.5), reject H Conclusion: There is evidence that at least one independent variable affects Y Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-6 Residuals in Multiple Regression Two variable model Y Residual = e i = (Y i Y i ) < Y i < Y i Sample observation Ŷ b bx bx x i X X x i The best fit equation is found by minimizing the sum of squared errors, e Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.
10 Chapter 4 4- Multiple Regression Assumptions Errors (residuals) from the regression model: Assumptions: < e i = (Y i Y i ) Independence of errors Error values are statistically independent Normality of errors Error values are normally distributed for any given set of X values Equal Variance (also called Homoscedasticity) The probability distribution of the errors has constant variance Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-8 Residual Plots Used in Multiple Regression These residual plots are used in multiple regression: < Residuals vs. Y i Residuals vs. X i Residuals vs. X i Residuals vs. time (if time series data) Use the residual plots to check for violations of regression assumptions Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-9 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 : β j = (no linear relationship) H : β j (linear relationship does exist between X j and Y) Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.
11 Chapter 4 4- Are Individual Variables Significant? H : β j = (no linear relationship) H : β j (linear relationship does exist between X j and Y) Test Statistic: t STAT b j S b j (df = n k ) Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-3 Regression Statistics Multiple R.73 R Square.548 Adjusted R Square.447 Standard Error Observations 5 Are Individual Variables Significant? Excel Output 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 Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-3 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 Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.
12 Chapter 4 4- H : β j = H : β j d.f. = 5-- = =.5 t / =.788 /=.5 Inferences about the Slope: t Test Example 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 t statistic ti ti for each variable falls in the rejection region (p-values <.5) /=.5 Decision: Reject H for each variable Conclusion: There is evidence that both Price and Advertising affect pie sales at =.5 Reject H Do not reject H Reject H -t α/ t α/ Statistics: A First 9 Prentice-Hall, Business Course, 5e Inc.. Chap 3-34 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 (X ) on pie sales: ± (.788)(.83) So the interval is ( , -.374) (This interval does not contain zero, so price has a significant effect on sales) Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-35 Confidence Interval Estimate for the Slope Confidence interval for the population slope β j 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 Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.
13 Chapter 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 Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-37 Dummy-Variable Example (with Levels) Let: Ŷ b b X b X Y = pie sales X = price X = holiday (X = if a holiday occurred during the week) (X = if there was no holiday that week) Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-38 Dummy-Variable Example (with Levels) Ŷ b b X b () (b b ) b X Ŷ b b X b () Y (sales) b + b b b Different intercept b X Same slope Holiday No Holiday If H : β = is rejected, then Holiday has a significant effect on pie sales X (Price) Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.
14 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 Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-4 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 Ŷ b b X b X b X b b X b X b (X X ) Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-4 Effect of Interaction Given: Y β βx βx β3xx ε Without interaction term, effect of X on Y is measured dby β With interaction term, effect of X on Y is measured by β + β 3 X Effect changes as X changes Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.
15 Chapter Interaction Example Suppose X is a dummy variable and the estimated regression equation is Ŷ = + X + 3X + 4X X Y 8 X = : Y = + X + 3() + 4X () = 4 + 6X 4 X = : Y = + X + 3() + 4X () = + X X.5.5 Slopes are different if the effect of X on Y depends on X value Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-43 Significance of Interaction Term Tested by utilizing the t-test on the coefficient associated with the interaction term. Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap 3-44 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 Used dummy variables Evaluated interaction effects Business Statistics: A First Course, 5e 9 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.
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