商务统计学英文版教学课件第13章

1、Introduction to Multiple RegressionChapter 13Introduction to Multiple RegreObjectivesIn this chapter, you learn: How to develop a multiple regression modelHow to interpret the regression coefficientsHow to determine which independent variables to include in the regression modelHow to use categorical

2、 independent variables in a regression modelObjectivesIn this chapter, youThe Multiple Regression ModelIdea: Examine the linear relationship between 1 dependent (Y) & 2 or more independent variables (Xi)Multiple Regression Model with k Independent Variables:Y-interceptPopulation slopesRandom ErrorDC

3、OVAThe Multiple Regression ModelIMultiple Regression EquationThe coefficients of the multiple regression model are estimated using sample dataEstimated (or predicted) value of YEstimated slope coefficientsMultiple regression equation with k independent variables:EstimatedinterceptIn this chapter we

4、will use Excel and Minitab to obtain the regression slope coefficients and other regression summary measures.DCOVAMultiple Regression EquationThTwo variable modelYX1X2Slope for variable X1Slope for variable X2Multiple Regression Equation(continued)DCOVATwo variable modelYX1X2Slope fA distributor of

5、frozen dessert pies wants to evaluate factors thought to influence demandDependent variable: Pie sales (units per week)Independent variables: Price (in $) Advertising ($100s)Data are collected for 15 weeksExample: 2 Independent VariablesDCOVAA distributor of frozen desserPie Sales ExampleSales = b0

6、+ b1 (Price) + b2 (Advertising)WeekPie SalesPrice($)Advertising($100s)13505.503.324607.503.333508.003.044308.004.553506.803.063807.504.074304.503.084706.403.794507.003.5104905.004.0113407.203.5123007.903.2134405.904.0144505.003.5153007.002.7Multiple regression equation:DCOVAPie Sales ExampleSales =

7、b0 + Excel Multiple Regression OutputRegression StatisticsMultiple R0.72213R Square0.52148Adjusted R Square0.44172Standard Error47.46341Observations15ANOVA dfSSMSFSignificance FRegression229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333CoefficientsStandard Errort StatP-val

8、ueLower 95%Upper 95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.305650.03979-48.57626-1.37392Advertising74.1309625.967322.854780.0144917.55303130.70888DCOVAExcel Multiple Regression OutpMinitab Multiple Regression OutputThe regression equation isSales = 307 - 2

9、5.0 Price + 74.1 AdvertisingPredictor Coef SE Coef T PConstant306.50 114.30 2.68 0.020Price -24.98 10.83 -2.31 0.040Advertising 74.13 25.97 2.85 0.014S = 47.4634 R-Sq = 52.1% R-Sq(adj) = 44.2%Analysis of VarianceSource DF SS MS F PRegression 2 29460 14730 6.54 0.012Residual Error12 27033 2253Total 1

10、4 56493DCOVAMinitab Multiple Regression OuThe Multiple Regression Equationb1 = -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 advertisingb2 = 74.131: sales will increase, on average, by 74.131 pies per wee

11、k for each $100 increase in advertising, net of the effects of changes due to pricewhere Sales is in number of pies per week Price is in $ Advertising is in $100s.DCOVAThe Multiple Regression EquatiUsing The Equation to Make PredictionsPredict sales for a week in which the selling price is $5.50 and

12、 advertising is $350:Predicted sales is 428.62 piesNote that Advertising is in $100s, so $350 means that X2 = 3.5DCOVAUsing The Equation to Make PrePredictions in Excel using PHStatPHStat | regression | multiple regression Check the “confidence and prediction interval estimates” boxDCOVAPredictions

13、in Excel using PHSInput valuesPredictions in PHStat(continued) Predicted Y valueConfidence interval for the mean value of Y, given these X valuesPrediction interval for an individual Y value, given these X valuesDCOVAInput valuesPredictions in PHSPredictions in MinitabInput valuesPredicted Values fo

14、r New ObservationsNewObs Fit SE Fit 95% CI 95% PI 1 428.6 17.2 (391.1, 466.1) (318.6, 538.6)Values of Predictors for New ObservationsNewObs Price Advertising 1 5.50 3.50 Confidence interval for the mean value of Y, given these X values Prediction interval for an individual Y value, given these X val

15、uesDCOVAPredictions in MinitabInput vaThe Coefficient of Multiple Determination, r2Reports the proportion of total variation in Y explained by all X variables taken togetherDCOVAThe Coefficient of Multiple DeRegression StatisticsMultiple R0.72213R Square0.52148Adjusted R Square0.44172Standard Error4

16、7.46341Observations15ANOVA dfSSMSFSignificance FRegression229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.305650.03979-48.57626-1.37

17、392Advertising74.1309625.967322.854780.0144917.55303130.7088852.1% of the variation in pie sales is explained by the variation in price and advertisingMultiple Coefficient of Determination In ExcelDCOVARegression StatisticsMultiple Multiple Coefficient of Determination In MinitabThe regression equat

18、ion isSales = 307 - 25.0 Price + 74.1 AdvertisingPredictor Coef SE Coef T PConstant306.50 114.30 2.68 0.020Price -24.98 10.83 -2.31 0.040Advertising 74.13 25.97 2.85 0.014S = 47.4634 R-Sq = 52.1% R-Sq(adj) = 44.2%Analysis of VarianceSource DF SS MS F PRegression 2 29460 14730 6.54 0.012Residual Erro

19、r12 27033 2253Total 14 5649352.1% of the variation in pie sales is explained by the variation in price and advertisingDCOVAMultiple Coefficient of DeterAdjusted r2r2 never decreases when a new X variable is added to the modelThis can be a disadvantage when comparing modelsWhat is the net effect of a

20、dding a new variable?We lose a degree of freedom when a new X variable is addedDid the new X variable add enough explanatory power to offset the loss of one degree of freedom?DCOVAAdjusted r2r2 never decreasesShows the proportion of variation in Y explained by all X variables adjusted for the number

21、 of X variables used (where n = sample size, k = number of independent variables)Penalizes excessive use of unimportant independent variablesSmaller than r2Useful in comparing among modelsAdjusted r2(continued)DCOVAShows the proportion of variatRegression StatisticsMultiple R0.72213R Square0.52148Ad

22、justed R Square0.44172Standard Error47.46341Observations15ANOVA dfSSMSFSignificance FRegression229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.97509

23、10.83213-2.305650.03979-48.57626-1.37392Advertising74.1309625.967322.854780.0144917.55303130.7088844.2% 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 variablesAdjusted r2 in ExcelDCOVARegression Sta

24、tisticsMultiple Adjusted r2 in MinitabThe regression equation isSales = 307 - 25.0 Price + 74.1 AdvertisingPredictor Coef SE Coef T PConstant306.50 114.30 2.68 0.020Price -24.98 10.83 -2.31 0.040Advertising 74.13 25.97 2.85 0.014S = 47.4634 R-Sq = 52.1% R-Sq(adj) = 44.2%Analysis of VarianceSource DF

25、 SS MS F PRegression 2 29460 14730 6.54 0.012Residual Error12 27033 2253Total 14 5649344.2% 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 variablesDCOVAAdjusted r2 in MinitabThe regrF Test for Overa

26、ll Significance of the ModelShows if there is a linear relationship between all of the X variables considered together and YUse F-test statisticHypotheses: H0: 1 = 2 = = k = 0 (no linear relationship) H1: at least one i 0 (at least one independent variable affects Y) Is the Model Significant?DCOVAF

27、Test for Overall SignificancF Test for Overall SignificanceTest statistic: where FSTAT has numerator d.f. = k and denominator d.f. = (n k - 1) DCOVAF Test for Overall SignificancRegression StatisticsMultiple R0.72213R Square0.52148Adjusted R Square0.44172Standard Error47.46341Observations15ANOVA dfS

28、SMSFSignificance FRegression229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.305650.03979-48.57626-1.37392Advertising74.1309625.96732

29、2.854780.0144917.55303130.70888(continued)F Test for Overall Significance In ExcelWith 2 and 12 degrees of freedomP-value for the F TestDCOVARegression StatisticsMultiple F Test for Overall Significance In MinitabThe regression equation isSales = 307 - 25.0 Price + 74.1 AdvertisingPredictor Coef SE

30、Coef T PConstant306.50 114.30 2.68 0.020Price -24.98 10.83 -2.31 0.040Advertising 74.13 25.97 2.85 0.014S = 47.4634 R-Sq = 52.1% R-Sq(adj) = 44.2%Analysis of VarianceSource DF SS MS F PRegression 2 29460 14730 6.54 0.012Residual Error12 27033 2253Total 14 56493With 2 and 12 degrees of freedomP-value

31、 for the F TestDCOVAF Test for Overall SignificancH0: 1 = 2 = 0H1: 1 and 2 not both zero = .05df1= 2 df2 = 12 Test Statistic: Decision:Conclusion:Since FSTAT test statistic is in the rejection region (p-value .05), reject H0There is evidence that at least one independent variable affects Y0 = .05F0.

32、05 = 3.885Reject H0Do not reject H0Critical Value: F0.05 = 3.885F Test for Overall Significance(continued)FDCOVAH0: 1 = 2 = 0Test Statistic:Two variable modelYX1X2Yi Yix2ix1iThe best fit equation is found by minimizing the sum of squared errors, e2Sample observationResiduals in Multiple RegressionRe

33、sidual = ei = (Yi Yi)DCOVATwo variable modelYX1X2Yi YixMultiple Regression AssumptionsAssumptions:The errors are normally distributedErrors have a constant varianceThe model errors are independentei = (Yi Yi)Errors (residuals) from the regression model:DCOVAMultiple Regression AssumptionResidual Plo

34、ts Used in Multiple RegressionThese residual plots are used in multiple regression:Residuals vs. YiResiduals vs. X1iResiduals vs. X2iResiduals vs. time (if time series data)Use the residual plots to check for violations of regression assumptionsDCOVAResidual Plots Used in MultipUse t tests of indivi

35、dual variable slopesShows if there is a linear relationship between the variable Xj and Y holding constant the effects of other X variablesHypotheses:H0: j = 0 (no linear relationship)H1: j 0 (linear relationship does exist between Xj and Y)Are Individual Variables Significant?DCOVAUse t tests of in

36、dividual variH0: j = 0 (no linear relationship between Xj and Y)H1: j 0 (linear relationship does exist between Xj and Y)Test Statistic:(df = n k 1)Are Individual Variables Significant?(continued)DCOVAH0: j = 0 (no linear relatioRegression StatisticsMultiple R0.72213R Square0.52148Adjusted R Square0

37、.44172Standard Error47.46341Observations15ANOVA dfSSMSFSignificance FRegression229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.30565

38、0.03979-48.57626-1.37392Advertising74.1309625.967322.854780.0144917.55303130.70888t Stat for Price is tSTAT = -2.306, with p-value .0398t Stat for Advertising is tSTAT = 2.855, with p-value .0145(continued)Are Individual Variables Significant? Excel OutputDCOVARegression StatisticsMultiple Are Indiv

39、idual Variables Significant? Minitab OutputThe regression equation isSales = 307 - 25.0 Price + 74.1 AdvertisingPredictor Coef SE Coef T PConstant306.50 114.30 2.68 0.020Price -24.98 10.83 -2.31 0.040Advertising 74.13 25.97 2.85 0.014S = 47.4634 R-Sq = 52.1% R-Sq(adj) = 44.2%Analysis of VarianceSour

40、ce DF SS MS F PRegression 2 29460 14730 6.54 0.012Residual Error12 27033 2253Total 14 56493t Stat for Price is tSTAT = -2.31, with p-value .040t Stat for Advertising is tSTAT = 2.85, with p-value .014DCOVAAre Individual Variables Signid.f. = 15-2-1 = 12 = .05t/2 = 2.1788Inferences about the Slope: t

41、 Test ExampleH0: j = 0H1: j 0The 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 = .05From the Excel output: Reject H0 for each variableDecision:Conclusion:Reject H0Reject H0a/2=.025-t/2Do not reject H

42、00t/2a/2=.025-2.17882.1788For Price tSTAT = -2.306, with p-value .0398For Advertising tSTAT = 2.855, with p-value .0145DCOVAd.f. = 15-2-1 = 12Inferences aConfidence Interval Estimate for the SlopeConfidence interval for the population slope j Example: Form a 95% confidence interval for the effect of

43、 changes in price (X1) on pie sales:-24.975 (2.1788)(10.832)So the interval is (-48.576 , -1.374)(This interval does not contain zero, so price has a significant effect on sales)CoefficientsStandard ErrorIntercept306.52619114.25389Price-24.9750910.83213Advertising74.1309625.96732where t has (n k 1)

44、d.f.Here, t has (15 2 1) = 12 d.f.DCOVAConfidence Interval Estimate Confidence Interval Estimate for the SlopeConfidence interval for the population slope jExample: Excel output also reports these interval endpoints: Weekly sales are estimated to be reduced by between 1.37 to 48.58 pies for each inc

45、rease of $1 in the selling price, holding the effect of advertising constantCoefficientsStandard ErrorLower 95%Upper 95%Intercept306.52619114.2538957.58835555.46404Price-24.9750910.83213-48.57626-1.37392Advertising74.1309625.9673217.55303130.70888(continued)DCOVAConfidence Interval Estimate Using Du

46、mmy VariablesA dummy variable is a categorical independent variable with two levels:yes or no, on or off, male or femalecoded as 0 or 1Assumes the slopes associated with numerical independent variables do not change with the value for the categorical variableIf more than two levels, the number of du

47、mmy variables needed is (number of levels - 1)DCOVAUsing Dummy VariablesA dummy vDummy-Variable Example (with 2 Levels)Let:Y = pie salesX1 = priceX2 = holiday (X2 = 1 if a holiday occurred during the week) (X2 = 0 if there was no holiday that week)DCOVADummy-Variable Example (withSame slopeDummy-Variable Example (with 2 Levels)(continued)X1 (Price)Y (sales)b0 + b2b0 HolidayNo HolidayDifferent interceptHoliday (X