商务统计学英文版教学课件第13章.ppt

1、Introduction to Multiple RegressionChapter 13ObjectivesIn this chapter,you learn:nHow to develop a multiple regression modelnHow to interpret the regression coefficientsnHow to determine which independent variables to include in the regression modelnHow to use categorical independent variables in a

2、regression modelThe Multiple Regression ModelIdea:Examine the linear relationship between 1 dependent(Y)&2 or more independent variables(Xi)ikik2i21i10iXXXY Multiple Regression Model with k Independent Variables:Y-interceptPopulation slopesRandom ErrorDCOVAMultiple Regression EquationThe coefficient

3、s of the multiple regression model are estimated using sample datakik2i21i10iXbXbXbbY Estimated(or predicted)value of YEstimated slope coefficientsMultiple regression equation with k independent variables:EstimatedinterceptIn this chapter we will use Excel and Minitab to obtain the regression slope

4、coefficients and other regression summary measures.DCOVATwo variable modelYX1X222110XbXbbYSlope for variable X1Slope for variable X2Multiple Regression Equation(continued)DCOVAA distributor of frozen dessert pies wants to evaluate factors thought to influence demandDependent variable:Pie sales(units

5、 per week)Independent variables:Price(in$)Advertising($100s)Data are collected for 15 weeksExample:2 Independent VariablesDCOVAPie Sales ExampleSales=b0+b1(Price)+b2(Advertising)WeekPie SalesPrice($)Advertising($100s)13505.503.324607.503.333508.003.044308.004.553506.803.063807.504.074304.503.084706.

6、403.794507.003.5104905.004.0113407.203.5123007.903.2134405.904.0144505.003.5153007.002.7Multiple regression equation:DCOVAExcel Multiple Regression OutputRegression StatisticsMultiple R0.72213R Square0.52148Adjusted R Square0.44172Standard Error47.46341Observations15ANOVA dfSSMSFSignificance FRegres

7、sion229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333 CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.305650.03979-48.57626-1.37392Advertising74.1309625.967322.854780.0144917.553031

8、30.70888ertising)74.131(Adv ce)24.975(Pri-306.526 SalesDCOVAMinitab Multiple Regression OutputThe regression equation isSales=307-25.0 Price+74.1 Advertising Predictor 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 0.014 S=47.4634 R-Sq=52.1%

9、R-Sq(adj)=44.2%Analysis of Variance Source DF SS MS F PRegression 2 29460 14730 6.54 0.012Residual Error 12 27033 2253Total 14 56493ertising)74.131(Adv ce)24.975(Pri-306.526 Sales DCOVAThe Multiple Regression Equationertising)74.131(Adv ce)24.975(Pri-306.526 Salesb1=-24.975:sales will decrease,on av

10、erage,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 week for each$100 increase in advertising,net of the effects of changes due to pricewhere Sales is in number of pies per week

11、 Price is in$Advertising is in$100s.DCOVAUsing The Equation to Make PredictionsPredict sales for a week in which the selling price is$5.50 and advertising is$350:Predicted sales is 428.62 pies428.62(3.5)74.131 (5.50)24.975-306.526 ertising)74.131(Adv ce)24.975(Pri-306.526 SalesNote that Advertising

12、is in$100s,so$350 means that X2=3.5DCOVAPredictions in Excel using PHStatnPHStat|regression|multiple regression Check the“confidence and prediction interval estimates”boxDCOVAInput valuesPredictions in PHStat(continued)Predicted Y valueConfidence interval for the mean value of Y,given these X values

13、Prediction interval for an individual Y value,given these X valuesDCOVAPredictions in MinitabInput valuesPredicted Values for New Observations NewObs 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 Observations NewObs Price Advertising 1 5.50 3.50value Y Pr

14、edicted Confidence interval for the mean value of Y,given these X values Prediction interval for an individual Y value,given these X valuesDCOVAThe Coefficient of Multiple Determination,r2nReports the proportion of total variation in Y explained by all X variables taken togethersquares of sum totals

15、quares of sum regressionSSTSSRr2DCOVARegression StatisticsMultiple R0.72213R Square0.52148Adjusted R Square0.44172Standard Error47.46341Observations15ANOVA dfSSMSFSignificance FRegression229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333 CoefficientsStandard Errort StatP-va

16、lueLower 95%Upper 95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.305650.03979-48.57626-1.37392Advertising74.1309625.967322.854780.0144917.55303130.70888.5214856493.329460.0SSTSSRr252.1%of the variation in pie sales is explained by the variation in price and adv

17、ertisingMultiple Coefficient of Determination In ExcelDCOVAMultiple Coefficient of Determination In MinitabThe regression equation isSales=307-25.0 Price+74.1 Advertising Predictor 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 0.014 S=47.46

18、34 R-Sq=52.1%R-Sq(adj)=44.2%Analysis of Variance Source DF SS MS F PRegression 2 29460 14730 6.54 0.012Residual Error 12 27033 2253Total 14 56493.5214856493.329460.0SSTSSRr252.1%of the variation in pie sales is explained by the variation in price and advertisingDCOVAAdjusted r2nr2 never decreases wh

19、en a new X variable is added to the modelnThis can be a disadvantage when comparing modelsnWhat is the net effect of adding a new variable?nWe lose a degree of freedom when a new X variable is addednDid the new X variable add enough explanatory power to offset the loss of one degree of freedom?DCOVA

20、nShows 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)nPenalizes excessive use of unimportant independent variablesnSmaller than r2nUseful in comparing among modelsAdjusted r2(continued)

21、11)1(122knnrradjDCOVARegression StatisticsMultiple R0.72213R Square0.52148Adjusted R Square0.44172Standard Error47.46341Observations15ANOVA dfSSMSFSignificance FRegression229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333 CoefficientsStandard Errort StatP-valueLower 95%Uppe

22、r 95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.305650.03979-48.57626-1.37392Advertising74.1309625.967322.854780.0144917.55303130.70888.44172r2adj44.2%of the variation in pie sales is explained by the variation in price and advertising,taking into account the

23、sample size and number of independent variablesAdjusted r2 in ExcelDCOVAAdjusted r2 in MinitabThe regression equation isSales=307-25.0 Price+74.1 Advertising Predictor 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 0.014 S=47.4634 R-Sq=52.1%

24、R-Sq(adj)=44.2%Analysis of Variance Source DF SS MS F PRegression 2 29460 14730 6.54 0.012Residual Error 12 27033 2253Total 14 56493.44172r2adj44.2%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 varia

25、blesDCOVAnF Test for Overall Significance of the ModelnShows if there is a linear relationship between all of the X variables considered together and YnUse F-test statisticnHypotheses:H0:1=2=k=0 (no linear relationship)H1:at least one i 0 (at least one independent variable affects Y)Is the Model Sig

26、nificant?DCOVAF Test for Overall SignificancenTest statistic:where FSTAT has numerator d.f.=k and denominator d.f.=(n k-1)1knSSEkSSRMSEMSRFSTATDCOVARegression StatisticsMultiple R0.72213R Square0.52148Adjusted R Square0.44172Standard Error47.46341Observations15ANOVA dfSSMSFSignificance FRegression22

27、9460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333 CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.305650.03979-48.57626-1.37392Advertising74.1309625.967322.854780.0144917.55303130.708

28、88(continued)F Test for Overall Significance In ExcelWith 2 and 12 degrees of freedomP-value for the F Test6.53862252.814730.0MSEMSRFSTATDCOVAF Test for Overall Significance In MinitabThe regression equation isSales=307-25.0 Price+74.1 Advertising Predictor Coef SE Coef T PConstant306.50 114.30 2.68

29、 0.020Price-24.98 10.83 -2.31 0.040Advertising 74.13 25.97 2.85 0.014 S=47.4634 R-Sq=52.1%R-Sq(adj)=44.2%Analysis of Variance Source DF SS MS F PRegression 2 29460 14730 6.54 0.012Residual Error 12 27033 2253Total 14 564936.53862252.814730.0MSEMSRFSTATWith 2 and 12 degrees of freedomP-value for the

30、F TestDCOVAH0:1=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 =.05F0.05=3.885Reject H0Do not reject H06.5386FSTATMSEMSRCrit

31、ical Value:F0.05=3.885F Test for Overall Significance(continued)FDCOVATwo variable modelYX1X222110XbXbbYYi Yix2ix1iThe best fit equation is found by minimizing the sum of squared errors,e2Sample observationResiduals in Multiple RegressionResidual=ei =(Yi Yi)DCOVAMultiple Regression AssumptionsAssump

32、tions:nThe errors are normally distributednErrors have a constant variancenThe model errors are independentei=(Yi Yi)Errors(residuals)from the regression model:DCOVAResidual Plots Used in Multiple RegressionnThese residual plots are used in multiple regression:nResiduals vs.YinResiduals vs.X1inResid

33、uals vs.X2inResiduals vs.time(if time series data)Use the residual plots to check for violations of regression assumptionsDCOVAnUse t tests of individual variable slopesnShows if there is a linear relationship between the variable Xj and Y holding constant the effects of other X variablesnHypotheses

34、:nH0:j=0(no linear relationship)nH1:j 0 (linear relationship does exist between Xj and Y)Are Individual Variables Significant?DCOVAH0: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?jbj

35、STATSbt0(continued)DCOVARegression StatisticsMultiple R0.72213R Square0.52148Adjusted R Square0.44172Standard Error47.46341Observations15ANOVA dfSSMSFSignificance FRegression229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333 CoefficientsStandard Errort StatP-valueLower 95%U

36、pper 95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.305650.03979-48.57626-1.37392Advertising74.1309625.967322.854780.0144917.55303130.70888t Stat for Price is tSTAT=-2.306,with p-value.0398t Stat for Advertising is tSTAT=2.855,with p-value.0145(continued)Are In

37、dividual Variables Significant?Excel OutputDCOVAAre Individual Variables Significant?Minitab OutputThe regression equation isSales=307-25.0 Price+74.1 Advertising Predictor 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 0.014 S=47.4634 R-Sq=

38、52.1%R-Sq(adj)=44.2%Analysis of Variance Source DF SS MS F PRegression 2 29460 14730 6.54 0.012Residual Error 12 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.014DCOVAd.f.=15-2-1=12n=.05t/2=2.1788Inferences about the Slope:t

39、 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 H0/2=.025-t/2Do not reject H00t/2/2=

40、.025-2.17882.1788For Price tSTAT=-2.306,with p-value.0398For Advertising tSTAT=2.855,with p-value.0145DCOVAConfidence Interval Estimate for the SlopeConfidence interval for the population slope j Example:Form a 95%confidence interval for the effect of changes in price(X1)on pie sales:-24.975 (2.1788

41、)(10.832)So the interval is (-48.576 ,-1.374)(This interval does not contain zero,so price has a significant effect on sales)jbjStb2/CoefficientsStandard ErrorIntercept306.52619114.25389Price-24.9750910.83213Advertising74.1309625.96732where t has (n k 1)d.f.Here,t has (15 2 1)=12 d.f.DCOVAConfidence

42、 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 increase of$1 in the selling price,holding the effect of advertising constant Coeff

43、icientsStandard ErrorLower 95%Upper 95%Intercept306.52619114.2538957.58835555.46404Price-24.9750910.83213-48.57626-1.37392Advertising74.1309625.9673217.55303130.70888(continued)DCOVAUsing Dummy VariablesnA dummy variable is a categorical independent variable with two levels:nyes or no,on or off,male

44、 or femalencoded as 0 or 1nAssumes the slopes associated with numerical independent variables do not change with the value for the categorical variablenIf more than two levels,the number of dummy variables needed is(number of levels-1)DCOVADummy-Variable Example (with 2 Levels)Let:Y =pie salesX1=pri

45、ceX2=holiday (X2=1 if a holiday occurred during the week)(X2=0 if there was no holiday that week)210XbXbbY21DCOVASame slopeDummy-Variable Example(with 2 Levels)(continued)X1(Price)Y(sales)b0+b2b0 101012010Xb b (0)bXbbYXb)b(b(1)bXbbY121121HolidayNo HolidayDifferent interceptHoliday(X2=1)No Holiday(X2

46、=0)If H0:2=0 is rejected,then“Holiday”has a significant effect on pie salesDCOVASales:number of pies sold per weekPrice:pie price in$Holiday:Interpreting the Dummy Variable Coefficient(with 2 Levels)Example:1 If a holiday occurred during the week0 If no holiday occurredb2=15:on average,sales were 15

47、 pies greater in weeks with a holiday than in weeks without a holiday,given the same price)15(Holiday 30(Price)-300 SalesDCOVAInteraction Between Independent VariablesnHypothesizes interaction between pairs of X variablesnResponse to one X variable may vary at different levels of another X variablen

48、Contains two-way cross product termsn)X(XbXbXbbXbXbXbbY213221103322110DCOVAEffect of InteractionnGiven:nWithout interaction term,effect of X1 on Y is measured by 1nWith interaction term,effect of X1 on Y is measured by 1+3 X2nEffect changes as X2 changes XXXXY21322110DCOVAX2=1:Y=1+2X1+3(1)+4X1(1)=4+

49、6X1 X2=0:Y=1+2X1+3(0)+4X1(0)=1+2X1 Interaction ExampleSlopes are different if the effect of X1 on Y depends on X2 valueX1Y =1+2X1+3X2+4X1X2 Suppose X2 is a dummy variable and the estimated regression equation is YDCOVAChapter SummaryIn this chapter we discussed:nHow to develop a multiple regression modelnHow to interpret the regression coefficientsnHow to determine which independent variables to include in the regression modelnHow to use categorical independent variables in a regression model