商务统计学英文版教学课件第12章.ppt

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1、Simple Linear Regression,Chapter 12,Simple Linear RegressionChapte,Objectives,In this chapter, you learn: How to use regression analysis to predict the value of a dependent variable based on a value of an independent variableTo understand the meaning of the regression coefficients b0 and b1To evalua

2、te the assumptions of regression analysis and know what to do if the assumptions are violatedTo make inferences about the slope and correlation coefficientTo estimate mean values and predict individual values,ObjectivesIn this chapter, you,Correlation vs. Regression,A scatter plot can be used to sho

3、w the relationship between two variablesCorrelation analysis is used to measure the strength of the association (linear relationship) between two variablesCorrelation is only concerned with strength of the relationship No causal effect is implied with correlationScatter plots were first presented in

4、 Ch. 2Correlation was first presented in Ch. 3,DCOVA,Correlation vs. RegressionA sc,Types of Relationships,Y,X,Y,X,Y,Y,X,X,Linear relationships,Curvilinear relationships,DCOVA,Types of RelationshipsYXYXYYXX,Types of Relationships,Y,X,Y,X,Y,Y,X,X,Strong relationships,Weak relationships,(continued),DC

5、OVA,Types of RelationshipsYXYXYYXX,Types of Relationships,Y,X,Y,X,No relationship,(continued),DCOVA,Types of RelationshipsYXYXNo r,Introduction to Regression Analysis,Regression analysis is used to:Predict the value of a dependent variable based on the value of at least one independent variableExpla

6、in the impact of changes in an independent variable on the dependent variableDependent variable: the variable we wish to predict or explainIndependent variable: the variable used to predict or explain the dependent variable,DCOVA,Introduction to Regression An,Simple Linear Regression Model,Only one

7、independent variable, XRelationship between X and Y is described by a linear functionChanges in Y are assumed to be related to changes in X,DCOVA,Simple Linear Regression Model,Linear component,Simple Linear Regression Model,Population Y intercept,Population SlopeCoefficient,Random Error term,Depend

8、ent Variable,Independent Variable,Random Error component,DCOVA,Linear componentSimple Linear,(continued),Random Error for this Xi value,Y,X,Observed Value of Y for Xi,Predicted Value of Y for Xi,Xi,Slope = 1,Intercept = 0,i,Simple Linear Regression Model,DCOVA,(continued)Random Error for th,Simple L

9、inear Regression Equation (Prediction Line),DCOVA,Simple Linear Regression Equat,The Least Squares Method,b0 and b1 are obtained by finding the values that minimize the sum of the squared differences between Y and :,DCOVA,The Least Squares Methodb0 an,Finding the Least Squares Equation,The coefficie

10、nts b0 and b1, and other regression results in this chapter, will be found using Excel or Minitab,Formulas are shown in the text for those who are interested,DCOVA,Finding the Least Squares Equa,b0 is the estimated mean value of Y when the value of X is zerob1 is the estimated change in the mean val

11、ue of Y as a result of a one-unit increase in X,Interpretation of the Slope and the Intercept,DCOVA,b0 is the estimated mean value,Simple Linear Regression Example,A real estate agent wishes to examine the relationship between the selling price of a home and its size (measured in square feet)A rando

12、m sample of 10 houses is selectedDependent variable (Y) = house price in $1000sIndependent variable (X) = square feet,DCOVA,Simple Linear Regression Examp,Simple Linear Regression Example: Data,DCOVA,Simple Linear Regression Examp,Simple Linear Regression Example: Scatter Plot,House price model: Sca

13、tter Plot,DCOVA,Simple Linear Regression Examp,Simple Linear Regression Example: Using Excel Data Analysis Function,1. Choose Data,2. Choose Data Analysis,3. Choose Regression,DCOVA,Simple Linear Regression Examp,Simple Linear Regression Example: Using Excel Data Analysis Function,(continued),Enter

14、Y range and X range and desired options,DCOVA,Simple Linear Regression Examp,Simple Linear Regression Example: Using PHStat,Add-Ins: PHStat: Regression: Simple Linear Regression,Simple Linear Regression Examp,Simple Linear Regression Example: Excel Output,The regression equation is:,DCOVA,Simple Lin

15、ear Regression Examp,Simple Linear Regression Example: Minitab Output,The regression equation isPrice = 98.2 + 0.110 Square FeetPredictor Coef SE Coef T PConstant 98.25 58.03 1.69 0.129Square Feet 0.10977 0.03297 3.33 0.010S = 41.3303 R-Sq = 58.1% R-Sq(adj) = 52.8%Analysis of VarianceSource DF SS MS

16、 F PRegression 1 18935 18935 11.08 0.010Residual Error8 13666 1708Total 9 32600,The regression equation is:,house price = 98.24833 + 0.10977 (square feet),DCOVA,Simple Linear Regression Examp,Simple Linear Regression Example: Graphical Representation,House price model: Scatter Plot and Prediction Li

17、ne,Slope = 0.10977,Intercept = 98.248,DCOVA,Simple Linear Regression Examp,Simple Linear Regression Example: Interpretation of bo,b0 is the estimated mean value of Y when the value of X is zero (if X = 0 is in the range of observed X values)Because a house cannot have a square footage of 0, b0 has n

18、o practical application,DCOVA,Simple Linear Regression Examp,Simple Linear Regression Example: Interpreting b1,b1 estimates the change in the mean value of Y as a result of a one-unit increase in XHere, b1 = 0.10977 tells us that the mean value of a house increases by .10977($1000) = $109.77, on ave

19、rage, for each additional one square foot of size,DCOVA,Simple Linear Regression Examp,Predict the price for a house with 2000 square feet:,The predicted price for a house with 2000 square feet is 317.85($1,000s) = $317,850,Simple Linear Regression Example: Making Predictions,DCOVA,Predict the price

20、 for a house,Simple Linear Regression Example: Making Predictions,When using a regression model for prediction, only predict within the relevant range of data,Relevant range for interpolation,Do not try to extrapolate beyond the range of observed Xs,DCOVA,Simple Linear Regression Examp,Measures of V

21、ariation,Total variation is made up of two parts:,Total Sum of Squares,Regression Sum of Squares,Error Sum of Squares,where: = Mean value of the dependent variableYi = Observed value of the dependent variable = Predicted value of Y for the given Xi value,DCOVA,Measures of VariationTotal var,SST = to

22、tal sum of squares (Total Variation)Measures the variation of the Yi values around their mean YSSR = regression sum of squares (Explained Variation)Variation attributable to the relationship between X and YSSE = error sum of squares (Unexplained Variation)Variation in Y attributable to factors other

23、 than X,(continued),Measures of Variation,DCOVA,SST = total sum of squares,(continued),Xi,Y,X,Yi,SST = (Yi - Y)2,SSE = (Yi - Yi )2,SSR = (Yi - Y)2,_,_,_,Y,Y,Y,_,Y,Measures of Variation,DCOVA,(continued)XiYXYiSST = (Yi -,The coefficient of determination is the portion of the total variation in the de

24、pendent variable that is explained by variation in the independent variableThe coefficient of determination is also called r-square and is denoted as r2,Coefficient of Determination, r2,note:,DCOVA,The coefficient of determinati,r2 = 1,Examples of Approximate r2 Values,Y,X,Y,X,r2 = 1,Perfect linear

25、relationship between X and Y: 100% of the variation in Y is explained by variation in X,DCOVA,r2 = 1Examples of Approximate,Examples of Approximate r2 Values,Y,X,Y,X,0 r2 1,Weaker linear relationships between X and Y: Some but not all of the variation in Y is explained by variation in X,DCOVA,Exampl

26、es of Approximate r2 V,Examples of Approximate r2 Values,r2 = 0,No linear relationship between X and Y: The value of Y does not depend on X. (None of the variation in Y is explained by variation in X),Y,X,r2 = 0,DCOVA,Examples of Approximate r2 V,Simple Linear Regression Example: Coefficient of Dete

27、rmination, r2 in Excel,58.08% of the variation in house prices is explained by variation in square feet,DCOVA,Simple Linear Regression Examp,Simple Linear Regression Example: Coefficient of Determination, r2 in Minitab,The regression equation isPrice = 98.2 + 0.110 Square FeetPredictor Coef SE Coef

28、T PConstant 98.25 58.03 1.69 0.129Square Feet 0.10977 0.03297 3.33 0.010S = 41.3303 R-Sq = 58.1% R-Sq(adj) = 52.8%Analysis of VarianceSource DF SS MS F PRegression 1 18935 18935 11.08 0.010Residual Error8 13666 1708Total 9 32600,58.08% of the variation in house prices is explained by variation in sq

29、uare feet,DCOVA,Simple Linear Regression Examp,Standard Error of Estimate,The standard deviation of the variation of observations around the regression line is estimated by,WhereSSE = error sum of squares n = sample size,DCOVA,Standard Error of EstimateThe,Simple Linear Regression Example:Standard E

30、rror of Estimate in Excel,DCOVA,Simple Linear Regression Examp,Simple Linear Regression Example:Standard Error of Estimate in Minitab,The regression equation isPrice = 98.2 + 0.110 Square FeetPredictor Coef SE Coef T PConstant 98.25 58.03 1.69 0.129Square Feet 0.10977 0.03297 3.33 0.010S = 41.3303 R

31、-Sq = 58.1% R-Sq(adj) = 52.8%Analysis of VarianceSource DF SS MS F PRegression 1 18935 18935 11.08 0.010Residual Error8 13666 1708Total 9 32600,DCOVA,Simple Linear Regression Examp,Comparing Standard Errors,Y,Y,X,X,SYX is a measure of the variation of observed Y values from the regression line,The m

32、agnitude of SYX should always be judged relative to the size of the Y values in the sample data,i.e., SYX = $41.33K is moderately small relative to house prices in the $200K - $400K range,DCOVA,Comparing Standard ErrorsYYXXS,Assumptions of RegressionL.I.N.E,LinearityThe relationship between X and Y

33、is linearIndependence of ErrorsError values are statistically independentParticularly important when data are collected over a period of timeNormality of ErrorError values are normally distributed for any given value of XEqual Variance (also called homoscedasticity)The probability distribution of th

34、e errors has constant variance,DCOVA,Assumptions of RegressionL.I.,Residual Analysis,The residual for observation i, ei, is the difference between its observed and predicted valueCheck the assumptions of regression by examining the residualsExamine for linearity assumptionEvaluate independence assum

35、ption Evaluate normal distribution assumption Examine for constant variance for all levels of X (homoscedasticity) Graphical Analysis of ResidualsCan plot residuals vs. X,DCOVA,Residual AnalysisThe residual,Residual Analysis for Linearity,Not Linear,Linear,x,residuals,x,Y,x,Y,x,residuals,DCOVA,Resid

36、ual Analysis for Linearit,Residual Analysis for Independence,Cycial Pattern:Not Independent,No Cycial Pattern Independent,X,X,residuals,residuals,X,residuals,DCOVA,Residual Analysis for Independ,Checking for Normality,Examine the Stem-and-Leaf Display of the ResidualsExamine the Boxplot of the Resid

37、ualsExamine the Histogram of the ResidualsConstruct a Normal Probability Plot of the Residuals,DCOVA,Checking for NormalityExamine,Residual Analysis for Normality,Percent,Residual,When using a normal probability plot, normal errors will approximately display in a straight line,-3 -2 -1 0 1 2 3,0,100

38、,DCOVA,Residual Analysis for Normalit,Residual Analysis for Equal Variance,Non-constant variance,Constant variance,x,x,Y,x,x,Y,residuals,residuals,DCOVA,Residual Analysis for Equal V,Simple Linear Regression Example: Excel Residual Output,Does not appear to violate any regression assumptions,DCOVA,S

39、imple Linear Regression Examp,Simple Linear Regression Example: Minitab Residual Output,DCOVA,Does not appear to violate any regression assumptions,Simple Linear Regression Examp,Used when data are collected over time to detect if autocorrelation is presentAutocorrelation exists if residuals in one

40、time period are related to residuals in another period,Measuring Autocorrelation:The Durbin-Watson Statistic,DCOVA,Used when data are collected o,Autocorrelation,Autocorrelation is correlation of the errors (residuals) over time,Violates the regression assumption that residuals are random and indepe

41、ndent,Here, residuals show a cyclic pattern, not random. Cyclical patterns are a sign of positive autocorrelation,DCOVA,AutocorrelationAutocorrelation,The Durbin-Watson Statistic,The possible range is 0 D 4 D should be close to 2 if H0 is true D less than 2 may signal positive autocorrelation, D gre

42、ater than 2 may signal negative autocorrelation,The Durbin-Watson statistic is used to test for autocorrelation,H0: positive autocorrelation doe not existH1: positive autocorrelation is present,DCOVA,The Durbin-Watson Statistic Th,Testing for Positive Autocorrelation,Calculate the Durbin-Watson test

43、 statistic = D (The Durbin-Watson Statistic can be found using Excel or Minitab),Decision rule: reject H0 if D dL,H0: positive autocorrelation does not existH1: positive autocorrelation is present,0,dU,2,dL,Reject H0,Do not reject H0,Find the values dL and dU from the Durbin-Watson table (for sample

44、 size n and number of independent variables k),Inconclusive,DCOVA,Testing for Positive Autocorre,Suppose we have the following time series data:Is there autocorrelation?,Testing for Positive Autocorrelation,(continued),DCOVA,Suppose we have the following,Example with n = 25:,Testing for Positive Aut

45、ocorrelation,(continued),Excel/PHStat output:,DCOVA,Example with n = 25:Durbin-Wa,Here, n = 25 and there is k = 1 one independent variableUsing the Durbin-Watson table, dL = 1.29 and dU = 1.45D = 1.00494 dL = 1.29, so reject H0 and conclude that significant positive autocorrelation exists,Testing fo

46、r Positive Autocorrelation,(continued),Decision: reject H0 since D = 1.00494 dL,0,dU=1.45,2,dL=1.29,Reject H0,Do not reject H0,Inconclusive,DCOVA,Here, n = 25 and there is k =,Inferences About the Slope,The standard error of the regression slope coefficient (b1) is estimated by,where:= Estimate of t

47、he standard error of the slope = Standard error of the estimate,DCOVA,Inferences About the SlopeThe,Inferences About the Slope: t Test,t test for a population slopeIs there a linear relationship between X and Y?Null and alternative hypotheses H0: 1 = 0(no linear relationship) H1: 1 0(linear relation

48、ship does exist)Test statistic,where: b1 = regression slope coefficient 1 = hypothesized slope Sb1 = standard error of the slope,DCOVA,Inferences About the Slope: t,Inferences About the Slope: t Test Example,Estimated Regression Equation:,The slope of this model is 0.1098 Is there a relationship bet

49、ween the square footage of the house and its sales price?,DCOVA,Inferences About the Slope: t,Inferences About the Slope: t Test Example,H0: 1 = 0H1: 1 0,From Excel output:,b1,Predictor Coef SE Coef T PConstant 98.25 58.03 1.69 0.129Square Feet 0.10977 0.03297 3.33 0.010,From Minitab output:,b1,DCOV

50、A,Inferences About the Slope: t,Inferences About the Slope: t Test Example,Test Statistic: tSTAT = 3.329,There is sufficient evidence that square footage affects house price,Decision: Reject H0,Reject H0,Reject H0,a/2=.025,-t/2,Do not reject H0,0,t/2,a/2=.025,-2.3060,2.3060,3.329,d.f. = 10- 2 = 8,H0