Multiple Linear RegressionMultiple Linear Regression ModelingIssues for MLRAnalysis of Real Estate DataSummary StatisticsStatistical InferencesResidual Analysis and Model GoodnessChapter 11. Linear RegressionMultiple Linear RegressionJ.C. WangOutlineMultiple Linear RegressionMultiple Linear Regression ModelingIssues for MLRAnalysis of Real Estate DataSummary StatisticsStatistical InferencesResidual Analysis and Model GoodnessMultiple Linear RegressionThe multiple linear regression analysis concentrates on amodel that has more than one independent (explantaory)variable. The independent variables are used to predict thedependent variable.Multiple Linear Regression ModelingIPurpose of multiple regression analysis is predictionIModel: y = b0+ b1x1+ ... + bnxn; where biare the slopes,y is a dependent variable and xiis an independentvariable.ICorrelation coefficient, rij.ICoefficient of determination, R2(or multiple R2).Multiple Linear Regression ModelingcontinuedIStandard error of the estimated regr. line, s.ITest hypothesis of slopes, p-value.ISlope confidence intervals.IResidual calculation.MLRReal Estate ExampleA realtor in a suburban town would like to study the relationshipbetween the size of a single-family house (as measured by thenumber of rooms) and the selling price of the house. The studyis to be carried out in two different neighborhood, one on theeast side (code=0) of the town and the other on the west side(code=1). A random sample of 8 houses was selected with thefollowing results:MLR: Real Estate ExamplecontinuedSelling Price # of Rooms Neighborhood98.2 6 0109.6 7 0119.3 8 0135.3 9 0108.5 6 1126.7 8 1138.8 9 1143.8 10 1Things to Consider in MLRIScatter plotsICorrelationIMLR equation, R2, CIs of slopes, and residualsCorrelationsCorrelation Coefficientsprice rooms hoodprice 1rooms 0.9682 1hood 0.4537 0.2750 1Icorrelation coefficient of price and rooms is 0.9682 andsign is positive.Icorrelation coefficient of price and hood is 0.4537 andsign is positive.Icorrelation coefficient of rooms and hood is 0.2750 andsign is positive.IThere is no multi-collinearity.Regression EquationRegression Coefficientscoeff Std Err t-Stat p-valueIntercept 38.99 6.3 6.2 0.0016rooms 10.21 0.82 12.5 6e−5hood 6.19 2.22 2.8 0.039Using the information in the above table, write the estimatedregression equation for this problem.price = 38.99 + 10.21 × rooms + 6.19 × hoodPredictionIUsing the estimated regression equation, predict theaverage selling price for a house with nine (9) rooms thatis located on the east side (code=0) of the town.price = 38.99 + 10.21 × 9 + 6.19 × 0 = 130.88IIs the previous prediction extrapolation? Why or why not?No, because number of rooms = 9 is between 6 and 10,and hood = 0 is between 0 and 1.iClicker question 11.4iClicker question 11.4Interpretation of SlopesIHolding hood constant, as rooms increases by one,price increases by 10.21IHolding rooms constant, as hood increases by one,price increases by 6.19Multiple R-SquaredExcel OutputRegression StatisticsR-square 0.9755Adjusted R-sq 0.9657Standard Error 3.0238Observations 8Interpret the meaning of the coefficient of multipledetermination in this problem. What measure did you use toanswer this questions?R2= 0.9755.98% of the variation in price is explained by this model.R2=SSRSST=1818.281863.99= 0.9755Standard Error of Estimated EquationExcel OutputRegression StatisticsR-square 0.9755Adjusted R-sq 0.9657Standard Error 3.0238Observations 8What is the standard error of the estimated regressionequation? Include the unit of measurement in your answer.s = 3.0238 thousands of dollarss =rSSEn − 3=r45.725= 3.0238 (=√MSE =√9.14)Significance of Regression EquationDetermine whether there is a significant relationship betweenselling price and the two explanatory variables at the 5% levelof significance.ANOVASource df SS MS F pRegression 2 1818.28 909.14 99.43 9e−5Residual 5 45.72 9.14Total 7 1864.00H0: β1= β2= 0 vs.Ha: at least one slope is not zeroF = 99.43, p-value = 9e−5, Reject H0.Significance of SlopesDetermine whether each explanatory variable makes asignificant contribution to the regression model. On the basis ofthese results, indicate the regression model that should beused in this problem at a 5% level of significance. Be sure tostate hypotheses, test statistic, p-value, and conclusion.Regression Coefficientscoeff Std Err t-Stat p-valueIntercept 38.99 6.30 6.2 0.0016rooms 10.21 0.82 12.5 6e−5hood 6.19 2.22 2.8 0.0390Significance of SlopescontinuedIrooms: H0: β1= 0 vs. Ha: β16= 0, t = 12.5, p-value =6e−5, Reject H0.Ihood: H0: β2= 0 vs. Ha: β26= 0, t = 2.8, p-value =0.039, Reject H0.ISince both slopes are significantly different from zero, bothslopes are making a significant contribution to this model.Confidence Intervals of SlopesSet up 95% confidence interval estimates of the populationslope for the relationship between selling price and number ofrooms and neighborhood.Regression CoefficientsLower 95% Upper 95%Intercept 22.80 55.19rooms 8.12 12.31hood 0.47 11.91I95% CI for rooms: (8.12, 12.31)I95% CI for hood: (0.47, 11.91)Residual PlotDescribe what you see on the residual plot.●●●●●●●●100 110 120 130 140−2 0 2 4fitted valuesresiduals148There is a random distribution of points.Goodness of Fitted ModelIs the linear model a good fit for this data?Consider the following:IR2= 0.9755, s = 3.0238IBoth rooms and hood are making a significantcontribution to the linear model.IThe residual plot shows a random distribution of points.Yes, this model appears to be appropriate for this data.Calculation of Predicted Value and ResidualFor this data set, the first set of observations is price = 98.2,rooms = 6, hood = 0. Calculate the residual, i.e., compute thedifference between the observed price and the predictedprice. Show all the formulas and calculations.ˆy1= 38.99 + 10.21 × 6 + 6.19 × 0 = 100.25e1= y1−ˆy1= 98.2 − 100.25 = −2.05iClicker question 11.5iClicker question