Stat 501 Oct. 8 “Extra” Sum of SquaresIn multiple regression, we may have interest in how much one set of variables (set 2) reduces the sum of squared errors given that another set of variables (set 1) already is in the model .The authors or our text refers to this as the “extra sum of squares” for the variables in set 2. The notation and calculation is SSR (set 2 of X variables | set 1 of X variables) = SSE(set 1) – SSE(set 1, set 2)This could be read as the “reduction in SSE due to set 2 after controlling for set 1.” Examples:SSR(X4|X1,X2,X3) = SSE(X1, X2, X3)-SSE(X1,X2,X3,X4)This is the reduction in SSE due to X4 after controlling for X1, X2, and X3. Put another way, this is how much SSE will decrease when X4 is added to a model that already includes X1, X2, X3.SSR(X1, X3| X2) =SSE(X2)-SSE(X1, X2, X3)This is how much SSE will decrease when X4 is added to a model that already includes X2 and X3.For the home sale price example, let Y = log (price), X1= log of home square area, X2= 1 if air conditioning present and 0 if not, X3 = number of cars that can fit in the garage, X4 = number of bedrooms. Following are results for the model that includes X1, X2, X3, and X4. Note that SSE(X1,X2,X3,X4) = 23.120. Regression Analysis: logprice versus logarea, Air, CarGarage, Bedrooms Predictor Coef SE Coef T PConstant 3.7959 0.3013 12.60 0.000logarea 1.08432 0.04370 24.82 0.000Air 0.08505 0.02656 3.20 0.001CarGarage 0.11163 0.01723 6.48 0.000Bedrooms 0.00175 0.01135 0.15 0.878Analysis of VarianceSource DF SS MS F PRegression 4 73.409 18.352 409.59 0.000Residual Error 516 23.120 0.045Total 520 96.529Following are results for the model that includes X1, X2, X3. Note that SSE(X1,X2,X3) = 23.121, so SSR(X4|X1,X2,X3) = SSE(X1, X2, X3)-SSE(X1,X2,X3,X4) = 23.121 – 23.120 = 0.001. Not very big, which is whyt bedrooms was not significant above. Regression Analysis: logprice versus logarea, Air, CarGaragePredictor Coef SE Coef T PConstant 3.7764 0.2733 13.82 0.000logarea 1.08760 0.03812 28.53 0.000Air 0.08538 0.02644 3.23 0.001CarGarage 0.11166 0.01722 6.49 0.000Analysis of VarianceSource DF SS MS F PRegression 3 73.408 24.469 547.14 0.000Residual Error 517 23.121 0.045Total 520 96.529Following are results for a model with only X2 included. Note that SSE = 84.526.So, SSR(X1, X3| X2) =SSE(X2)-SSE(X1, X2, X3) = 84.256-23.121 = 61.135.Adding X1 and X3 to a model that had only X2 will decrease the SSE by 72.135.Regression Analysis: logprice versus Air Predictor Coef SE Coef T PConstant 12.0965 0.0430 281.18 0.000Air 0.40512 0.04719 8.58 0.000Analysis of VarianceSource DF SS MS F PRegression 1 12.003 12.003 73.70 0.000Residual Error 519 84.526 0.163Total 520 96.529Testing Whether One or More Variables Can Be Dropped From the ModelA statistical test of whether a set of variables can be simultaneously eliminated as predictor variables is based on the F-statisticF = )full(MSEfullfor dferror - reducedfor dferror )Full(SSE)duced(ReSSE with df = (error df for reduced – error df for full) and error df for full. .The precise null hypothesis is that some specified set of coefficients is 0.The Reduced model is the model that results if this null hypothesis is true. The Full model is the model with all variables being considered included. Example:Suppose a potential model is i3i32i21i1oixxxy . This is the “full” model.If we wish to test H0: 031 the reduced model is i2i2oixy Basically, we’re testing to see if variables x1 and x3 are “significant” given that x2 will be in the model. The relevant F-statistic is )x,x,x(MSEfulldferrorreduceddferror)x|x,x(SSR)x,x,x(MSEfulldferrorreduceddferror)x,x,x(SSE)x(SSEF3212313213212Tasks we’ll complete in class:1. Consider the home price example with a full model of X1,X2,X3. Calculate an F for testing H0: 031 2. Consider the home price example with a full model of X1,X2,X3, X4. Calculate an F for testing H0:
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