PSU STAT 501  Nonconstant Variance and Weighted Least Squares (4 pages)
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Nonconstant Variance and Weighted Least Squares
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 Penn State University
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 Stat 501  Regression Methods
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Stat 501 November 29 Nonconstant Variance and Weighted Least Squares Example 1 Y Price market share of product x1 price price x3 Discount 1 if discount promotion in effect and 0 otherwise x3x4 1 if both discount and package promotions in effect and 0 otherwise N 36 consecutive months Regression results including residual plot that indicates values of x3 Predictor Constant x1 price x3 Discount x3x4 Coef 3 1959 0 3336 0 30808 0 17623 Source Regression Residual Error Total DF 3 32 35 SE Coef 0 3562 0 1523 0 06412 0 06597 P 0 000 0 036 0 000 0 012 MS 0 58761 0 02136 F 27 51 SS 1 76282 0 68343 2 44626 T 8 97 2 19 4 80 2 67 P 0 000 Interpretation There s a pattern of nonconstant variance related to whether x3 1 or x3 0 Here are the residual variances for the two groups Variable RESI1 x3 Discount 0 1 N 15 21 N 0 0 Variance 0 0105 0 0268 Weighting for a weighted least squares For weights we use 1 2 In this case the two i variances given just above estimated the i2 This leads to these weights Variable wts x3 Discount 0 1 Weight 95 238 37 313 Weighted analysis using weights in wts Predictor Constant x1 price x3 Discount x3x4 Coef 3 1743 0 3243 0 30834 0 17585 Source Regression Residual Error Total DF 3 32 35 SE Coef 0 3567 0 1529 0 06577 0 07612 SS 98 118 34 031 132 149 T 8 90 2 12 4 69 2 31 MS 32 706 1 063 P 0 000 0 042 0 000 0 027 F 30 75 P 0 000 IMPORTANT Minitab s AOV will be in terms of weighted SS Notice how different the SS values are from the SS values for the unweighted case IMPORTANT When plotting residuals after a weighted least squares use standardized residuals Example 2 Y Sale Price of Home X1 SqrFeet square feet size of home X2 Lot Size square feet lot size Regression results and residual plot Predictor Constant SqrFeet Lotsize Coef 102610 156 228 1 1012 SE Coef 12531 4 825 0 2940 T 8 19 32 38 3 75 P 0 000 0 000 0 000 Interpretation There s a strong pattern of nonconstant variance related to the mean Fit Two possibilities for weights 1 Assume variance is proportional to mean so use wi 1 fits 2 Assume standard deviation is proportional to mean so use wi 1 fits 2 Results after using possibility 1 residual polot on other side Predictor Constant SqrFeet Lotsize Coef 92713 153 264 0 9699 SE Coef 10565 4 731 0 2631 T 8 78 32 39 3 69 P 0 000 0 000 0 000 2 Results after using possibility 2 Predictor Constant SqrFeet Lotsize Coef 76556 147 404 0 7808 SE Coef 9010 4 657 0 2358 T 8 50 31 65 3 31 P 0 000 0 000 0 001 The second set of weights looks better based on plot of standardized residuals
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