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 Coef SE Coef T PConstant 3.1959 0.3562 8.97 0.000x1_price -0.3336 0.1523 -2.19 0.036x3_Discount 0.30808 0.06412 4.80 0.000x3x4 0.17623 0.06597 2.67 0.012Source DF SS MS F PRegression 3 1.76282 0.58761 27.51 0.000Residual Error 32 0.68343 0.02136Total 35 2.44626Interpretation: 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 x3_Discount N N* VarianceRESI1 0 15 0 0.0105 1 21 0 0.0268Weighting for a weighted least squares: For weights, we use 2iˆ1. In this case the two variances given just above estimated the 2i. This leads to these weights:Variable x3_Discount Weightwts 0 95.238 1 37.313Weighted analysis using weights in wtsPredictor Coef SE Coef T PConstant 3.1743 0.3567 8.90 0.000x1_price -0.3243 0.1529 -2.12 0.042x3_Discount 0.30834 0.06577 4.69 0.000x3x4 0.17585 0.07612 2.31 0.027Source DF SS MS F PRegression 3 98.118 32.706 30.75 0.000Residual Error 32 34.031 1.063Total 35 132.149IMPORTANT: 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 Coef SE Coef T PConstant -102610 12531 -8.19 0.000SqrFeet 156.228 4.825 32.38 0.000Lotsize 1.1012 0.2940 3.75 0.000Interpretation: 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)2Results after using possibility 1 (residual polot on other side):Predictor Coef SE Coef T PConstant -92713 10565 -8.78 0.000SqrFeet 153.264 4.731 32.39 0.000Lotsize 0.9699 0.2631 3.69 0.0002. Results after using possibility 2Predictor Coef SE Coef T PConstant -76556 9010 -8.50 0.000SqrFeet 147.404 4.657 31.65 0.000Lotsize 0.7808 0.2358 3.31 0.001The second set of weights looks better (based on plot of standardized
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