Homework07Solns - Stat 512 2 Solutions to Homework#7 Dr...

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Stat 512 – 2 Solutions to Homework #7 Dr. Simonsen Due October 19, 2005 by 4:30pm. 1. For this problem, the idea is to demonstrate the similarity between regression with dummy variables and ANOVA. To do this run the following SAS code. (If you download this assignment from the Homework page of the class website you can copy and paste the code into SAS.) (a) Compare the ANOVA table and parameter results from the GLM analysis and Parameterization #1. What do the coefficients associated with X 1 and X 2 (i.e. b 1 and b 2 ) estimate in terms of treatment means? What constraint system does this parameterization correspond to? The ANOVA tables from the two analyses are identical. The parameter estimates , , and ˆ μ 1 ˆ τ 2 ˆ τ from the GLM analysis are the same as the parameter estimates b 0 , b 1 , and b 2 from the regression model. The fourth parameter estimate which is set to zero in the GLM analysis does not appear in the regression analysis, since it is assumed equal to zero in that model. Parameterization #1 sets 3 ˆ τ μ 1 = β 0 + β 1 = μ + τ 1 , μ 2 = β 0 + β 2 = μ + τ 2 , and μ 3 = β 0 = μ . This system of equations can be solved for the β ’s to obtain β 0 = μ = μ 3 , β 1 = τ 1 = μ 1 μ 3 , and β 2 = τ 2 = μ 2 μ 3 . The coefficient associated with X is b 1 = 1 ˆ τ , which thus is an estimate for μ 1 μ 3 . The coefficient associated with X 2 is b 2 = , which thus is an estimate for μ 2 ˆ τ 2 μ 3 . This corresponds to the constraint system τ 3 = 0. The GLM Procedure Dependent Variable: response Sum of Source DF Squares Mean Square F Value Pr > F Model 2 38.88888889 19.44444444 15.91 0.0040 Error 6 7.33333333 1.22222222 Corrected Total 8 46.22222222 Standard Parameter Estimate Error t Value Pr > |t| Intercept 26.66666667 B 0.63828474 41.78 <.0001 trt 1 -5.00000000 B 0.90267093 -5.54 0.0015 trt 2 -1.66666667 B 0.90267093 -1.85 0.1144 trt 3 0.00000000 B . . . The REG Procedure Model: MODEL1 Dependent Variable: response Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 2 38.88889 19.44444 15.91 0.0040
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Error 6 7.33333 1.22222 Corrected Total 8 46.22222 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 26.66667 0.63828 41.78 <.0001 x1 1 -5.00000 0.90267 -5.54 0.0015 x2 1 -1.66667 0.90267 -1.85 0.1144 (b) Compare the ANOVA table and parameter results from the GLM analysis and Parameterization #2. What do the coefficients associated with X 1 and X 2 (i.e. b
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