Question 4 Some researchers (Daniel, 1999) were interested in comparing the effectiveness of three treatments for severe depression. For the sake of simplicity, we denote the three treatments A, B, and C. The researchers collected the data on a random sample of n= 36 severely depressed individuals. The variables recorded in the data set are as follows: Y = measure of the effectiveness of the treatment for individual (continuous variable) I1 = age (in years) of individual (continous variable) 12 = weight (in kgs) of individual (continous variable) Iz = factor predictor with three levels - denoted A, B and C - for three treatments, respectively; The following sequence of models was fitted to these data in R. 1 > fit. 1 fit.2 fit.3 fit. 4 fit.5-1m (y x1+x3) 6 > fit. 6 fit.7 fit.8 fit.9 fit. full anova (fit.1, fit. 2, fit.3,fit.4,fit.5, fit.6, fit.7, fit.8, fit.9, fit. full) 3 Analysis of Variance Table 2 4: Y 5: y + 7: 8: Y 9: y 4 5 Model 1: y x1 6 Model 2: X2 7 Model 3: x3 8 Model X1 + x2 9 Model x1 + x3 10 Model 6: y ~ x2 + x3 11 Model x1 x2 + x3 12 Model X1 + x2 + x3 + x1 :x2 + x2:x3 13 Model X1 + x2 + x3 + x1 :x2 + x2:x3 + x1: x3 14 Model 10: y x1 * x2 * x3 15 Res. Df RSS Df Sum of Sq F Pr (>F) 16 1 34 1970.6 17 2 34 3289.5 0 - 1318.96 18 3 33 4467.8 1 - 1178.31 194 33 1491.3 2976.49 20 5 32 1165.6 1 325.76 18.4986 0.0002457 *** 21 6 32 2760.0 0 - 1594.38 22 7 31 876.3 1 1883.67 106.9649 2.537e-10 *** 23 8 28 703.8 3 172.46 3.2644 0.0388420 * 24 9 26 432.5 2 271.29 7.7025 0.00 26056 ** 25 10 24 422.6 2 9.89 0.2808 0.7576070 26 --- OO NN w Model selection quantities were also computed: 1 Model Res. df SSE AIC BIC Rsq Adj.Rs 21 fit. full 24 422.6452 216.8321 237.4178 0.9217 0.8858 32 fit.9 26 432.5360 213.6649 231.0836 0.9198 0.8921 4 3 fit.8 28 703.8223 227.1918 241.4435 0.8695 0.8369 54 fit.7 31 876.2841 229.0817 238.5829 0.8376 0.8166 65 fit.6 32 2759.95 86 268.3838 276.3014 0.4884 0.4405 76 fit.5 32 1165.5747 237.35 18 245.2694 0.7840 0.7637 8 7 fit.4 33 1491.3391 244.2244 250.55 84 0.7236 0.7068 98 fit.3 33 4467.8333 283.7246 290.0587 0.1719 0.1217 10 9 fit.2 34 3289.5273 270.7029 275.4534 0.3903 0.3723 11 10 fit.1 34 1970.5682 252.2557 257.0062 0.6347 0.6240 In the output, Res.df is the residual degrees of freedom in the model, and the table also includes the residual sum of squares (RSS, which is also termed the sum of squares of the residuals, SSRes) AIC, BIC, R and Raj quantities. On the basis of the analyses above, identify the most appropriate model to repre- sent the variation in response, and write down precisely (in terms of B parameters) the form of the conditional expectation, E[Y:|X], for the selected model. Question 4 Some researchers (Daniel, 1999) were interested in comparing the effectiveness of three treatments for severe depression. For the sake of simplicity, we denote the three treatments A, B, and C. The researchers collected the data on a random sample of n= 36 severely depressed individuals. The variables recorded in the data set are as follows: Y = measure of the effectiveness of the treatment for individual (continuous variable) I1 = age (in years) of individual (continous variable) 12 = weight (in kgs) of individual (continous variable) Iz = factor predictor with three levels - denoted A, B and C - for three treatments, respectively; The following sequence of models was fitted to these data in R. 1 > fit. 1 fit.2 fit.3 fit. 4 fit.5-1m (y x1+x3) 6 > fit. 6 fit.7 fit.8 fit.9 fit. full anova (fit.1, fit. 2, fit.3,fit.4,fit.5, fit.6, fit.7, fit.8, fit.9, fit. full) 3 Analysis of Variance Table 2 4: Y 5: y + 7: 8: Y 9: y 4 5 Model 1: y x1 6 Model 2: X2 7 Model 3: x3 8 Model X1 + x2 9 Model x1 + x3 10 Model 6: y ~ x2 + x3 11 Model x1 x2 + x3 12 Model X1 + x2 + x3 + x1 :x2 + x2:x3 13 Model X1 + x2 + x3 + x1 :x2 + x2:x3 + x1: x3 14 Model 10: y x1 * x2 * x3 15 Res. Df RSS Df Sum of Sq F Pr (>F) 16 1 34 1970.6 17 2 34 3289.5 0 - 1318.96 18 3 33 4467.8 1 - 1178.31 194 33 1491.3 2976.49 20 5 32 1165.6 1 325.76 18.4986 0.0002457 *** 21 6 32 2760.0 0 - 1594.38 22 7 31 876.3 1 1883.67 106.9649 2.537e-10 *** 23 8 28 703.8 3 172.46 3.2644 0.0388420 * 24 9 26 432.5 2 271.29 7.7025 0.00 26056 ** 25 10 24 422.6 2 9.89 0.2808 0.7576070 26 --- OO NN w Model selection quantities were also computed: 1 Model Res. df SSE AIC BIC Rsq Adj.Rs 21 fit. full 24 422.6452 216.8321 237.4178 0.9217 0.8858 32 fit.9 26 432.5360 213.6649 231.0836 0.9198 0.8921 4 3 fit.8 28 703.8223 227.1918 241.4435 0.8695 0.8369 54 fit.7 31 876.2841 229.0817 238.5829 0.8376 0.8166 65 fit.6 32 2759.95 86 268.3838 276.3014 0.4884 0.4405 76 fit.5 32 1165.5747 237.35 18 245.2694 0.7840 0.7637 8 7 fit.4 33 1491.3391 244.2244 250.55 84 0.7236 0.7068 98 fit.3 33 4467.8333 283.7246 290.0587 0.1719 0.1217 10 9 fit.2 34 3289.5273 270.7029 275.4534 0.3903 0.3723 11 10 fit.1 34 1970.5682 252.2557 257.0062 0.6347 0.6240 In the output, Res.df is the residual degrees of freedom in the model, and the table also includes the residual sum of squares (RSS, which is also termed the sum of squares of the residuals, SSRes) AIC, BIC, R and Raj quantities. On the basis of the analyses above, identify the most appropriate model to repre- sent the variation in response, and write down precisely (in terms of B parameters) the form of the conditional expectation, E[Y:|X], for the selected model