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Analysis of Regression Models and Residual Diagnostics in Various Real-World Applications

Study on regression models and diagnostic techniques in statistics.

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    Analysis of Regression Models and Residual Diagnostics in Various Real - World Applications 8.4 Elasticity of moissanite. Moissanite is popular abrasive material because of its extreme hardness. Another important property of moissanite is elasticity. The elastic properties of the material were investigated in the Journal of Applied Physics (September 1993). A diamond anvil cell was used to compress a mixture of moissanite, sodium chloride, and gold in a ratio of 33.99:1 by volume. The compressed volume, y, of the mixture (relative to the zero - pressure volume) was measured at each of 11 different pressures (GPa). The results are displayed in the table (p.397). A MINITAB printout for the straight - line regression model ๐ธ ( ๐‘ฆ ) = ๐›ฝ 0 + ๐›ฝ 1 ๐‘ฅ and a MINITAB residual plot are displayed at left. MOISSANITE COMPRESSED VOLUME y, % PRESSURE x, GPa 100 0 96 9.4 93.8 15.8 90.2 30.4 87.7 41.6 86.2 46.9 85.2 51.6 83.3 60.1 82.9 62.6 82.9 62.6 81.7 68.4 a. Calculate the regression residuals. The regression residuals are calculated and given below. RESI1 1.38508 - 0.21233 - 0.77653 - 0.64485 - 0.28219 - 0.42754 - 0.22625 0.04630 0.28529

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    0.28529 0.56774 b. Plot the residuals against x. Do you detect a trend? The plot of residuals against the x values is given below. From the above graph a trend can be observed in the residuals. c. Propose an alternative model based on the plot part b. From the above plot it was observed that there is trend in the residuals. Therefore a quadratic model is considered for the above data. d. Fit and analyze the model you proposed in part c. A quadratic model is fitted for the above data and the regression coefficients are given below. Regression Analysis: COMPRESSED VOLUME y, % versus PRESSURE x, GPa, x^2 Analysis of Variance

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    Source DF Adj SS Adj MS F - Value P - Value Regression 2 370.248 185.124 1694.30 0.000 PRESSURE x, GPa 1 39.322 39.322 359.88 0.000 x^2 1 2.909 2.909 26.62 0.001 Error 8 0.874 0.109 Lack - of - Fit 7 0.874 0.125 * * Pure Error 1 0.000 0.000 Total 10 371.122 Model Summary S R - sq R - sq(adj) R - sq(pred) 0.330549 99.76% 99.71% 99.13% Coefficients Term Coef SE Coef T - Value P - Value VIF Constant 99.503 0.268 370.88 0.000 PRESSURE x, GPa - 0.3473 0.0183 - 18.97 0.000 17.25 x^2 0.001311 0.000254 5.16 0.001 17.25 Regression Equation COMPRESSED VOLUME y, % = 99.503 - 0.3473 PRESSURE x, GPa + 0.001311 x^2 From the above results it can be observed that the model is significant and is able to explain 99.76% variation in compressed volume. The regression coefficients of the variables pressure and square of pressure are significant. The residual plot against the observations is also given below; it can be observed that the plot has improved.

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    8.12 Fair market value of Hawaiian properties. Prior to 1980, private homeowners in Hawaii had to lease the land their homes were built on because the law (dating back to the islandsโ€™ feudal period) required that land be owned only by the big estates. After 1980, however, a new law instituted condemnation proceedings so that citizens could buy their own land. To comply with the 1980 law, one large Hawaiian estate wanted to use regression analysis to estimate the fair market value of its land. Its first proposal was the quadratic model ๐ธ ( ๐‘ฆ ) = ๐›ฝ 0 + ๐›ฝ 1 ๐‘ฅ + ๐›ฝ 2 ๐‘ฅ 2 HAWAII PROPERTY LEASED FEE VALUE y, thousands of dollars SIZE x, thousands 1 70.7 13.5 2 52.7 9.6 3 87.6 17.6 4 43.2 7.9 5 103.8 11.5 6 45.1 8.2

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