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PPT9 Residual Analysis 1 F2010

# PPT9 Residual Analysis 1 F2010 - McGill University Advanced...

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Residual Analysis Using residuals to detect departure from assumptions
Residual Plots Plots of the residuals in a regression analysis can be used to detect departures from the assumption of: 1.Normality of the error terms 2.Constant variance 3.Independence of the error terms

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Definition of Residual ( 29 0 1 1 Error term in a true multiple regression model ( ) ... i i i k k y E y y x x ε β = - = - + + + ( 29 0 1 1 Residual term in an estimated mutiple regression model ˆ ... i i i i k k e y y y b b x b x = - = - + + +
Example 8.1 Page 251 ATHLETE FAT INTAKE CHOLESTEROL 1 1290 1182 2 1350 1172 3 1470 1264 4 1600 1493 5 1710 1571 6 1840 1711 7 1980 1804 8 2230 1840 9 2400 1956 10 2930 1954

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Calculating Residuals The regression equation is CHOLESTEROL = 578.9 + 0.5403 FAT INTAKE X 1 = 1290 = 578.9 + 0.5403(1290) = 1275.92 1 ˆ y 1 1 ˆ - 1182 -1275.92 -93.92 i e y y = = = Residual
Residuals: Linear Model Fitted Value Residual 2200 2000 1800 1600 1400 1200 200 100 0 -100 -200 Residuals Versus the Fitted Values (response is CHOLESTEROL) e 1 = -93.92

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??Pattern?? Do you see a pattern in the residual plot? Residuals are positive for athletes with intermediate levels of fat intake. Residuals are negative for athletes with low or high levels of fat intake. This suggests the use of a second order (Quadratic) model.
Quadratic Model The regression equation is: CHOLESTEROL = - 1216.14 + 2.3989 FAT INTAKE - 0.000450 FAT^2 X 1 = 1290 2 ˆ 1216.14 2.3989 .00045 y x x = - + - 2 1 ˆ 1216.14 2.3989(1290) .00045(1290) 1129.56 y = - + - = 1 1 1 ˆ 1182 1129.56 52.44 e y y = - = - = Residual

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Residuals: Quadratic Model Fitted Value Residual 2000 1900 1800 1700 1600 1500 1400 1300 1200 1100 50 25 0 -25 -50 -75 Residuals Versus the Fitted Values (response is CHOLESTEROL) e 1 = 52.44
Detecting Lack of Fit 1. Plot residuals e i on the vertical axis, against each of the independent variables x 1 , x 2 , … x k on the horizontal axis. 2. Plot the residuals on the vertical axis against the predicted value, , on the horizontal axis. 3. In each plot look for trends, dramatic changes in variability, and/or more than 5% of residuals that lie outside 2 std. dev. of 0. Any of these patterns indicates a problem with model fit. ˆ y

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Lack of Fit Tests Minitab Help Screen
Regression: Chol (Y) on Fat (X)

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Options: Lack of Fit Tests – Data Subsetting
Regression Analysis: Chol (y) versus Fat (x) The regression equation is Chol (y) = 579 + 0.540 Fat (x) Predictor Coef SE Coef T P Constant 578.9 167.0 3.47 0.008 Fat (x) 0.54030 0.08593 6.29 0.000 S = 133.438 R-Sq = 83.2% R-Sq(adj) = 81.1% Lack of fit test Possible curvature in variable Fat (x) (P-Value = 0.002 ) Possible lack of fit at outer X-values (P-Value = 0.039) Overall lack of fit test is significant at P = 0.002

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Fitted Line Plot
Fitted Line Plot: Linear plot of Chol (Y) vs Fat (X)

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Linear Plot: R 2 = 83.2%
Fitted Line Plot – Quadratic Model

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Quadratic Plot: R 2 = 97.7%
The Quadratic Model P-value < .001 Impressive!

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PPT9 Residual Analysis 1 F2010 - McGill University Advanced...

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