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BCOR 1020 Business Statistics Lecture 26 – April 24, 2007

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Overview Chapter 12 – Linear Regression –Violations of Assumptions –Unusual Observations –Example(s)

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Chapter 12 – Violation of Assumptions Three Important Assumptions: 1.The errors are normally distributed. 2.The errors have constant variance (i.e., they are homoscedastic) 3.The errors are independent (i.e., they are nonautocorrelated). The error i is unobservable. The residuals e i from the fitted regression give clues about the violation of these assumptions.

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Chapter 12 – Violation of Assumptions Histogram of Residuals: Check for non-normality by creating histograms of the residuals or standardized residuals (each residual is divided by its standard error). Standardized residuals range between -3 and +3 unless there are outliers.

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Chapter 12 – Violation of Assumptions Normal Probability Plot: The Normal Probability Plot tests the assumption H 0 : Errors are normally distributed H 1 : Errors are not normally distributed If H 0 is true, the residual probability plot should be linear.

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Chapter 12 – Violation of Assumptions Tests for Heteroscedasticity: Plot the residuals against X. Ideally, there is no pattern in the residuals moving from left to right.

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Chapter 12 – Violation of Assumptions Tests for Heteroscedasticity: The “fan-out” pattern of increasing residual variance is the most common pattern indicating heteroscedasticity.

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Chapter 12 – Violation of Assumptions Example: Consider the plots of the residuals for the dataset Ship Cost… (overhead & handout) Let’s quickly assess whether the regression assumptions are reasonable.

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Chapter 12 – Unusual Observations Standardized Residuals: Excel Use Excel’s Tools > Data Analysis > Regression Standardized Residuals: MegaStat MegaStat give same general output as Excel.

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Chapter 12 – Unusual Observations Studentized Deleted Residuals: Studentized deleted residuals are another way to identify unusual observations. A studentized deleted residual whose absolute value is 2 or more may be considered unusual. A studentized deleted residual whose absolute value is 3 or more is an outlier.

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Chapter 12 – Other Regression Problems Outliers: To fix the problem, - delete the data - delete the data - formulate a multiple regression model that includes the lurking variable Outliers may be caused by - an error in recording data - impossible data - an observation that has been influenced by an unspecified “lurking” variable that should have been controlled but wasn’t.

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Example Consider Data Set B on p.545 of your text in which bivariate data is compiled to determine whether there is a relationship Number of Employees (X) and Revenue (Y) for n = 24 large automotive companies in1999. A scatter plot of the data follows: Note the high R 2 value – 86% of the total variation in y is accounted for by the regression line. The correlation, r =.9261 is significant. Now let’s generate the MegaStat regression output…

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Example – Residuals… Studentized Deleted ObservationRevenuePredicted ResidualLeverageResidual 135.9037.36-1.460.043-0.082-0.080 2154.60135.3919.210.2231.1981.211 312.8027.15-14.350.050-0.810-0.803 413.8022.88-9.080.054-0.514-0.505 551.0068.03-17.030.052-0.962-0.960 6144.40106.0438.360.1232.2532.510 710.606.883.720.0770.2130.208 8161.30181.88-20.580.461-1.542-1.595 948.7035.0213.680.0450.7700.763 1012.709.503.200.0720.1830.179 1112.6027.94-15.340.049-0.866-0.860 129.1020.78-11.680.056-0.661-0.653 1313.8020.35-6.550.057-0.371-0.364 1416.1010.545.560.0710.3170.311 1527.508.9518.550.0731.0601.063 1651.5040.8410.660.0420.5990.590 1737.5048.52-11.020.042-0.619-0.610 1841.4042.97-1.570.042-0.088-0.086 1928.6058.58-29.980.045-1.687-1.767 2011.405.056.350.0800.3640.357 2199.7056.8742.830.0442.4092.744 2211.9024.59-12.690.052-0.717-0.709 2376.3091.62-15.320.089-0.883-0.878 2426.8022.244.560.0550.2580.252 Observations #6 and #21 have unusually large studentized residuals.

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Clickers Based on the portion of the regression output below, how much of the total variation in the y-variable is accounted for by the regression line? (A) 0% (B) 18% (C) 86% (D) 93% Regression Analysis r²0.858n24 r0.926k1 Std. Error18.182Dep. Var.Revenue

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Clickers Based on the regression output below, What can we conclude about the hypothesis test At the 10% level of significance? (A) Reject H 0 in favor of H 1. (B) Fail to reject H 0 in favor of H 1. (C) Not enough information is given. Regression outputconfidence interval variables coefficientsstd. error t (df=22)p-value95% lower95% upper Intercept0.81535.4168 0.151.8817-10.418512.0490 Employees0.30480.0265 11.5168.74E-110.24990.3597

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Clickers The graph below would best be used to check which regression assumption? (A) The errors are normal. (B) The errors have constant variance. (C) The errors are independent.

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FCQ…

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