Section2.4posted_1

Section2.4posted_1 - ST OR 155 Section 2 T uesday Febr uar...

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STOR 155, Section 2 Tuesday, February 2, 2010 Section 2.4
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Section 2.4: Cautions about correlation and regression Residuals Outliers and influential observations Lurking variables Correlations based on averaged data Restricted-range problem (omitted)
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Residuals In a regression, there are three numbers connected with every individual: x, the value of the explanatory variable, y, the observed value of the response variable, and ŷ , the predicted value of the response variable. The observed and predicted values y and ŷ are almost never equal. The difference y – ŷ is called the residual for that individual (or for that x ). residual for a value x in the dataset = y – ŷ.
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x = parent s’ averag e height y = daughte r’s height = ŷ predicte d daughte r’s height resid ual 63.5 60 61.90 - 1.90 67.0 66 65.27 + 0.73 65.5 65 63.83 + 1.17 69.5 66 67.68 - 1.68 67.5 67 65.75 + 1.25 65.5 63 63.82 - 0.82 70.0 69 68.16 + 0.84 63.0 63 61.42 + 1.58 63.0 61 61.42 - 0.42 67.5 65 65.75 - 0.75 Residuals: Example Parents’ heights as predictor of daughter’s height for n = 10 women. Regression line is For x = 63.5 : 61.90 = 0.69 + 0.96 63.5. Residual for first x is y – ŷ = 60 – 61.90 = −1.90. ŷ = 0.69 + 0.96 x
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For x = 63.5 : 61.90 = 0.69 + 0.96 63.5. Residual for first x is y – ŷ = 60 – 61.90 = −1.90. Residuals The residuals are the vertical distances that regression wants to minimize.
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Residuals The residuals appear as vertical distances on the plot of the regression line. Or, we can plot them by themselves. Why would we want to plot them by themselves?
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Residual plots r 2 is high and the fit looks good in the scatterplot. But the residual plot does a better job of showing that the fit is much better for values of x near the center than at the two ends. Scatterplot and regression Residual plot (Ex. 2.20, p. 128 shows this also.)
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Outliers and influential observations Earlier we saw the effect that outliers can have on the correlation .
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