Lecture 15 Part 2_Leverage etc

# Lecture 15 Part 2_Leverage etc - Lecture 15 Part 2 Leverage...

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1 Lecture 15 Part 2, Leverage and Influence STAT 102 • Outliers, Leverage and influence in simple linear regression (Review) • Outliers and influential observations in multiple linear regression; Leverage plots Notes re the text: This topic is covered in Section 6.7. The emphasis there is on the use of quantitative measures like “DFITS” and “Cook’s D” to identify leverage. We do not recommend these methods. Instead we suggest in these overheads a more graphical perspective, based on Leverage plots.

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2 Outliers and influential points in simple regression Does the age at which a child begins to talk predict a score on a test of mental ability at a later age? gesell.JMP contains data on the age at first word (x) and their Gesell Adaptive score (y), an ability test taken at a later age. Child 18 is an outlier in the x direction, so it is a leverage point and potentially influential. Child 19 is a regression outlier. 50 60 70 80 90 100 110 120 130 Score 18 19 5 10 15 20 25 30 35 40 45 Age
3 Outliers in Simple Linear Regression Three types of outliers in scatterplots: – Outlier in x direction – Outlier in y direction – Outlier from regression line of scatterplot (residual has large magnitude) Several possibilities need to be investigated when an outlier is observed: – There was an error in recording the value. – The point is not representative of the population of interest. – The observation is valid. Identify regression outliers from the scatterplot and residual plot

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4 Leverage and Influential Points An observation has high leverage if it is an outlier in the x direction. An observation is influential if removing it would markedly change the slope of the least squares line. Observations that have high leverage and moderate to large residuals tend to be influential. Observations with little or no leverage ( ) cannot be influential xx
5 Outliers and influential points in simple linear regression To assess whether a point is influential, fit the least squares line with and without the point and see how much of a difference it makes. [ In JMP, exclude the corresponding row and re-fit the line. ] Child 18 has high leverage, and turns out to be influential; Child 19 has low leverage and hence turns out to not be influential. 50 60 70 80 90 100 110 120 130 Score 18 19 5 10 15 20 25 30 35 40 45 Age

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6 Full data : Rsquare=0.41 Score = 109.87 - 1.127 Age W/O 19 : Rsquare=0.57 Score = 109.30 - 1.193Age # 19: possible outlier, but not influential W/O 18 : Rsquare=0.11 Score = 105.63 - 0.779Age #18: High leverage and influential.
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Lecture 15 Part 2_Leverage etc - Lecture 15 Part 2 Leverage...

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