Motivational Examples Three Types of Unusual Observations Lecture 15 Outline: Motivational Examples Three Types of Unusual Observations 12/3/2018 ST3131, Lecture 15

(Problem 4.3, Page 116) Computer Repair Data (1) For the data on Page 27, n=14, p=1 (a) Fit a linear regression model relating Minutes to Units 12/3/2018 ST3131, Lecture 15

Check each of the standard regression assumptions and indicate which assumption(s) seems to be violated. Assumptions about the form of the model Assumptions about the measurement errors Assumptions about the predictor variables Assumptions about the observations 12/3/2018 ST3131, Lecture 15

(1) For the data on Page 117, n=24, p=1 (a) Fit a linear regression model relating Minutes to Units 12/3/2018 ST3131, Lecture 15

Check each of the standard regression assumptions and indicate which assumption(s) seems to be violated. Assumptions about the form of the model Assumptions about the measurement errors Assumptions about the predictor variables Assumptions about the observations 12/3/2018 ST3131, Lecture 15

Leverage, Influence, and Outliers Assumption about the observations requires that each observation should play a similar role in the regression fit. That is, it requires that a fit is not overly determined by one or few observations. If there are such points, it is necessary to find them out. High Leverage points, Influence points and outliers are such points. High Leverage points /Outliers in the Predictor variables Pii are called the leverage of observation Xi . It reflects how far Xi is from the sample mean of the predictor variables. The Function is called Potential Function of observation Xi . Observations with Larger are called High Leverage points. Points with greater than are usually regarded as high leverage points. High Leverage points are also called outliers in the Predictor variables (in X-direction). 12/3/2018 ST3131, Lecture 15

New York Rivers Data (Page 10) A linear regression fit relating Nitrogen to Com./Indus. A plot of the leverage values (index plot, dot plot, or box plot) will reveal Points with high leverage observations. From the raw data , It can be found that Observation 5 (Hackensack river) is an urban river close To New York City while other rivers are in the countryside (in the rural area) 12/3/2018 ST3131, Lecture 15

Outliers in the Response variable(in Y-direction) Observations with large standardized residuals are outliers in the Responses variable. These outliers’ response values are far from the Sample center of the response variable (in Y-direction), so are their Standardized residuals from 0. Observations with absolute standardized Residuals greater than 2 or 3 are usually called outliers A plot of the standardized residuals (index plot, dot plot, or box plot or plot against fitted values) will usually reveal outliers in the Y-direction. 12/3/2018 ST3131, Lecture 15

Influential Points A point is an Influential point if its deletion, singly or in combination With others (2 or 3) , causes substantial changes in the fitted model ( Estimation, fitted values, t-test, etc) 12/3/2018 ST3131, Lecture 15

After-class Questions: How to measure the influence of an observation? How to detect an influential observation? Are high-leverage points influential? Are outliers influential? 12/3/2018 ST3131, Lecture 15