Statistics 350 Lecture 10

Today Last Day: Start Chapter 3 Today: Section 3.8 Homework #3: Chapter 2 Problems (page 89-99): 13, 16,55, 56 Due: February 7 Read Sections Mid-Term next Friday…..Sections ; ; (READ)

Overview of Remedial Measures for Model Violations Suppose you do a scatter plot of Y vs. X, and decide to attempt to regress Y on X Next, you test the simple linear regression model assumption using the plots discussed last day If you decide that model (2.1) is not appropriate, then there are three options: 1 2 3

Overview of Remedial Measures for Model Violations If there is a problem with non-linearity of the regression function: If there is a problem with non-constant error variance:

Overview of Remedial Measures for Model Violations If there is a problem with lack of independence among errors: If there is a problem with non-normality of error terms:

Overview of Remedial Measures for Model Violations If there are omitted explanatory variables: If there are outliers: What to do with outliers once you find them is the tricky part. Outliers are the most interesting observations in your data, and every effort should be made to determine why they occurred. This isn't always possible. General recommendations:

Transformations Sometimes simple transformations of X and/or Y may make the simple linear regression model appropriate for the transformed data Especially when:

Transformations If the problem is non-linearity of the regression function, a transformation of X often helps This is particularly true when the distribution of the error terms is close to normal and the variance appears to be approximately constant

Transformations Note:

Transformations Transforming X will not help address non-normality of the errors More helpful to attempt to transform Y Let Y`=h(Y) for some function h( ), and perform a simple linear regression using the Y`…must still do usual model validation for the new fitted model

Transformations Transformation of Y affects the shape of the regression function, the variance of the errors and the distribution of the error terms Is a good idea when all these violation appear On the other hand, when only one of these violations is evident, transforming Y can cause problems with the other two For example, if the straight line seems to fit well, but the variances of residuals are changing, the transformation of Y will create curvature in the regression function, causing a bad fit.

Transformations Suppose true linear response is on the log scale: This implies that: In these situation, variance tends to increase with the mean because of the

Transformations Nothing stops you from transforming both sides Do this when:

Transformations Common transformations: