1. Regression Analysis Week 8 DIAGNOSTIC AND REMEDIAL MEASURES Residuals The main purpose examining residuals Diagnostic for Residuals Test involving residuals

2. Residuals The observed error:  e i = Y i – Ŷ For regression model, the true error ε i are assumed to be independent normal random variables, with mean 0 and variance σ 2. If the model is appropriate for the data, the e i should then reflect the properties assumed for the ε i.

3. Residuals (2) Properties of Residuals Mean Variance The residuals ei are not independent random variables as they involve the fitted values Ŷi which are based on the sample estimates b o, b 1, b 2,..., b p-1. X’e = 0 and Ŷ’e = 0

4. Residuals (3) Standardized Residuals: This residuals are useful in identifying outlying observations. There are still other measures based on residuals (see ch 11)

5. The main purpose in examining residuals To identify whether 1. The regression function is not linear 2. The error terms do not have constant variance 3. The error terms are not independent 4. The error terms are not normally distributed 5. The model fits all but one or a few outlier observations 6. One or several important independent variables have been omitted from the model

6. Diagnostics Look at the distribution of each variable Look at the relationship between pairs of variables Plot the residuals versus ◦Each explanatory variable ◦Time ◦Fitted values ◦Omitted variables

7. Diagnostics (2) Are the residuals approximately normal? ◦Look at a histogram, box plots, stem and leaf plots or dot plots ◦Normal quantile plot Is the variance constant? ◦Plot the squared residuals vs anything that might be related to the variance

8. Scatter Plot Matrix

9. Remedial measures Transformations such as Box-Cox Analyze without outliers More in NKNW Ch 11

10. Tests Involving Residuals Tests for Randomness Tests for Constancy of Variance Tests for Outliers Tests for Normality More in NKNW Ch 11