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

2. The observed error: • ei = Yi – Ŷ • For regression model, the true error εi are assumed to be independent normal random variables, with mean 0 and variance σ2. If the model is appropriate for the data, the ei should then reflect the properties assumed for the εi. Residuals

3. 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 bo, b1, b2, ..., bp-1. • X’e = 0 and Ŷ’e = 0 Residuals (2)

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

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

6. 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 Diagnostics

7. 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 Diagnostics (2)

8. Scatter Plot Matrix

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

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