Lecture 8 - Residual Analysis - Definition

Residual Analysis
In Lecture 7 we discussed how to use the method of least-square to perform simple linear regression on a set of data. We also discussed the four assumptions we make about our data in order to use the method of least-squares for the regression:
1. Linearity
2. Independence of errors
3. Normality of error
4. Equal variance of errors

The error is also known as the residual and is the difference between the observed Y_i value, for any particular X_i, and the value for Y_i predicted by our regression model which is usually symbolized by Ŷ_i (read "y hat sub i"). The residual is symbolized by the greek letter epsilon (lower case) - ε_i.

ε_i = Y_i - Ŷ_i

We perform a four-part residual analysis on our data to evaluate whether each of the four assumptions hold and, based on the outcome, we can determine whether our linear regression model is the correct model.

It's called a residual analysis because 3 of the 4 assumptions (independence, normality and equality of variance) directly relate to the errors (the residuals) and the other assumption (linearity) is tested by assessing the residuals.