Assumptions of Least Squares Regression

If we concentrate on the actual observations this becomes y_i =β₀+β₁x_i + e_i, i=1, .., n

The standard assumption for least squares regression is e_i ~ N(0,s) , i=1, .., n

This translates into three assumptions:

1) The model is good (that is, the relationship is linear and not, say, quadratic, exponential or something else)
2) The residuals have a normal distribution
3) The residuals have equal variance (are homoscadastic)

We can check these assumptions using two graphs:

• Residual vs. Fits plot: this is just what it says, a scatterplot of the residuals (on y-axis) vs. the fitted values.
• Normal plot of residuals, again, just what it says

Both of these graphs can be done by MINITAB as part of the regression command.

How to check the Assumptions of Least Squares Regression

1) Good Model
For this assumption draw the Residuals vs. Fits plot and check for any pattern
Example:

2) Residuals have a Normal Distribution
For this assumption draw the Normal Probability plot and see whether the dots form a straight line
Example:

3) Residuals have Equal Variance
For this assumption again draw the Residuals vs. Fits plot and check whether the variance (or spread) of the dots changes as you go along the x axis.
Example:

Example Let's check the assumptions for the Alcohol vs. Tobacco dataset, without Northern Ireland
Stat > Regression > Regression, Response= Tobacco, Predictors= Alcohol, Storage: Residuals and Fits
Graph > Scatterplot, With Regression > Y variables= RESI1, X variables= FITS1
Graph > Probability Plot > Simple, Variable= RESI1

Instead of doing them ourselves we can just let MINITAB draw the graph as part of the regression command:
Stat > Regression > Regression, Response= Tobacco, Predictors= Alcohol, Graphs > Normal Plot and Residuals vs. Fits Plot

Note a final decision on whether the assumptions are justified is always made based on the Residual vs. Fits Plot and the Normal plot of Residuals

For more on this see page 540 of the textbook.