Regression lines give us useful information about the data they are collected from. They show how one variable changes on average with another, and they can be used to find out what one variable is likely to be when we know the other - provided that we ask this question within the limits of the scatter diagram. To project the line at either end - to extrapolate - is always risky because the relationship between x and y may change or some kind of cut off point may exist. For instance, a regression line might be drawn relating the chronological age of some children to their bone age, and it might be a straight line between, say, the ages of 5 and 10 years, but to project it up to the age of 30 would clearly lead to error. Computer packages will often produce the intercept from a regression equation, with no warning that it may be totally meaningless. Consider a regression of blood pressure against age in middle aged men. The regression coefficient is often positive, indicating that blood pressure increases with age. The intercept is often close to zero, but it would be wrong to conclude that this is a reliable estimate of the blood pressure in newly born male infants!
Plots of residuals, , similar to the ones discussed in Simple Linear Regression Analysis for simple linear regression, are used to check the adequacy of a fitted multiple linear regression model. The residuals are expected to be normally distributed with a mean of zero and a constant variance of . In addition, they should not show any patterns or trends when plotted against any variable or in a time or run-order sequence. Residual plots may also be obtained using standardized and studentized residuals. Standardized residuals, , are obtained using the following equation: