We can determine the conformity of our data the MR linearity assumption via direct examination of residual scores. Recall that a residual or "error" score is equal to (Y - Y') and that we would expect this to be approximately zero once the linear regression determined. Thus, a plot of residual scores against [Yacute] that shows a peculiar pattern would suggest the violation of linearity. Two of the most common type of nonlinear relationships are represented in this way:

These cases can be modeled through polynomial regression. Polynomial regression simply adds terms to the original equation to account for nonlinear relationships. Squaring the original variable accounts for a quadratic trend, a cubed term accounts for the cubic trend, and so on. After the linear component, each term adds another "bend" in the prediction line. As with traditional MR, polynomial regression can be interpreted by looking at R Squared and changes in R Squared.

A couple notes of caution: in order to look at any higher order effect (such as the cubic), all lower order effects (the quadratic and the linear) must be placed into the equation; a degree of freedom is lost for each additional term added into the equation; and lastly, the N:k Ratio changes for each added term.

Recall that trend tests in ANOVA were accomplished with subjects grouped into discrete categories. With polynomial regression, the subjects are not grouped, but rather each individual has a unique score on a continuous variable. Therefore, MR trend information is much more complete than information available with typical ANOVA.