Orthogonal contrasts

This is pretty technical but you might come across it in papers using analysis of variance (ANOVA) to look at the relationships between two (or more) continuous variables.

Details #

Tweaking what I wrote for “othogonal”:
In ANOVA methods you might be looking at how some variables appear to relate to another. A typical use of “orthogonal contrasts” might be in looking at how the number of sessions attended might relate to how much clients’ scores improve on a change measure. The number of sessions attended is clearly one variable, however, improvement might relate in some non-linear way to the number of sessions and a question that has interested therapy researchers is how much the change might be related to some power of the number of sessions: to the linear count, or to the square of the count say.

One catch here is “multicollinearity”: that two predictor variables, such as the linear count of sessions and square of that count are not independent of each other. In fact as the count must be zero or more the linear and squared count are always strongly positively correlated and so you have “multicollinearity” which means that is difficult or impossible using those two variables to separate their effects on, say, improvement on some change measure.

Using “orthogonal contrasts” removes the multicollinearity problem by creating two different variables, still representing the linear and the squared session counts, that are “orthogonal”, i.e. uncorrelated. These are transforms of the linear and the squared counts that allow an ANOVA to separate these possible power relationships, they transform the raw count data in a way that allows the statistician to estimate any linear and quadratic relationship between improvement and number of sessions attended.

In principle orthogonal contrasts can be extended to cubic or even higher powers of a variable but I’ve never seen that used in our field and can’t think when such “higher polynomial” models would be likely to be plausible or potentially useful models to test.