Posh word meaning “at right angles to”, “perpendicular to”, “independent of”
Details #
Can come up in a number of ways but the main ones are in exploratory factor analysis (“orthogonal rotation”) and in analysis of variance (ANOVA; “orthogonal contrasts”). In both it is about one thing being full independent of another.
In exploratory factor analysis (EFA) it comes up as, as with principal component analysis (PCA) the maths starts up by reorganising multivariate data, often answer s on items in a multi-item questionnaire into a set of fully independent factors (or components). To make this more understandable there are ways of rotating those factors/components. Orthogonal rotations preserve the independence of the factors/components whereas oblique rotations allow them to correlate. Orthogonal systems are nice and simple, like North/South and East/West on maps or the x and y axes of a conventional plot but where you might be expecting distinguishable but correlated dimensions, e.g. anxiety and depression when looking at data from the Hospital Anxiety and depression scales say, you might prefer an oblique rotation.
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. Orthogonal contrasts allow an ANOVA to separate these possible power relationships as they partition the data in a way that allows the statistician to estimate any linear and quadratic relationship between improvement and number of sessions attended. That might show that the linear relationship is for more improvement with longer durations of therapy but that the quadratic term adds a more rapid improvement across the shorter durations than the longer ones. 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.
Those are both pretty technical ideas and when I have time, I hope to put up some more extensive unpacking of both ideas in separate entries here.
Try also #
Analysis of variance (ANOVA)
Exploratory factor analysis (EFA)
Hospital Anxiety and Depression scales (HADS)
Principal component analysis (PCA)
Chapters #
Both uses of “orthogonal” go rather beyond the level we were aiming for in the OMbook but EFA and rotations would belong in chapter 3 and ANOVA in chapters 5, 7 and 8.
Online resources #
Not yet.
Dates #
First created 19.xii.23, cosmetic tweaks and expansion of the orthogonal contrasts section 4.viii.24.
