I'm an undergrad psychology student writing a paper on personality and I have come across the concept of orthogonality numerous times. I understand that it refers to the statistical independence of each variable, my question is what is the importance of this? Is orthogonality needed for analysis or is there another reason?
You just walked into a minefield! Seriously, I'm not a statistical maven, but there are major theoretical issues here, and it's good to see an undergraduate noticing that assumptions are being made in the development of our theoretical models.
As best I understand the history here, it begins with the idea that "carving nature at the joints" involves identifying the underlying dimensions of the domain being studied - analogous to Cartesian geometric coordinates (i.e., length/width/breadth or X/Y/Z). If you buy this idea, then it becomes obvious that the goal of a personality model is to identify its basic dimensions (orthogonal traits). Some methods of rotating the matrix of a principal components analysis require the factors to be orthogonal. This includes the near-ubiquitous Varimax rotation. And many factor analytic students of personality prefer those models, because they believe that only thus will the "pure" factorial dimensions be found. (Yes, there are also rotation methods that allow for correlated factors, and some researchers - such as Cattell - prefer them.)
So two assumptions have already been made before we start analyzing data: (1) personality is best described dimensionally (as traits) rather than categorically (as types), and (2) those traits should be as independent of one another as possible. A great deal of effort was expended developing scales that accomplish this; the earliest measures, for example, featured rather strong negative correlations between extraversion and neuroticism, which Eysenck labored to reduce.
Could either of these assumptions be incorrect? Certainly it's possible. With regard to the second one, for instance, just suppose that one personality trait was associated with a set of dopamine circuits, while another was associated with a set of norepinephrine circuits. (Yeah, that's way oversimplistic! But bear with it as an illustration.) Well, those neurotransmitter systems are correlated, in that both are synthesized from the same essential amino acid and they operate in similar brain regions. So of course the personality traits would be correlated as well.
I did say you'd found your way into a minefield, right?
Traits also vary in intensity. Some are cardinal and appear to be permanent, unchangeable. Others are weak traits and may change overtime, influenced by environment, or life experiences. They may have been learned overtime. Needless to say, behavior will not always truly reflect the traits. Those are two things separate, although they may correlate. For instance, sociopath's are able to hide their cardinal traits from others, for awhile. Eventually they show their true selves.
So, my point is, that traits correlation may be useful for some things, but the correlation will change according to how strong or weak are the traits being compared
It's primarily a conceptual issue, only secondarily a statistical one. The question is whether you want dimension reduction procedures such as factor analysis to come up with dimensions that are as "pure" as possible. The real question, of course, is whether what you get, then, has anything to do with what is observed in real life, where "noise" and a degree of "messy-ness" are the rule rather than the exception. On the plus side though, I suppose dimension purity facilitates the interpretation so it really depends on the concept that you are looking at.
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