Regression: what item of subscales is strongest IV controlling for sum score of scale

Question

We want to explore the cross sectional association of 5 symptoms of psychopathology with functional impairment.

If we just take people who score high on symptom 1, then high on symptom 2, etc., and look at the means of social impairment in these groups, one particular symptom (let's call it symptom 1) stands out that is associated with very high functional impairment (twice as high as the others).

The problem is that this is confounded with the sum score of all symptoms, because people who score high on symptom 1 have high scores on all other symptoms as well. This is different from all other symptoms. People who score high on symptom 2 only have average symptom load on the other symptoms (the same for symptom 3, symptom 4, etc).

I wonder (1) what test would be appropriate to test this, and (2) how one would control for the fact that for one symptom the sum of other symptoms seems particularly high.

Answer 1

There are a couple of options, depending on what you are trying to do.

If you are just interested in the 5 subscores then just look at those 5 as IVs and don't include the total.

On the other hand, it sounds like you want to see if (say) a particularly high score on subscale 1 is an important IV. This is a topic in psychometrics. In a test with multiple subtests, there are three qualities of the pattern of scores that can be important: Elevation (the overall mean), scatter (the deviation of the scores around the mean) and shape (the pattern of the scatter). There is, as far as I know, no standard method of operationalizing shape. Perhaps the simplest is to simply rank order the subtests from highest to lowest and then use this ordering as a categorical variable. E.g. someone who is high on subtests 1 and 3, middle on 2 and 4 and low on 5 might get 13245. Other people have used cluster analysis (see Watkins and Glutting).

Cattell proposed:

$r_s = \frac{\sum x'y'}{n \sigma_{x'} \sigma_{y'}}$

where the primes indicate standardization (see the link for more).

Yet another idea is to subtract from each subtest the mean of all the subtests (perhaps weighted).