Test of symptom severity equality / symptom configuration equality

We have a questionnaire measuring 9 clinical symptoms. The sum scores simply adds all these symptoms up. Symptoms are ordered (0,1,2,3), and pretty strongly skewed (lots of 0s).

1) First, we want to test whether the symptom configuration deviates from the Null-Hypothesis of all symptoms being equally severe (equally common in the population). Our hypothesis is that the data do deviate from this H0. The H0 would posit that each symptom contributes with 11.1% to the sum-score of each participant. If we average each symptom over all participants, average the sum score over all participants, and then divide the sum score by each symptom mean score, we get these values:

24.6 % 15.7 % 14.3 % 11.3 % 10.6 % 9.7 % 8.5 % 3.9 % 1.6 %

As you can see, symptoms vary drastically in severity. The pattern clearly deviates from the H0 pattern that would be

11.1 % 11.1 % 11.1 % 11.1 % 11.1 % 11.1 % 11.1 % 11.1 % 11.1 %

However, we need a statistical test for this and do not know what to use (R or SPSS).

2) As a second question, we want to investigate if the symptom configuration stays stable over time. We have a second measurement point, in which we follow the same procedure to get 9 relative severity scores, one for each symptom. The scores are very similar to the ones from the first measurement point (large symptoms stay large, small symptoms stay small. Once again we wonder how we would best measure stability statistically.

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Are the symptoms really independent? I would start with checking whether the symptoms do not correlate with each other; I doubt that they don't, but that depends on the symptoms. – January Nov 11 '12 at 9:09
The symptoms are correlated, some of them highly (.6). But I don't see why that should interfere what we want to test. – Torvon Nov 11 '12 at 14:58
Very simply -- since you can't treat the symptoms as independent variables, most of the straightforward solutions don't apply. – January Nov 11 '12 at 16:32
I don't quite understand. The fact that variables are correlated does not prohibit performing most tests. E.g. I can use a t-test to compare mean differences of 2 symptoms in a population no matter if they are correlated or not, right? What I need here is some sort of multivariate t-test, I guess. – Torvon Nov 11 '12 at 20:21

1 Answer

This problem calls for Friedman test. It is multivariate extension of Wilcoxon test (or: nonparametric repeated-mesaures ANOVA).

You can still use Wilcoxon test in a multiple comparison manner pairwise, analogically to post-hoc tests in ANOVA: it will show you whether one symptom is more severe than the other.

Be sure to penalize it for the multiple-comparison effects; since your data are correlated, the Bonferroni, Holms and other corrections not taking this fact into consideration will give you too conservative test.

You can (always) make multiple comparison using bootstraps (see: Westfall P.G,Young S.S - Resampling-Based Multiple Testing Examples and Methods for p-Value Adjustment(1993))

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Thank you. If the DV were metric, what test would be appropriate? Hotelling? – Torvon Nov 12 '12 at 20:09
Yes. Or significance of Wilks lambda. Or many others. – Adam Ryczkowski Nov 12 '12 at 20:30
Do I understand correctly that you would use the same test to assess whether 9 variables at two measurement points differed from each other? If the test is significant, it means that all differences between the 9 variables (y1time1 to y1time2, y2time1 to y2time2, etc.) are significant? Thank you! – Torvon Nov 16 '12 at 1:21
Yes, I would. The test will also work, if the variables are measured on more than two levels, as long as the variables can be treated as ordinal. – Adam Ryczkowski Nov 16 '12 at 7:27