I am trying to understand a research result. Imagine 10 different psychological questionnaires 1,2,3,...10 of different length (i.e. number of questions) that are all supposedly measuring the same construct. Each scale has between 10 and 30 questions.
I now compute the Jaccard Index J ranging from 0 to 1 for each pair of scales:
J = a/(a + b + c), where
a = number of items shared by 2 scales,
b = number of items unique to the first scale,
c = number of items unique to the second scale.
Afterwards, I compute the average Jaccard Index for each scale (with all other scales).
Curiously, in my empirical example, this averaged coefficient for each scale correlates 0.5 with the number of items per scale — the longer the scale, the higher the Jaccard Index.
I want to understand if this is due to chance, or some property of the Jaccard index, and would very much appreciate feedback.