Imagine two variables, X and Y, each with 20 observations. Pearson's r(x,y) = 0.5. This correlation arises because a) averaged across items, subjects score with a certain distance from the mean subject score, and b) averaged across subjects, items score with a certain distance from the mean item score. I will show how these two variance components add to the correlation coefficient.
The subject variance is defined as the variance across subjects in terms of their average item score. The average score per subject ranges from 3.0 to 10.5, with a mean of 6.5 and a standard deviation (i.e. subject variance) of 2.218.
The item variance is defined as the variance across items in terms of their average subject score. The average score per item ranges from 5.5 to 7.5, with a mean of 6.5 and a standard deviation (i.e. subject variance) of 1.414.
The subject variance is larger than the item variance. Hence I expect the subject variance to contribute more to the correlation compared to the item variance.
1000 permuted random re-orderings of columns (items) gives me a mean correlation of 0.312, which should be an indicator of the subject variance.
1000 permuted random re-orderings of rows (subjects) gives me a mean correlation of 0.009 , which should be an indicator of the item variance.
A matrix of covariance between items across subjects
arises as a) subjects vary in how much their average item score
diverges from the average symptom score across all subjects, and b)
items vary in how much their average score across subjects diverges from
the average item score across all items.
between is determined by variance A and variance B;
variance is the dispersion from the mean. By scrambling items, we should be
left with the correlation caused by dispersion of each subject from the mean,
across all items. If we scramble the subjects, we should be left with the
correlation caused by dispersion of each item from the mean, across all
subjects. Technically these two basic correlations should together explain the
full correlation (by some calculation). By permuted scrambling, we generate a
distribution of edge strengths (or, actually, two: one for cor and one for pcor)
against which each of the real edge strengths can then be compared. Significant
edges are set to 1, non-significant edges are set to 0. The result is a
thresholded non-weighted network. Network strength is calculated as the percentage
of edges reaching significance.