Statistical Test to test two sets of variables

Question

If I have a two sets of variables and I want to find out if there is any significant relationship between them is there a statistical test to do this? e.g. Set 1: Columns A-D are paired data and Set 2: Columns W-Z are paired data and I want to see if the relationship between these two sets is significant.

I can think of independently testing every combination of Set 1 and Set 2 using something like wilcoxon-ranksum, but this doesn't seem efficient and it seems like I would miss out on "group" information if I were able to evaluate on the sets themselves.

Answer 1

Your goals are not sufficiently clear to be to give a complete answer. First, maybe you mean that data in A-D are 'blocked' (not 'paired'). That might mean that there are $n$ subjects, each producing scores A-D.

Then you might want to begin by looking at scores A-D to see how they may be correlated. You can easily look at the correlation between A & B, between A & C, and so on, as illustrated in R below:

set.seed(2020)
a = rnorm(20, 100, 10)
b = a + rnorm(20,0,3)
c = a + rnorm(20,0,3)
d = a + rnorm(20,0,3)
MAT = cbind(a,b,c,d)
round(cor(MAT),3)
      a     b     c     d
a 1.000 0.984 0.996 0.965
b 0.984 1.000 0.982 0.950
c 0.996 0.982 1.000 0.968
d 0.965 0.950 0.968 1.000

pairs(MAT)

Similarly, for W-Z.

It is not clear what kinds of comparisons you might want to make between A through D and W through Z. You might look at differences A-W, B-X, and so on. Or you might get compare a summary score of E of A,B,C,D and with a summary score V of W,X,Y,Z.

Such paired comparisons could be made with a paired t test or paired Wilcoxon (signed-rank) test, depending on the nature of the data.

Also, there are multi-level ANOVA designs that could take a more detailed look at A through D compared with W through Z.

Please revise your question, if you would like more detailed guidance toward your main objectives, so some of us can see what you have in mind.