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EDIT: Sorry, I wrote the previous post too hastily, not defining what specific variables I mean.

Thanks for the articles here, here, and here. I understand that it is possible to generate and retrieve sets of data for which the correlations are not transitive. However, I do not know how to apply it to a specific group of variables.

Generally, in psychology, measurement is performed using questionnaire methods that usually operate on a narrow range of results that are non-zero integers, for example in [1,2,3...20]. In such a juxtaposition of several variables, a distribution similar to the normal is usually obtained.

Could this (I mean a set of variables that are normally distributed positive integers) make any difference to a potential transition of correlation, or is it completely irrelevant?

This is interesting for me because the situation in which I would observe on such data, the lack of transitivity of correlation happens to me much less frequently than it would result from the "law of the lack of transitivity of correlation."

===============Previous text Let us assume three quantitative variables: A, B, and C. So far my experience in determining the correlation has been such that: if A correlates positively with B and A correlates positively with C, then B and C correlate positively with each other. Additionally, usually the strength of the correlation coefficients (roughly) was comparable.

While reading about the limitations of structural modeling, I came across something that I cannot understand, the possibility for: A positively correlate with B, A positively correlate with C, and B and C do not correlate with each other, or (which is an even more incomprehensible concept for me) C and B correlated negatively with each other. So we have two problems:

  • A correlates with B, A correlates with C, B and C do not correlate
  • A correlates+ with B, A correlates+ with C, B and C correlates negatively

Are Pearson's correlations of continuous variables transitive? What influences it? How is it possible that the matrix of three variables is finally non-positive? And how to understand this relationship? What key terms should I use to dig into this?

P.S. Maybe there is some code in R to generate such a non-positive matrix of three variables that would allow me to analyze these relationships and understand them better?

kwadratens
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    Your first paragraph is incorrect: it is possible for the correlations of $A,B$ and $A,C$ to be positive while the correlation of $B,C$ is negative. For instance, let $A=(0,1,-1),$ $B=(2,1,-3),$ and $C=(-1,1,0).$ R code for generating variables with specified correlations is posted at https://stats.stackexchange.com/a/313138/919. – whuber Jul 03 '22 at 15:49
  • The method you are presenting is awesome, great code. Nevertheless - it has a disadvantage from my point of view: no randomness. The new variable is not random, it is just a transformation of the previous variable, so I just get the same variable every time I generate it. For testing with the Monte Carlo method, unfortunately, it will not be useful to me. – kwadratens Jul 07 '22 at 12:09
  • Re "no randomness." That's exactly the opposite of what it does. As I stated in that answer, "Pick a vector X, of the same length as Y, according to any distribution you like." That is as random as one can possibly get! – whuber Jul 07 '22 at 12:12
  • I understand the input can be random but the output isn't, right? There is only one result for a single set of input data. The number of correlating data sets is infinite, but the result of the function is one. – kwadratens Jul 07 '22 at 14:59
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    When the input is random, so is the output. When you repeat with more input from the same distribution, the output usually will be different. – whuber Jul 07 '22 at 15:18

1 Answers1

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Suppose $B$ and $D$ are independent, where $D$ has bigger variance.

Let $A=B+D$ and $C=B-D$. Then:

  • $A$ and $B$ will be positively correlated.
  • $B$ and $C$ will be positively correlated.
  • $A$ and $C$ will be negatively correlated.
Matt F.
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  • Can it be assumed then that these variables are not independent, but simply transformations of another variable? And would it mean that the matrix of the variables thus obtained is reducible? – kwadratens Jul 07 '22 at 12:11