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Is it fair to think of this geometrically, as a, b and c being three lines, with a and c being perpendicular and b lying in between, such that the correlation between the pairs a&b and b&c are positive, yet a&c have zero correlation?
|/_ for a quick representation
That is a good question and exposes that I'm a little out of my depth. But I've spent a happy half hour writing some crude dice rolling simulations, so what follows is partially checked (I'd like to draw some scatter plots too!)
Consider a data generating process using a red d6 and a green d6, where d6 is jargon for the ordinary cubical die with 6 faces. We regard the red and green dice as generating the red and green random variables. A third, yellow random variable is generated by adding together the red and green rolls.
Then red and yellow have a correlation of 0.7 (Will checking with pencil and paper discover that this is 1/√2 ?). Yellow and green also have a correlation of 0.7. Red and green have a correlation of 0.00506. Now I'm regretting writing a dice rolling simulation, rather than a computation using distributions. That has to be really 0.
But lines don't really work. Two of the scatter plots have lines at a definite slope, but red versus green is just a filled in square showing zero correlation.
I'd really like to get the third correlation to be negative rather than zero, to make the point about non-transitivity more strongly. Can I do that with dice? Yes.
Roll five dice, A,B,C,D,E. Generate three random variables
Red = A + B + C
Yellow = A + B + D + E
Green = - C + D + E
Red and yellow share A and B giving them a correlation of 0.57. Yellow and green share D and E giving them a correlation of 0.59 (it has to be the same, but I'm out of time to do the computation exactly)
Meanwhile Red and Green share C but with C subtracted from Green, for a correlation of -0.3
That is shocking. Red correlates positively with yellow. Yellow correlates positively with green. But red and green have a negative correlation.
Now we have reached the point where I really need scatter plots. I think the Red/Yellow plot and the Yellow/Green plot are basically the same (there is an offset because the red mean is 10.5 and the green mean is 3.5, but I don't think that matters). Red/Green contrasts by sloping down rather than up. It doesn't lie between Red/Yellow and Yellow/Green at all.
Non-transitive dice make my head hurt, but thank you for going to such lengths to answer my question!
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