The Simpson paradox [Le paradoxe de Simpson].




The Simpson paradox is based on the following phenomenon:


Let {n1,d1,n2,d2,n3,d3,n4,d4} be eight integer numbers such that:
                     n1     n2
                    ---- < ----
                     d1     d2
                     n3     n4
                    ---- < ----
                     d3     d4
Then:
                                 \
                     n1     n2    |
                    ---- < ----   |
                     d1     d2    |      n1     n3     n2     n4
                                  | ==> ---- + ---- < ---- + ----
                     n3     n4    |      d1     d3     d2     d4
                    ---- < ----   |
                     d3     d4    |
                                 /
But unfortunately (and obviously) nothing can be said about:
                     n1 + n3     n2 + n4
                    --------- ? ---------
                     d1 + d3     d2 + d4
where '?' means '<', '=' or again '>'.


The {n1,d1,n2,d2,n3,d3,n4,d4} space is a eight-dimensional one and then difficult to visualize. In order to use a bidimensional space, for {n1,d1,n2,d2,n3,d3,n4,d4} inside [1,5], the two following coordinates are computed:
                    X = (((n1-1)*5 + (n3-1))*5 + (d1-1))*5 + (d3-1)
                    Y = (((n2-1)*5 + (n4-1))*5 + (d2-1))*5 + (d4-1)
Then, the white points {X,Y} display the case where:
                     n1 + n3     n2 + n4
                    --------- > ---------
                     d1 + d3     d2 + d4
the so called paradoxal case...

For example:
                     1     2
                    --- < ---
                     3     4
                     4     2
                    --- < ---
                     3     1
when:
                     1 + 4     2 + 2
                    ------- > -------
                     3 + 3     4 + 1
By the way {Black, Red, Green} colors are used for the 3x3x3-2=25 other cases...



See some related pictures (including this one):




More information on this subject is available in the july 2013 issue of PLS (Pour La Science) with the Jean-Paul Delahaye's paper L'embarrassant paradoxe de Simpson.

See my own discovery of the "double" Simpson paradox.


(CMAP28 WWW site: this page was created on 04/18/2013 and last updated on 04/11/2020 11:21:42 -CEST-)



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