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 '>'.

Nota: this wrong way for adding two rational numbers is the so-called Median-Mean.

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
```

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 09/20/2022 12:16:36 -CEST-)

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