The Simpson paradox [Le paradoxe de Simpson].
The Simpson paradox [Le paradoxe de Simpson].
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 '>'.
X = (((n1-1)*4 + (n3-1))*4 + (d1-1))*4 + (d3-1)
Y = (((n2-1)*4 + (n4-1))*4 + (d2-1))*4 + (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...
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...
