Simpson's Paradox
Imagine that you are the head of the local health board.
Top of your agenda is Grot's Disease.
Although often fatal, there is no recognised treatment.
What is to be done about it. Many persons swear by Blogg's
Panacea, while others let the disease take its natural course.
Should you spend health board money on this alternative therapy?
You commission a survey, and the final analysis is
Treatment and outcomes of Grot's Disease
 Natural  Treat

Live  108  153 

Die  123  120 
Well there are two surprises.
First your scientists have given you a clear answer.
Second the alternative therapy does help.
Obviously for the 120 patients who took it and died anyway
it was a failure. However one is interested in the ratio
of those who live to those who die.
If a patient takes Blogg's Panacea his odds are better than
50:50, since 153 > 120 but without
it the odds are worse than 50:50, since 108 < 123.
You decide to spend a large amount of public money to make
Blogg's rather expensive Panacea available on the
National Health.
You are surprised that this decision proves unpopular with
your local feminist cooperative. They have checked
the figures for women only.
Female patients

Natural

Treat

Live

57

32

Die

100

57

They complain that you have fobbed them off with a treatment
that only benefits men, and is useless for women. The
survival rate for women without treatment is
57/(57+100)=36.3%. The survival rate for women with
treatment is 32/(32+57)=36.0%. They are not rabid
feminists and do not accuse you of murdering women because
of the 0.3% fall in survival rates.
Never the less you cannot fall back on the usual politicians' waffle
about the benefit being disappointing.
On the strict arithmetic Blogg's Panacea is actually
harmful.
Unfortunately, instead of blowing over, the trouble gets
worse.
One would expect that since the treatment does not benefit
women the initial favourable figures must come from a large
benefit to men.
The local men's group have subtracted the women's figures
from the totals for both sexes to get the men's figures.
You do not have to look too hard to see the problem.
Male patients

Natural

Treat

Live

51

121

Die

23

63

Twice 23 is 46 which is less than 51. Untreated, more than
twice as many men survive as die. However, twice 63 is 126
which is more than 121. With the treatment, fewer than twice
as many survive as die. 121+63=184 men took the treatment.
If it had been merely ineffective 130 should have survived.
The treatment is benefical to men and women, yet harmful to
men, and harmful to women. This is Simpson's paradox: Any
statistical relationship between two variables may be
reversed by including additional factors in the analysis.
You stop wasting public money on a quack treatment, and at a
humiliating press conference you promise that the health
board will never be caught
by Simpson's paradox ever again.
As you sit in your office, relaxing, relieved that the
incomprehensible awfulness of Simpson's Paradox is now
safely in the past, you receive a phone call from the Home
Office. The Home Secretary's own son died off Grott's
disease. It couldn't really be helped, he had low blood
pressure. You can see from the figures that Blogg's Panacea
is of limited benefit to men with low blood pressure.
Men, low B.P.

Natural

Treat

Live

4

51

Die

6

57

Obviously the outcome for untreated men with low blood
pressure is that 40% live and 60% die. Treament helps, but
not alot. It does not improve the odds are far as 50:50.
But why in gods name did you stop treating men with high
blood pressure? The figures there are much more impressive.
Men, high B.P.

Natural

Treat

Live

47

70

Die

17

6

There were one third the deaths, even though slightly more
patients were treated than went untreated. At first you
think the Home Secretary is talking about special cases, of
exceptionally high or low blood pressure, but quickly
checking the arithmetic you realise that this is not the
case. The data for men has simply been divided into data for
men with blood pressure less than a threshold and men with
blood pressure greater than or equal to the same threshold.
Blogg's Panacea is harmful to all men, but benefical to some
and a real life saver to all the others. Simpson's paradox has
struck again, reversing the statistical relationship between
two variables on the inclusion of an additional factor in
the analysis.
As you travel home on the train you are horrified to see
your photograph in the newspaper under the headlline
Health Board gets it wrong again. The Home Office
has leaked the story to the press before they even told
you. White faced, you ask if you might take a look at your
neighbours paper. The Home Office hasn't leaked the story of
men's blood pressures. The story is all about womens' ages.
Grott's Disease in young women

Natural

Treat

Live

49

25

Die

19

8

You are getting rather experienced at reading these
tables. About half as many young women were treated with
Blogg's Panacea as did without. In the live row,
25 is a touch over half of 49, while in the die
row, 8 is somewhat less than half 19. It is an unimpressive
treatment but better than nothing.
Grott's Disease in older women

Natural

Treat

Live

8

7

Die

81

49

There is not much hope in this table for the older woman
with Grott's Disease. Blogg's Panacea boosts your chance of
survival from 1 in 11 to 1 in 8. Not impressive either, but
it does help in both cases. Simpson's paradox strikes again.
When it says that
Any statistical relationship between two variables may be
reversed by including additional factors in the analysis.
it includes relationships that have already been reversed by
the inclusion of previous factors
Simpson's Paradox is deeply troubling. We use the phrase
in the final analysis as though there really
were a final analysis.
Simpson's Paradox casts doubt on this.
What I need to do next is to go to the library to try to
find
Dawid, A.P. 1979. Conditional Independence in Statistical
Theory Journal of the Royal Statistical Society
A41(1):131
and
Simpson, C. 1951. The Interpretation of Interaction in
Contingency Tables. Journal of the Royal Statistical
Society B13:238241
I've been started off on this by the book
Computation, Causation, and Discovery.
Edited by Clark Glymour and Gregory F. Cooper
The MIT Press
ISBN 0262571242