Transcript drug

SIMPSON’S PARADOX
(Pearson et al. 1899; Yule 1903; Simpson 1951)
• Any statistical relationship between two
variables may be reversed by including
additional factors in the analysis.
Application: The adjustment problem
• Which factors should be included in the
analysis.
EXAMPLES OF SIMPSON’S REVERSAL
• e.g., UC Berkeley's alleged sex bias in graduate admission (Science
- 1975). Overall data showed a higher rate of admission among
male applicants, but, broken down by departments, data showed a
slight bias in favor of admitting female applicants.
• e.g., "reverse regression" (1970-80): Should one, in salary
discrimination cases, compare salaries of equally qualified men and
women, or, instead, compare qualifications of equally paid men and
women. (Opposite conclusions.)
• Practical Dilemma: Why break down by department?
How about by some other variable Z?
Find Z such that P(y|do(x)) = ∑z P(y|x,z)P(z)
• Solution: The back-door algorithm (Chapter 3).
PEARSON’S SHOCK:
“SPURIOUS CORRELATION”
We are thus forced to the conclusion that a mixture of heterogeneous
groups, each of which exhibits in itself no organic correlation, will
exhibit a greater or less amount of correlation. This correlation may
properly be called spurious, yet as it is almost impossible to guarantee
the absolute homogeneity of any community, our results for correlation
are always liable to an error, the amount of which cannot be foretold.
To those who persist on looking upon all correlation as cause and
effect, the fact that correlation can be produced between two quite
uncorrelated characters A and B by taking an artificial mixture of two
closely allied races, must come as rather a shock. [Pearson, Lee &
Brandy-Moore (1899)]
1. Causation = perfect correlation
2. “Not all correlations are correlations” (Aldrich 1994)
SIMPSON’S PARADOX
(1951 – 1994)
M R R
T 18 12 30
T
7 3 10
25 15 40
+
M
T
T
R R
2 8 10
9 21 30
11 29 40
T – Treated
R – Recovered
M – Males
=
T
T
T – Not treated
R – Dead
M – Females
Easy question (1950-1994)
•When / why the reversal?
Harder questions (1994)
•Is the treatment useful? Which table to consult?
•Why is Simpson’s reversal a paradox?
R
20
16
36
R
20 40
24 40
44 80
SIMPSON’S REVERSAL
Group behavior:
Pr(recovery | drug, male) > Pr(recovery | no-drug, male)
Pr(recovery | drug, female) > Pr(recovery | no-drug, female)
Overall behavior:
Pr(recovery | drug) < Pr(recovery | no-drug)
TO ADJUST
OR NOT TO ADJUST?
Treatment
X
Recovery
Gender
Z
Treatment
X
Recovery
Y
Mediating
factor
Z
Y
THE INEVITABLE CONCLUSION:
THE PARADOX STEMS FROM
CAUSAL INTERPRETATION
TWO PROOFS:
1. Surprise surfaces only when we speak about “efficacy,” not
about evidence for prediction.
2. When two causal models generate the same statistical data and
In one we decide to use the drug yet in the other not to use it,
our decision must be driven by causal and not by statistical
considerations.
Thus, there is no statistical criterion to warn us against consulting
the wrong table.
Q. Can Temporal information help?
A No!, see Figure 6.3 (c).
WHY TEMPORAL INFORMATION
DOES NOT HELP
Treatment
Treatment
Treatment
C
C
C
F
F
F
Gender
Blood
Pressure
E
E
E
Recovery
Recovery
Recovery
(a)
•
•
(b)
(c)
C
F
E
(d)
In (c), F may occur before or after C, and the correct answer is to
consult the combined table.
In (d), may occur before or after C, and the correct answer is to
consult the F-specific tables
WHY SIMPSON’S
PARADOX EVOKES SURPRISE
1. People think causes, not proportions.
2. "Reversal" is possible in the calculus
of proportions but impossible in the
calculus of causes.
CAUSAL CALCULUS
PROHIBITS REVERSAL
Group behavior:
do{drug}
>
Pr(recovery | drug, female) >
Pr(recovery | drug, male)
do{drug}
do{no-drug}
Pr(recovery | no-drug, male)
Pr(recovery | no-drug, female)
do{no-drug}
Assumption:
Pr (male | do{drug} ) = Pr (male | do{no-drug})
Overall behavior:
do{drug}
do{no-drug}
Pr(recovery | drug) > Pr(recovery | no-drug)
THE SURE THING PRINCIPLE
Theorem 6.1.1
An action C that increases the probability of an
event E in each subpopulation must also increase
the probability of E in the population as a whole,
provided that the action does not change the
distribution of the subpopulations.