Download Correlation and Causation

Philosophical Devices
Week 9 – Correlation & Causation
--------------------------------------------------------------------------------------------------------PROBABILISTIC INDEPENDENCE & DEPENDENCE
 Independence: p is probabilistically independent of q iffdf assuming q doesn’t alter the
probability of p – i.e. iff Pr(p|q) = Pr(p)
Now, we know that, by the definition of conditional probability,
Pr(p|q) = Pr(p & q)/Pr(q)
Further, if p is probabilistically independent of q, we also know that
Pr(p|q)= Pr(p)
By the transitivity of identity (i.e. because the relation picked out by ‘=’ is transitive), we
therefore can conclude that
Pr(p) = Pr(p & q)/Pr(q)
Multiplying both sides by Pr(q) gives us
Pr(p)  Pr(q) = Pr(p & q)
This is a necessary condition for probabilistic independence: any p and any q that satisfy the
above are probabilistically independent of one another.
Further, because order is irrelevant for both conjunction and multiplication, this shows that
independence is symmetric – i.e. if p is probabilistically independent of q, then q is
probabilistically independent of p!
Examples of probabilistically independent propositions:
1.
2.
3.

Dependence: p is probabilistically dependent on q iffdf assuming q does alter the
probability of p – i.e. iff Pr(p|q) ≠ Pr(p)

Positive Dependence: p and q are positively probabilistically dependent iffdf Pr(p & q)
> Pr(p)  Pr(q)
 Only occurs if either assuming p raises the likelihood of q or assuming q raises
the likelihood of p – i.e. either Pr(q|p)>Pr(q) or Pr(p|q)>Pr(p)

Negative Dependence: p and q are negatively probabilistically dependent iffdf Pr(p &
q) < Pr(p)  Pr(q)
 Only occurs if either assuming q decreases the likelihood of p or assuming p
decreases the likelihood of q – i.e. either Pr(q|p)<Pr(q) or Pr(p|q)<Pr(p)
Positive dependent propositions:
1.
2.
Negative dependent propositions:
1.
2.
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Correlation & Causation
3.
3.
--------------------------------------------------------------------------------------------------------CORRELATION & CAUSATION

Correlations are objective probabilistic dependences between distinct properties of
individuals
(Note: an interesting side issue here is what sort of thing properties are, and whether an
individual’s possessing a property is the sort of thing to which we can legitimately attach an
objective chance value)

Positive correlation: Two properties Φ and  are positively correlated iffdf there is a
positive probabilistic dependence between their possession – i.e., letting ‘F’ be the
event of some individual possessing Φ and ‘G’ be some individual possessing ,
Pr(F|G)>Pr(F)
The implication of a correlation is that one property occurs more frequently when the other is
present. The same holds the other way too! (Question: why?)

Negative correlation: Two properties Φ and  are negatively correlated iffdf there is a
negative probabilistic dependence between their possession – i.e., letting ‘F’ be the
event of some individual possessing Φ and ‘G’ be some individual possessing ,
Pr(F|G)<Pr(F)
A frequently used (and correct) truism: correlation is not causation
Now, amongst normal humans, there is a clear correlation between feeling nauseous the next
day and drinking to excess tonight. And this correlation can be explained by a causal
connection between the two: tonight’s heavy partying causes tomorrow’s nausea.
There is also (again, amongst normal humans) a correlation between nauseous and feeling
feverish, but this correlation isn’t one of causation – the nausea doesn’t cause the fever, nor
the fever the nausea. Instead, both are symptoms of some common cause (e.g. having eaten
spoiled food, having drunk to excess)
Examples of correlation without causation:
1.
2.
3.
So, correlation doesn’t entail causation. However, arguably, correlation does imply a weaker,
disjuctive claim:
(CC) If A is correlated to B, then either (i) A causes B, (ii) B causes A, or (iii) there is
some C which causes both A and B
The support for this isn’t iron-clad, but still quite intuitive: if (CC) weren’t true, then the
correlation would be a startlingly odd coincidence. (Note that the credence one should attach
to the truth of (CC) is inversely proportional to the chances of there being such a coincidence,
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Correlation & Causation
such that the greater likelihood that the correlation is a coincidence, the less credence we
should put in (CC)).
Better, once we grant that something like (CC) is true, we can use correlations to tell us
something about causation – namely, if we can rule out the possibility of a common cause,
then we can infer a direct causal connection between the correlated properties, and if we rule
out the possibility of a direct causal connection, then we can infer a common cause!
--------------------------------------------------------------------------------------------------------TRACKING DOWN CAUSES?
Common causes tend to ‘screen off’ correlations between their joint effects, such that the
correlation disappears once we control for the common cause.
Let’s go back to our above example. There is a correlation between A (being nauseous) and B
(feeling feverish), such that Pr(A|B)>Pr(A).
Now, to see whether C (having eaten bad Chinese food) is a common cause for both, we can
look separately at those people who have eaten bad Chinese and those who haven’t. If, when
we look at these two groups, we discover that
(i) in cases where people ate bad Chinese, adding the fact that they feel feverish
doesn’t make nausea more likely – i.e. Pr(A|B & C) = Pr(A|C)
(ii) in cases where people didn’t eat bad Chinese, adding the fact that they feel
feverish doesn’t make nausea more likely – i.e. Pr(A|B & ¬C) = Pr(A|¬C)
then we know that eating bad Chinese ‘screens off’ the nausea-fever correlation. This is
pretty good evidence that it is a common cause for our initial correlation.
We say a correlation is ‘spurious’ when it doesn’t correspond to any direct causal connection,
and the common cause is called a ‘confounding’ property.
Suppose that our efforts at screening off fail. That doesn’t mean we’ve shown that there is a
direct causal connection between the correlates!

The Easy Way Forwards: random experiment. Pick a sample group, then arrange for a
random subset of this group to ‘have’ the putative cause (this is the test group), and
some to not (the control group).
- Randomization ensures that the cause is not systematically correlated with any
other potentially causally potent properties
- Any resulting correlation implies a genuine causal connection!
- Variation: random clinical trial. Take a group of people who suffer from some
disease, then randomly select a subset to receive a potential treatment (test group),
while the rest do not (control group)
- Lurking problem: statistical fluctuations.
o Solution: bigger sample sizes!

The Hard Way Forwards: survey everything. Check all confounding properties that
could possibly be responsible for a spurious correlation, show that none of these
screen off.
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- Requires lots & lots of work
- Stronger than randomization?
--------------------------------------------------------------------------------------------------------SIMPSON’S PARADOX
Suppose a University advertises positions in the Department of History and in the Department of Geography.
Five men apply for the positions in History and one is hired, and eight women apply and two are hired. The
success rate for men is twenty percent, and the success rate for women is twenty-five percent. The History
Department has favoured women over men. In the Geography Department eight men apply and six are hired,
and five women apply and four are hired. The success rate for men is seventy-five percent and for women it is
eighty percent. The Geography Department has favoured women over men. Yet across the University as a whole
13 men and 13 women applied for jobs, and 7 men and 6 women were hired. The success rate for male
applicants is greater than the success rate for female applicants. (Bickle et al., 1975)
Explanation: more women are applying for jobs that are
harder to get. Of the women applying for jobs, more are
applying for jobs in History than in Geography, but
History hired only 3 out of 13 applicants, whereas
Geography hired 10 out of 13 applicants. Hence the
success rate was much higher in Geography, where there were more male applicants.
Men
History
1/5 <
Geography 6/8 <
University 7/13 >
Women
2/8
4/5
6/13