March 20, 2013
Correlation and Causation
We must be very careful in interpreting
correlation coefficients. Just because two
variables are highly correlated does not mean
that one causes the other.
In statistical terms, we say that correlation
does not imply causation.
How are the following
causal relationships
flawed conclusions?
(Or are they flawed?)
There are many good examples of
correlation which are nonsensical when
interpreted in terms of causation.
Example 1:
Ice cream sales and the number of shark
attacks on swimmers are positively correlated.
Can I conclude that a rise in ice cream sales is
going to cause more shark attacks?
Of course ice cream does not cause shark
attacks! Ice cream sales and shark attacks both
increase during the summer. So, the two
variables are positively correlated, but there is
no causal relationship between the two!
Example 3:
Children with bigger feet spell better.
Again, a child's shoe size and her ability to
spell are both related to a child's age ...
Children with bigger feet spell better because
they are older, their greater age bringing about
bigger feet and, not quite so certainly, better
spelling. Thus the two variables are positively
correlated and there is no causal relationship.
Example 2:
The number of cavities an elementary school
child has had and the child's vocabulary size
have a strong positive correlation.
Can I conclude that if my niece gets a cavity,
the cavity will cause her to increase her
vocabulary?
The number of cavities a child has had and the
size of a child's vocabulary are both related to
a child's age ... as a child gets older, the child is
both more likely to get cavities and increase
her vocabulary, thus the two variables are
positively correlated. Again, there is no
causal relationship.
Example 4:
In areas of the South, those counties with
higher divorce rates generally have lower death
rates.
My sister lives in Tennessee. Can I conclude
that if she gets divorced, her divorce will
cause her to live longer?
No. Again this example has to do with age &
demographics. Couples who are older are less
likely to get divorced and are more likely to die
than are couples from counties with younger
demographic profiles. Thus, the two variables
are negatively correlated, and there is no
causal relationship.
March 20, 2013
Example 5:
Joey did exceptionally poorly last semester, so I
punished him. He did much better this
semester. Clearly, punishment is effective in
improving students' grades.
Example 6:
Nations that add fluoride to their water have a
higher cancer rate than those that do not.
Can I conclude that fluoride causes cancer?
I highly doubt it!
Often exceptional performances (exceptionally
bad or exceptionally good) are followed by
more "normal" performances, so the change in
performance might better be explained by
regression towards the mean.
I don't think so! Nations that add fluoride to
their water are generally wealthier and more
health-conscious, and thus a greater
percentage of their citizens live long enough to
develop cancer, which is, to a large extent, a
disease of old age. Thus, the two variables are
positively correlated, and there is no causal
relationship.
Example 7:
The more firemen fighting a fire, the more
damage there is going to be. Therefore
firemen cause damage.
Example 8:
The frequency of accidents on a road fell after
a speed camera was installed. Therefore, the
speed camera has improved road safety.
Firemen don't cause damage. Firemen are sent
according to the severity of the fire ... the
larger the fire, the more firemen are sent.
Large fires demand more firemen and large
fires cause more damage.
Probably not ...
Speed cameras are often installed after a road
incurs an exceptionally high number of
accidents, and this value usually falls
(regression to the mean) immediately
afterwards.
Example 9:
Since the 1950s, both the atmospheric CO2
level and crime levels have increased sharply.
Hence, atmospheric CO2 causes crime.
Atmospheric CO2 does not cause crime. In
fact, both the increase in atmospheric CO2
levels and the increase in crime levels are more
likely to have been caused by an increase in
population since the 1950s.
Let's examine the possible
relationships among two
correlated variables: direct
causation, common response, and
confounding.
These three relationships are the
three relationships which can be
taken (or mistaken) for causation!
March 20, 2013
Direct Causation:
changes in X cause changes in Y.
For example, football weekends cause
heavier traffic, more food sales, etc.
Confounding:
the effect of X on Y is hopelessly mixed up with
the effects of other explanatory variables on Y.
For example, if we are studying the effects of Tylenol on
reducing pain, and we give a group of pain-sufferers
Tylenol and record how much their pain is reduced, we
are confounding the effect of giving them Tylenol with
giving them any pill. Many people report a reduction in
pain by simply being given a sugar pill with no
medication in it at all, this is called the placebo effect. To
establish causation, a designed experiment must be run.
Common Response:
both X and Y respond to changes in some
unobserved variable.
most of the examples we just looked at are
examples of common response.
· Ice cream sales and shark attacks both
increase during summer.
· The number of cavities and children's
vocabulary are both related to a child's age.
· Skirt lengths and stock prices are both
controlled by the general attitude of the
country, liberal or conservative.
Statistics and Causation
1. A strong relationship between two
variables does not always mean that changes
in one variable cause changes in the other.
2. The relationship between two variables is
often influenced by other variables lurking in
the background.
3. The best evidence for causation comes
from randomized comparative experiments.
Statistics
Section 6.2 - Correlation does NOT imply Causation
1. When I’m stressed, I get muscle cramps. However, when I’m stressed, I also drink lots of
coffee and lose sleep. So it’s hard to tell whether my cramps are actually caused by coffee, lack of
sleep, stress, or some combination of the above.
“Lots of coffee” and “lack of sleep” are examples of:
a)
b)
c)
d)
direct causation
common response
confounding
regression towards the mean
2. A pro baseball player has an exceptionally poor batting performance during a night game.
The very next day the batting coach spends several hours working with the player. The next
game the same baseball player has a much better batting performance. The change in
performance most likely can be explained by:
a)
b)
c)
d)
direct causation
common response
confounding
regression towards the mean
3. Chris runs home from school every day. Chris finds that when he runs faster, he gets home
sooner. The change in travel time most likely can be explained by:
a)
b)
c)
d)
direct causation
common response
confounding
regression towards the mean
4. Surf board sales rise when lemonade sales rise. A conclusion about the causality in the
preceding example is flawed because it is most likely a result of:
a)
b)
c)
d)
direct causation
common response
confounding
regression towards the mean