Association between variables
Statistics is filled with associations between variables, some positive, some negative.
x = how much hair a man has at age 40
x = how much soda you drink
x = your high school GPA
x = how many times you go to church
x = how much your car costs
y = how much hair his father had at age 40
y = how many cavities you have
y = your college GPA
y = how long you live
y = how low a golfing average do you have
What statisticians hope to prove is that changes in the explanatory variable will cause changes in the response
variables. How many times have you heard lines like this:
• If you smoke, you will get cancer.
• If you eat chocolate, you will break out.
• If you don’t study, you won’t pass.
• If you practice your piano, you will get better.
All of these are cause and effect statements and very powerful ones, if true. The fact is that none of them can be
proven true. People smoke and don’t get cancer. People never study and pass courses. People eat chocolate and
their faces are clear. People can practice the piano until the cows come home and still be awful.
There are three types of associations in which we are interested. In each diagram below, the dashed line
represents an observed association between variables.
Causation is very difficult to prove and there are not that many examples that I can give you and the ones I
mention are kind of obvious.
• Driving faster causes you to use more gasoline.
• Talking more on your telephone causes you to pay more money.
It is very difficult to prove causation although you hear about studies all the time … drinking more wine causes
people to live longer or eating more broccoli leads to lower blood pressure.
Common response refers to a “lurking variable” z. When z changes, x changes and y changes. That creates an
association between x and y although there may not even be a casual link between x and y.
• There is a positive association between one’s high school GPA and one’s college GPA. Students who
are low in high school are usually low in college and students high in high school are usually high in
college. But there is no causation. There are lurking variables that effect each causing a common
response. What are some of them?
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Stu Schwartz
• Suppose a study was done and the results showed that people who drove more expensive cars were
better golfers. Could we then say that driving expensive cars cause you to score lower on a golf
course? Of course not. There are confounding variables affecting both the cost of your car and the
score on a golf course making it a common response situation. Can you give a couple?
Confounding is a little more complex. We think x causes y but there are other lurking variables which also can
have an influence on y. We have no idea how much x has to do with y.
In common response, the lurking variables affect both x and y, but in confounding, the lurking variables affect
just y.
• People who go to church (synagogue, mosque) tend to live longer. Suppose a study was done and
there was evidence to this fact. But how long one lives has many lurking variables. Name some of
them.
So while this study might have merit, there is certainly no causation and it is difficult to know how much
influence going to church might actually have on the duration of one’s life.
• People who own pets tend to have lower blood pressure. Suppose a study was done and there was
evidence to this fact. But blood pressure have many luring variables. Name some of them.
So again, while this study might have merit, there is certainly no causation and it is difficult to know how much
influence having pets might have on one’s blood pressure..
Most of the studies I have mentioned would need some experiment to test them. We will talk about how this is
done in chapter 5. But many of life’s big questions cannot easily be settled by an experiment.
• Gun control will reduce crime. A simple causation statement. But how do we ethically test it?
• Smoking causes lung cancer. There certainly is a strong association. Also a consistent one – not just
Americans or Asians or Christians or women. The association is strong among all groups. People who smoke
more often and more cigarettes a day get long cancer more often. People who stop don’t get it as often.
Experiments with animals shows that tars from cigarette smoke cause cancer. And still, there is no agreement
that smoking causes cancer. Why?
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Stu Schwartz
For HW, complete the following page.
Directions:
Suppose studies were to be done on the following. Determine if you believe the association would be positive,
negative, or none.
Then decide if the relationship would most likely be a) causation, b) common response, or c) confounding.
If it is common response, identify lurking variable(s) affecting both.
If it is confounding, identify the lurking variable(s) affecting the response variable.
Put your answers on the sheet following this page.
1. When you are on a diet, the amount of calories you eat daily vs. the amount of weight you lose.
2. The amount of ice cream consumed on a beach vs. the number of people who go in the water.
3. The number of pets you own vs. the amount you spend on pet food.
4. How much you pay for a house vs. how much you pay for a car.
5. How much you study vs. your GPA.
6. The number of policeman that are visible on a stretch of road vs. the speed you travel.
7. How a student does in algebra vs. the student does in geometry.
8. A person’s height vs. the amount of money that person has.
9. The total amount of money spent on concessions at Phillies games and the number of wins the Phillies have.
10. The amount of drinking people do and the incidence of liver disease.
11. The number of people in a family vs. the number of cars the family owns.
12. The number of problems on a math test vs. the amount of time it takes students to complete the exam.
13. The average speed it takes cars to travel from Philadelphia to New York on the turnpike vs. the average
amount of times it takes.
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Stu Schwartz
Study
Association
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Relationship
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Confounding
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Stu Schwartz