4.2 Cautions about Correlation and Regression - PowerPoint Presentation

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4.2 Cautions about Correlation and RegressionSlide2

Correlation and regression are powerful tools, but have limitations.

Correlation and regression describe only linear relationship.

Correlation r and the least-squares regression are not resistant. Slide3

Extrapolation

The use of a regression line for prediction far outside the domain of the explanatory variable x that you used to obtain the line or curve. Slide4

Such predictions are often inaccurate

Suppose that you have data on a child’s growth between 3 and 8 years of age. You find a strong linear relationship between age x and height y. If you fit a regression line to these data and use it to predict height at age 25 years, you will predict that the child will be 8 feet tall.Slide5

Lurking Variable

A variable that is not among the explanatory or response variables in a study and yet may influence the interpretation of relationships among those variables.Slide6

Remember the link between cancer and dental plaque? It could be that bad mouth hygiene is an indicator of other life style factors associated with cancer.Slide7

Lurking variables continued

Lurking variables are often unrecognized and unmeasured. Detecting their effect is challenging.

Many lurking variables change systematically over time.

 one useful method of detecting lurking variables is to

plot both the response variable and the regression residuals against the time order of the observation. (See Example 4.12 on

pg

228)Slide8
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Explaining AssociationSlide12

Causation

The best evidence for causation comes from experiments that actually change x while holding all other factors fixed. If y changes, then we have a good reason to think that x caused the change in y.

Even well-established causal relations may not generalize to other settings.

Sugar substitute caused bladder tumor in rats. Should we avoid this particular sugar substitute?Slide13

Common Response

The observed association between the variables x and y is explained by a lurking variable z. Both x and y change in response to changes in z. This common response creates an association even though there may be no direct causal link between x and y.

Students who are smart and who have learned a lot tend to have both high SAT scores and high college grades. The positive correlation is explained by this common response to students’ ability and knowledge.Slide14

Confounding

In short, “mixing of influences.”

Two variables are confounded when their effects on a response variable cannot be distinguished from each other. Slide15

Example of Confounding

It

is likely that more education is a cause of higher income—many highly paid

professions require

advanced education. However, confounding is also present.

People who

have high ability and come from prosperous homes are more likely to get

many years

of education than people who are less able or poorer. Of course, people who

start out

able and rich are more likely to have high earnings even without much

education. We

can’t say how much of the higher income of well-educated people is

actually caused

by their education.Slide16

Establishing Causation without Experiments

The association is strong.

The association is consistent

.

Higher doses are associated with stronger

responses.

The alleged cause precedes the effect in time

.

The alleged cause is plausible

.

See Example 4.18 Does Smoking Cause Lung Cancer (pg. 236)Slide17

4.33 FIGHTING FIRES

Someone

says, “There is

a strong

positive correlation between

the number

of firefighters at a fire and the amount of damage the

fire does

. So sending

lots of

firefighters just causes more damage.” Why is this reasoning wrong?Slide18

4.36 BETTER READERS

A

study of elementary school children, ages 6 to 11, finds a

high positive

correlation between shoe size

x

and score

y

on a test of reading

comprehension. What

explains this correlation?Slide19

Try this at home

Exercises 4.38, 4.41, 4.43, 4.45