Grade by Grade
D. Statistical Analysis · Two-Variable Data Analysis and Validity

Correlation and Causation: Types of Relationships

D2.2About 70 minutes
Learning goals
  • I can explain why a strong correlation does not prove that one variable causes the other.
  • I can recognize and distinguish cause-and-effect, reverse cause-and-effect, common-cause, accidental and presumed relationships.
  • I can identify possible lurking (hidden) variables in a correlation.
  • I can explain how a randomized controlled experiment can show cause and effect.

Words to know

cause-and-effect relationship
A change in X directly produces a change in Y.
reverse cause-and-effect relationship
The cause runs the other way: Y produces the change in X.
common-cause factor
A third variable that causes both X and Y, making them correlated even though neither causes the other.
accidental relationship
A correlation that happens by coincidence, with no causal connection at all.
presumed relationship
A correlation that seems logical, but for which no cause-and-effect link or common cause can be found.
lurking (extraneous) variable
A variable not included in the analysis that affects the variables being studied. When it is mixed up with the explanatory variable, it is called a confounding variable.
randomized controlled experiment
A study in which subjects are randomly assigned to treatment and control groups so that the groups differ, on average, only in the treatment.

Ice cream and forest fires

Across the months of the year in Ontario, ice cream sales and the number of forest fires tend to be strongly positively correlated. Should the province ban ice cream to protect the forests?

Of course not. Hot, dry summer weather increases both. This example comes straight from the Ontario curriculum, and so does the main lesson of today: correlation is not causation. In Grade 9 you met this idea; now you will learn to name precisely what kind of relationship you are looking at.

Five types of relationships

When two variables X and Y are correlated, there are several possible explanations. The Ontario curriculum names three (cause-and-effect, common-cause and accidental); data management textbooks usually add two more.

  1. Cause and effect. A change in X directly causes a change in Y. Example from the curriculum: the age of a tree and its diameter. As a tree ages, it adds a growth ring each year, so it gets wider.
  2. Reverse cause and effect. The cause runs the other way. Example: towns with more police officers have more crime. More police do not cause crime; areas with more crime tend to hire more police.
  3. Common cause. A third variable Z causes both. Example: ice cream sales and forest fires (Z = hot weather).
  4. Accidental (coincidental). There is no causal connection at all; the correlation is a fluke. Example from the curriculum: the consumer price index and the number of known planets. Both have risen over the decades as time passed, but planet discoveries do not affect prices.
  5. Presumed. The correlation seems logical, but there is no clear cause-and-effect link and no common cause has been found. Example: a correlation between playing chess and good marks seems reasonable, but the evidence for a direct cause is unclear.
Cause and effectXYtree age → diameterReverse cause and effectXYY actually causes XCommon causeZXYhot weather → bothAccidental (coincidence)XY?no real link at allPresumed relationshipXYseems logical, but no clearcause or common cause foundsolid arrow = causes · dashed line = correlated only
Arrows show causes. In a common-cause relationship, X and Y are correlated (dashed line) only because Z drives both.

Lurking variables

A lurking (or extraneous) variable is one you did not measure but that affects the variables you did measure. In the ice cream example, temperature is the lurking variable. When a lurking variable is mixed up with the explanatory variable so you cannot separate their effects, it is called a confounding variable.

Time itself is a common lurking variable. Many quantities grow over the years (population, prices, phone ownership, the number of known planets). Any two of them will be correlated over time, whether or not they are connected. Be very suspicious of correlations between two time series.

Worked example

Example 1: Classify the relationship

Classify each relationship and justify.

  1. Hours spent practising piano and number of mistakes in a recital piece (negative).
  2. Number of firefighters at a fire and the amount of damage (positive).
  3. Shoe size and reading level of students in Grades 1 to 8 (positive).
  4. People who own more umbrellas tend to own more boots (positive).
  1. 1. Cause and effect: practice directly improves accuracy. (Other factors also matter, but practice is a genuine cause.)
  2. 2. Common cause: bigger fires need more firefighters and cause more damage. Firefighters do not cause the damage. (You could also call it reverse cause: the size of the fire determines the number of firefighters sent.)
  3. 3. Common cause: age increases both foot size and reading level.
  4. 4. Common cause, most likely: living in a rainy or snowy area, or having a larger household, could lead to owning more of both.

Cause and effect; common cause (fire size); common cause (age); common cause (climate or household).

Worked example

Example 2: Reverse cause

A study finds that people who use a fitness app have lower resting heart rates. The app company claims the app lowers heart rates. Give another explanation.

  1. People who already exercise a lot (and therefore already have low resting heart rates) are more likely to download a fitness app.
  2. So fitness may cause app use, rather than app use causing fitness: reverse cause and effect.
  3. It could also be a common cause: interest in health leads people both to exercise and to use the app.

The correlation could be reverse cause and effect or a common cause. To test the company's claim you would need an experiment.

How can we show cause?

Observed data (from surveys, records or sensors) can show correlation but rarely prove cause, because lurking variables can never all be ruled out. The gold standard is a randomized controlled experiment:

  1. Randomly assign subjects to a treatment group (gets X) and a control group (does not, or gets a placebo).
  2. Random assignment spreads the lurking variables evenly between the groups, on average.
  3. If the groups then differ in Y by more than chance would explain, X is the likely cause.

When experiments are impossible or unethical (you cannot make people smoke to study lung cancer), researchers build a case from many observational studies: the cause comes before the effect, the link is found consistently in different groups, more of X goes with more of Y (a dose-response pattern), and there is a plausible mechanism. This is how the link between smoking and lung cancer was established.

Volunteers(e.g. 40 people)RandomassignmentTreatment groupgets XControl groupno X (or placebo)Compare Ybetween groups
In a randomized controlled experiment, chance decides who gets the treatment, so the only systematic difference between the groups is X itself.
Worked example

Example 3: Designing an experiment

Design an experiment to test whether a 20-minute nap after lunch improves the afternoon quiz scores of Grade 12 students.

  1. Recruit volunteers (with consent; for students under 18, parent or guardian permission).
  2. Randomly assign half to nap 20 minutes after lunch and half to sit quietly reading for 20 minutes (the control).
  3. Give both groups the same quiz at the same time of day. The markers should not know who napped (blinding).
  4. Compare the mean or median scores of the two groups. Repeat on several days or with more students if possible.

Random assignment and a control group let you attribute a difference in scores to the nap, not to lurking variables such as how much sleep students got the night before.

Questions to ask about any correlation

  1. Could X cause Y? Is there a plausible mechanism?
  2. Could Y cause X instead?
  3. Is there a third variable that could cause both?
  4. Could it be a coincidence (especially two trends over time, or a tiny sample)?
  5. Was it an experiment with random assignment, or only observation?
Just for fun

I used to think correlation implied causation. Then I took a data management course, and now I don't. Was it the course? Correlation says maybe; only a randomized controlled experiment can tell.

Hands-on activity

Test a cause at home: sugar and water temperature

You need: 4 identical clear glasses, sugar cubes (or level teaspoons of sugar), a measuring cup, cold tap water, warm tap water (not boiling; check that it is comfortable to touch), a spoon and a timer.

  1. Fill two glasses with 250 mL of cold water and two with 250 mL of warm water.
  2. To be fair, randomly decide the order you test the glasses (for example, write the glass labels on paper slips and draw them).
  3. Drop one sugar cube in a glass, stir gently at the same speed each time, and time how long until it fully dissolves.
  4. Compare the times for warm and cold water.
  5. Explain why this is an experiment that can show cause, while noticing "sugar dissolves faster in summer" would only be a correlation.

What to look for: sugar dissolves noticeably faster in warm water. Controlled variables (same amount of water, same cube size, same stirring) and random order are what make the comparison fair.

Your turn

1

Suppose that over the past 15 years, the number of smartphones in Canada and the number of craft breweries both rose steadily, giving r close to 1. Classify the relationship.

2

Students who eat breakfast have higher marks. Name a possible common-cause factor.

3

Cities with more hospitals have more deaths per year. Explain.

4

Heavier cars use more fuel per 100 km. Classify the relationship.

5

People who are depressed tend to sleep poorly. Why is the direction of cause unclear?

6

Why can't an observational study usually prove cause and effect?

7

What does random assignment do in an experiment?

More practice: warm-up

1

Name the curriculum's example of a cause-and-effect relationship.

2

Name the curriculum's example of a common-cause relationship.

3

Name the curriculum's example of an accidental relationship.

4

In a common-cause relationship, does X cause Y?

5

What group in an experiment does not receive the treatment?

More practice: on your own

1

The more time teens spend on social media, the lower their self-esteem scores, in a survey. Give three different explanations.

2

Neighbourhoods with more trees have lower summer temperatures and higher house prices. Draw (describe) the causal diagram and classify the price–temperature relationship.

3

Over 50 years, the number of people with university degrees and the average height of adults both increased. Classify and explain.

4

Design a simple experiment to test whether background music helps students memorize a list of words.

5

A study of 1000 people found coffee drinkers live longer. A newspaper says "coffee adds years to your life." What is wrong with the headline?

Challenge

1

Explain why the "presumed" category is useful, and give an example of your own.

2

A medical trial gives a new drug to 100 volunteers who ask for it and compares them with 100 who do not want it. Explain the flaw and how to fix it.

3

Why can't we run a randomized experiment to find out whether poverty affects health? How can statisticians still study the question?

Check your understanding

Quiz

1.

Ice cream sales and forest fires are positively correlated. This is an example of:

2.

Tree age and trunk diameter are positively correlated. This is:

3.

The consumer price index and the number of known planets have both increased over time. The relationship is:

4.

Areas with more police officers have more crime because high-crime areas hire more officers. This is:

5.

What is the main purpose of random assignment in an experiment?

6.

Which study design best supports a cause-and-effect conclusion?

7.

A strong correlation exists between X and Y. Which statement must be true?

For parents and teachers

Big idea (D2.2): a correlation can arise from cause and effect, reverse cause, a common cause, coincidence, or a presumed link that has not been explained. Only a well-designed randomized controlled experiment (or a large, consistent body of evidence) can establish cause.

How to help: When you hear a health or news claim ("people who do X live longer"), ask your teen: "What else could explain that?" The sugar-dissolving activity is quick, safe and shows what makes a fair experiment. Supervise the warm water.

Common difficulties:

  • Confusing common cause with reverse cause. Ask: is there a third variable?
  • Thinking a very large r proves cause.
  • Forgetting that time can be a hidden common cause for two trends.
  • Believing a large survey can replace an experiment.