Math 115: Introduction to Statistics – 8. Associations and Causation

Where We’ve Been

In 8a we built the tools to describe a relationship between two categorical variables — contingency tables, conditional proportions, standardized bar plots.

If two variables are associated, the natural next question is: does one cause changes in the other?

Today’s question: when can we draw a causal conclusion from data, and when can we only say “associated”?

Explanatory and Response Variables

When we suspect one variable may affect another:

Explanatory variable: the variable that might influence or explain the other
- Also called: independent variable, predictor
Response variable: the variable that might be affected
- Also called: dependent variable, outcome

Not every study has clear explanatory/response roles — sometimes we just have two variables and ask if they’re associated.

Two Scope-of-Inference Questions

We’ve already met one scope-of-inference question (Topic 5):

Generalization — can we apply findings beyond the sampled cases?
- Answered by: how we sampled (random vs. convenience)

Today’s new question:

Causation — can we conclude one variable causes changes in the other?
- Answered by: how we conducted the study

These are two separate questions.

Observational Study vs. Experiment

Observational study: researchers observe without intervening

Cases observed “as they are”
No control over the explanatory variable
Example: a survey asking about diet and health

Experiment: researchers deliberately manipulate the explanatory variable

Cases assigned to treatment conditions
Researchers control which case gets which treatment
Example: randomly assigning patients to a drug or a placebo

Observational or Experiment?

For each scenario, decide: observational study or experiment?

A researcher surveys 500 college students about hours of sleep per night and current GPA. Observational. The researcher just records what students report — no one is assigned a sleep amount.
A pharmaceutical company randomly assigns 200 patients with chronic migraines to receive either a new drug or a placebo for 12 weeks. Experiment. The researcher controls who gets which treatment via random assignment.
A school district compares math test scores between students enrolled in a flipped-classroom section and those in a traditional section. Students chose their own section. Observational. The district records scores by section but didn’t assign students to a section — students self-selected.

Your turn. For each scenario, decide whether it’s an observational study or an experiment — the key question is whether the researcher intervened by assigning cases to treatment groups. Advance when you are ready to see the answers. Scenario one — a survey of five hundred college students about sleep and GPA. The researcher just records what students report, without assigning anyone a sleep amount. That’s observational. Scenario two — a pharmaceutical company randomly assigns two hundred chronic-migraine patients to either a new drug or a placebo. The researcher controls who gets which treatment, and the assignment is random. That’s an experiment. Scenario three — a school district compares test scores by classroom section, but students chose their own section. The district recorded the scores; it didn’t assign students to a section. That makes it observational, even though it’s a comparison.

Ice Cream and Drowning

A surprising finding: ice cream sales and drowning deaths are strongly positively associated.

Ice cream sales high → drownings high
Ice cream sales low → drownings low

Does eating ice cream cause drowning?

No! Both go up together because of a third variable that drives them both:

Hot weather → more ice cream sales
Hot weather → more swimming → more drownings

Confounding Variables

A confounding variable is associated with both the explanatory and response variables.

flowchart TD
    A[Temperature<br/>confounder] --> B[Ice cream sales]
    A --> C[Drowning deaths]

    style A fill:#ffcccc,stroke:#cc0000,stroke-width:3px
    style B fill:#cce5ff,stroke:#0066cc,stroke-width:2px
    style C fill:#cce5ff,stroke:#0066cc,stroke-width:2px

Why it matters: a confounder creates an alternative explanation for the association. In an observational study, we usually can’t rule confounders out — so we can only say variables are associated, not that one causes the other.

Spot the Confounder

For each observed association, name a plausible confounder that could explain it without any direct causal link.

Children who eat breakfast tend to score higher on standardized tests. Family income / home environment. Households with more resources are more likely to ensure breakfast and provide tutoring, books, and stable study conditions — driving both the breakfast and the score up.
Children with bigger feet tend to be better readers. Age. Older children have bigger feet and more years of reading instruction. Compare children of the same age and the foot–reading association disappears.
Cities with more police officers tend to have more reported crime. City size. Big cities have more officers and more crime. The association is driven by population — not by officers causing crime.

Your turn — three real associations, three plausible confounders to name. For each one, the question is the same. What third variable could be creating the association without any direct causal link? Pause here and try each one, then advance when you are ready to see the answers. First — children who eat breakfast tend to score higher on standardized tests. Does eating breakfast directly cause higher scores? Maybe in part — but family resources are a plausible alternative explanation. Households that can reliably provide breakfast may also provide tutoring, books, and stable study conditions, all of which push scores up. The breakfast association could be real, or family environment could be driving much of it, and an observational study can’t tell the two apart. Next — children with bigger feet tend to be better readers. Does foot growth somehow cause better reading? Obviously not — age is the confounder. Older kids have bigger feet and more years of reading instruction. If you only compared seven-year-olds to seven-year-olds, the foot-reading association would vanish — it was age all along. Last one — cities with more police officers tend to have more reported crime. Are officers somehow causing crime? A plausible confounder is city size — big cities have more officers and more crime, so much of the police-crime association may simply reflect population.

Random Assignment

The fix for confounding: random assignment.

Random assignment: cases are randomly assigned to treatment groups

Researchers control who gets which treatment
Assignment is by chance — not by patient preference, doctor recommendation, or any other characteristic

Key benefit: on average, random assignment balances the groups across every possible confounder — including ones we haven’t thought of, because chance doesn’t systematically sort patients into one group or the other.

Stent Study

The SAMMPRIS clinical trial (Chimowitz et al., NEJM 2011) tested whether a stent — a small mesh tube placed in a narrowed artery — prevents strokes in at-risk patients.

451 at-risk patients assigned to receive a stent (treatment) or standard medical care alone (control)
Outcome (“stroke” or “no event”) recorded after 365 days
Explanatory: group (treatment / control)
Response: outcome (stroke / no event)

Stent Study — Random Assignment

Each of the 451 patients was randomly assigned to treatment or control — by chance, not by patient or doctor choice.

Chance doesn’t pick favorites: an older patient is no more likely than a younger one to receive a stent — same for sicker patients, smokers, and so on.

Result: on average, the two groups end up similar in age, stenosis severity (artery narrowing), other health conditions — in every way except the stent itself.

What If Observational?

If patients had chosen whether to get a stent, severe stenosis would push both ways:

More likely to choose a stent (doctors recommend it)
More likely to have a stroke (artery risk, independent of treatment)

flowchart TD
    A[Stenosis severity<br/>confounder] --> B[Stent treatment]
    A --> C[Stroke outcome]

    style A fill:#ffcccc,stroke:#cc0000,stroke-width:3px
    style B fill:#cce5ff,stroke:#0066cc,stroke-width:2px
    style C fill:#cce5ff,stroke:#0066cc,stroke-width:2px

Random assignment breaks this link.

Stent Study Results

Stent study outcomes
Group	Stroke	No event
control	28	199
treatment	45	179

Stroke proportions: control 12%, treatment 20%.

It turns out this difference is statistically significant.

Because random assignment ruled out confounders, we can conclude the stent caused more strokes — a result that surprised researchers and changed clinical practice.

Random Assignment vs. Random Sampling

These sound similar but answer different questions:

	Random Sampling	Random Assignment
What	How cases are selected from a population	How cases are assigned to treatment groups
Purpose	Enables generalization	Enables causal conclusions
Controls for	Sampling bias	Confounding variables

A study can do either, both, or neither.

Recycling — Observational

Do signs on recycling bins reduce contamination? Students compared two bins for a month:

Student Center (public): no sign
Lichty Hall (residence): sign

Lower contamination at the signed bin.

But location is a confounder — public vs. residential bins may behave differently regardless of signs.

Recycling — Experiment

Same question, better design: 60 bins across campus; randomly assign 30 to display a sign for the month, 30 without. Record contamination per bin.

Suppose the same finding: lower contamination on the sign bins.

Random assignment balanced location types — a residential bin was no more likely to get the sign than a public one. The signs caused the reduction.

Judging Scope of Inference

For each study, what can we conclude? Causal or association? Generalize, or not?

A pharmaceutical company draws a random sample of 500 patients with chronic migraines from a national registry. Each is randomly assigned to a new drug or a placebo for 12 weeks. Causal + generalizable. Random assignment rules out confounders → causation. Random sampling → generalize to the registry population.
A psychology researcher recruits 50 students from her own intro class as volunteers. She randomly assigns each student to read a fiction or non-fiction passage, then measures empathy. Causal, not generalizable. Random assignment → causation. Convenience sample (her students) → conclusions don’t generalize beyond them.
A health survey randomly samples 1,000 adults from a city and compares exercise frequency (high vs. low) to self-reported stress levels. Association + generalizable. Random sample → generalizes to the city’s adults. No random assignment (researchers didn’t set exercise levels) → confounders possible (e.g., income, age) → only association.

Your turn — three studies, three calls. For each one, decide what we can conclude — causal or association only, generalize or not. Pause here, work each one through, then advance when you are ready to see the answers. First — a pharmaceutical company draws a random sample of five hundred chronic-migraine patients from a national registry, then randomly assigns each patient to a new drug or to a placebo for twelve weeks. Both kinds of randomness are present. Random sampling means the findings generalize to the registry population. Random assignment means we can call the effect causal. Best of both worlds. Next — a psychology researcher recruits fifty students from her own intro class, then randomly assigns each student to read a fiction or non-fiction passage, and measures empathy afterward. Random assignment is in place, so we can claim a causal effect within the sample. But the sample is a convenience sample — her own students — so we can’t generalize beyond her class. For the last scenario, a health survey randomly samples one thousand adults from a city and compares exercise frequency — high or low — against self-reported stress levels. Random sampling means the findings generalize to the city’s adult population. The researchers never dictated exercise levels, though, so there was no random assignment. That leaves confounders like income or age potentially driving the association, and the strongest conclusion we can draw is that exercise frequency and stress are associated.

Summary

Two separate questions about what we can conclude from a study:

	Random Sampling	No Random Sampling
Random Assignment	Causal + Generalizable	Causal, not generalizable
No Random Assignment	Association + Generalizable	Association only

Generalization → how did we sample?
Causation → how did we assign cases to groups?

Here’s the punchline of the lecture, laid out as a two-by-two. The columns describe how cases were sampled — random or not. The rows describe how cases were assigned to treatment groups — randomized or observational. Each cell tells us what kind of conclusion the combination supports. The top-left cell is the gold standard: random sampling combined with random assignment lets us both generalize and claim causation. The top-right cell is more typical of clinical trials, where random assignment supports a causal claim but a convenience sample limits generalization. The bottom-left cell is the typical large survey: random sampling generalizes the findings, but without random assignment we can only say the variables are associated. And the bottom-right cell is what we usually have to work with: neither kind of randomness, so we can only say the variables are associated within the cases we happened to observe. With two questions and two design choices, you have the framework you’ll use every time you read a study from now on.

References

Introduction to Modern Statistics (2e) textbook by Mine Çetinkaya-Rundel and Johanna Hardin
Sections 1.1, 1.2, 2.2, 2.3, 4.1, 4.2, 4.3
Chimowitz MI, et al. Stenting versus aggressive medical therapy for intracranial arterial stenosis. N Engl J Med. 2011;365(11):993–1003. PMID 21899409