Difficulty: Introductory | Prerequisites: Basic understanding of variables (explanatory vs response). If you have not covered Section 1.2, review that first.
This section sits right at the start of the course and addresses a question you will return to all semester: when can you say one thing causes another? Most of the data you encounter in everyday life (news headlines, health claims, social media stats) shows association, not causation. Understanding the difference, and knowing what kind of study design lets you make causal claims, is foundational to every statistical method that follows. You should already be comfortable identifying explanatory and response variables from Section 1.2.
Two variables can be associated (their values move together) without one causing the other, often because a confounding variable drives both. Observational studies can reveal associations but cannot establish causation. Randomised experiments, which use control, randomisation, and replication, are the gold standard for establishing causal relationships.
Association
Two variables are associated if values of one variable tend to be related to values of the other variable. In simple terms, when one goes up (or changes), the other tends to move in a predictable way.
Causation / causal association
Two variables are causally associated if changing the value of the explanatory variable directly influences the value of the response variable. Think of it as: tweaking the input reliably changes the output.
Confounding variable (confounding factor, lurking variable)
A third variable that is associated with both the explanatory variable and the response variable, offering a plausible alternative explanation for the observed association. In simple terms, this is the hidden factor that might be the real reason two things look connected.
Observational study
A study in which the researcher does not actively control the value of any variable, but simply observes the values as they naturally exist. Think of it as watching and recording without interfering.
Experiment
A study in which the researcher actively controls one or more of the explanatory variables. In simple terms, the researcher deliberately changes something to see what happens.
Experimental units / subjects
The individuals in an experiment. When the experimental units are human, they are called subjects.
Factor
An explanatory variable in an experiment. Think of it as the thing the researcher is testing.
Levels
The specific values or categories of a factor being compared. For example, two different doses of a drug are two levels of the factor "drug dose."
Treatment
A specific experimental condition applied to experimental units. Each treatment is formed by combining a specific level of each factor.
Nuisance variables (lurking variables)
Other variables that influence the response variable but are not the focus of the study.
Statistically significant
A response to a treatment is statistically significant if it is larger than you would expect by chance alone (due to random variation among the subjects). Think of it as: the result is too big to be a coincidence.
Confounded
Two variables are confounded when their effects on a response variable cannot be distinguished from each other.
Placebo
A fake treatment that resembles the active treatment as closely as possible. Its purpose is to control for the psychological effect of receiving any treatment at all.
Placebo effect
The tendency of patients to respond favourably to any treatment, even a sham one, because of trust in the doctor, expectation of improvement, or the natural course of the illness.
Control group
The group of experimental units that receives no treatment or a placebo treatment. It serves as the baseline for comparison.
Double-blind experiment
An experiment in which neither the subjects nor the experimenter knows which individuals received which treatment until the study is complete. This avoids placebo effects and interpretation bias.
Completely randomised experiment (randomised comparative experiment)
An experiment in which all experimental units are allocated at random among all treatments.
Matched pairs experiment
An experiment that compares exactly two treatments by either pairing similar subjects and randomly assigning one to each treatment, or giving both treatments to the same subject in random order.
Bias (in an experiment)
A systematic favouring of certain outcomes, often introduced when treatment groups differ in some way other than the treatment itself.
Randomisation
The process of allocating experimental units to treatments by chance, ensuring treatment groups are probabilistically alike on all lurking variables.
Replication
Applying each treatment to enough subjects to reduce chance variation in the results and ensure that findings are not due to uncontrolled factors.
The central rule of this section: association is not necessarily causal.
Two variables can move together (positive association) or move in opposite directions (negative association) without one causing the other. The classic example from the lecture: countries with more TVs per 1,000 people tend to have higher life expectancy (r = 0.74). Buying more TVs does not make people live longer. A third variable, national wealth, drives both TV ownership and access to healthcare.
How to spot whether a headline implies causation:
Causal language uses words like "improves," "causes," "reduces," "leads to." Example: "Daily Exercise Improves Mental Performance" implies causation (exercise = explanatory, mental performance = response).
Associational language uses words like "linked to," "tend to," "related to." Example: "Cat owners tend to be more educated than dog owners" implies association only (no claim that owning a cat makes you smarter).
For more examples of associations that are clearly not causal (spurious correlations), the lecture references Tyler Vigen's website.
A confounding variable is associated with both the explanatory and the response variable, and it offers an alternative explanation for the observed relationship. Whenever confounding variables are present (or may be present), a causal association cannot be determined.
Classic examples of confounding:
Ice cream sales and drowning deaths both rise in summer. The confounding variable is temperature/season, not a causal link between ice cream and drowning.
People who own a yacht are more likely to buy a sports car. The confounding variable is wealth.
People with shorter hair tend to be taller. The confounding variable is sex (on average, men tend to have shorter hair and tend to be taller).
Air pollution is higher in places with more paved ground. The confounding variable is urbanisation (cities have both more pavement and more pollution sources).
Beef and pork consumption worldwide have both risen over the past 100 years. The confounding variable is population growth.
The confounding variable framework:
Think of three nodes: Explanatory Variable, Response Variable, and Confounding Variable. The confounding variable has arrows pointing to both the explanatory and the response variable. Because it is linked to both, you cannot tell whether the explanatory variable is doing the work or whether the confounder is responsible for the pattern you see.
The distinction matters because it determines whether you can make causal claims.
Observational study: the researcher watches and records without manipulating anything. Confounding variables are almost always present, so observational studies generally cannot establish causation.
Example: Observing which male crickets female crickets choose to mate with, then checking the health of the chosen males.
Example: Surveying smokers and non-smokers, then comparing cancer rates. The researcher does not assign anyone to smoke.
Example: A political scientist surveys voters about which candidate they supported. She does not assign anyone a candidate.
Experiment: the researcher actively controls one or more explanatory variables (the factors).
Example: Deliberately infecting some male crickets with parasites and seeing whether females avoid them.
Example: Giving students brand A or brand B of coffee in random order, then asking which they prefer. The researcher controls which coffee each student tries.
Example: Having subjects walk on a treadmill and measuring metabolic rate before, immediately after, and 12 hours after.
The key test: did the researcher assign the explanatory variable, or did it occur naturally? If assigned, it is an experiment. If observed as-is, it is an observational study.
Students often think that a strong correlation (e.g. r = 0.74) proves causation. It does not. Strength of association says nothing about whether the relationship is causal.
Students confuse "observational study" with "casual observation." An observational study can be rigorous and carefully designed; the defining feature is that the researcher does not manipulate variables.
Students sometimes believe that any study with a control group is an experiment. A study can compare groups without the researcher having assigned individuals to those groups (that would be an observational study with a comparison group, not an experiment).
Students often think confounding variables are mistakes in the study. They are not errors; they are features of the real world that make causal interpretation difficult.
"Association is not necessarily causal" is the single most repeated statement in this section. Expect to see it on the exam.
Be ready to identify confounding variables in a described scenario. The exam will likely give you a relationship and ask you to name a plausible confounder.
You will be asked to classify studies as observational or experimental. Use the key test: did the researcher assign the explanatory variable?
Know the three principles of experiments (control, randomise, replicate) and be able to identify whether a described experiment follows them.
Understand why a study without a control group is problematic (the gastric freezing example is a favourite exam scenario).
Be able to distinguish between a completely randomised experiment and a matched pairs experiment, and know when each is more appropriate.
True or False: A correlation of r = 0.90 between two variables proves that one causes the other.
False. Correlation measures association, not causation.
Fill in the blank: A variable associated with both the explanatory and the response variable is called a __________ variable.
Confounding (also accepted: lurking, confounding factor).
True or False: In an observational study, the researcher assigns subjects to treatment groups.
False. The researcher observes values as they naturally exist.
True or False: Randomisation helps make treatment groups alike on lurking variables.
True.
Fill in the blank: The three principles of experimental design are control, __________, and replication.
Randomisation.
Q: Countries with more TVs per 1,000 people tend to have higher life expectancy. Does this mean buying more TVs increases life expectancy? Identify a confounding variable.
A: No. National wealth is a confounding variable. Wealthier countries can afford both more TVs and better healthcare, which increases life expectancy.
Q: A researcher surveys a large sample of adults, records whether they smoked regularly and whether they have cancer, and finds smokers have higher cancer rates. Is this an experiment or an observational study? Can the researcher conclude that smoking causes cancer from this study alone?
A: This is an observational study, because the researcher did not assign anyone to smoke. The researcher cannot conclude causation from this study alone, because confounding variables may be present.
Q: A group of college students visits a nursing home weekly, talks with residents, and serves them herbal tea. After several months, residents are healthier. What is the explanatory variable? What is a lurking variable?
A: The explanatory variable is the herbal tea. A lurking variable is the visits and social interaction from the college students, which could also explain the improvement.
Q: In the Hydroxyurea study, 150 sickle-cell sufferers received the drug and another 150 received a placebo. Identify the experimental units, the treatments, the factor, and the response variable.
A: Experimental units (subjects): the 300 sickle-cell sufferers. Treatments: Hydroxyurea and placebo. Factor: type of pill received. Response variable: number of episodes of pain reported.
Q: Researchers give a new allergy medication to 50 volunteers (no control group) and 44 report improvement. How could this study be improved?
A: By using a control group (option 3 from the lecture). Without a control group, the improvement could be due to the placebo effect or natural recovery, and the treatment effect is confounded with these lurking variables.
This section connects directly to hypothesis testing later in the course: the concept of "statistically significant" introduced here will be formalised with p-values and confidence intervals. Understanding confounding is also essential for regression analysis, where you will learn to control for confounders mathematically. The distinction between observational studies and experiments underpins the entire field of causal inference, which extends well beyond this course into epidemiology, economics, and social science.
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