Why a Third Variable Matters
Suppose a regression analysis finds that people who exercise more tend to have lower blood pressure. That association is useful to describe, but it does not by itself show that exercise caused the difference. Age, for example, might influence both how much people exercise and their blood pressure. If the groups being compared differ in age, their overall association can reflect both the exercise–blood-pressure relationship and the age differences between them.
A variable that helps explain both the predictor and the outcome is therefore important when interpreting an explanatory question. A simple causal diagram makes the concern visible: it records proposed causal links without claiming that data alone have proved those links.
Reading a Simple Causal Diagram
In a causal diagram, variables are represented by labels and arrows indicate proposed causal influence. Let \(C\) stand for a possible common cause. The basic pattern is:
The arrows say that changes in \(C\) could affect \(X\), and could also affect \(Y\). They do not say that \(X\) causes \(Y\). If the research question is whether \(X\) affects \(Y\), the diagram may include that proposed link as well:
Here \(X \longrightarrow Y\) is the causal effect of interest, while the other two arrows show why a comparison of different \(X\) values may also compare different values of \(C\). The diagram is a summary of substantive assumptions: it is not a result that a regression calculation can establish. Deciding whether the arrows are plausible requires knowledge of the variables, their timing, and how the data were collected.
Worked Example: Exercise, Age, and Blood Pressure
Imagine a hypothetical observational data set of 100 adults. The predictor \(X\) is whether a person exercises regularly, and the outcome \(Y\) is systolic blood pressure in millimetres of mercury (mm Hg). Consider age group as a possible common cause.
| Age group | Exercise group | Number of people | Mean blood pressure |
|---|---|---|---|
| Younger | Regular exercise | 30 | 118 mm Hg |
| Younger | No regular exercise | 20 | 120 mm Hg |
| Older | Regular exercise | 10 | 138 mm Hg |
| Older | No regular exercise | 40 | 140 mm Hg |
The proposed diagram is \(C \longrightarrow X\) and \(C \longrightarrow Y\), where \(C\) is age group. The table is consistent with age group influencing exercise habits: 30 of 50 younger people exercise regularly, compared with 10 of 50 older people. It is also consistent with age group influencing blood pressure: the means are higher in the older group.
Across all ages, the mean for the regular-exercise group is
The mean for the group without regular exercise is
The overall difference is \(123-133.3=-10.3\) mm Hg: the regular-exercise group has a mean about 10.3 mm Hg lower. But within the younger group the difference is \(118-120=-2\) mm Hg, and within the older group it is \(138-140=-2\) mm Hg. The groups have different age compositions, so the overall comparison combines within-age differences with differences between age groups. These hypothetical numbers do not establish what exercise causes; they show why the overall association alone does not isolate that effect.
How a Common Cause Can Shape an Association
If \(C\) affects both \(X\) and \(Y\), people with different values of \(X\) may also tend to have different values of \(C\). Since \(C\) affects \(Y\), part of the observed difference in \(Y\) across values of \(X\) may be associated with those differences in \(C\). The association between \(X\) and \(Y\) can therefore reflect more than the proposed causal effect of \(X\) on \(Y\).
The direction and size of this influence depend on the particular variables and data. A common cause can make an association appear stronger, make it appear weaker, or even contribute to an association that points in the opposite direction from a within-group pattern. The exercise example illustrates how a pooled comparison can differ from comparisons within age groups. The key point is not that every observed association is caused by a third variable; it is that a plausible common cause gives a reason to be cautious about a causal interpretation.
Worked Example: Ice-Cream Sales and Drownings
Consider a hypothetical set of weeks in which higher ice-cream sales \(X\) occur alongside more drownings \(Y\). A tempting interpretation would be that higher ice-cream sales cause more drownings. A simple alternative is that warmer weather \(C\) contributes to both more ice-cream sales and more swimming, which creates more opportunities for drownings.
The proposed common-cause pattern is
For a numerical illustration, suppose there are 10 hypothetical cool weeks and 10 hypothetical hot weeks. Average weekly sales are 20 units in cool weeks and 80 units in hot weeks; average weekly drownings are 1 in cool weeks and 5 in hot weeks. With equal numbers of weeks in each group, the overall average sales are
and the overall average number of drownings is
The higher averages occur in hot weeks for both variables, so a positive overall association is plausible. Yet this comparison does not show that buying ice cream causes drownings. The weather-based explanation supplies a common cause that could produce the association even if changing ice-cream sales would not change the number of drownings. The diagram is a useful alternative explanation to investigate, not proof that weather is the only cause.
Common Causes Are Not Every Third Variable
A variable does not become a common cause merely because it is measured alongside \(X\) and \(Y\), or because it is associated with both. The common-cause idea makes a specific claim about causal direction: \(C\) influences \(X\) and \(Y\). Timing can help assess whether this story is plausible. A variable recorded only after the outcome cannot ordinarily be the cause of that earlier outcome, though the timing of measurement alone does not establish all causal relationships.
It also matters whether the variable is instead part of the pathway from \(X\) to \(Y\). For example, if a treatment changes a biological process, which then changes an outcome, that process is an intermediate variable in the proposed causal story. Its role is different from a variable that precedes and influences both treatment and outcome. These roles should not be treated as interchangeable when interpreting an explanatory question.
Worked Example: Shift Work, Caffeine, and Sleep
Suppose a hypothetical study asks whether daily caffeine consumption \(X\) affects sleep duration \(Y\). Some people work night shifts, while others work daytime schedules. Shift work \(C\) could lead people to consume more caffeine and could also affect when, or how long, they sleep. A plausible diagram is
The relevant question is not simply whether shift work is recorded in the data. It is whether the proposed causal links make sense for the study’s people, measurements, and timing. If they do, caffeine users and non-users might differ in their work schedules, and those schedule differences could contribute to a difference in sleep duration. A regression association between caffeine and sleep would then be difficult to interpret as the effect of caffeine alone.
By contrast, if the proposed story is that caffeine changes sleep duration partly through a later change in alertness, alertness would be an intermediate step in that story, not automatically a common cause of caffeine use and sleep. Drawing the arrows forces the analyst to state which role is being proposed instead of treating every third variable as equivalent.
What the Diagram Does and Does Not Establish
A causal diagram is a compact way to make assumptions explicit. It helps ask whether a third variable could influence both the predictor and outcome, and whether the observed association might therefore have more than one explanation. It does not prove that any arrow is correct, measure the size of a causal effect, or guarantee that a statistical adjustment will remove the concern.
If a possible common cause is measured, comparing people with similar values of that variable can help clarify the association. But that comparison is not automatically decisive. Other common causes might remain unmeasured, measurements might be inaccurate, and the chosen groups might not be comparable. The first task is to identify and justify the causal structure; a regression formula cannot replace that reasoning.
This concern is specific to explanatory interpretation. A regression may still describe an association accurately, or help make predictions, even when the association cannot be interpreted as the causal effect of \(X\) on \(Y\). The intended question matters: describing a pattern and claiming an effect are different goals.
Common Mistake
- Wrong reasoning: “The regression includes the predictor and outcome, so their association must be the predictor’s effect on the outcome.”
- Correction: A regression association does not by itself identify a causal effect. Ask whether a plausible \(C\) could influence both \(X\) and \(Y\), and show those proposed links in a simple diagram. Then interpret the regression in light of the design and assumptions, not just the fitted relationship.
Check Your Understanding
For each question, state the proposed common cause and explain what it could imply for an explanatory interpretation.
- A hypothetical analysis finds that people who spend more time outdoors buy more sunscreen and report more sunburns. Draw a simple diagram for a possible common cause.
- In the exercise and blood-pressure example, why is the overall difference not the same as either within-age difference?
- A study asks whether a training program affects test scores. Explain the difference between a variable that influences both program participation and scores and a variable that lies on the proposed pathway from the program to scores.
- Does drawing \(C \longrightarrow X\) and \(C \longrightarrow Y\) prove that \(C\) causes both variables? Explain.
- Why might an association still be useful for prediction even if a common cause makes its causal interpretation uncertain?