Why a Shared Cause Matters
Suppose a data set shows that people who use a particular health service have a different rate of illness from people who do not. That observed difference is an association; by itself, it does not establish that using the service caused the difference. The two groups might differ in another way that affects both service use and illness.
For example, age may influence whether a person uses a service and also influence the risk of illness. If users and nonusers have different age distributions, their overall illness rates partly reflect that difference in age composition. A common cause can therefore create an apparent association, or change the size of an association that is also related to an exposure’s causal effect.
The word “common” refers to the fact that \(U\) is a cause of both variables. It does not mean that \(U\) is simply correlated with both of them. Causal order and subject-matter knowledge matter: a variable measured alongside \(X\) and \(Y\) is not a common cause merely because it is associated with each.
Confounding is a feature of the relationship being studied and the causal structure assumed for it. It does not tell us, on its own, whether the observed association is positive or negative, or how large the difference will be. Nor does the presence of a common cause guarantee a large change in every particular data set. The size of the effect depends, among other things, on how strongly \(U\) relates to each variable and how the groups differ in their distribution of \(U\).
How Different Group Composition Can Create an Association
A useful way to see the mechanism is to compare outcome rates within levels of a common cause and then compare the overall rates. Suppose \(U\) has categories \(u\), and define \(R_x=P(Y=1\mid X=x)\), the observed outcome rate among people with exposure status \(x\). The overall rate is a weighted average of the rates within categories of \(U\):
The weights \(P(U=u\mid X=x)\) are the proportions in each category of \(U\) among people with exposure status \(x\). Thus, two overall rates can differ because the within-category outcome rates differ, because the category proportions differ, or both.
If the within-category rates are the same for exposed and unexposed people, write \(r_u=P(Y=1\mid X=1,U=u)=P(Y=1\mid X=0,U=u)\). Subtracting the two weighted averages gives:
The overall difference can therefore be nonzero even though there is no exposed–unexposed difference within any category of \(U\). The group composition alone can produce the crude association. This calculation describes an observed-rate decomposition; a causal interpretation additionally requires appropriate causal assumptions.
Worked Example: Age Creates an Apparent Health-Service Association
Imagine a hypothetical population in which age affects both use of a screening service and the risk of a particular illness. For simplicity, divide people into younger and older age groups. Assume the following outcome rates: 2% in younger people and 12% in older people, whether or not they use the service. Among 100 service users, 80 are older and 20 are younger; among 100 nonusers, 20 are older and 80 are younger. These constructed rates and counts are hypothetical.
Check the common-cause structure. The assumed diagram is \(X \leftarrow U \rightarrow Y\), where \(X\) is service use, \(Y\) is illness, and \(U\) is age group. The example assumes older age makes service use more likely and raises illness risk. It assumes no within-age difference in illness rates by service use.
Calculate the rate among users. Of the 80 older users, the expected number of cases at a 12% rate is \(80(0.12)=9.6\). Of the 20 younger users, it is \(20(0.02)=0.4\). The weighted rate is:
Calculate the rate among nonusers. The corresponding weighted rate is:
The crude comparison is \(0.10-0.04=0.06\), or a 6-percentage-point higher illness rate among users. Yet within each age group the rates are identical: 12% for older people and 2% for younger people. In this hypothetical example, the overall association arises from the different proportions of older people in the two service-use groups, not from a within-age association.
The conclusion is limited to the stipulated example. In real data, rates may differ within age groups, age may be measured less precisely, and other common causes may also be present. A crude difference alone cannot sort those possibilities out.
A Common Cause Can Also Change an Existing Association
Confounding need not create an association from nothing. If exposed and unexposed people differ in their distribution of \(U\), an observed comparison can combine differences within levels of \(U\) with differences in group composition. The crude association may then be larger or smaller than the within-category differences in the example. It is not valid to infer the direction of that change from the label “confounder” alone.
Worked Example: A Shared Cause Enlarges a Risk Difference
Consider a hypothetical exposure and outcome, with a binary background risk factor \(U\). Suppose the outcome rates are 2% without the exposure and 7% with it among people at lower background risk. Among people at higher background risk, the corresponding rates are 20% and 25%. In each group the exposed rate is 5 percentage points higher. Suppose 20% of exposed people are at lower background risk, compared with 80% of unexposed people. These are stipulated hypothetical population values.
Check the assumed structure. The diagram is \(X \leftarrow U \rightarrow Y\), and the example assumes that \(U\) influences exposure status and outcome risk. It also stipulates a 5-percentage-point within-group difference in both categories of \(U\). The calculation describes these given rates and compositions; it does not establish that the same structure holds in a real setting.
Compute the crude exposed rate. Among exposed people, 20% are in the lower-risk group and 80% in the higher-risk group:
Compute the crude unexposed rate. Among unexposed people, the weights are reversed:
The crude difference is \(0.214-0.056=0.158\), or 15.8 percentage points. For comparison, give both exposure groups the same 50% lower-risk and 50% higher-risk composition. The weighted rates are:
With a common 50–50 composition, the difference is 5 percentage points, matching the stipulated difference in each category. The crude difference is larger because the exposed group contains a greater proportion of people at higher background risk. In an actual analysis, whether a comparison with a common composition has a causal interpretation depends on whether the relevant common causes have been measured and whether the causal assumptions are appropriate.
Applied Reasoning About Common Causes
A small diagram helps make a causal claim explicit without treating an observed correlation as a cause. For example, prior preparation may influence both whether students receive tutoring and their later test scores. A simplified diagram is:
The diagram does not say that tutoring has no effect on scores. It says that, even before asking about any effect of tutoring, students who receive tutoring may differ from students who do not because of prior preparation. A comparison of their average scores can reflect both that difference and any difference associated with tutoring itself.
Worked Example: Prior Preparation and Average Test Scores
Suppose, in a hypothetical set of students, those with higher prior preparation average 84 points with tutoring and 80 without tutoring. Those with lower prior preparation average 64 points with tutoring and 60 without. Among 100 tutored students, 80 have higher prior preparation; among 100 students without tutoring, 20 have higher prior preparation. The averages and group sizes are stipulated for illustration.
Check the assumed common cause. Prior preparation is assumed to make tutoring more likely and to be associated with higher later scores. It therefore acts as a common cause in the simplified diagram. The example stipulates a 4-point tutoring-associated difference within each preparation group.
Calculate the average among tutored students. The weighted mean is:
Calculate the average without tutoring. The comparison group has a different preparation mix:
The crude difference is \(80-64=16\) points. To compare students using the same 50–50 preparation mix, calculate:
The crude 16-point difference and the 4-point difference at a common preparation mix answer different descriptive comparisons. The example makes clear why a difference in average scores cannot, by itself, be attributed to tutoring. The stipulated values do not prove a causal effect; real-world interpretation would depend on the design and causal assumptions.
What Makes a Variable a Plausible Common Cause?
To assess a proposed common cause, ask what causal story would make it influence both the exposure and outcome. Consider whether the proposed cause precedes exposure and outcome in the relevant process, and whether there is a defensible mechanism for each arrow. For instance, a background health condition might affect the chance of receiving a treatment and the chance of a later outcome. In contrast, a variable measured after exposure is not automatically a cause of that earlier exposure.
A common cause can be measured accurately, measured only approximately, or not measured at all. If a relevant cause is omitted, the observed exposure–outcome association may remain confounded. If it is measured coarsely, comparisons within the recorded categories may still mix people whose underlying values differ. The numerical examples used clear categories so that the arithmetic could be followed; real variables are not always so tidy.
The definition is causal, but observational data usually record associations rather than arrows. Subject-matter reasoning, timing, and study design are needed to support the claim that \(U\) influences both \(X\) and \(Y\). A regression coefficient does not by itself establish those causal arrows, and adjusting a regression for a variable does not automatically remove confounding.
Common Mistake
Correction: Association with both variables is not enough to establish that the variable causes both. Check the proposed causal timing and mechanisms. Even for a plausible common cause, measurement quality, other unmeasured causes, and the purpose of the comparison matter. The model’s adjusted association is not automatically a causal effect.
The practical value of recognizing common causes is that it changes how an observed association is interpreted. Instead of treating a crude difference as a direct exposure–outcome relationship, ask whether the groups differ in causes of the outcome that also influence exposure. That question motivates careful adjustment decisions, but the decision about which variables to adjust for requires a fuller causal analysis.
Check Your Understanding
Use the definitions and examples above to answer the following questions.
- In \(X \leftarrow U \rightarrow Y\), what does each arrow represent, and what does \(U\) have to do with both \(X\) and \(Y\)?
- Can a crude exposure–outcome association occur when the outcome rates are equal for exposed and unexposed people within every category of a common cause? Explain briefly.
- In the age and screening-service example, what caused the overall rate difference even though the rates matched within each age group?
- In the test-score example, why does the 16-point crude difference not by itself establish that tutoring causes a 16-point increase?
- Why is observing a correlation between a candidate variable and both exposure and outcome insufficient to call it a common cause?