When Two Variables Share a Common Response
A scatterplot or regression can show that two variables tend to vary together. As in “Association Versus Causation in Regression,” that pattern alone does not establish that changing one variable causes the other to change. A third variable may help explain why the two variables are associated.
One important possibility is that both variables respond to the same third variable. For example, among children, shoe size and reading ability may both tend to increase with age. That does not mean larger shoes improve reading. As children grow older, their feet tend to grow, and they also tend to gain reading experience and skill. Age is a possible common response: both shoe size and reading ability may respond to it.
Suppose \(X\) is shoe size, \(Y\) is reading ability, and \(Z\) is age. The possible pattern can be represented as \(Z \to X\) and \(Z \to Y\). The arrows describe possible influence from age to each measurement; they do not say that shoe size causes reading ability or vice versa. The association between \(X\) and \(Y\) may partly reflect their shared relationship with \(Z\).
A common response is a specific kind of third-variable explanation. In “Observational Studies and Confounding,” you learned that a third variable related to both an explanatory variable and a response can mix together their relationships. Here the emphasis is on the shared pathway: both variables may respond to the same factor. These descriptions can overlap. Naming age as a possible common response does not establish that it explains all, or even most, of the observed association.
How to Evaluate a Common-Response Explanation
A useful explanation names the third variable and connects it separately to each of the two variables. Simply saying “there may be another factor” is too vague. Ask whether the proposed third variable could plausibly influence, or be related to, both measurements in the setting being studied.
Identify the two variables and describe how they vary together in the observed cases.
Name a factor that is relevant to the individuals or objects in the study, rather than offering a generic possibility.
Describe how the third variable could be related to or influence each of the two variables. A complete explanation needs both links.
Say that the third variable may help explain the association. Do not state that it definitely accounts for the pattern unless the study supports that claim.
Timing and the way variables are measured can help assess whether the explanation is plausible. For example, age is recorded for each child and could be connected to both physical growth and reading development. By contrast, a proposed factor that cannot plausibly relate to one of the measurements is not a strong common-response explanation.
When a study has measurements of the third variable, it may be useful to look at the association between the other two variables among cases with similar values of that factor. If the association is weaker within age groups than across all children, that pattern is consistent with age helping explain the overall association. It still does not prove that age is the only explanation. If the third variable was not measured, its role can be proposed as a possibility, but not directly checked from those data.
Worked Examples: Finding a Shared Factor
Worked Example: Shoe Size and Reading Ability
A fictional school district records shoe size and a reading assessment score for 180 children ages 6 through 12. In the full group, children with larger shoe sizes tend to have higher reading scores. The study is observational: the researchers record measurements and do not assign shoe sizes or ages.
State. Determine whether the positive association between shoe size and reading score supports the claim that larger shoe size improves reading ability.
Plan. Consider a common-response explanation. Identify a third variable that could be related to both measurements, and explain each connection. Because this is an observational study, describe the association rather than treating it as a demonstrated causal effect.
Do. Age is a plausible third variable. Older children generally have had more time for physical growth, which may be related to larger shoe size. They also generally have had more time in school and more reading instruction, which may be related to higher reading scores. Thus, age may be associated with both shoe size and reading ability. This shared relationship could contribute to the positive association in the full group. The information given does not show whether age accounts for all of it.
Conclude. “Among the 180 children ages 6 through 12, shoe size and reading score were positively associated. Age may help explain this pattern because it is related to physical growth and may also be related to reading development. The observational association does not show that larger shoe size improves reading ability.” This identifies the cases, the pattern, and both links to the possible common response without presenting the explanation as proven.
Worked Example: Ice-Cream Sales and Swimming Rescues
A fictional coastal town reviews records for 48 summer days. On days with more ice-cream sales, the town also tends to record more swimming rescues. A visitor suggests that buying ice cream makes people more likely to need rescue.
State. Assess whether the association between daily ice-cream sales and rescue counts supports that causal claim.
Plan. Identify a possible third variable that could be related to both daily measurements. Check that it offers a plausible connection to ice-cream sales and to swimming activity or rescue counts. Then state what the observational records can and cannot establish.
Do. Weather, especially temperature, is a plausible common response. Warmer days may encourage more people to buy ice cream. Warm, sunny days may also draw more people to the beach, creating more opportunities for swimming and therefore more opportunities for rescues. The records show that sales and rescue counts tend to be higher on the same days, but they do not show that purchasing ice cream causes rescues. The records also do not establish that temperature fully explains the association.
Conclude. “Across the 48 summer days, ice-cream sales and swimming rescue counts were positively associated. Warmer weather may help explain the association because it could increase both ice-cream purchases and beach activity. These observational records do not show that buying ice cream causes a rescue.” A complete explanation makes the two links to weather explicit rather than just naming “the season.”
Worked Example: Screen Time and Bedtime
A fictional survey asks 240 middle-school students to report their recreational screen time on a typical evening and the time they usually go to bed. Students reporting more screen time also tend to report later bedtimes. A family counselor wonders whether screen use alone explains the pattern.
State. Identify a possible common response that could contribute to the association, and explain how it might relate to both variables.
Plan. Think of a factor that could affect students’ opportunities for screen use and their bedtime. Describe the connection to each measurement and avoid claiming that the factor is known to explain the association.
Do. Household evening schedules are a plausible third variable. In some households, students may have more unscheduled time in the evening, which could allow more recreational screen use. The same schedules may also permit or lead to later bedtimes. The data are self-reported and observational, so this explanation is possible rather than established. Other factors could also contribute, and screen use itself might be related to bedtime.
Conclude. “Among the 240 students surveyed, reported evening screen time and usual bedtime were positively associated. Household evening schedules may contribute to the pattern if they are related to both students’ screen use and their bedtimes. The survey alone does not show that screen use causes later bedtimes or establish that household schedules explain the association.” This conclusion keeps the possible common response separate from a claim about direct causation.
What a Common Response Does—and Does Not—Show
A common-response explanation helps you think beyond the two variables in a regression. It can show why an association might appear without a direct causal effect from one measured variable to the other. The explanation is especially useful when the third variable has a clear connection to the study context, such as age for children’s growth and development or weather for outdoor activity.
However, a plausible pathway is not a finding by itself. If age, temperature, or household schedules were not measured, the study cannot use those data to compare cases with similar values of that factor. Even when the third variable was measured, an observed pattern within groups does not, on its own, establish the complete causal story. Other variables, measurement limitations, or direct relationships may remain relevant.
The same third variable can also affect the interpretation of a regression slope. The slope describes the fitted association in the data. If both \(X\) and \(Y\) are related to \(Z\), their overall association may partly reflect how the cases differ in \(Z\). Be careful not to interpret the slope as the effect of changing \(X\) while all other influences are held constant; that claim needs support beyond a simple observational regression.
Common Mistakes and AP Exam Tips
A strong response connects the proposed third variable to both measured variables. It also distinguishes what the data show from what the explanation suggests. These are common errors to avoid:
- Naming only one connection. Saying “age affects reading” is not enough to explain the shoe-size and reading association. Explain how age may also be related to shoe size.
- Claiming that the third variable definitely explains the pattern. Use “may contribute” or “could help explain” unless the study provides stronger evidence. A plausible mechanism is not proof.
- Turning association into causation. The association between shoe size and reading ability does not establish that changing shoe size changes reading ability.
- Confusing the roles of the variables. A common response is a third variable, not simply another name for \(X\) or \(Y\). Identify the two associated variables and the shared factor separately.
- Ignoring the cases and context. Specify whose measurements show the association—for example, the children in the fictional school district or the days in the town’s records.
For full credit, state the observed association in context, name a plausible third variable, and explain how it could be related to each measured variable. Then limit the claim: the third variable may help explain the association, but the observational pattern alone does not prove that explanation or a direct causal effect.
Check Your Understanding
For each situation, name a possible common response and explain its possible connection to both variables.
- Among young children, height and the number of words they can read are positively associated. Give a plausible common response and explain both links.
- Across several weeks, park attendance and cold-drink sales tend to be higher on the same days. Identify a possible common response and explain why it could relate to both measurements.
- In a survey, students who report more evening chores also tend to report less time for hobbies. Name a possible third variable and explain how it might relate to both.
- Write one sentence that distinguishes a possible common-response explanation from proof that the third variable caused the observed association.
- Why is naming a third variable without explaining its connection to both measured variables an incomplete answer?