When a Third Variable Changes the Story
In Observational Studies and Their Limits and Retrospective Versus Prospective Studies, you learned that researchers can observe variables without assigning treatments. When two variables are associated in an observational study, it is important to ask whether another variable could help explain the pattern. That third variable may be connected to both variables being studied.
Suppose a town notices that weeks with higher ice-cream sales also tend to have more drownings. The association alone does not mean that buying ice cream causes drownings. A third variable—temperature, for example—may be related to both. Hot weather can increase ice-cream sales, and it can also lead more people to swim, creating more opportunities for drowning.
The explanatory variable is the variable used to help explain or predict changes in another variable. The response is the outcome being studied. A possible confounder is a third variable: it is not the explanatory variable or the response, but it may be connected to both.
Not every unmeasured variable is a confounder. To be a plausible confounder, the variable needs a sensible connection to both variables in the study. Simply naming something that was not measured is not enough; explain how it could be related to the explanatory variable and how it could be related to the response.
How Confounding Mixes Relationships
Imagine that researchers compare the response for different values or groups of an explanatory variable. If those groups also differ in a third variable related to the response, the overall comparison mixes together more than one pattern. The apparent relationship between the explanatory variable and response may be stronger, weaker, or even different in direction from the relationships seen among comparable cases.
One way to investigate a possible confounder is to divide the data into groups, or strata, based on its values or categories. Then compare the explanatory-variable groups within each stratum. This does not automatically remove every source of bias or establish cause and effect, but it can reveal whether the overall pattern is partly connected to differences in the third variable.
For numerical responses, researchers might compare group averages; for categorical responses, they might compare percentages. The key idea is to make like-with-like comparisons when possible. For example, comparing people who experienced similar temperatures may be more informative than comparing all people in one large group with all people in another.
Worked Example: Ice-Cream Sales and Drownings
Worked Example: Ice-Cream Sales and Drownings
A town reviews weekly records and finds that weeks with higher ice-cream sales also tend to have more drownings. Identify the explanatory variable, the response, and a possible lurking variable. Explain how it could account for the association.
If the question is whether ice-cream sales are associated with drownings, ice-cream sales are the explanatory variable, and the number of drownings is the response. A plausible lurking variable is temperature or, more broadly, hot-weather conditions.
Hot weather may be related to higher ice-cream sales because people are more likely to buy cold treats when it is hot. Hot weather may also be related to more swimming or more visits to beaches and pools. With more people in or near the water, the number of drowning incidents could increase. Thus temperature and related swimming activity may contribute to the observed association between ice-cream sales and drownings.
The data described do not establish that ice-cream sales cause drownings. They also do not prove that temperature fully explains the pattern. A careful conclusion is that the two recorded variables are associated in the weeks studied, and that hot weather and swimming activity are plausible confounding factors that could help explain the association.
The example illustrates why a plausible explanation should be specific. “There might be another variable” is too vague. Naming temperature and tracing its possible connection to both sales and drownings gives a reason the association could arise without ice-cream sales causing drownings.
Looking Within Groups Can Reveal a Mixed Pattern
A comparison within strata can help show how an overall association may be mixed with a third variable. Consider a fictional observational review of dehydration signs among outdoor-program participants. Researchers record whether participants received a water reminder, whether the day was hot or cool, and whether participants showed signs of dehydration. Temperature may be related to reminder status and to dehydration risk.
The invented counts are shown below. The reminder groups have different numbers of participants, so percentages within each temperature-and-reminder group are useful for comparing the shares with dehydration signs.
| Day type | Reminder status | Participants with signs | Total participants | Percent with signs |
|---|---|---|---|---|
| Cool | Received reminder | 2 | 100 | 2% |
| Cool | No reminder | 4 | 200 | 2% |
| Hot | Received reminder | 12 | 100 | 12% |
| Hot | No reminder | 120 | 800 | 15% |
On cool days, the percentages are equal: \(2/100=0.02\), or 2%, among participants who received a reminder, and \(4/200=0.02\), also 2%, among participants who did not. On hot days, the percentage is lower among participants who received a reminder: \(12/100=0.12\), or 12%, compared with \(120/800=0.15\), or 15%, among participants who did not.
Now combine the temperature groups. Among participants who received a reminder, \(2+12=14\) of \(100+100=200\) showed signs, for \(14/200=0.07\), or 7%. Among participants who did not receive a reminder, \(4+120=124\) of \(200+800=1000\) showed signs, for \(124/1000=0.124\), or 12.4%. The overall percentage is lower among participants who received reminders.
However, the temperature mix differs between the groups. Half of the reminder group was on hot days: \(100/200=50\%\). In the no-reminder group, \(800/1000=80\%\) were on hot days. Since signs were more common on hot days in both reminder categories, the different proportions of hot days affect the overall comparison. Temperature is a plausible confounder if it is also connected to whether a reminder was received.
These data do not show that reminders caused the lower overall percentage. Reminder status was not randomly assigned in this fictional review, and the two groups differed in temperature mix. The within-temperature comparisons are also descriptive: the cool-day percentages are equal, and the hot-day percentage is lower for the reminder group. A full explanation keeps those two statements separate rather than claiming that the reminder percentage is lower within both temperature groups.
Worked Example: Fire Size and Fire Damage
Worked Example: Fire Size and Fire Damage
A community report finds that fires with more firefighters at the scene tend to have greater property damage. A reader concludes that sending more firefighters causes more damage. Identify a possible confounding variable and explain the problem with that conclusion.
The explanatory variable is the number of firefighters, and the response is the amount of property damage. A plausible confounding variable is the size or severity of the fire when help is requested. Larger fires may lead dispatchers to send more firefighters, so initial fire severity is related to the explanatory variable. Larger fires may also cause more property damage, so severity is related to the response.
The observed association can therefore arise because severe fires tend to involve both more firefighters and more damage. The report does not show that adding firefighters causes damage. In fact, firefighters may be sent to larger fires precisely because those fires are already more dangerous. To examine the relationship more carefully, researchers could compare fires of similar initial severity, while recognizing that other relevant differences might still remain.
This example also shows why the timing and meaning of a variable matter. “Fire size when help is requested” is a more useful possible confounder than a vague phrase such as “the circumstances.” It gives a clear reason both the number of firefighters and the damage could differ.
What Confounding Does—and Does Not—Tell Us
Finding a plausible confounder does not prove that it accounts for the whole observed association. It identifies a reasonable alternative explanation that should be considered. The association might be partly explained by the third variable, or other variables may also matter. Data collection and study design determine which explanations can be examined.
Researchers can sometimes measure a possible confounder and compare groups within its categories, as in the temperature example. They can also plan an experiment, when appropriate, and use random assignment to help create treatment groups that are similar on average with respect to other variables. Random assignment does not guarantee that every group will match perfectly, especially in a small experiment, but it helps reduce confounding as an explanation for a treatment-group difference.
In an observational study, researchers do not assign the explanatory variable. Even if they measure and account for some possible confounders, unmeasured or poorly measured variables may remain. As in Observational Studies and Their Limits, an association in observational data alone does not establish that one variable caused another. The next tutorial, Association Versus Causation in Study Conclusions, develops how to phrase that distinction.
Common Mistakes and AP Exam Tips
- Calling any unmeasured variable a confounder. A full answer explains how the proposed variable is related to both the explanatory variable and the response.
- Mixing up the explanatory variable and response. State which variable is being used to explain or predict and which is the outcome. Then identify the possible confounder separately.
- Claiming that the confounder definitely caused the association. Say it is a plausible explanation or may contribute. Observational data may not establish how much it explains.
- Assuming the overall comparison tells the whole story. If a relevant third variable is available, compare groups within its categories as well. Describe each comparison accurately; do not claim that every subgroup has the same pattern.
- Turning an association into a causal claim. “More ice-cream sales caused more drownings” is not supported by the association. A careful answer describes the association and explains how temperature and swimming activity could be related to both variables.
For a strong AP response, name the possible confounder and make the two links explicit. For example: “Hot weather could be related to higher ice-cream sales and to more swimming, which could increase opportunities for drowning. Therefore, temperature and swimming activity could help explain the association; the observed relationship does not show that ice-cream sales caused drownings.”
Check Your Understanding
For each situation, identify a possible confounding variable and explain how it may be related to both variables being studied.
- A town finds that neighborhoods with more bicycles also have more reported bike injuries. Name one possible confounding variable and describe both connections.
- A school observes an association between time spent using a study app and exam scores. Students who choose to use the app may differ from students who do not. What kind of third variable could confound the association?
- In the dehydration example, compare the percentages with signs on cool days and on hot days. Which comparison is equal, and which is lower among participants who received a reminder?
- A report finds that hospitals with more intensive-care beds have more deaths. Give a plausible confounder and explain why the association alone does not show that beds cause deaths.
- What two relationships must you explain before calling a third variable a plausible confounder?