What a Relationship Plot Can—and Cannot—Show
A scatterplot may show a clear pattern: as a predictor increases, the response tends to increase or decrease. Earlier material on describing direction, form, and strength helps us state what that pattern looks like. But a visible association is not, by itself, evidence that changing the predictor would change the response. A plot summarizes how observed values vary together; it does not reveal why they vary together.
This distinction matters whenever a conclusion uses causal language such as “increases,” “reduces,” or “leads to.” A plot of study time and exam scores might show that students who study longer tend to score higher. That pattern alone cannot establish that study time caused the score differences. Students who study longer may also differ in preparation, course access, or other relevant ways.
One way to make the causal question precise is to imagine, for each unit \(i\), two potential responses: \(Y_i(1)\), the response if the unit receives or experiences an exposure, and \(Y_i(0)\), the response if that same unit does not. The individual causal effect is \(Y_i(1)-Y_i(0)\). In ordinary data, we observe only one of these outcomes for a unit, not both at once. A causal analysis therefore needs a defensible way to compare the observed groups or conditions. A visual pattern alone does not supply that justification.
Confounding: A Third Variable Shapes Both
A confounder is a variable related to both the predictor or exposure and the response, and not simply a consequence of the exposure. If the confounder is distributed differently across exposure groups, an overall association can partly or entirely reflect that difference. The pattern can be misleading even when the exposure has no effect within comparable confounder groups.
Worked Example: Exercise, Risk Groups, and Blood Pressure
Hypothetical data. Suppose a sample contains people in two pre-existing health-risk groups. The table gives the number of people and their average systolic blood pressure in each exercise group. Assume, for this illustration, that within each risk group the average is the same for regular exercisers and non-exercisers.
| Risk group | Regular exercise | No regular exercise |
|---|---|---|
| Lower risk | 80 people; mean 120 mmHg | 20 people; mean 120 mmHg |
| Higher risk | 20 people; mean 150 mmHg | 80 people; mean 150 mmHg |
The overall mean among regular exercisers is a weighted average of the two group means:
Among people with no regular exercise, the overall mean is:
The observed overall difference is \(126-144=-18\) mmHg: regular exercisers have an average systolic pressure 18 mmHg lower in this hypothetical sample. Yet within the lower-risk group the difference is \(120-120=0\) mmHg, and within the higher-risk group it is \(150-150=0\) mmHg. The overall pattern arises because the exercise groups contain different proportions of the two risk groups. The risk group is a plausible confounder if it is related to exercise habits and blood pressure.
This calculation does not prove that exercise has no causal effect in real settings. It shows why the plot or overall comparison cannot settle that question: an unadjusted association may reflect group composition. Whether risk group captures all relevant differences, and how the data were collected, also matter.
The weighted-average calculation is a useful justification for the general point. If group \(g\) has mean response \(\bar{y}_g\) and makes up a proportion \(w_g\) of an exposure group, that exposure group’s overall mean is \(\sum_g w_g\bar{y}_g\). Two exposure groups can have different overall means because their within-group means differ, because their weights differ, or both. A plot of the overall data does not automatically separate these explanations.
Selection: Who Appears in the Data Matters
A relationship plot describes the units included in the data set. If inclusion depends on the predictor, the response, or factors related to either, the observed pattern may not represent the pattern in the wider population. In some cases, restricting attention to selected units can itself create an association that was not present in the full population.
Worked Example: Selection Creates an App–Symptom Pattern
Hypothetical population. Imagine 1,000 people, half of whom use a health app and half of whom experience a particular symptom. Suppose app use and the symptom are independent in the full population, so the four combinations occur equally often: 250 people in each cell. Now imagine an online survey that receives responses only from people who use the app or have the symptom. People who neither use the app nor have the symptom are not included.
| Symptom | No symptom | Total | |
|---|---|---|---|
| Uses app | 250 | 250 | 500 |
| Does not use app | 250 | 0 selected | 250 |
In the full population, half of app users have the symptom: \(250/500=0.50\). Half of non-users also have it: \(250/500=0.50\). There is no association in these proportions. In the selected survey, however, all 250 included non-users have the symptom, so their symptom proportion is \(250/250=1.00\). Among app users it remains \(250/500=0.50\). The selected sample therefore shows a strong negative association between app use and the symptom.
This pattern is created by the selection rule in the example, not by an effect of app use. In practice, selection mechanisms are rarely this exact, but participation, clinic attendance, or availability of complete records can depend on characteristics related to both variables. A plot cannot reveal the selection process on its own. The sampling and participation process must be considered before extending an observed pattern beyond the people represented.
Selection can also affect which part of a relationship is visible. For example, a plot limited to people who meet a program’s entry criteria may show a different pattern from a plot of everyone eligible for the program. This is not merely a question of having fewer points: it is a question of whether the included points were selected in a way that changes the comparison.
Direction: A Plot Does Not Determine Which Way the Arrow Points
A relationship plot assigns variables to axes: one is called the predictor and the other the response. That choice helps organize a description or a statistical analysis, but the axis labels do not establish causal direction. If two variables are associated, at least three broad explanations may be relevant: the predictor affects the response, the response affects the predictor, or other variables influence both.
Worked Example: Late-Night Phone Use and Sleep
Hypothetical observation. In a group of 20 people, suppose 10 people with higher late-night phone use average 5.5 hours of sleep, while 10 people with lower late-night phone use average 7.0 hours. The observed difference in mean sleep duration is:
A plot could show that higher phone use is associated with shorter sleep. One possible explanation is that late-night phone use reduces time available for sleep. But another possibility is that people who have trouble sleeping spend more time using their phones while awake. A third is that a factor such as shift work affects both phone use and sleep duration. The observed difference of 1.5 hours describes this hypothetical group; it does not distinguish among these explanations.
The fact that phone use is placed on the horizontal axis and sleep duration on the vertical axis does not resolve the direction either. To support the claim that reducing phone use would increase sleep, evidence must address temporal order and alternative explanations, not just the orientation or shape of the plot.
What Stronger Causal Evidence Requires
No single feature of a relationship plot establishes causation. Causal conclusions depend on how the data were generated and on assumptions that make the compared conditions informative. A randomized experiment, when feasible and ethically appropriate, assigns an exposure by chance. This tends to balance pre-existing characteristics across assigned groups, so a difference in outcomes can more credibly be attributed to the assigned condition. Random assignment does not guarantee perfect execution: noncompliance, missing outcomes, or other departures from the design can complicate interpretation.
In observational data, the exposure is not assigned at random. A causal interpretation then requires a strong argument that the compared groups are sufficiently comparable, including with respect to relevant confounders, and that selection has not created a misleading comparison. It also matters that the proposed cause precedes the response. These are questions about design, subject-matter knowledge, and assumptions; they are not answered by how tight or smooth a plotted pattern appears.
Plots remain valuable. They can reveal unexpected structure, show whether patterns differ across groups, and identify regions with little data. Earlier material on group-specific trends and sparse regions is relevant here: even a well-described pattern can be limited to a particular subgroup or observed range. These observations help formulate and scrutinize causal questions, but they do not replace evidence about the process that generated the data.
Common Mistake: Treating a Strong Pattern as a Causal Result
Wrong reasoning: “The points form a clear downward trend, so increasing the predictor must reduce the response.” This confuses a strong association with a causal effect. The trend might reflect confounding, selective inclusion, reverse direction, or a combination of these.
Correction: First describe what the plot establishes: the observed predictor and response vary together in the displayed data. Then ask how the units entered the data, whether a third variable could influence both, whether the proposed cause precedes the response, and what design supports the comparison. Use causal wording only when the design and assumptions justify it. Otherwise, use measured language such as “is associated with” or “tends to occur alongside.”
Check Your Understanding
Answer each question by distinguishing what the observed pattern shows from what would be needed for a causal conclusion.
- A plot shows that people who exercise more have lower average blood pressure. Give one possible confounder and explain how it could relate to both variables.
- In the selection example, why does the survey show an app–symptom association even though the full population has equal symptom proportions among app users and non-users?
- A plot places late-night phone use on the horizontal axis and sleep duration on the vertical axis. Does that choice establish that phone use causes sleep duration to change? Explain.
- State the difference between an association and a causal effect in your own words.
- Name one feature of a study design that can strengthen a causal interpretation, and one limitation that could still complicate that interpretation.