Does the Relationship Keep the Same Rate of Change?
A linear model represents a relationship with a constant rate of change: for each one-unit increase in \(x\), the predicted response changes by the same amount. Real relationships do not always behave that way. As in “Comparing Fit Using Residual Plots,” residuals help us see what a fitted line has left unexplained. Here, we use that evidence to decide whether a line seems adequate or whether a curved relationship deserves further investigation.
A scatterplot can show an overall positive or negative association and still bend. A line may capture the broad direction while missing how the relationship changes across the range. The residual plot can make that mismatch easier to see: a systematic curve in residuals means the line tends to overpredict in some parts of the range and underpredict in others. As “Curved Residual Pattern Means Nonlinear Relationship” explains, that pattern is evidence against treating the relationship as adequately linear.
The word investigate matters. A curved residual pattern is evidence that the line misses structure; it does not, by itself, identify the best alternative model or prove the true relationship has a particular shape. Consider the scatterplot, the scale and range of the variables, how the data were collected, and what is plausible in context. A change in how a process works, a boundary on possible responses, or a known rise-and-fall pattern may make curvature sensible. If no context supports the apparent bend, check whether it could instead reflect a few unusual observations, groups mixed together, or limited data.
This decision is also limited to the observed data and the purpose of the analysis. A small bend may not matter for a rough description, but it could matter when accurate predictions across the range are important. As “Scope of Inference for a Regression Model” and “Reliability Within the Range of Data” emphasize, the cases and explanatory-variable values represented by the data affect what conclusions a model can support.
A Practical Decision Process
Use the following sequence to decide whether to keep a linear model provisionally or look more closely at a curve. It combines visual evidence with context instead of relying on a single feature or statistic.
Describe the overall relationship between \(x\) and \(y\). Ask whether it looks approximately straight or whether its direction or steepness appears to change across the observed range.
Look for residuals scattered around zero without a clear bend. A run of positive residuals at both ends and negative residuals in the middle, or the reverse, can reveal curvature the line has missed.
Consider whether the bend is sustained and visible across several values of \(x\). Check for an isolated unusual point, distinct clusters, or a change in spread that could complicate the diagnosis.
If the curve is clear and plausible, investigate a curved model. If the pattern is weak and the linear model’s residuals otherwise look reasonable, a line may be adequate for the stated purpose. Qualify your conclusion to these data.
A residual’s sign tells you which way the line misses. A positive residual means the observed response is greater than the predicted response, so the line underpredicts that case. A negative residual means the line overpredicts. For instance, positive residuals at both ends and negative residuals in the middle indicate underprediction at the ends and overprediction in the middle. That pattern suggests the relationship bends upward relative to the fitted line. The reverse pattern suggests a bend downward relative to the line.
Do not decide based on \(r\) or \(r^2\) alone. A strong linear association can coexist with noticeable curvature: the line can summarize the broad direction while still leaving a systematic pattern. In “Comparing Models With Different Explanatory Variables,” you learned to compare \(r^2\) and \(s\) for models with different predictors; those summaries do not replace checking whether residuals show structure. Here, the key question is whether a line’s errors appear systematically related to where a case falls in the range.
Worked Example: A Clear Bend in Pump Output
Worked Example: Choosing a Model for Pump Settings
Original AP-style question. A technician records the pump setting \(x\), on a scale from 1 to 7, and water flow \(y\), in liters per minute, for seven settings on an invented test system. The scatterplot rises overall but appears to flatten and then steepen. A fitted line’s residuals, in order of settings 1 through 7, are \(5, 0, -3, -4, -3, 0,\) and \(5\) liters per minute. The pump controls are known to change their effect across the settings. Should the linear model be retained as an adequate description, or should a curved relationship be investigated?
State. We are deciding whether a linear model adequately describes water flow across the seven tested pump settings.
Plan. We will use the scatterplot to assess the original relationship and the residuals to check whether the fitted line leaves a systematic pattern. We will then consider whether the observed shape is plausible for this pump. The conclusion will apply to the tested settings, not automatically to all possible pump settings.
Do. The residuals are positive at both ends and negative in the middle. Thus, the line underpredicts flow at settings 1 and 7 and overpredicts it around the middle settings. This is not simply residuals falling on both sides of zero: their order follows a clear bend. The scatterplot’s changing steepness is consistent with that residual pattern. In addition, the stated control mechanism makes it plausible that the effect of changing the setting is not constant.
Conclude. The linear model does not appear to capture the observed water-flow pattern adequately across these seven settings. The curved scatterplot and systematic residual bend, together with the context of the pump controls, support investigating a curved relationship. This does not establish the exact form of the relationship or guarantee that another model will predict better; that would require fitting and evaluating an appropriate alternative.
Worked Example: A Line Can Be Adequate Despite Ordinary Scatter
Worked Example: Reviewing a Walking-Time Model
Original AP-style question. A recreation planner records walking distance \(x\), in kilometers, and completion time \(y\), in minutes, for 24 invented walks on similar terrain. The scatterplot shows a roughly straight positive association from 1 to 8 kilometers. In the residual plot, points fall on both sides of zero, mostly between \(-4\) and \(+4\) minutes, with no sustained curve or trend and no obvious change in spread. Is there evidence here that the planner should replace the linear model with a curved one?
Solution. The described plots do not provide clear evidence that a curved model is needed. The scatterplot is approximately straight over the observed distance range, and the residuals show no obvious systematic bend. Their spread also appears reasonably similar across the range. The time values have a straightforward interpretation in context, and no stated feature of the walks suggests that the rate of increase must change over this interval.
The appropriate conclusion is not that the relationship has been proven to be perfectly linear. Residual plots can reveal visible patterns, but they cannot rule out every small departure from a line. Rather, the evidence described gives no strong reason to investigate curvature for a basic description of these 24 walks. If the planner needed highly accurate predictions, the importance of prediction error and the intended use would also matter.
Conclusion. A linear model appears reasonable for describing completion time and distance for these walks, because the scatterplot is roughly straight and the residuals show no clear systematic pattern. That conclusion is limited to the observed walks and distances from 1 to 8 kilometers.
Worked Example: Check What Might Be Creating the Bend
Worked Example: Investigating a Residual Pattern in Battery Use
Original AP-style question. An inventoried set of 20 devices is used to model battery energy remaining, in percentage points, against hours of screen use. The linear model’s residual plot appears positive for roughly the first 3 hours, negative from about 4 to 7 hours, and positive again near 8 hours. Most residuals are within 3 percentage points of zero, but one device has a residual of \(+11\) percentage points. The devices include two different battery types. What should the analyst conclude about the linear model?
Solution. The positive-negative-positive sequence across the range resembles a curved residual pattern: the line underpredicts in the earlier and later portions and overpredicts in the middle. That is a reason to question whether one straight-line model adequately describes all 20 devices. However, the residual plot alone does not tell us whether the apparent bend represents one common curved relationship.
The analyst should check whether the two battery types form distinct groups in the scatterplot or residual plot. If they do, the apparent pattern could partly reflect combining devices with different behavior rather than a single smooth curve. The \(+11\)-percentage-point residual also merits investigation as an unusual observation, but it should not simply be deleted. As “What to Do With an Unusual Point” explains, an unusual observation is not automatically an error; check the device record and compare the fit with and without the point as a diagnostic.
Conclusion. The linear model should be treated cautiously: the residual signs suggest possible curvature, but the battery-type groups and the large residual should be investigated before choosing an alternative. If the bend remains across the data after those checks and is plausible for the battery-use process, investigating a curved model would be reasonable. The evidence does not justify declaring one specific alternative best without evaluating it.
Common Mistakes and AP Exam Tips
- Keeping a line because the overall association is strong. A strong positive or negative association does not show that the pattern is straight. A full-credit answer checks for a systematic residual curve, even when the scatterplot has a clear overall direction.
- Calling every curve proof of a particular alternative. A residual bend is evidence that the line misses structure. It does not, on its own, establish the exact curve or show that a proposed alternative will fit better. Say that the pattern supports investigating another form.
- Ignoring which way the line misses. Describe the residual signs in order and interpret them: positive residuals mean underprediction, and negative residuals mean overprediction. This makes the evidence for curvature specific rather than vague.
- Letting one unusual point decide the model choice. A single large residual may affect what the plot looks like, but it is not the same as a sustained curve. Investigate the point and look at the overall pattern; do not remove a valid observation merely to make the residual plot look straighter.
- Forgetting the context. A visible bend may make sense when the process has a changing rate or a natural limit. Context helps judge whether curvature is plausible, but it does not replace evidence from the plots.
- Making an unlimited claim. Do not say that a line is “the true model” or that a curve will predict better in every setting. Tie the decision to the observed data, range, and purpose.
A strong AP response names the evidence and makes a measured decision. For example: “The residuals are positive at both ends and negative in the middle, so the line underpredicts at the ends and overpredicts in the middle. This systematic pattern suggests that a linear model misses curvature, so a curved relationship should be investigated for these observations.” If the residuals show no clear structure, say that the linear model appears reasonable for the data and purpose rather than claiming that the relationship is certainly linear.
Check Your Understanding
Use the scatterplot, residual-plot, and context evidence to answer each question.
- A line’s residuals are negative at both ends of the \(x\)-range and positive in the middle. What does the line tend to do in those regions, and what does the pattern suggest?
- A scatterplot shows a strong positive association, but its residual plot has a clear bend. Why is the strength of the association not enough to justify keeping the line?
- A residual plot has a slight apparent curve based on a small number of cases, with one unusually large residual. Name two checks to make before choosing another model.
- What is the difference between concluding that a linear model is reasonable and claiming that the relationship is certainly linear?
- Why should a model-choice conclusion identify the observed data or range and the purpose of the analysis?