When Residuals Bend
In “Reading a Residual Plot for Model Fit,” you learned that residuals should form a roughly pattern-free band around zero when a straight line describes the relationship reasonably well. A clear curve is different: the residuals tend to be positive in some parts of the plot and negative in others, following a bend rather than scattering randomly.
The sign of a residual helps explain the curve. Since a residual is the observed response minus the predicted response, a positive residual means the observation is above the fitted line: the line underpredicts there. A negative residual means the observation is below the line: the line overpredicts there. When these signs change systematically across the explanatory-variable range, the line is missing a feature of the relationship.
A U-shaped pattern often means the line tends to underpredict at low and high values of \(x\), while overpredicting in the middle. An upside-down U often suggests the opposite: overprediction at the ends and underprediction in the middle. Real plots may not be perfectly smooth or symmetric. Look for a consistent bend across several nearby points, not a curve formed by one unusual observation.
The practical recommendation is specific: do not use the fitted linear model as is when its residual plot shows clear curvature. The line leaves a systematic pattern in its errors, so its predictions may be biased in different parts of the observed \(x\)-range. This does not, by itself, identify the cause of the bend or prove which alternative model is best. Describe what the plot shows and investigate a more suitable way to represent the relationship.
Reading the Shape and Its Consequences
Start at one end of the residual plot and follow the points across the horizontal axis. Ask whether the residuals move from mostly positive to mostly negative and then back again, or show another sustained bend. Then translate the signs into the model’s behavior: where is the fitted line below the observations, and where is it above them?
Curvature also warns against treating the fitted line’s slope as a good summary of how the response changes throughout the full range. A line has one constant slope. A curved relationship may rise quickly in one region and more slowly in another, or even change direction. The line can summarize an overall trend while still missing those local changes.
This is a model-fit conclusion, not a causal conclusion. A residual plot does not establish why the relationship bends, whether a particular variable causes the response, or whether a particular curved model is correct. As emphasized in “Common Errors Linking Regression to Causation,” claims about cause depend on study design, not on the shape of a plot.
Worked Examples: Curvature in Residuals
Worked Example: A U-Shaped Residual Pattern
Original AP-style question. In an invented greenhouse exercise, \(x\) is the number of hours of supplemental light per day and \(y\) is seedling growth in centimeters. The observations are \((1,16)\), \((2,15)\), \((3,16)\), \((4,19)\), and \((5,24)\). Find the least-squares line and residuals, then explain what their pattern suggests.
State. We want to determine whether the residual plot supports using a straight line to describe growth across these light levels.
Plan. Calculate the least-squares line, use it to find each predicted growth value, and subtract each prediction from the observed growth. This is a descriptive assessment of the fitted model, not an inference procedure, so there are no inference conditions to check.
Do. The means are \(\bar{x}=3\) hours and \(\bar{y}=18\) centimeters. The sum of squared \(x\)-deviations is \(S_{xx}=10\), and the sum of the products of corresponding deviations is \(S_{xy}=20\). Thus the slope is \(b=S_{xy}/S_{xx}=20/10=2\) centimeters per hour. The intercept is \(a=\bar{y}-b\bar{x}=18-2(3)=12\) centimeters, so the fitted line is \(\hat{y}=12+2x\). At \(x=1\), for example, the predicted growth is \(12+2(1)=14\) centimeters, and the residual is \(16-14=2\) centimeters. The remaining predictions and residuals are:
| Hours of light, \(x\) | Observed growth, \(y\) (cm) | Predicted growth, \(\hat{y}\) (cm) | Residual, \(y-\hat{y}\) (cm) |
|---|---|---|---|
| 1 | 16 | 14 | 2 |
| 2 | 15 | 16 | -1 |
| 3 | 16 | 18 | -2 |
| 4 | 19 | 20 | -1 |
| 5 | 24 | 22 | 2 |
Conclude. The residuals are positive at the low and high ends of the observed light range and negative in the middle. The line underpredicts growth at the ends and overpredicts it in the middle, creating a clear U-shaped pattern. This suggests a nonlinear relationship, so the fitted straight-line model should not be used as is.
Worked Example: An Upside-Down Curve
Original AP-style question. In an invented equipment test, \(x\) is a machine setting and \(y\) is the operating temperature in degrees Celsius. Five observations are \((1,31)\), \((2,39)\), \((3,45)\), \((4,49)\), and \((5,51)\). Find the regression line and describe the residual pattern.
State. We will check whether a straight-line model leaves a systematic curved pattern in its residuals.
Plan. Find the least-squares line, calculate the residual for each setting, and interpret the signs and their order. This is a descriptive model check, so no inference conditions apply.
Do. Here \(\bar{x}=3\) settings and \(\bar{y}=43\) degrees Celsius. The deviation sums are \(S_{xx}=10\) and \(S_{xy}=50\), giving slope \(b=50/10=5\) degrees Celsius per setting. The intercept is \(a=43-5(3)=28\) degrees Celsius. Thus \(\hat{y}=28+5x\). At setting \(x=1\), the prediction is \(28+5(1)=33\) degrees Celsius and the residual is \(31-33=-2\) degrees Celsius. The full set of predictions and residuals is:
| Setting, \(x\) | Observed temperature, \(y\) (°C) | Predicted temperature, \(\hat{y}\) (°C) | Residual, \(y-\hat{y}\) (°C) |
|---|---|---|---|
| 1 | 31 | 33 | -2 |
| 2 | 39 | 38 | 1 |
| 3 | 45 | 43 | 2 |
| 4 | 49 | 48 | 1 |
| 5 | 51 | 53 | -2 |
Conclude. Residuals are negative at the ends and positive in the middle, a curved pattern in the reverse direction from the first example. The line overpredicts at the ends and underpredicts in the middle. This is evidence that the linear model does not capture the pattern adequately and should not be used as is.
Worked Example: Describing Curvature Without Recomputing the Line
Original AP-style question. In an invented calibration exercise, \(x\) is a device setting and \(y\) is the measured operating cost in dollars. A residual plot for a fitted line has residuals of \(6, 0, -3, -6, -3, 0,\) and \(6\) dollars at settings \(x=1,2,3,4,5,6,\) and \(7\), respectively. Describe the pattern and make a recommendation.
State. The task is to interpret the residual plot, not to refit the regression line.
Plan. Compare the residual signs across settings and translate them into over- or underprediction. Because the residuals are already supplied, no regression calculations or inference conditions are needed.
Do. At settings 1 and 7, residuals are \(6\) dollars, so the fitted line underpredicts cost by 6 dollars. At settings 3, 4, and 5, the residuals are \(-3\), \(-6\), and \(-3\) dollars, so the line overpredicts cost there. The values at settings 2 and 6 are zero. Moving across the settings, the residuals make a sustained bend: they start high, become negative in the middle, and return high at the other end.
Conclude. This is a clear U-shaped residual pattern, not a pattern-free band around zero. The linear model has systematic errors across the settings and should not be used as is. The plot does not tell us which alternative model to choose; that requires further analysis and attention to the context.
Common Mistakes and AP Exam Tip
- Calling any positive residual a curve. One positive residual does not establish curvature. Describe a sustained pattern across multiple \(x\)-values, and distinguish it from one isolated point.
- Reversing overprediction and underprediction. Since residual = observed response minus predicted response, a positive residual means the prediction was too low; a negative residual means it was too high.
- Saying the line fits because it has a positive or negative slope. The slope describes the fitted line’s overall direction, not whether the residuals show a systematic bend. Inspect the residual plot separately.
- Claiming the plot proves a particular alternative model. A curve suggests that a straight line is inadequate as it stands; it does not prove that a specific curved equation is appropriate or identify the reason for the pattern.
- Overstating the conclusion. Say that the curved pattern suggests a nonlinear relationship and that the linear model should not be used as is. Do not claim that the plot alone proves the model is wrong for every possible use.
For full-credit communication, name the shape, describe where residuals are positive and negative, interpret those signs as underprediction or overprediction, and make a qualified recommendation. For example: “Residuals are mostly positive at low and high settings and mostly negative in the middle, forming a U-shaped pattern. The line underpredicts at the ends and overpredicts in the middle, so it does not describe the relationship adequately as a straight line.”
Check Your Understanding
Use the residual signs and their order across \(x\) to assess whether a line leaves a curved pattern.
- Residuals are mostly positive at low and high \(x\)-values and mostly negative in the middle. What shape do they suggest, and where does the line underpredict?
- Residuals are negative at both ends of the \(x\)-range and positive in the middle. What does this suggest about the line’s predictions?
- Why is one large positive residual not enough, by itself, to conclude that the relationship is curved?
- A residual plot shows a clear bend. What recommendation should you make about using the fitted linear model?
- Does a curved residual pattern prove which alternative model is best? Explain briefly.