Why Residual Plots Matter in an Independent Review
A fitted model can produce plausible parameter estimates and still leave systematic structure in its residuals. A residual-versus-time plot asks whether prediction errors change over the observation sequence or elapsed time. A residual-versus-prediction plot asks whether errors vary with the magnitude of the prediction. These plots are review tools: they can reveal patterns that deserve investigation, but they do not identify a cause by themselves.
The first review question is what residual and prediction were plotted. A raw residual has the units of the measured outcome, while a standardized residual is scaled by an estimate of its expected error. The two can show different patterns when the model allows the residual variance to depend on prediction. The prediction type also matters. State whether the horizontal axis uses population or individual predictions, and keep that choice consistent within the plot being interpreted.
What Pattern Is Expected?
Under a correctly specified conditional mean model, residuals should be centered around zero across the relevant times and prediction range. More precisely, before accounting for parameter estimation, the model implies \(E(e_i\mid\text{model inputs})=0\). If its variance model is also appropriate, then \(Var(e_i\mid\text{model inputs})=V_i\); dividing by \(\sqrt{V_i}\) gives standardized residuals with conditional mean zero and variance one.
These are reference expectations, not guarantees that every finite sample will look perfectly patternless. Model parameters are estimated from the same data, observations may be repeated within a subject, and a small number of points can make a smooth or a cluster look more persuasive than it is. Review the point cloud and its context before interpreting a visual pattern. A smooth curve or binned average can help summarize the plot, but it is descriptive evidence rather than a formal test.
A useful review separates three questions: Is the residual center changing? Is the spread changing? Is the residual distribution asymmetric or dominated by a few unusual points? The time plot is especially informative about ordering and temporal structure. The prediction plot is especially informative about whether errors depend on the predicted magnitude. Neither plot alone establishes which model component, if any, is responsible.
Worked Example: A Residual Trend Across Time
Suppose a hypothetical concentration analysis displays six raw residuals, in mg/L, against elapsed time. The plotted values are: at 1 hour, 0.18; at 2 hours, 0.12; at 3 hours, 0.03; at 4 hours, -0.08; at 5 hours, -0.19; and at 6 hours, -0.26. Assume the residual convention is observed minus predicted, so a negative residual means the prediction exceeds the observation.
Compare the average residual at the first three times with that at the last three:
The difference in these group averages is \(-0.177-0.110=-0.287\) mg/L. The sequence also moves from positive to negative residuals. This is a clear descriptive time pattern: the model tends to underpredict earlier and overpredict later in these hypothetical observations. It motivates checking time-related model behavior, the observations and event alignment, and the plotted residual definition. It does not, on its own, prove which feature is wrong or that the pattern would persist in new data.
Reading Residuals Against Predictions
A residual-versus-prediction plot is useful for checking whether the residual center or spread changes as predicted outcomes increase. A smooth curve that moves above or below zero can indicate a conditional mean pattern. A fan or funnel-shaped cloud can indicate changing variance. But a funnel in raw residuals is not automatically evidence that the variance model is wrong: the model may intentionally specify a variance that grows with the prediction.
For example, under a proportional-error variance model, the residual standard deviation may be proportional to the prediction. Raw residuals can then spread out at larger predictions even if the variance model is correct. Standardizing by the model-based standard deviation is one way to inspect whether the remaining spread is roughly stable. The residual definition and scaling should be stated, and the plot should not be interpreted as though raw and standardized residuals were interchangeable.
Worked Example: A Raw-Residual Funnel and Variance Scaling
Consider hypothetical raw residuals, in mg/L, at two prediction levels. At a prediction of 2 mg/L, the four residuals are \(-0.2,\ 0.1,\ 0.2,\ -0.1\) mg/L. At a prediction of 10 mg/L, they are \(-1.0,\ 0.5,\ 1.0,\ -0.5\) mg/L. Calculate the root mean square residual around zero at each level as a simple description of spread.
The raw residual spread is larger at the higher prediction. Suppose, for illustration, the model specifies a standard deviation equal to \(0.1\) times the prediction. The corresponding standard deviations are \(0.2\) mg/L and \(1.0\) mg/L. Dividing each raw residual by its specified standard deviation gives \(-1,\ 0.5,\ 1,\ -0.5\) at both prediction levels. Thus the raw plot shows a funnel, while the standardized values have the same spread in these two groups. The example shows why a raw funnel must be compared with the stated error model before it is called a variance-model failure. With only four observations per level, these calculations illustrate a pattern; they do not establish that the variance specification is adequate.
Asymmetry, Clusters, and Display Choices
A residual plot can also show an uneven distribution around zero: for example, many modest negative residuals and a few large positive ones. Such asymmetry can be relevant to the assumed residual distribution or may reflect an unusual observation, data issue, or mixture of observations with different behavior. A residual-versus-prediction plot can help locate whether the asymmetry occurs across the prediction range or only in one region. It does not replace a check of the records or an assessment of the residual distribution itself.
How points are displayed can create or conceal apparent patterns. When many observations share similar times or predictions, points overlap; a dense region may appear to contain fewer observations than it does. Connecting points in arbitrary row order can create artificial zigzags. A truncated vertical axis can exaggerate small differences, and a poorly chosen scale can compress meaningful variation. Check axis labels, units, limits, point counts, and whether the plotted records are the intended analysis observations.
Repeated measurements also require care. A run of positive residuals from one subject followed by a run of negative residuals from another can look like a population-wide time trend if subject identity is hidden. Within-subject dependence means that plotted points are not necessarily independent pieces of evidence. Inspecting subject identifiers or using a display that distinguishes subjects can help determine whether a pattern is broadly shared or concentrated in a few individuals. This is a diagnostic clarification, not a substitute for the model’s assumptions.
Worked Example: A Positive Tail at Similar Predictions
Suppose eight hypothetical observations with predictions near 5 mg/L have raw residuals \(-0.3,\ -0.2,\ -0.1,\ 0,\ 0.1,\ 0.2,\ 0.3,\ 1.0\) mg/L. The residual sum is \(1.0\) mg/L, so the mean is \(1.0/8=0.125\) mg/L. The ordered values have median \((0+0.1)/2=0.05\) mg/L.
Most values lie between \(-0.3\) and \(0.3\) mg/L, but one is \(1.0\) mg/L. The positive tail and the difference between mean and median are reasons to inspect the high residual. Check the observation, prediction, units, timing, and whether it is part of a subject-specific cluster. The plot may suggest asymmetry or an influential point, but this small example does not show whether the observation is erroneous or whether the residual distribution is generally asymmetric.
A Structured Review
The following sequence keeps a visual observation separate from a proposed explanation. It also helps ensure that the plot corresponds to the submitted analysis rather than to a display or data-processing mistake.
Check the observation set, residual definition, prediction type, time variable, units, and any residual scaling. State these choices in the review.
Record whether the center shifts, spread changes, asymmetry appears, or a small group of points dominates. Use specific regions of time or prediction rather than labels such as “bad fit.”
Inspect axis scales, overplotting, ordering, missing records, and subject clustering. If helpful, compare raw and appropriately standardized residuals.
Describe the model feature or data issue that the pattern motivates checking, without claiming that the plot alone has identified the cause.
Comparing the Two Plots and Their Limits
| Plot | Primary question | Pattern to investigate | Important caution |
|---|---|---|---|
| Residual versus time | Does the residual center or spread change over observation time? | Drift, runs of one sign, or time-varying spread | Subject clustering, event timing, and row ordering can shape the display. |
| Residual versus prediction | Does the residual center or spread change with predicted magnitude? | Curvature, a shifted center, or a funnel | A raw funnel can be expected under a prediction-dependent variance model. |
These plots answer different descriptive questions. A residual-versus-time display preserves temporal order, while a residual-versus-prediction display organizes observations by predicted magnitude. A histogram or quantile plot instead summarizes distributional shape without showing where values occur in time or prediction space. Such displays can complement the two plots, but they do not remove the need to check the specific temporal and prediction-dependent patterns at issue here.
Visual diagnostics have limits. A pattern can be difficult to see in a small sample, and a smooth curve can suggest structure where there is little information. Conversely, points that look patternless do not prove that a model is correct. These plots are best treated as evidence for targeted follow-up: verify the inputs and display, compare the pattern with the model’s stated mean and variance assumptions, and document what remains uncertain.
Common Mistake
Wrong reasoning: “The raw residuals fan out at high predictions, so the model’s residual error implementation must be wrong.” This conclusion ignores whether the specified error model expects residual variance to increase with prediction.
Correction: First identify whether the plot uses raw or standardized residuals and write down the model-implied residual scale. Compare the observed spread with that scale. A raw funnel may be consistent with a proportional variance model; a funnel that remains in appropriately standardized residuals is a more direct reason to investigate the variance specification, though it is still not proof of a particular cause.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Residuals are defined as observed minus predicted. What does a negative residual mean?
- Raw residuals grow wider as predictions increase, but standardized residuals have similar spread across the prediction range. What should an independent reviewer conclude?
- A time plot shows an apparent sequence of positive residuals followed by negative residuals. Name two checks that could help determine whether this reflects a broad temporal pattern or a display artifact.
- For residuals \(-0.2,\ 0.1,\ 0.1\) mg/L, calculate the mean residual and interpret its sign.
Solutions. (1) The prediction exceeds the observation for that record. (2) The raw pattern may be consistent with a variance that increases with prediction; confirm the residual scaling and stated variance model before raising a variance concern. (3) Check subject identities for clustering and verify the time variable, row ordering, plotted records, and axis scale. (4) The mean is \((-0.2+0.1+0.1)/3=0/3=0\) mg/L. The three residuals have no average signed error in this small set, although that does not mean their individual errors are zero or that the model is adequate.