From Diagnostic Plots to a Defensible Summary
A diagnostic plot is useful only if its evidence is communicated accurately. A reader needs to know which plots were examined, what visible features appeared, and how far those features support a conclusion. “The assumptions look fine” is too vague: it does not identify the evidence, and it can sound as if a visual check proves conditions that cannot be proved from a sample.
A good summary is brief but specific. It names the diagnostic displays, describes a pattern in relation to a reference such as zero or a straight line, and limits the conclusion to what the plots can show. Residual plots provide evidence about a fitted model; they do not reveal the unobserved errors directly.
For a regression fitted by ordinary least squares, the fitted value for observation \(i\) is \(\hat{y}_i\), and its residual is \(e_i=y_i-\hat{y}_i\). A residual-versus-fitted plot places \(e_i\) against \(\hat{y}_i\). Other useful displays include residuals against a predictor or observation order, a histogram of residuals, and a normal quantile plot. Each answers a different diagnostic question, so a report should name the display rather than refer vaguely to “the plots.”
A Structure for Reporting What You See
A practical summary can be built in three parts. First, name the plots. Second, describe the visible evidence using words such as “approximately,” “no clear pattern,” “curved,” or “increasing spread.” Third, state the scope of the conclusion: the display did or did not show a conspicuous pattern, but it cannot establish that the corresponding condition is true in the population.
For example, identify the residual-versus-fitted plot, normal quantile plot, and residuals-versus-order plot that were examined.
Say whether residuals are scattered around zero, whether a curve or changing spread is visible, and whether a quantile plot departs from its reference line.
Report what the displays suggest, and note that visual diagnostics cannot prove assumptions or rule out every problem.
Descriptions should be observational before they are interpretive. “The residuals fan outward as fitted values increase” reports a visible feature. “The variance assumption is false” is a stronger conclusion and is not justified by that sentence alone. Likewise, “the residuals follow a curve” is more informative than “the model is bad”; a curve is evidence of a systematic feature that the fitted mean pattern may not capture, but its practical importance depends on context and the rest of the analysis.
Why Least-Squares Residuals Need a Coherence Check
When a fitted line includes an intercept, ordinary least squares imposes useful constraints on its residuals. Their sum is zero, and their inner product with the fitted values is zero. These constraints do not mean that residuals are random, independent, normally distributed, or pattern-free. They do matter when constructing or checking a numerical illustration: a list that violates them cannot be the residuals from the stated least-squares fit with an intercept.
The examples below use hypothetical observations. Where residuals are listed, their values are checked against the least-squares constraints before a plot pattern is described. This avoids confusing a schematic sketch with actual output from a fitted regression.
Worked Example: Reporting Curvature in a Residual-versus-Fitted Plot
Suppose a hypothetical simple linear regression uses predictor values \(x=-4,-3,\ldots,4\). The fitted values are \(\hat{y}=20+2x\), and the constructed residuals are \(e=x^2-\frac{20}{3}\). Determine whether the residual values are compatible with an ordinary least-squares fit with an intercept, then write a concise diagnostic statement.
The predictor values have \(\sum x=0\) and \(\sum x^2=60\). Therefore the residuals sum to \( \sum e=\sum x^2-9\left(\frac{20}{3}\right)=60-60=0\). Also, \(\sum xe=\sum x^3-\frac{20}{3}\sum x=0\), because the values of \(x\) are symmetric about zero. Since \(\hat{y}=20+2x\), \( \sum \hat{y}e=20\sum e+2\sum xe=0\). The residuals satisfy both least-squares constraints.
In order from \(x=-4\) to \(x=4\), the residuals are \(28/3, 7/3, -8/3, -17/3, -20/3, -17/3, -8/3, 7/3, 28/3\). The middle residuals are negative and the residuals at both ends are positive. A residual-versus-fitted plot would therefore display a curved, roughly U-shaped pattern rather than a patternless cloud around zero.
Possible summary: “The residual-versus-fitted plot showed a clear U-shaped pattern, with negative residuals near the middle fitted values and positive residuals at both ends. This is evidence that the fitted straight-line mean pattern may not describe the observed relationship adequately; the plot alone does not identify the appropriate alternative model.”
Describe Other Displays Without Overstating Them
The same reporting discipline applies to plots aimed at different features. A normal quantile plot is judged by how its ordered residuals compare with a reference line. A histogram gives a rough view of the residual distribution, but its shape depends on the binning and may hide detail. A residuals-versus-order plot can reveal runs, cycles, or clusters in the order in which observations were recorded. State which display supplies the evidence, and avoid using one display as if it answered every diagnostic question.
Worked Example: Describing Tail Departures in a Normal Quantile Plot
In a hypothetical regression with 24 observations, an analyst examines a normal quantile plot and a residual histogram. The middle points of the quantile plot lie fairly close to the reference line, while several points at both ends bend away from it. The histogram is roughly mound-shaped but has a few separated values in its tails. What should the report say?
The quantile plot gives the clearest stated evidence: the central points are close to the line, but the tails depart from it. The histogram is broadly compatible with a single mound but also shows a few tail values. Since no precise probability calculation is being made here, the conclusion should remain visual and qualified.
Possible summary: “The normal quantile plot was approximately linear through the center, with departures at both tails; the residual histogram was roughly mound-shaped with a few separated tail values. These displays raise a possible concern about tail behavior, but a visual check with 24 observations cannot establish the error distribution or determine the effect on a particular inference.”
This wording does not say that the residuals are normal or nonnormal with certainty. It states what the plots show and identifies why the conclusion is limited.
Worked Example: Reporting an Alternating Pattern Across Observation Order
Eight hypothetical observations were collected in a meaningful sequence. A fitted regression has fitted values \(\hat{y}_i=5+0.5i\), for \(i=1,\ldots,8\), and residuals \(1,-1,-1,1,1,-1,-1,1\). Check the least-squares constraints and describe the residuals-versus-order plot.
The residual sum is \(1-1-1+1+1-1-1+1=0\). Their weighted sum by observation index is \(1-2-3+4+5-6-7+8=0\). Thus, because the fitted values are \(5+0.5i\), their inner product with the residuals is \( \sum_i\hat{y}_i e_i=5(0)+0.5(0)=0\).
The residual sequence alternates in sign in pairs: positive, negative, negative, positive, positive, negative, negative, positive. This is a visible ordered pattern, even though the residuals sum to zero and are orthogonal to the fitted values. Those least-squares constraints do not remove or rule out structure across observation order.
Possible summary: “The residuals-versus-order plot showed a repeated alternating pattern rather than an unstructured sequence around zero. Because the observations have a meaningful order, this pattern merits attention; the plot alone does not establish its cause or prove a particular dependence structure.”
Make the Summary Specific but Concise
A diagnostic summary is not a catalogue of every point on every plot. It should prioritize the features that could change how the fitted model is interpreted or used. If no obvious issue appears, say so directly but narrowly. If several displays point to the same concern, name the displays and state the shared evidence. If they differ, report that difference rather than forcing a single verdict.
For example, a residual-versus-fitted plot may show no obvious curve while a normal quantile plot has tail departures. The appropriate report is not simply “the diagnostics were satisfactory.” It distinguishes evidence about the mean pattern from evidence about distributional shape. Similarly, a single large residual is not the same as a systematic pattern across fitted values or observation order. Identify whether the evidence is isolated or repeated.
Avoid unsupported certainty. “No clear trend was visible” is appropriately limited; “the errors are independent” is not established merely because an order plot looks unstructured. “The points were approximately linear in the normal quantile plot” is a visual description; it is not proof of normality. This wording matters because readers may use the summary to judge whether subsequent regression inference is reasonable.
Common Mistakes
- Giving a verdict without evidence. “The assumptions are fine” does not say which plots were examined or what they showed. Name the displays and describe their visible features.
- Turning absence of a visible pattern into proof. A plot with no conspicuous structure provides limited reassurance, not proof that a condition holds.
- Confusing residual constraints with diagnostic success. A zero residual sum and zero residual–fitted inner product are properties of least squares with an intercept; they do not show that residuals are pattern-free.
- Using broad labels instead of descriptions. “The model is bad” is less useful than naming a curved pattern, tail departure, isolated point, or ordered sequence.
- Claiming a cause from a plot. A pattern can suggest a feature needing investigation, but a plot alone generally does not identify why it occurred.
Check Your Understanding
For each question, focus on both the diagnostic evidence and the limits of what can be concluded.
- Write one sentence that reports a residual-versus-fitted plot with a widening vertical spread but no clear curvature, without claiming that a variance assumption is proven or disproven.
- A normal quantile plot is close to its reference line except for one point at the upper end. What should a careful summary mention, and what should it avoid concluding?
- Why does a residual sum of zero not imply that residuals have no pattern across observation order?
- Give two reasons to name the specific diagnostic plots in a written summary rather than writing “the residual diagnostics looked good.”
- What is the difference between saying “no conspicuous pattern was visible” and saying “the model assumption is true”?