Why Reporting the Choice Matters
A fitted curve is not a complete account of a nonlinear regression analysis. Readers also need to know what made the chosen form reasonable, whether other forms remain credible, how uncertain the fitted relationship is, and where the data provide little support. A concise report can communicate these points without presenting the selected curve as the only possible description of the relationship.
Suppose \(X\) is a predictor and \(Y\) is an outcome. The regression relationship of interest can be written as \(m(x)=E(Y\mid X=x)\), the mean outcome at predictor value \(x\). A nonlinear form specifies a particular shape for this relationship. The fitted curve, \(\hat m(x)\), estimates that mean under the chosen form. This tutorial concerns how to report a choice among such forms and its limits, not how to conduct a full diagnostic analysis.
What a Useful Account Includes
A clear explanation does not need to reproduce every modelling step. It should give readers enough context to understand what the choice means and how strongly the evidence supports it. The following elements are useful:
- State the selected form and scope. Name the curve family and give the predictor range covered by the observed data. If the form was chosen for prediction rather than description, say so.
- Summarize the evidence. Report the evidence that informed the choice, such as comparative prediction error or the shape of the fitted curves. Comparisons should use the same outcome scale and the same evaluation approach.
- Name plausible alternatives. Identify forms that remain reasonable, especially if their performance or fitted relationships are similar. A selection among candidates does not establish that every unselected form is wrong.
- Describe uncertainty. Give uncertainty for the fitted relationship where relevant, and clarify whether it is conditional on the selected form. Avoid treating uncertainty within one form as if it also accounts for uncertainty about which form to use.
- Locate weak support. Note predictor regions with few observations, wide uncertainty, or materially different conclusions across plausible forms.
These points are related but not interchangeable. A curve can have a narrow uncertainty band under its chosen form while competing forms imply different curves. Conversely, candidate forms may agree closely in the middle of the observed predictor range but diverge near its edge. Reporting both kinds of evidence gives a more faithful account than reporting a single fit statistic.
Worked Example: Summarizing Comparative Evidence
Worked Example: Summarizing Comparative Evidence
Consider a hypothetical analysis of 84 observations, with predictor values from 0 to 10 and an outcome measured in milligrams per litre. The analyst compares four candidate forms using repeated validation splits and records the mean root mean squared prediction error (RMSE). RMSE is expressed in the outcome’s units; smaller values indicate lower prediction error on this measure.
| Candidate form | Mean validation RMSE | Across-split standard deviation |
|---|---|---|
| Linear | 5.2 mg/L | 0.6 mg/L |
| Quadratic | 4.1 mg/L | 0.5 mg/L |
| Saturating | 4.0 mg/L | 0.7 mg/L |
| Monotone spline | 4.2 mg/L | 0.6 mg/L |
The saturating form has the smallest mean validation RMSE. Relative to the linear form, the difference is \(5.2-4.0=1.2\) mg/L. Relative to the quadratic form, it is only \(4.1-4.0=0.1\) mg/L; relative to the monotone spline, it is \(4.2-4.0=0.2\) mg/L.
The evidence therefore favors the saturating form over the linear form on this prediction measure, but offers little separation between the saturating, quadratic, and monotone spline forms. The standard deviations describe variation in scores across the validation splits; they are not confidence intervals for the differences between forms. It would be too strong to say that the saturating form is definitively superior to all alternatives.
A measured report could read: “Across predictor values from 0 to 10, we used a saturating curve as the primary description. It had the lowest mean validation RMSE (4.0 mg/L), compared with 4.1 mg/L for the quadratic form, 4.2 mg/L for a monotone spline, and 5.2 mg/L for the linear form. The small differences among the three nonlinear forms leave their relative predictive performance uncertain.” This account names the choice, outcome-scale evidence, and remaining uncertainty without claiming more than the comparison supports.
Uncertainty Within a Form Is Not Form-Choice Uncertainty
An uncertainty interval for a fitted mean is generally calculated under a specified model. For example, a pointwise interval at \(x_0\) concerns the mean \(m(x_0)\) under the assumptions used for that fitted form. It does not automatically allow for the possibility that a different candidate form would give a different mean at \(x_0\). If reporting a pointwise interval, state the predictor value, the outcome units, and that it is conditional on the chosen form. A pointwise interval also does not, by itself, give simultaneous coverage for the entire curve.
The distinction can be seen directly. Let forms \(A\) and \(B\) give estimates \(\hat m_A(x_0)\) and \(\hat m_B(x_0)\). A standard error calculated within form \(A\) describes uncertainty associated with \(\hat m_A(x_0)\) under that model. It contains no term for the difference between the two forms:
This is not a formula for combining uncertainty across candidate forms. It is a reminder about what the within-form interval represents: changing the candidate form can change the estimate, and that change is not captured merely by computing an interval under form \(A\). If plausible forms produce materially different conclusions, report that sensitivity separately.
Worked Example: Reporting a Pointwise Interval and Alternatives
Suppose a hypothetical saturating regression gives an estimated mean outcome of 12.4 mg/L at \(x=6\), with a standard error of 1.1 mg/L. For an approximate 95% pointwise interval, use the normal multiplier 1.96 under the stated model and standard-error approximation:
Rounded to three significant figures, the interval is approximately 10.2 to 14.6 mg/L. Under the assumptions of the fitted model and its standard-error calculation, this is an approximate 95% confidence interval for the mean outcome at \(x=6\), conditional on using the saturating form. It is not a 95% probability statement about a fixed mean being inside this particular interval.
Suppose a quadratic fit estimates 11.8 mg/L at the same predictor value and a monotone spline estimates 13.1 mg/L. Those estimates are not included as a special allowance in the interval calculated from the saturating fit. A useful account would report the interval and note that the alternative fitted means at \(x=6\) range from 11.8 to 13.1 mg/L. That range describes sensitivity across the stated candidates; it is not itself a confidence interval.
Worked Example: Describing a Weakly Supported Region
Worked Example: Describing a Weakly Supported Region
In another hypothetical data set, the predictor ranges from 0 to 10. The counts in four ranges are 25 observations from 0 to 2, 38 from 2 to 5, 17 from 5 to 8, and 4 from 8 to 10. Thus, the upper part of the observed range is represented by relatively few observations. A fitted saturating curve estimates a mean of 14.1 mg/L at \(x=5\), with a pointwise 95% interval from 12.8 to 15.4 mg/L. At \(x=9\), it estimates 18.2 mg/L, with an interval from 12.0 to 24.4 mg/L.
The upper-range interval is much wider. Its half-width is \(24.4-18.2=6.2\) mg/L, compared with \(14.1-12.8=1.3\) mg/L at \(x=5\). This contrast supports a cautious statement about greater uncertainty near \(x=9\), alongside the fact that only four observations lie between 8 and 10.
Suppose a plausible alternative curve estimates 15.0 mg/L at \(x=9\), compared with 18.2 mg/L for the selected curve. The difference is \(18.2-15.0=3.2\) mg/L. The report should say that the estimated mean in the upper range is uncertain and sensitive to the curve choice. It should not present 18.2 mg/L as a precise, well-supported conclusion simply because it is the selected model’s point estimate.
The observed range ends at \(x=10\). Predictions beyond 10 would be extrapolations, not conclusions directly supported by observations at those predictor values. Even within the observed range, the sparse upper region deserves a more qualified interpretation than the densely represented middle range.
How to Keep the Report Concise and Accurate
A compact report can follow a consistent order: state the chosen form and why it was preferred; give the strongest comparative evidence; name important alternatives; report uncertainty at relevant predictor values; and identify regions or claims that are weakly supported. The amount of detail should reflect the practical question. If the analysis is intended for prediction, comparative validation performance may be central. If the goal is to describe the relationship, the reported curve shape and its uncertainty across the observed range may be more important.
Keep comparisons on a common basis. For example, validation RMSE values should refer to the same outcome scale and evaluation setup. A difference between scores is meaningful only in relation to their scale and variability; a small numerical advantage does not automatically establish a practically important improvement. If candidate rankings vary across validation splits, say that the ranking is unstable rather than presenting only the best score.
Also distinguish a candidate’s plausibility from a claim that it is true. The selected form is a working representation supported by the available data and analysis. Evidence can support one form more than another without proving the selected shape is uniquely correct. Where reasonable forms lead to similar fitted relationships, that agreement can be reported. Where they diverge, describe where and in what way.
Common Mistake: Treating the Selected Curve as Certain
Wrong reasoning: “The saturating curve had the lowest validation RMSE, so the relationship is saturating and the reported curve is correct.”
Correction: The selected curve performed best on the stated comparison, but the size of its advantage and the performance of plausible alternatives matter. In the first example, its mean RMSE was only 0.1 mg/L lower than the quadratic form and 0.2 mg/L lower than the monotone spline. Those results support using the saturating curve as a primary description, while leaving meaningful uncertainty about which nonlinear form is preferable.
A related mistake is treating a narrow pointwise interval as evidence that the form choice is secure. That interval describes uncertainty conditional on the selected form. It does not resolve disagreement between candidate forms or make a sparsely observed region well supported.
Check Your Understanding
Use the ideas in this tutorial to answer the following questions.
- A selected curve has mean validation RMSE 3.8 units, while two alternatives have values of 3.9 and 4.0 units. What can you conclude, and what would be too strong to claim?
- What does a pointwise confidence interval for the mean at one predictor value describe, and what does it not automatically account for?
- A fitted curve has a much wider pointwise interval near the upper edge of the observed predictor range than in the middle. Give two pieces of context that would help readers understand this uncertainty.
- Why should a report name plausible alternatives even when one form was selected?
- If observations end at \(x=10\), how should predictions at \(x=12\) be described?