Comparing Nonlinear Fits on a Common Outcome Scale

Why a Common Scale Matters

Two nonlinear models can tell different practical stories even when both produce plausible curves. One may level off, while another continues to rise; their fitted values may be close near the center of the data and far apart near an edge. To see what these differences mean, compare the curves at the same predictor values and express both predictions in the same outcome units.

A comparison is difficult to interpret if one curve is shown in millimetres and another in log millimetres, or if the curves are drawn over different predictor ranges. The first task is therefore not to declare a winner. It is to make the candidate fits comparable, show where the observations lie, and describe how uncertain the fitted values are.

Definition: A comparison of nonlinear fits is on a common outcome scale when every candidate curve represents the same outcome quantity, measured in the same units, at the same predictor values. The curves should also be displayed over a predictor range that makes their observed-data support clear.

Set Up a Fair Comparison

Suppose the observed outcome is \(Y\), measured in its original units, and the predictor is \(x\). Each candidate model gives a fitted mean response, written here as \(\hat{m}_j(x)\) for candidate \(j\). This is the model's estimated mean outcome at predictor value \(x\); it is not a prediction for every individual observation.

Before comparing curves, identify the quantity they are intended to estimate. In a standard mean-regression setting, a candidate model represents the conditional mean \(E(Y\mid x)\). Its fitted mean curve and uncertainty intervals are meaningful under the model's assumptions: observations are independent, the chosen mean function adequately describes the conditional mean, and the variance assumptions used to calculate uncertainty are reasonable. If an interval relies on a normal approximation, that approximation must also be adequate for the fitted estimate. These assumptions apply to each candidate's inference; placing curves on the same axes does not make an unsuitable model suitable.

For a useful visual comparison, use identical axis limits and units, evaluate each fit on a shared grid of predictor values, and show the observed predictor values or their distribution. A shared grid prevents a curve from appearing different merely because it was evaluated at different locations. Showing the data support helps readers distinguish a difference in the observed range from a difference that occurs where data are sparse.

Use the same predictor definition and units for every candidate.
Express every fitted value as the same outcome quantity in the same units.
Show a common predictor range and indicate where observations occur.
Describe uncertainty for fitted means separately from uncertainty for individual outcomes.

Worked Example: Reading Two Curves in the Same Units

Suppose a hypothetical laboratory records an enzyme response \(Y\), in activity units, across doses \(x\), in milligrams per litre. The observed doses range from 1 to 6 mg/L. Two candidate models estimate the mean response in activity units: a quadratic curve and a saturating curve.

The fitted curves are

\hat{m}_{Q}(x)=12+4x-0.4x^2, \qquad \hat{m}_{S}(x)=10+14(1-e^{-0.35x}).

Both formulas return activity units, so their fitted means can be compared directly. Evaluate them at the same doses:

Dose \(x\) (mg/L)	Quadratic fitted mean (activity units)	Saturating fitted mean (activity units)	Quadratic minus saturating (activity units)
1	12 + 4(1) - 0.4(1²) = 15.60	10 + 14(1 - e^-0.35) = 14.13	1.466
3	12 + 4(3) - 0.4(3²) = 20.40	10 + 14(1 - e^-1.05) = 19.10	1.299
5	12 + 4(5) - 0.4(5²) = 22.00	10 + 14(1 - e^-1.75) = 21.57	0.433

At these doses, the quadratic curve is above the saturating curve, but the fitted difference narrows from about 1.466 activity units at 1 mg/L to 0.433 activity units at 5 mg/L. The comparison describes the fitted means, not a causal effect of changing dose. Both curves cover the observed dose range, but a plot should still show how many observations support different parts of that range, especially near its edges.

Show Uncertainty Without Overstating It

A pointwise confidence interval describes uncertainty in a fitted mean at one specified predictor value. It does not describe the likely range of individual outcomes, and a collection of pointwise intervals is not automatically a confidence band that covers the whole curve at once. Those are different inferential targets.

When two fitted means at the same \(x\) are compared, their separate standard errors do not, by themselves, give the standard error of their difference. Both estimates may depend on the same observations. If the estimates are positively associated, treating them as independent can overstate the uncertainty in their difference; with negative association, it can understate it.

Result: If fitted mean estimates \(\hat{m}_1(x)\) and \(\hat{m}_2(x)\) have standard errors \(SE_1\) and \(SE_2\), and their estimated covariance is \(\operatorname{Cov}(\hat{m}_1,\hat{m}_2)\), then

SE\!\left(\hat{m}_1(x)-\hat{m}_2(x)\right) = \sqrt{SE_1^2+SE_2^2-2\operatorname{Cov}(\hat{m}_1,\hat{m}_2)}.

This follows from the variance rule for a difference. For any two random quantities \(A\) and \(B\), expand the squared deviation of \(A-B\) from its mean:

\begin{aligned} \operatorname{Var}(A-B) &=E\!\left[\left\{(A-E(A))-(B-E(B))\right\}^2\right]\\ &=\operatorname{Var}(A)+\operatorname{Var}(B)-2\operatorname{Cov}(A,B). \end{aligned}

Taking the square root gives the standard error formula. For two different candidate models fit to the same data, the covariance may not be supplied by standard model output. A paired resampling procedure that refits both models to each resampled data set can estimate how their fitted means vary jointly. The procedure must preserve the paired nature of the two fits.

Worked Example: Uncertainty in a Difference Between Fits

At a hypothetical dose of 3 mg/L, suppose two candidate models estimate mean enzyme responses of 20.4 and 19.1 activity units. Their standard errors are 0.8 and 0.7 activity units. A paired resampling analysis estimates the covariance between their fitted means as 0.336 activity-units squared. Use a normal approximation to form a 95% confidence interval for the difference, defined as model 1 minus model 2.

The estimated difference is

\hat{d}=20.4-19.1=1.3\text{ activity units}.

Using the covariance in the variance formula gives

SE(\hat{d}) = \sqrt{0.8^2+0.7^2-2(0.336)} = \sqrt{0.64+0.49-0.672} = \sqrt{0.458} \approx 0.676756.

A 95% normal-approximation interval is the estimated difference plus or minus 1.96 standard errors:

1.3\pm1.96\sqrt{0.458} \approx 1.3\pm1.32644 = (-0.0264,\ 2.6264)\text{ activity units}.

The interval includes zero. Under the stated approximation and resampling-based covariance estimate, these data do not establish a nonzero difference in fitted mean response at 3 mg/L at the 5% two-sided level. This does not show that the candidate curves are identical: it indicates that the estimated difference at this particular dose is uncertain. The conclusion is limited to this dose and this uncertainty calculation.

For context, the individual pointwise intervals would be \(20.4\pm1.96(0.8)=(18.832,\ 21.968)\) and \(19.1\pm1.96(0.7)=(17.728,\ 20.472)\), in activity units. Their overlap is not the calculation for the difference interval. The difference interval must account for the covariance between the fitted means.

Put Transformed-Outcome Fits Back on the Intended Scale

Sometimes a candidate model is fitted to a transformed outcome, such as \(\log(Y)\), while another is fitted directly to \(Y\). Their predictions should not be plotted together until both represent the same outcome quantity on the same scale. Simply exponentiating a fitted mean on the log scale generally does not produce the arithmetic conditional mean on the original scale.

For example, if \(\log(Y)=\eta(x)+\varepsilon\), with normally distributed errors of mean zero and constant variance \(\sigma^2\), then \(Y\) has a lognormal conditional distribution. Its arithmetic conditional mean is

E(Y\mid x)=\exp\!\left(\eta(x)+\frac{\sigma^2}{2}\right).

The correction depends on the error variance. By contrast, \(\exp(\eta(x))\) is the conditional median under this model. Which quantity to plot depends on the scientific question; comparing a median from one model with a mean from another is not a common-scale comparison.

Worked Example: Comparing a Log-Scale Fit With a Direct Fit

At a hypothetical dose, a log-outcome model gives a fitted log mean \(\hat{\eta}=\log(20)\), and its estimated conditional error variance on the log scale is \(0.09\). A direct model fitted to the original outcome scale estimates a mean response of 21.4 activity units. Assume the log-outcome model's conditional errors are normal with mean zero and the stated variance.

Exponentiating the fitted log mean alone gives

\exp(\hat{\eta})=\exp(\log(20))=20,

which estimates the conditional median under the lognormal model, not its arithmetic mean. To compare arithmetic mean responses, apply the variance correction:

\widehat{E(Y\mid x)} = \exp\!\left(\log(20)+\frac{0.09}{2}\right) = 20\exp(0.045) \approx 20(1.04603) \approx 20.92\text{ activity units}.

The comparable fitted arithmetic means are therefore 20.92 activity units from the log-outcome model and 21.4 activity units from the direct model, a difference of about \(-0.48\) activity units when calculated as log-outcome model minus direct model. An uncertainty interval for the corrected mean would also need to account for uncertainty in the fitted log mean and in the variance correction; the point estimate alone does not supply that interval.

Common Mistakes and Practical Interpretation

The purpose of a common-scale plot is to make differences interpretable, not to manufacture a single ranking from visual appearance. A curve may be higher in one region and lower in another. Even where one is higher, the difference may be small relative to uncertainty. Readers should be able to see both the fitted values and where the observations provide support.

Comparing different outcome quantities. A log-scale fitted mean and an original-scale arithmetic mean are not directly comparable. Convert each prediction to the same target quantity before plotting.
Treating separate bands as a test of the difference. Overlap or non-overlap of two separate pointwise confidence intervals is not a substitute for an interval or test for the difference that accounts for dependence.
Hiding weak data support. A smooth curve can look persuasive where few observations occur. Show the observed predictor values or their distribution alongside the curves.
Comparing fitted means with individual outcomes. Confidence intervals for a mean curve describe uncertainty about the mean, not the spread of individual responses. Use an appropriate prediction interval when the target is an individual outcome.
Reading a plot as proof of causation or model truth. A plotted difference is a difference between fitted associations under candidate models. Its interpretation depends on the study design, model assumptions, and data support.

A clear comparison therefore names the outcome quantity, gives units, uses a shared predictor grid and axes, shows observed-data coverage, and distinguishes pointwise uncertainty for each curve from uncertainty in their difference. These choices allow practical contrasts to be seen without implying more certainty than the analysis provides.

Check Your Understanding

Answer each question using the ideas in this tutorial.

Two candidate curves use the same predictor values, but one is expressed in log units and one in the original outcome units. What must be clarified before they can be compared?
Why does the standard error of the difference between two fitted means generally require their covariance?
At one predictor value, two separate 95% pointwise confidence intervals overlap. What can and cannot be concluded from that fact alone?
Under a lognormal outcome model, what does \(\exp(\eta(x))\) represent, and how is the arithmetic conditional mean obtained?
Name two features a plot should include to help readers judge whether candidate curves are meaningfully comparable and supported by the data.