Comparing Candidate Forms Near Predictor Boundaries

Why Compare Curves at the Boundaries?

Two nonlinear curves can trace nearly the same path through the middle of the observed predictor values and still make quite different predictions near the smallest or largest observed value. These differences matter when a description or prediction concerns observations near an edge of the data. A plot dominated visually by the central observations may not make the contrast easy to see.

This tutorial focuses on comparing candidate curves within the observed predictor range, especially at its two ends. The aim is not to decide that one curve must be correct from an endpoint check alone. Instead, inspect what each candidate implies at the boundaries, compare those implications with the observed responses there, and report meaningful differences in context.

A Consistent Basis for Comparison

A candidate regression form represents the mean response as a function of the predictor. One general notation is \(E(Y_i\mid x_i)=f(x_i;\theta)\), where \(Y_i\) is the response for observation \(i\), \(x_i\) is its predictor value, and \(\theta\) denotes the parameters of the chosen form. After fitting, \(\hat{f}(x)\) is the fitted mean response at \(x\). For a descriptive comparison of fitted curves, we do not need to assume that errors have a particular probability distribution. We do need to compare curves fitted to the same data, for the same response and predictor, and on the same response scale.

The usual regression interpretation also relies on the model representing the conditional mean appropriately; in particular, the errors have mean zero conditional on the predictor. If a curve is used to describe expected response, its fitted values should be read as estimated mean responses, not as guarantees about individual outcomes. The boundary checks below are descriptive checks of candidate fits, not standalone hypothesis tests.

A useful comparison has several parts. First, calculate each curve’s fitted values at \(x_{\min}\) and \(x_{\max}\). Next, compare those values with the observations at or nearest to each boundary. Finally, inspect the curve’s change over a short interval next to each boundary. If derivatives are available, the derivative gives the curve’s instantaneous slope; otherwise, a finite difference over a specified interval gives an interpretable average change. Always retain the predictor and response units.

Why Small Interior Differences Can Grow at an Edge

The preceding example has a short algebraic explanation. The difference between the two fitted curves is exactly the cubic term:

When \(z=0\), this difference is zero. For \(|z|\leq 0.2\), its absolute value is at most \(8(0.2)^3=0.064\). But at either boundary, \(|z|=1\), so the absolute difference is \(8(1)^3=8\). The same term that barely changes the fitted curve near the center can have a much larger effect at the edges.

The slopes also differ. Since \(z=(x-5)/5\), \(dz/dx=1/5\). Differentiating each fitted curve with respect to \(x\) gives

At the largest observed predictor value, \(z=1\). The quadratic slope is \((8-8)/5=0\) outcome units per predictor unit, whereas the cubic slope is \((8-8+24)/5=4.8\) outcome units per predictor unit. Thus the fitted values at the upper boundary differ, and the curves imply quite different local behavior there. The derivative calculation describes the fitted curves at the boundary; it does not establish what happens beyond the observed range.

Compare Fitted Values With Boundary Observations

A boundary comparison is more informative when it includes the observed response values. For an observation \(i\), the residual is \(e_i=y_i-\hat{f}(x_i)\): the observed response minus the fitted response. A positive residual means the curve underpredicts that observed response; a negative residual means it overpredicts. Comparing endpoint residuals shows how closely each candidate fits those particular observations, but it does not summarize its fit over all observations.

The comparison also depends on which observations actually occur at an endpoint. If several observations share the smallest or largest predictor value, examine their responses rather than treating one response as representative of all of them. If there is no observation at a particular exact value because predictor values are continuous, the minimum and maximum observed values are still the boundaries: evaluate the fitted curves at those observed predictor values and compare them with the corresponding observations.

Inspect Changes Just Inside Each Boundary

Endpoint fitted values do not tell the whole story. Two curves can pass close to the same endpoint value but approach it at different rates. A practical check is to calculate each fitted curve’s change over a short interval that remains inside the observed range. Divide the change in fitted response by the change in predictor to express an average slope over that interval. State the interval, since a nonlinear curve’s average slope can depend on the interval selected.

Using Boundary Checks Responsibly

Boundary comparisons help explain why candidate forms that look similar overall may not be interchangeable for a particular purpose. If predictions near the maximum predictor value are important, for example, a large difference between fitted values there deserves explicit attention. The same is true at the minimum. If boundary predictions are not the focus, the difference can still help characterize the curve, but it should not be presented as decisive without considering the rest of the evidence.

Keep these checks disciplined:

• Use the same observed predictor range and response scale for every candidate being compared.
• Report endpoint fitted values with units and, where useful, the observed responses and residuals.
• When comparing local change, state the boundary-adjacent interval or the slope’s units.
• Describe differences as features of the fitted curves, not proof that one curve is the true relationship.
• Do not interpret behavior outside \(x_{\min}\) and \(x_{\max}\) as observed boundary behavior; values outside that range are beyond the data.

A common mistake is to look at nearly overlapping curves in the center and conclude that the candidate forms must give similar predictions throughout the range. The reasoning fails because terms such as \(z^3\) can be tiny when \(z\) is near zero but much larger when \(|z|\) approaches one. The correction is to calculate and compare fitted values at both observed boundaries and inspect nearby changes, then describe any discrepancy in the units and context of the response.