Testing Added Segments at Prespecified Knots

Why Test Added Segments?

A continuous piecewise linear regression can represent a relationship whose slope changes at specified values of a predictor. Adding hinge terms makes that flexibility possible, but a more flexible model will never have a larger residual sum of squares than its simpler counterpart. The relevant question is therefore not whether the expanded model fits at least as well; it is whether the reduction in residual error is large relative to the additional parameters used.

A nested-model partial F-test answers this question for a set of knots chosen in advance. It compares a straight-line model with a model that adds hinge terms at those fixed locations. The test concerns whether the added terms, taken together, improve the fit beyond what random error would plausibly produce under the simpler model.

The Two Nested Models

Suppose \(n\) observations \((x_i,y_i)\) are available, and \(q\) distinct knots \(k_1,\ldots,k_q\) have been specified. The hinge function at knot \(k_j\) is \((x_i-k_j)_+=\max(0,x_i-k_j)\). The reduced model is an ordinary straight line. The full model adds one hinge term for each knot:

In the full model, the fitted relationship is continuous at each knot. The coefficient \(\gamma_j\) is the change in slope at knot \(k_j\), holding the other terms in the model in place. For example, with two knots, the slope before the first knot is \(\beta_1\), between the knots it is \(\beta_1+\gamma_1\), and after the second knot it is \(\beta_1+\gamma_1+\gamma_2\).

The test compares residual sums of squares. Let \(SSE_R\) be the residual sum of squares from the reduced model and \(SSE_F\) the residual sum of squares from the full model. With full column rank, the reduced model has two estimated coefficients and the full model has \(q+2\). Thus the full-model residual degrees of freedom are \(n-q-2\), while the number of added parameters is \(q\).

Under \(H_0\) and the usual normal linear-model assumptions, this statistic has an \(F\) distribution with \(q\) numerator degrees of freedom and \(n-q-2\) denominator degrees of freedom. The numerator is the reduction in residual sum of squares per added coefficient; the denominator is the full model’s residual mean square.

Assumptions and Why the Test Works

For the exact reference distribution, treat the predictor values and knots as fixed. Assume the errors are independent, have mean zero and common variance \(\sigma^2\), and are normally distributed. Equivalently, conditional on the predictor values, \(\varepsilon_i\sim N(0,\sigma^2)\). The model must also be correctly specified for the comparison, and the columns of the full model matrix must be linearly independent. In practical terms, the observed predictor values must provide enough variation to distinguish the straight-line terms from the hinge terms.

A short derivation explains the statistic. Under the null, both models describe the conditional mean, but the full model has \(q\) additional independent directions in which it can fit the response. In nested least squares, the reduction \(SSE_R-SSE_F\) is the squared length of the fitted component contributed by those added directions. Under normal errors, divided by \(\sigma^2\), this reduction has a chi-square distribution with \(q\) degrees of freedom. The full-model residual sum of squares, divided by \(\sigma^2\), has a chi-square distribution with \(n-q-2\) degrees of freedom. These quantities are independent because they arise from orthogonal components of the normal error vector. Therefore,

This argument also shows why the test uses the full model’s residual mean square as its estimate of error variance. A large \(F\) value means the added terms reduce error substantially compared with the residual variation that remains in the full model. The p-value is the probability, assuming \(H_0\) and the model assumptions, of obtaining an \(F\) statistic at least as large as the observed value.

Interpreting the Result Carefully

The test’s conclusion is conditional on the knot locations in the model. A significant result supports the claim that the fixed-knot piecewise model improves on a straight line at those locations, under the assumptions. It does not by itself prove that the relationship truly changes slope, establish a causal effect, or show that every added segment is useful. Regression describes association unless the study design and additional assumptions justify causal interpretation.

A nonsignificant result likewise does not prove that the true slope is identical throughout the predictor range. The available sample may have limited information to detect slope changes, particularly if few observations lie near or between knots. It is more accurate to report that the test did not provide sufficient evidence of improvement than to claim that no slope change exists.

The test also depends on how the knots were chosen. Its usual interpretation assumes the knot locations were prespecified independently of the observed outcomes. Selecting locations after inspecting which ones produce the largest reduction in SSE changes the testing problem; the nominal p-value from the fixed-knot comparison no longer accounts for that selection. Keep the distinction clear: this procedure tests added hinge terms at given locations, not an unknown breakpoint search.

Common Mistake: Testing Each Hinge Separately

A tempting approach is to fit the full model and test each \(\gamma_j\) separately, then treat the collection of individual tests as equivalent to the nested-model test. They are not the same question. The partial F-test evaluates the joint null that all \(q\) hinge coefficients are zero, accounting for the reduction in SSE across the added terms as a group. Separate coefficient tests address individual terms conditional on the others and do not replace that joint comparison.

Another error is to interpret a decrease in SSE alone as evidence of an improved model. Adding parameters cannot increase least-squares SSE, even when the extra terms only fit noise. The F statistic scales that decrease by both the number of added parameters and the remaining full-model error. The resulting p-value, together with the model assumptions and context, supports the decision.