Restricted Cubic Splines at Boundary Knots

Why Restrict a Cubic Spline?

Longitudinal biomarkers may change at different rates over follow-up: a trajectory can rise quickly, flatten, or turn downward. A single straight line may miss this pattern, while an unrestricted cubic polynomial can bend sharply, especially beyond the times represented in the data. Restricted cubic splines offer a compromise: they allow smooth curvature over the observed range and require the trajectory to be linear outside designated boundary knots.

That restriction matters in a joint model because the longitudinal submodel represents an underlying biomarker trajectory, not just the values recorded at visits. Predictions beyond the observation window can affect interpretation of the latent trajectory and any event-model quantity that uses it. A linear tail is a modeling constraint, not evidence that the biomarker truly continues linearly after follow-up.

A Basis With Linear Tails

Let the ordered knots be \(\kappa_1<\kappa_2<\cdots<\kappa_K\), with \(K\geq 3\). The boundary knots are \(\kappa_1\) and \(\kappa_K\); the other knots are interior knots. Define the positive-part cube by \((u)_+^3=\max(u,0)^3\). One common restricted cubic spline basis is

and

The resulting trajectory can be written as

There are \(K\) coefficients: an intercept, a linear term, and \(K-2\) nonlinear terms. Each \(R_j(t)\) has units of time squared, because a cubed time difference is divided by a time difference. Thus its coefficient has units of biomarker per time squared, so every term in \(f(t)\) has biomarker units.

In a longitudinal submodel, \(f(t)\) can serve as a population mean trajectory. A subject-specific latent trajectory can also include random coefficients on the same basis. In either case, the restriction concerns the shape: outside the boundary knots the spline contribution is linear in time.

Why the Tails Are Linear

The linear-tail property follows directly from the basis. If \(t\leq\kappa_1\), every positive-part term in every \(D_j(t)\) is zero. Consequently, every \(R_j(t)=0\), and \(f(t)=\beta_0+\beta_1t\) on the left tail.

For \(t\geq\kappa_K\), expand the two cubes in \(D_j(t)\). For \(j=1,\ldots,K-1\),

The \(3t^2\) term is identical for every \(j\). Subtracting \(D_{K-1}(t)\) therefore cancels the quadratic terms and leaves a linear function:

Thus the full trajectory on the right tail is linear. This establishes the main algebraic result: the restriction is built into the choice of basis, rather than imposed later by trimming a fitted curve.

The basis is also smooth at the knots. Each positive-part cube and its first two derivatives agree from the left and right at its knot: the function, first derivative, and second derivative are all zero there. Sums and differences of these terms preserve that continuity. The spline therefore has no jumps in its value, slope, or curvature at a knot.

Use in a Joint Model and Interpretation

A spline can specify the time-dependent mean in a longitudinal submodel. For example, a Gaussian measurement model might have \(Y_{ij}=m_i(t_{ij})+\epsilon_{ij}\), where \(Y_{ij}\) is the measurement for subject \(i\) at visit \(j\), \(m_i(t)\) is that subject’s latent trajectory, and \(\epsilon_{ij}\) is measurement error. A spline basis can be used in \(m_i(t)\) to allow a nonconstant rate of change. The observation model and any assumptions about random effects or measurement errors still need to be specified separately; the spline basis alone does not supply them.

If the subject-specific trajectory has random spline coefficients, then its tail remains linear conditional on those coefficients. The subject-specific slope may differ from the population slope. In a joint model, any association between a latent trajectory and event risk is interpreted through the specified event submodel; a linear-tail restriction does not itself establish an event association or a causal effect.

Choosing and Checking the Boundary Restriction

The restriction is useful when a stable, simple extrapolation is preferable to an unconstrained cubic tail. It does not guarantee that the extrapolation is accurate. Beyond the boundary knot, the fitted trajectory cannot bend to reflect a genuine plateau, acceleration, or reversal. The location of the boundary knots therefore encodes a substantive modeling choice: it determines where flexibility ends and linear behavior begins.

Check the fitted trajectory and its first derivative across the observed follow-up range, especially near the boundary knots. Look for abrupt-looking changes in slope, implausible curvature inside the knots, or tail slopes that yield unrealistic predictions when carried beyond the data. A smooth curve can still fit poorly, and a plausible in-range curve can still have a biologically implausible extrapolation. Sparse observations near a boundary also make it difficult to assess how the fitted curve behaves there.

Restricted cubic splines differ from ordinary cubic polynomial regression, which uses a global polynomial and may curve increasingly at extremes. They also differ from an unrestricted piecewise cubic spline, whose tail need not be linear unless boundary constraints are imposed. B-spline bases are another way to represent spline functions; basis choice changes the coordinates used for the function, while restrictions determine the permitted shape. Penalized splines address the balance between fit and smoothness through a penalty; they are a distinct modeling approach, not a consequence of linear-tail restrictions.

The boundary restriction can be valuable because it prevents the spline’s cubic components from producing arbitrary curvature indefinitely. Its limitation is equally important: it replaces unconstrained tail curvature with a particular extrapolation assumption. In a joint model, communicate that assumption alongside the fitted trajectory and avoid presenting the tail as if it were directly observed.