Computing Spline Slopes and Cumulative Exposure

Why slopes and cumulative exposure require the fitted trajectory

In a joint model, the longitudinal component describes how a subject’s underlying biomarker changes over time. A fitted value at one time describes the trajectory’s level; its derivative describes its instantaneous rate of change; and its integral over an interval summarizes accumulated exposure. These are related summaries, but they answer different questions. A subject can have a high biomarker level and a decreasing slope, for example, while still having a large cumulative exposure.

For a spline trajectory, the calculations are defined by the fitted function, not by connecting the observed measurements with straight lines. This matters because the fitted spline can be smooth between observations and can incorporate subject-specific random effects. Taking differences of noisy measurements can produce a very different estimate of change.

If the biomarker is measured in units \(U\) and time in years, then \(m_i(t)\) has units \(U\), \(m_i'(t)\) has units \(U\) per year, and cumulative exposure has units \(U\)-years. Average exposure has the same units as the biomarker. These unit checks are useful safeguards against calculation errors.

Calculating from a spline representation

Write the subject-specific trajectory as a linear combination of spline basis functions:

where \(\mathbf{s}(t)\) is a vector of basis functions and \(\mathbf{a}_i\) is the corresponding vector of coefficients for subject \(i\). In a random-effects representation, these coefficients combine population coefficients and subject-specific random effects. Once the coefficients are specified, differentiating and integrating the trajectory means differentiating and integrating each basis function.

These expressions assume the fitted coefficients are held fixed for the subject-specific calculation. They also require the spline to be differentiable at the time where a slope is requested. A spline may have different smoothness at its knots depending on its degree and construction. For a piecewise-defined spline, calculate the derivative within each interval and check the behavior at the boundaries. A continuous trajectory need not have a continuous derivative unless the spline construction imposes that condition.

A useful special case is the truncated-power term \((t-\kappa)_+^q\), where \((x)_+=\max(x,0)\), \(\kappa\) is a knot, and \(q\) is a positive integer. Away from the knot, differentiation gives:

For \(q=1\), the derivative is 0 to the left of the knot and 1 to the right, and is not defined at the knot. For \(q>1\), the derivative at the knot exists and is 0. Integration follows directly by integrating the polynomial on the portion of the interval after the knot:

This identity also handles intervals entirely before the knot: both positive-part terms are zero. If an interval crosses a knot, the same expression accounts for the portion after it. With a general piecewise-polynomial basis, an equivalent reliable approach is to split the integral at every knot and integrate the relevant polynomial on each resulting interval.

From the trajectory to cumulative exposure

Cumulative exposure is the area under the fitted trajectory over a specified time interval. It depends on both the biomarker level and the duration of follow-up. Therefore, always state the integration limits, and do not compare cumulative exposures from different follow-up durations without accounting for that difference. Average exposure divides the area by the interval length and can be easier to compare when interval lengths differ.

The integral can also be written as a linear function of the spline coefficients. Define the integrated basis vector \(\mathbf{S}_{a,T}=\int_a^T\mathbf{s}(u)\,du\). Then the cumulative exposure is \(\mathbf{S}_{a,T}^{\mathsf T}\mathbf{a}_i\). This makes the calculation convenient in software and clarifies how subject-specific random effects affect the summary: their coefficients contribute through the same integrated basis terms as the fixed coefficients.

Why these calculations are valid—and what they assume

The calculations above follow from linearity: a finite linear combination of differentiable basis functions can be differentiated term by term, and a finite linear combination of integrable basis functions can be integrated term by term. For the truncated-power basis, the derivative and integral are ordinary polynomial calculations on either side of the knot. As a result, the slope and exposure are exact for the stated fitted function, apart from rounding.

These summaries inherit the assumptions and limitations of the fitted trajectory. In particular, the spline form and knot locations determine the shape between observations; estimates near boundaries may be especially sensitive to the fitted form. The calculations also treat the estimated coefficients as fixed. In practice, coefficient estimates are uncertain, so reported slopes and exposures should not be presented as if known exactly when inferential uncertainty matters.

For a linear summary \(g_i=\mathbf{c}^{\mathsf T}\mathbf{a}_i\), such as a slope at a fixed time or an integrated exposure, uncertainty can be propagated from the coefficient covariance matrix \(\mathbf{V}_i\) using \(\operatorname{Var}(g_i)=\mathbf{c}^{\mathsf T}\mathbf{V}_i\mathbf{c}\), when that covariance is available and the summary is linear in the coefficients. Here \(\mathbf{c}=\mathbf{s}'(t)\) for a slope and \(\mathbf{c}=\mathbf{S}_{a,T}\) for cumulative exposure. This calculation does not remove uncertainty from model specification or from estimated random effects; it quantifies coefficient uncertainty under the fitted model.

Diagnostics, alternatives, and common pitfalls

A strong check is to compare an analytic derivative or integral with a numerical calculation from the same fitted function. For a derivative at an interior point, a centered finite difference \(\{m_i(t+h)-m_i(t-h)\}/(2h)\) should approach the analytic derivative as \(h\) becomes small, provided the function is smooth around \(t\). For integration, numerical quadrature over the same limits should agree with the exact integral to the chosen numerical tolerance. These checks can reveal coding mistakes, but the finite-difference or quadrature result is an approximation rather than a replacement for an available exact calculation.

If the derivative is requested at a knot, inspect the spline’s continuity there. For example, the first derivative of a truncated linear term has a jump at its knot, so there is no single two-sided derivative at that point. By contrast, a truncated quadratic term has a continuous first derivative there. Do not report a unique slope at a knot unless the model’s basis ensures that the derivative exists.

The principal alternative to differentiating the fitted trajectory is to estimate change from nearby observed values. That approach can be useful descriptively, but it is sensitive to measurement error, visit spacing, and the choice of observations. It does not generally estimate the derivative of the fitted subject-specific trajectory. Similarly, summing observed biomarker values is not an area-under-the-trajectory calculation unless the time spacing and numerical integration rule are explicitly accounted for.

• Wrong: “The cumulative exposure is the fitted value at the final visit multiplied by follow-up time.” Correction: Exposure is the integral over the whole trajectory; the final value alone cannot determine the area.
• Wrong: “The slope is the difference between any two observed measurements divided by their time gap.” Correction: That is a finite difference of observations, not necessarily the derivative of the fitted trajectory.
• Wrong: “A cumulative exposure of 70.5 has the same units as the biomarker.” Correction: If time is in years, the units are \(U\)-years; dividing by duration gives average exposure in \(U\).
• Wrong: “The spline derivative is automatically defined at every knot.” Correction: Check the spline’s smoothness and the particular basis term at that knot.

In joint modeling, slopes and cumulative exposure can serve as scientifically meaningful summaries of a fitted longitudinal process. They can also help describe how an individual’s trajectory changes over a specified interval. Their interpretation depends on the interval, time scale, biomarker scale, and model used; none of these summaries by itself establishes a causal effect on an event outcome.