Nonlinear Biomarker Effects in Landmark Models

Why Allow a Curved Biomarker Effect?

A landmark model predicts subsequent event risk among individuals who are event-free at a specified landmark time and have the information required to enter the landmark risk set. The biomarker recorded or summarized at that time need not have a straight-line association with subsequent risk. Risk might rise sharply at high values, flatten over part of its range, or be lowest near a middle value. A model that forces a straight-line effect can miss these patterns and produce misleading predictions.

The goal is to represent the association flexibly enough to capture meaningful curvature while retaining stable, interpretable predictions. This tutorial considers a fixed-horizon logistic model and a landmark Cox model. In both, the nonlinear term describes how the fitted risk changes across biomarker values; it does not by itself establish a causal effect.

Landmark Models with a Flexible Biomarker Term

For a fixed horizon, define \(Y=1\) if the event occurs after \(t_L\) and by \(t_L+\Delta\), and \(Y=0\) if it does not occur by that horizon. For individuals eligible at \(t_L\), a logistic model can be written as

where \(Z\) denotes other predictors, \(T\) is the event time, and \(\operatorname{logit}(p)=\log\{p/(1-p)\}\). The predicted horizon risk for specified values \(x,z\) is

A landmark Cox model instead describes the hazard at residual time \(u\) after the landmark. One form is

Here \(h_0(u)\) is the baseline hazard. Under proportional hazards, the biomarker’s contribution is constant over residual time, though its association with the biomarker value can be nonlinear. A contrast between \(x_1\) and \(x_0\), holding \(Z\) fixed, has hazard ratio \(\exp\{f(x_1)-f(x_0)\}\). It is the contrast in fitted function values—not an individual spline coefficient—that determines this hazard ratio.

Representing Curvature

A transformation is a compact choice of shape. For example, \(f(X)=\beta_1\log(X)\) is appropriate only for positive \(X\), and specifies a particular relationship between the log biomarker and the model’s linear predictor. A quadratic term, \(f(X)=\beta_1X+\beta_2X^2\), permits curvature but may impose an unrealistic shape, especially near the ends of the observed range.

Splines use several basis functions to make a flexible curve. A restricted cubic spline is cubic between knots and constrained to be linear beyond its boundary knots. With three ordered knots \(\kappa_1<\kappa_2<\kappa_3\), a simple restricted cubic spline representation has a linear term and one nonlinear basis function:

where \(a_+=\max(a,0)\), and \(c\) is a centering value. With more knots, the basis contains additional nonlinear terms. The knot locations and number of knots determine how much flexibility is available; they should be chosen with the data range and sample size in mind, not by searching for a curve that happens to fit every fluctuation.

Interpreting the Fitted Curve

With a linear biomarker term, a one-unit increase has the same change in the log odds at every biomarker value. With a nonlinear function \(f(x)\), the local change depends on \(x\). In a differentiable model, the slope of the log odds is \(f'(x)\); the change in predicted probability also depends on the current probability, since the logistic curve is steepest near one half and flatter near zero and one. For a clinical interpretation, predictions at clinically meaningful values and contrasts across the observed range are usually clearer than interpreting individual basis coefficients.

For a Cox model, \(f(x_1)-f(x_0)\) compares log hazards at two biomarker values, conditional on the other modeled predictors. The associated hazard ratio compares instantaneous event rates among individuals still at risk at the same residual time. It is not a ratio of fixed-horizon risks. A fixed-horizon risk from a Cox model also uses the estimated baseline cumulative hazard; under the model, survival through residual time \(\Delta\) is \(S(\Delta\mid x,z)=\exp[-H_0(\Delta)\exp\{f(x)+\gamma^\mathsf{T}z\}]\).

Choosing a Representation and Checking Its Adequacy

A spline is often a useful default when a straight-line effect is doubtful but there is not a strong scientific reason to impose a particular transformation. A transformation can be more concise and may be easier to explain when its shape is substantively justified. Categories are easy to display but discard within-category information, make results depend on cutpoints, and create artificial jumps at category boundaries. Categorizing a continuous biomarker is therefore not a general remedy for nonlinearity.

A flexible curve is not automatically a better model. Check how many observations and events support different parts of the biomarker range, inspect fitted risks or hazards across that range, and look for unstable bends near sparse tails. Compare a linear specification with a prespecified transformation or spline using appropriate model comparisons, while recognizing that improved fit alone does not guarantee accurate prediction. If the curve changes markedly with small changes in knot placement, the data may not support that degree of flexibility.

Predictions outside the observed range are especially fragile. A restricted cubic spline is linear beyond its boundary knots by construction, but that mathematical constraint does not provide empirical evidence that the relationship continues linearly there. Report the range used to fit and interpret the curve. Also distinguish uncertainty in the estimated curve from variation in individual outcomes; a point prediction is not a guarantee for a particular patient.

Deriving a Risk Contrast from the Model

The interpretation of a nonlinear logistic model follows directly from its odds. Holding other predictors fixed, the fitted odds at biomarker value \(x\) are \(\exp\{\beta_0+f(x)+\gamma^\mathsf{T}z\}\). For values \(x_1\) and \(x_0\), dividing these odds cancels the intercept and the terms for \(z\):

This derivation explains why a single spline coefficient is not generally an odds ratio for a one-unit biomarker increase. The contrast depends on both values being compared and on the entire fitted function. The same subtraction of function values yields a Cox log-hazard contrast, but the resulting hazard ratio is not an odds ratio or a fixed-horizon risk ratio.