Continuous Biomarkers And Treatment-Effect Gradients

Why Model a Continuous Biomarker?

A biomarker measured on a continuous scale may carry information about how treatment effect changes across patients, not merely whether a patient falls above or below a chosen threshold. A model of that pattern can estimate treatment effects at clinically relevant biomarker values while retaining the scale and information of the measurement. The aim is to describe and estimate the gradient, not to turn every estimated difference into a treatment rule.

The earlier material on treatment-by-biomarker interaction models establishes the role of an interaction: treatment effects may vary with a biomarker. Here the key task is to represent that variation when the biomarker is continuous, and to interpret model-based effects at specific values.

In a randomized trial, this conditional mean contrast can represent a causal treatment effect at \(z\), provided treatment assignment is randomized, treatment and outcome are well defined, there is no interference between participants, and both treatment groups are represented at the biomarker values being interpreted. The biomarker must be measured before treatment; conditioning on a post-treatment measurement can change the estimand and introduce bias.

A Linear Treatment-Effect Gradient

For a continuous outcome on its original measurement scale, a simple model is

where \(c\) is a chosen centering value, such as a clinically familiar value or the sample mean biomarker level. The term \(\beta_Z(z-c)\) allows the outcome to vary with the biomarker in the control group, while \(\beta_{TZ}T(z-c)\) allows the treatment contrast to vary with it. Subtracting the model prediction under control from the prediction under treatment gives

Thus \(\beta_T\) is the modeled treatment effect at \(z=c\), and \(\beta_{TZ}\) is the change in treatment effect per one-unit increase in the biomarker. Centering changes the interpretation of the treatment main-effect coefficient, but not the fitted treatment-effect curve.

A predicted effect is a contrast, so its uncertainty depends on the covariance between the estimated coefficients. At a specified \(z\), let \(x=z-c\). The estimated contrast is \(\hat{\beta}_T+x\hat{\beta}_{TZ}\), with estimated variance

A confidence interval can then be formed using the contrast estimate and its standard error, with an appropriate critical value for the fitted model. Reporting only the interaction coefficient does not give the uncertainty for the treatment effect at a particular biomarker value.

Allowing Curvature With Flexible Functions

A straight-line gradient is useful when the change in effect per biomarker unit is plausibly constant across the observed range. If the pattern may bend, a flexible function can represent curvature without dividing the biomarker into groups. For example, choose a set of basis functions \(B_1(z),\ldots,B_K(z)\), such as spline basis functions, and fit

Here \(f(z)\) describes the biomarker pattern in the control group and \(g(z)\) describes how the treatment contrast changes with the biomarker. The resulting conditional treatment effect is \(\tau(z)=\tau_0+\sum_{k=1}^{K}\gamma_kB_k(z)\). Including the biomarker basis terms in both treatment groups is important: a biomarker can be prognostic even when it does not modify treatment effect.

A simple example of a piecewise-linear basis is the hinge function \(h(z)=\max(0,z-k)\), where \(k\) is a prespecified knot. A treatment-effect curve \(\tau(z)=\tau_0+\gamma_1(z-c)+\gamma_2h(z)\) has slope \(\gamma_1\) below the knot and slope \(\gamma_1+\gamma_2\) above it. Spline models extend this idea with multiple basis terms and usually impose constraints that keep the fitted curve smooth. Knots and model complexity should be chosen with clinical knowledge and available information in mind, rather than adjusted repeatedly until a favorable pattern appears.

Choosing Values, Checking Fit, and Interpreting Results

Predictions are easiest to interpret when reported at values selected for clinical relevance before examining the results—for example, a typical value and values representing meaningful low and high measurements. Report the biomarker value and units, the estimated treatment effect and outcome units, and uncertainty for each contrast. Ensure the selected values lie within the range with adequate observations in both randomized groups. A fitted curve at sparsely observed extremes can be mathematically calculable but scientifically unreliable.

A model-based crossing of a clinically important effect threshold can summarize a curve, but it is not automatically an eligibility cutoff or a validated decision rule. For instance, in the previous example, solving \(\hat{\tau}(z)=1.0\) above the knot gives \(0.15+0.15z=1.0\), or \(z\approx5.67\). That is the point where this estimated curve reaches 1.0 points under its assumptions. Uncertainty in the curve and the clinical consequences of decisions would need consideration before using such a value to guide treatment.

Model assessment should examine whether the assumed shape is credible and whether data support it. Useful checks include plotting outcomes and fitted treatment contrasts against the biomarker, examining residual patterns, checking the number of participants in each treatment group across the biomarker range, and comparing prespecified linear and flexible specifications. Large changes in predictions under modest changes in spline complexity or knot placement indicate that the gradient is weakly determined. A significant interaction test does not guarantee a well-estimated effect at every biomarker value.

For a binary outcome or a model using a nonlinear link, an interaction coefficient is generally on the model's link scale. It should not be described directly as a difference in event probabilities. Instead, calculate predicted outcomes under each treatment at the biomarker value and subtract them on the clinically relevant scale, such as a risk difference. This distinction prevents a correct model coefficient from being given the wrong clinical interpretation.

Common Pitfalls and Limitations

• Assuming the interaction must be linear. A straight line is a modeling assumption, not a consequence of using a continuous biomarker. Check plausible alternatives and report the specification.
• Using the treatment main effect as the effect for everyone. In a linear interaction model, \(\beta_T\) is the treatment effect at \(z=c\), not at all biomarker values.
• Ignoring prognostic variation. Omitting a needed biomarker main-effect function can misrepresent outcome patterns and make the treatment model less credible.
• Extrapolating beyond support. Predictions outside the observed biomarker range, or in regions with few participants in one arm, rely heavily on functional-form assumptions.
• Treating a fitted gradient as individual response certainty. A conditional average effect describes a group defined by biomarker value; it does not reveal both potential outcomes for an individual.
• Searching among many curves or thresholds. Data-driven choices can overstate evidence for effect modification. Prespecification, transparent reporting, and independent validation help limit this problem.