Why the Shape of the Current-Value Link Matters
A current-value association says that the event hazard at time \(t\) depends on the subject’s underlying biomarker value at that time. But that dependence need not be a straight line on the log-hazard scale. A one-unit increase might have a constant multiplicative effect, an effect that changes with the biomarker level, or a pattern that is better represented by a smooth curve. Those alternatives can imply notably different hazard ratios across a clinically relevant range.
The purpose of a link-sensitivity analysis is not to identify a curve that looks complicated or to select the model with the most flexible fit automatically. It is to determine whether conclusions about relative hazards depend materially on the assumed shape of the association, especially over biomarker values that the data actually support.
Define the Links on a Common Scale
Let \(m_i(t)\) denote subject \(i\)’s latent, underlying biomarker value at time \(t\), as specified by the longitudinal component of the joint model. Let \(w_i\) denote covariates in the event model, and let \(h_0(t)\) be the baseline hazard. A current-value link can be written as:
Here \(f\) is the function linking the current biomarker value to log hazard. For clarity, the examples below write \(f\) directly as a log-hazard contribution. Equivalent parameterizations put coefficients outside the function; whatever parameterization is used, contrasts in \(f\), rather than the coefficient alone, determine the hazard ratio.
For two biomarker values \(a\) and \(b\), the conditional hazard ratio is
This expression compares hazards at the same time and covariate values, changing only the current latent biomarker value. The baseline hazard and covariate contribution cancel. It is the appropriate common scale for comparing link shapes; raw coefficients from different transformations are generally not directly comparable.
Assumptions for a Meaningful Comparison
The comparison concerns the association structure, not a collection of unrelated model changes. Fit alternative links using the same participants and event information, the same longitudinal trajectory specification, the same covariates, and the same baseline-hazard treatment. In the joint model, the latent current value is estimated rather than treated as a perfectly observed measurement; the longitudinal and event components should be handled consistently under each link specification.
The interpretation also relies on the model being an adequate description of the data over the range examined. In particular, the longitudinal model must provide credible estimates of \(m_i(t)\), the event model must appropriately represent the hazard conditional on that value and included covariates, and censoring must be adequately addressed by the analysis assumptions. The link describes a conditional association; it does not by itself establish that changing the biomarker would cause a change in event risk.
From Curvature to Hazard-Ratio Differences
A short Taylor expansion explains why a linear link can be a useful local approximation but inadequate over a broad range. If \(f\) is twice differentiable near a reference value \(c\), then for a nearby value \(m\),
The first derivative describes the local change in log hazard per biomarker unit. The second derivative describes curvature: if it is nonzero, the local slope changes with the biomarker value. Since hazard ratios exponentiate differences in \(f\), even moderate curvature can produce meaningful differences over a wide enough range. Conversely, a visible bend on a graph may have little practical consequence if the implied hazard ratios barely change.
For a quadratic link using \(z=(m-c)/s_m\),
The quadratic contribution can have different implications on either side of the reference value. The sign of \(\alpha_2\) alone does not tell the full clinical story: the values being contrasted and the linear contribution matter too.
Worked Example: Linear Versus Quadratic Effects Across a Range
Suppose a hypothetical biomarker is measured in mg/L, and the reference value is \(c=6\) mg/L with scale \(s_m=2\) mg/L, so \(z=(m-6)/2\). Consider a linear link \(f(m)=0.30z\) and a quadratic link \(f(m)=0.30z+0.20z^2\). Treat these values as hypothetical fitted log-hazard contributions.
At 4 mg/L, \(z=-1\); at 6 mg/L, \(z=0\); and at 8 mg/L, \(z=1\). Under the linear link, the log hazard-ratio contrast from 4 to 8 mg/L is
Under the quadratic link, the squared terms at both endpoints are 1, so they cancel in this particular contrast:
The endpoint comparison alone therefore misses the curvature. Compare 6 with 4 mg/L instead. The linear link gives \(\exp\{0-(-0.30)\}=\exp(0.30)=1.350\). The quadratic link gives \(f(4)=-0.30+0.20=-0.10\), so its contrast is \(0-(-0.10)=0.10\), and \(\operatorname{HR}(6:4)=\exp(0.10)=1.105\). In this hypothetical calculation, the estimated increase in hazard from 4 to 6 mg/L is much smaller under the quadratic link. This comparison illustrates why a useful assessment checks multiple contrasts across the range, not just its endpoints.
Smooth Transformations and Their Implied Comparisons
A smooth transformed link can represent changing effects without requiring that one quadratic term describe the entire range. A prespecified transformation, such as a logarithm, has a particular scientific interpretation. A spline instead estimates a smooth shape from the data, subject to its basis and smoothness choices. In either case, interpret the function through predicted contrasts and hazard ratios at stated biomarker values.
Worked Example: A Log-Transformed Current Value
Suppose a positive biomarker is measured in mg/L and a hypothetical link is \(f(m)=0.80\log(m/6)\). The reference is 6 mg/L, and the logarithm is natural. The ratio comparing 12 with 6 mg/L is
so \(\operatorname{HR}(12:6)=\exp(0.5545)=1.741\). The comparison of 6 with 3 mg/L has the same log contrast, \(0.80\log(6/3)=0.5545\), and the same hazard ratio, 1.741. This link therefore assigns equal log-hazard changes to equal proportional changes in the biomarker, such as a doubling, rather than to equal absolute increases.
For comparison, the local slope at 6 mg/L is \(f'(m)=0.80/m\), or \(0.1333\) per mg/L at 6 mg/L. A linear link using that slope would predict a 3 mg/L increase from 6 to 9 mg/L to change log hazard by \(0.1333(3)=0.4000\), giving an HR of \(\exp(0.4000)=1.492\). The logarithmic link instead gives \(0.80\log(9/6)=0.3244\) and an HR of 1.383. The difference illustrates that matching a local slope does not make two link functions equivalent across a range.
For a spline, the fitted function can be evaluated at selected values just as in these examples. The estimates of the spline coefficients themselves may be difficult to interpret, so report contrasts, their uncertainty, and the biomarker range over which they are supported. If the model provides a standard error for a log-hazard contrast \(\widehat{\Delta}\), a usual large-sample 95% Wald interval is \(\widehat{\Delta}\pm1.96SE(\widehat{\Delta})\); exponentiating both endpoints gives an interval for the hazard ratio. This interval reflects uncertainty under the fitted model and does not account for every possible uncertainty about choosing the link specification.
Worked Example: Reading a Smooth Fitted Link
Suppose a hypothetical spline model estimates \(f(m)-f(6)\) as \(-0.10\), \(0\), \(0.25\), and \(0.70\) at biomarker values 4, 6, 8, and 10 mg/L, respectively. The estimated hazard ratio for 10 versus 4 mg/L is
The fitted log-hazard increases by \(0.10\) from 4 to 6 mg/L, by \(0.25\) from 6 to 8 mg/L, and by \(0.45\) from 8 to 10 mg/L. The implied HRs for these successive increases are \(\exp(0.10)=1.105\), \(\exp(0.25)=1.284\), and \(\exp(0.45)=1.568\). Thus the fitted association steepens over these intervals.
Suppose the standard error of the 10-versus-4 log contrast is hypothetically \(0.25\). Its Wald interval is \(0.80\pm1.96(0.25)=[0.31,1.29]\). Exponentiation gives an estimated HR of 2.226 with a 95% interval of approximately \([1.363,3.632]\). Under the model and its large-sample approximation, this interval is a measure of uncertainty for the conditional hazard ratio contrast. It is not a probability statement about whether the fixed true HR lies in this particular interval.
How to Compare Specifications
Use the same longitudinal trajectory model, covariates, baseline-hazard specification, analysis set, and treatment of censoring. Change the current-value link, not several assumptions at once.
Identify a reference and contrasts within the observed or well-supported range of latent biomarker values. Avoid drawing conclusions from regions with little information.
For each link, calculate \(f(b)-f(a)\), exponentiate it, and compare the resulting hazard ratios for the same values \(a\) and \(b\).
Inspect uncertainty for the contrasts and the fitted shape. Likelihood-based comparisons require the same data and compatible likelihoods; a likelihood-ratio test additionally requires nested models and its regularity conditions.
Ask whether differences among the estimated hazard ratios would change the scientific interpretation over the relevant biomarker range. Statistical evidence of curvature and practical importance are related but distinct questions.
A linear link is nested within a quadratic link when the quadratic coefficient is set to zero. If the remaining model is identical and regularity conditions hold, a likelihood-ratio statistic comparing these models has one degree of freedom under the null. More flexible spline comparisons may involve multiple parameters and require attention to how the spline is constructed and constrained. A test can flag departure from linearity without showing that the departure is consequential; conversely, imprecise estimates can leave meaningful curvature uncertain.
Diagnostics, Alternatives, and Limitations
Plot the estimated \(f(m)\), or its contrast from a reference value, across the biomarker range, preferably with uncertainty bands. Check where observations and event information support the curve, whether the tails are unstable, and whether a few subjects with extreme trajectories drive the apparent shape. Compare predicted hazard ratios at clinically interpretable values, not only coefficient estimates or a global fit statistic. A smooth curve can look persuasive while being weakly identified where there are few measurements or events.
| Link specification | Useful feature | Main caution |
|---|---|---|
| Linear | Parsimonious; the log-hazard change per unit is constant. | May miss changing effects across a broad biomarker range. |
| Quadratic | Adds a simple, interpretable curvature term. | Can imply an implausible shape near or beyond the data-supported range. |
| Smooth transformation or spline | Can represent gradual changes in slope without imposing one global straight line. | More flexible shapes can be uncertain, especially in sparse regions, and depend on transformation or smoothing choices. |
These alternatives have different limitations. A linear link is often stable but restrictive; a quadratic link is more flexible but imposes a particular global shape; a spline can adapt more broadly but needs enough information and careful uncertainty assessment. A prespecified transformation may be motivated by the measurement scale, but it also encodes a specific pattern of change. None can repair a poorly specified longitudinal trajectory or guarantee reliable extrapolation beyond observed biomarker values.
A frequent mistake is to compare the numerical sizes of \(\alpha\) from a linear model and a coefficient from a log-transformed model, then declare the larger coefficient to represent a stronger association. The units and transformations differ, so those coefficients are not on a common scale. The correction is to calculate \(f(b)-f(a)\) for the same biomarker values under both models, exponentiate each contrast, and compare the resulting conditional hazard ratios with their uncertainty.
Another pitfall is to treat a statistically detectable quadratic term as proof that nonlinearity matters in practice, or to interpret a curved fitted line beyond the data-supported range. Use both the uncertainty in fitted contrasts and the range of observed information. Sensitivity analysis is most informative when it shows whether the substantive conclusions remain stable across plausible link choices.
Check Your Understanding
Answer each question using contrasts on the log-hazard scale, and then compare hazard ratios.
- For \(f(m)=0.25(m-5)\), calculate the conditional HR for 7 versus 5 units of the biomarker.
- For \(f(m)=0.20z+0.10z^2\), where \(z=(m-6)/2\), calculate the conditional HR for 6 versus 4 mg/L.
- Why should coefficients from a linear and a log-transformed link not be compared directly?
- A fitted spline bends sharply at the highest biomarker values, where few subjects have measurements. What is a key diagnostic concern?
Solutions. (1) The log contrast is \(0.25(7-5)=0.50\), so the HR is \(\exp(0.50)=1.649\). (2) At 4 mg/L, \(z=-1\) and \(f(4)=-0.20+0.10=-0.10\); at 6 mg/L, \(z=0\) and \(f(6)=0\). The log contrast is \(0.10\), so the HR is \(\exp(0.10)=1.105\). (3) The coefficients correspond to different transformations and scales. Compare fitted hazard ratios for the same biomarker contrast instead. (4) The bend may be weakly supported and highly uncertain; check the data support and uncertainty there, and do not treat the tail shape as reliable evidence without adequate information.