What a Current-Value Link Represents
In a joint model, the event hazard can depend on the subject’s underlying marker trajectory. A current-value link specifies that the hazard at time \(t\) depends on the latent marker level at that same time. “Current” refers to the time at which the hazard is being considered; it does not mean the most recently observed marker measurement. The distinction matters because a measurement can be noisy, or no measurement may have been taken at the event time.
Write \(m_i(t;\mathbf b_i)\) for subject \(i\)’s latent marker value at time \(t\), as determined by that subject’s longitudinal trajectory and random effects \(\mathbf b_i\). Let \(\mathbf w_i\) denote covariates in the event model, \(h_0(t)\) the baseline hazard, and \(\alpha\) the association parameter. A basic current-value hazard model is:
The hazard is conditional on the random effects and the listed covariates. The marker trajectory is supplied by the longitudinal submodel; the event model does not substitute a noisy observed value for \(m_i(t;\mathbf b_i)\). The coefficient \(\alpha\) is a log-hazard association per marker unit. Thus, its numerical value depends on the marker’s measurement units, while \(\exp\{\alpha m_i(t;\mathbf b_i)\}\) is dimensionless. A positive \(\alpha\) means that a higher current marker level corresponds to a higher conditional hazard, with other terms held fixed; a negative \(\alpha\) means the corresponding association is downward.
Hazard Ratio for a Specified Marker Difference
A coefficient is most interpretable when translated into a comparison with a stated marker difference. Consider two conditional hazard evaluations at the same time \(t\), with the same covariates and baseline hazard, and with latent marker values \(m_A(t)\) and \(m_B(t)\). Their hazard ratio is:
This follows by dividing the two hazard expressions: the common \(h_0(t)\) and covariate factor cancel, leaving the exponential of the difference between the two marker terms. If the specified difference is \(\Delta=m_A(t)-m_B(t)\), the conditional hazard ratio is \(\exp(\alpha\Delta)\). This concise comparison relies on the model as written: the subjects have the same other hazard predictors, and the current-value term is linear on the log-hazard scale.
The ratio is a comparison of instantaneous hazards at time \(t\), not a ratio of event probabilities over a follow-up interval. It is conditional on the model’s random effects and covariates. If the marker difference between two subjects changes over time, their hazard ratio can change over time too, even though the model uses a single constant \(\alpha\). Therefore, a constant association parameter does not by itself imply that every pair of subjects has a constant hazard ratio across follow-up.
Worked Example: Interpreting the Association Parameter
Suppose a hypothetical marker is measured in milligrams per liter and the fitted current-value coefficient is \(\alpha=0.35\) per milligram per liter. Compare two otherwise identical subjects whose latent marker levels at the same time differ by \(0.5\) milligrams per liter.
The higher-marker subject’s conditional hazard ratio is:
At that time, the model therefore associates a marker level \(0.5\) milligrams per liter higher with a conditional hazard about \(1.191\) times as large, or about 19.1% higher, holding the other hazard predictors fixed. This is an association in the specified model, not evidence that changing the marker would cause the hazard to change.
Worked Example: Computing Hazards and Their Ratio
Consider a hypothetical model with \(h_0(t)=0.012\) per month at a particular time, \(\gamma=0.20\) for a binary covariate \(w=1\), and \(\alpha=0.35\) per marker unit. Compare two otherwise identical subjects with latent marker levels \(m_A(t)=4\) and \(m_B(t)=6\). Assume both have \(w=1\), and that the model contains no additional subject-specific hazard terms.
For subject A:
For subject B:
The marker difference for A relative to B is \(4-6=-2\) units. Thus:
The same comparison in the opposite direction is \(\exp(0.70)\approx2.014\). The small difference that can arise from taking the reciprocal of the rounded value is worth noticing: \(1/2.014\approx0.4965\), whereas \(\exp(-0.70)\approx0.4966\). The exponential values are reciprocals before rounding; rounded displays need not be exact reciprocals. In context, subject A’s conditional hazard is about half subject B’s at this time, not because their event probabilities have been shown to be in that ratio, but because that is the ratio of their model-based instantaneous hazards.
Worked Example: A Negative Association Parameter
Suppose a hypothetical study uses a marker measured in points and the fitted current-value association parameter is \(\alpha=-0.60\) per point. At a given time, compare otherwise identical subjects whose latent marker values differ by \(0.8\) points, with the higher value belonging to subject A.
The conditional hazard ratio for A relative to B is:
The model associates A’s \(0.8\)-point higher latent marker value with a conditional hazard about \(0.6188\) times B’s, or about 38.1% lower, at that time. Reversing the comparison gives \(\exp(0.48)\approx1.616\). The negative sign describes the direction of association; it does not mean the hazard itself is negative, since the exponential hazard model remains positive when \(h_0(t)>0\).
Assumptions and Scope of the Interpretation
The hazard-ratio interpretation is exact for the stated model and comparison, but it is not assumption-free. The latent trajectory must have a meaningful definition on the time scale used in the event model, and the event hazard must be specified conditionally on that trajectory and the relevant random effects. The log-hazard contribution of the current marker must be linear with coefficient \(\alpha\), with the same coefficient applying to the comparison being made. The subjects being compared must share the other hazard predictors, or those differences must be explicitly included in the ratio.
The model also treats \(\alpha\) as an association parameter, not a causal effect. Subjects with different marker values may differ in other ways, and conditioning on modeled random effects does not automatically remove all confounding or justify an intervention interpretation. In addition, the ratio compares instantaneous hazards among subjects still at risk at time \(t\); it is not a statement about the overall proportion who will experience the event.
This conditional interpretation is distinct from a marginal hazard comparison obtained after averaging over random effects. Marginalizing over subject-specific effects can change hazard relationships, so \(\exp(\alpha\Delta)\) should not be described as a population-averaged hazard ratio without further justification. This distinction is central to conditional and marginal hazard interpretations.
Checking the Model and Avoiding Misinterpretation
Before interpreting \(\alpha\), verify the marker units, the sign, and the precise comparison. A coefficient per one marker unit can imply a very different ratio for a half-unit difference or a ten-unit difference. Also check that the reported contrast is between latent marker levels at the same time and that other hazard predictors are held fixed. When the marker difference varies over time, report the time-specific contrast rather than presenting a single ratio as though it necessarily applied throughout follow-up.
Diagnostics should assess both parts of the joint model. Check whether the longitudinal submodel describes the marker trajectories adequately, especially in regions of time that contribute to the event analysis. Check event-model fit and whether the assumed log-linear current-value contribution is plausible; systematic discrepancies in fitted versus observed event patterns or unstable estimates of \(\alpha\) can signal problems. Such checks do not prove the current-value structure is correct, but they can reveal that the model is failing to capture important features of the data.
A current-value link is one association structure, not a universal description. It focuses on the marker level at the event time and may not represent other ways a trajectory could be related to event risk. A model that instead links the hazard to random effects directly makes a different conditional association specification. Neither approach is automatically preferable: the scientific question, longitudinal model, and adequacy of assumptions matter. More complex structures may be needed, but the simple current-value interpretation should not be stretched beyond the model that defines it.
Specify the time, the direction of the contrast, and the marker difference \(\Delta\) in its measurement units.
Confirm the comparison uses the same baseline hazard and covariate values, and respects the model’s conditional interpretation.
Use \(\exp(\alpha\Delta)\), report the direction and magnitude, and call it a conditional hazard ratio rather than an event-probability ratio.
Check Your Understanding
Answer each question using the conditional current-value model, and then compare with the brief solutions.
- If \(\alpha=0.25\) per marker unit, what is the hazard ratio for a marker difference of 3 units, higher versus lower?
- If \(\alpha=-0.40\) per marker unit, what is the hazard ratio for a 1.5-unit higher marker value, relative to the lower value?
- Why is the current-value predictor not generally the most recent observed marker measurement?
- Suppose two subjects have different covariate values as well as different marker levels. Can their hazard ratio generally be calculated as \(\exp(\alpha\Delta)\) alone?
Solutions. (1) \(\exp(0.25\times3)=\exp(0.75)\approx2.117\); the higher value corresponds to about 2.117 times the conditional hazard, with other predictors fixed. (2) \(\exp(-0.40\times1.5)=\exp(-0.60)\approx0.5488\); the higher value corresponds to about 0.5488 times the conditional hazard. (3) The current-value predictor is the latent trajectory value at the time of interest; the latest observation may have measurement error and may have occurred earlier. (4) No. The covariate contribution must also be included unless the covariates are equal or their difference is otherwise accounted for.