Thresholded Current-Value Associations

Why Put a Threshold in a Current-Value Link?

A current-value association allows the event hazard to depend on a subject’s latent marker value at time \(t\). A linear link assumes that the same increase in the marker has the same multiplicative effect on the hazard throughout the marker’s range. That assumption may be implausible when a marker is associated with hazard only after it passes a clinically meaningful level. A thresholded link represents that idea: values at or below a specified threshold make no contribution to the association term, while values above it contribute according to their excess over the threshold.

The threshold model concerns the hazard conditional on the latent marker trajectory, as in the earlier material on current-value associations. It does not replace the longitudinal model, and it does not mean that an observed noisy measurement is itself the hazard covariate. The chosen link states how the latent current value enters the event model.

Definition and Model

A standard thresholded current-value association is specified in a shared-random-effects model as

Here \(h_i(t\mid b_i)\) is subject \(i\)’s hazard at time \(t\), conditional on that subject’s random effects \(b_i\); \(h_0(t)\) is the baseline hazard; \(w_i\) is a vector of observed covariates with coefficient vector \(\gamma\); and \(m_i(t)\) is the latent marker value at time \(t\) from the longitudinal model. The threshold \(c\) is expressed in the marker’s units. The association parameter \(\alpha\) is the change in log hazard per one marker unit of excess above \(c\); its units are the reciprocal of the marker units.

In this specification, the association contribution is zero for every marker value at or below \(c\). Above \(c\), it is \(\alpha\{m_i(t)-c\}\). If \(\alpha>0\), greater excess above the threshold corresponds to a greater conditional hazard; if \(\alpha<0\), it corresponds to a lower conditional hazard. The model does not imply a jump in hazard at \(c\), because the excess is zero at the threshold and approaches zero continuously from above.

Deriving the Hazard Contrast

To compare two marker values \(m_A\) and \(m_B\), consider hazards at the same time under the same baseline hazard and covariate values. The comparison is conditional on the modeled latent marker values and holds other terms in the hazard model fixed. Dividing the two hazards cancels both the baseline hazard and the shared covariate contribution:

The subscript \(A:B\) specifies the order: hazard at \(m_A\) divided by hazard at \(m_B\). A value above 1 means the conditional hazard at \(m_A\) is higher; a value below 1 means it is lower. When both values are below the threshold, both excesses are zero and the ratio is 1. When both are above, the difference in excesses equals \(m_A-m_B\), so the contrast is the usual exponential contrast for a linear link. When the values straddle the threshold, only the amount above the threshold contributes.

This result follows directly from the model, rather than from a separate approximation. For example, writing the two hazards explicitly gives \(h_A=h_0(t)\exp\{\gamma^\top w+\alpha q(m_A;c)\}\) and \(h_B=h_0(t)\exp\{\gamma^\top w+\alpha q(m_B;c)\}\). Their ratio is

This is a conditional hazard ratio, not a ratio of event probabilities over a follow-up interval. It also does not automatically equal a population-averaged hazard ratio obtained after integrating over random effects.

Calculating a Hazard, Not Just a Contrast

A hazard ratio describes a relative instantaneous event rate at a specified time, conditional on the model quantities. To calculate a hazard itself, the baseline hazard and all other covariate terms are needed. The next example makes the units and arithmetic explicit.

Assumptions, Diagnostics, and Alternatives

The hazard-ratio calculations rely on the stated model and a controlled comparison. In particular, the two hazards must be compared at the same time with the same baseline hazard and the same values of other included covariates. The marker values must be latent current values on the scale used to define \(c\), and the same threshold and association parameter must apply to both. The interpretation also relies on the link being correctly specified: the association contribution is exactly zero below \(c\) and linear in excess above \(c\). These conditions do not establish that a threshold model is scientifically appropriate; they identify what the model says if it is used.

Useful checks include plotting the fitted latent trajectories against the threshold and inspecting how much information is available on each side. If few subjects or few event times occur above \(c\), the upper-side association may be weakly informed. If almost all values lie on one side, the data may provide little evidence about the change in behavior at the threshold. Compare observed marker measurements with fitted latent trajectories, while remembering that the hazard link uses the latent values rather than raw noisy measurements. Also check whether the threshold was chosen for a substantive reason or selected after examining the outcomes; data-driven selection can make ordinary uncertainty summaries misleading unless that selection is accounted for.

The thresholded link is not the only possible association structure. A standard linear current-value link uses \(\alpha m_i(t)\), so the same marker increment has the same log-hazard effect throughout the range. The thresholded excess link instead gives no contribution below \(c\) and a linear effect above it. A step link, such as a term proportional to \(I\{m_i(t)>c\}\), represents a discontinuous change on crossing the threshold: all values above the threshold receive the same added term, regardless of how far above they lie. That is a different scientific claim from an effect proportional to excess.

A common mistake is to treat the hinge as a simple indicator and conclude that every value above \(c\) has the same hazard multiplier. That is incorrect: the multiplier here is \(\exp\{\alpha(m-c)\}\), so it varies with the amount above the threshold. A second frequent error is reversing the requested hazard-ratio order. Always write the numerator and denominator before subtracting excesses, as in \(\alpha[q(m_A;c)-q(m_B;c)]\).

The model has limitations. A single fixed threshold may be too simple if the association changes gradually, changes more than once, or has a nonzero slope below the threshold. The assumed linear increase above \(c\) may also fail. Estimates can be sensitive to the chosen threshold, especially when data near or above it are sparse. Finally, a subject-specific conditional hazard contrast should not be described as a causal effect or as an unconditional population contrast without further assumptions and analysis.