Current-Value Associations by Event Cause

Why Link the Current Biomarker Value to Each Cause?

In a competing-risks joint model, a biomarker may be associated with more than one possible event, and the strength or direction of that association need not be the same for every cause. A cause-specific current-value association represents the relationship between a subject’s underlying biomarker level at time $t$ and the instantaneous rate of a particular event at that time. It uses the latent trajectory, rather than treating a single noisy measurement as the subject’s true current level.

The key parameter is cause-specific: it describes how the hazard for one event cause changes with the latent current biomarker value, conditional on the model’s covariates and latent effects. It does not, by itself, describe a causal effect, an absolute event probability, or the probability that a particular cause will occur by a given time.

Model Definition and Interpretation

Definition: Let $i$ index subjects and $k=1,\ldots,K$ index event causes. Let $m_i(t)$ be subject $i$’s latent, underlying biomarker value at time $t$, as specified by the longitudinal submodel. A cause-specific current-value model is $$ h_{ik}(t\mid \mathcal{F}_{i}(t), b_i) = h_{0k}(t)\exp\left\{\gamma_k^{\mathsf T}w_i+\alpha_k m_i(t)\right\}, $$ for times when subject $i$ is still event-free and at risk. Here $h_{0k}(t)$ is the baseline hazard for cause $k$, $w_i$ is a vector of covariates, $\gamma_k$ is its cause-specific coefficient vector, $b_i$ denotes the subject’s latent random effects, and $\alpha_k$ is the cause-specific current-value association parameter. The hazard is conditional on the specified covariates and latent effects; $\mathcal{F}_{i}(t)$ denotes the subject’s observed history through time $t$.

The latent value $m_i(t)$ is the underlying trajectory value supplied by the longitudinal model. For example, if the longitudinal model has a fixed-effects component $x_i(t)^{\mathsf T}\beta$ and a random-effects component $z_i(t)^{\mathsf T}b_i$, then $m_i(t)=x_i(t)^{\mathsf T}\beta+z_i(t)^{\mathsf T}b_i$. The observed biomarker measurement can differ from $m_i(t)$ because of measurement error and within-subject variability represented by the longitudinal model.

The model’s proportional-hazards form makes the interpretation direct. At a fixed time, for two otherwise identical at-risk subjects whose latent current values differ by one biomarker unit, the ratio of their cause-$k$ hazards is $\exp(\alpha_k)$. “Otherwise identical” means the same values of $w_i$, the same baseline time, and the same other conditioning quantities. A positive $\alpha_k$ corresponds to a hazard ratio above one; a negative $\alpha_k$ corresponds to a hazard ratio below one.

\frac{h_{ik}(t\mid m_i(t)+1,w_i,b_i)} {h_{ik}(t\mid m_i(t),w_i,b_i)} = \frac{h_{0k}(t)\exp\{\gamma_k^{\mathsf T}w_i+\alpha_k[m_i(t)+1]\}} {h_{0k}(t)\exp\{\gamma_k^{\mathsf T}w_i+\alpha_km_i(t)\}} = \exp(\alpha_k).

This is a conditional hazard ratio, not a ratio of cumulative risks. It applies among subjects still event-free and at risk at time $t$. When another event cause occurs, the subject is no longer at risk for the first event, as described in the earlier material on event types and risk sets in joint competing-risks data.

Assumptions Behind the Association

The model is useful only when its link and conditioning structure are suitable for the data. Its main assumptions include the following:

Appropriate longitudinal trajectory. The longitudinal submodel must provide a meaningful estimate of the underlying biomarker trajectory, including its time pattern and between-subject variation. If that trajectory is misspecified, the current values entering the hazard model can also be systematically wrong.
Log-linear current-value link. Conditional on the modeled covariates and latent effects, a one-unit change in $m_i(t)$ changes the log cause-specific hazard by the constant $\alpha_k$. Thus the hazard ratio for one unit does not vary with the current value unless the model is extended.
Proportional-hazards structure. For fixed $m_i(t)$ and covariates, the cause-$k$ hazard is multiplied over time by $h_{0k}(t)$, without an additional time-varying coefficient on the current value.
Correct event and risk-set information. The event cause and event or censoring times must be recorded and handled consistently with the cause-specific risk sets. The hazard describes the instantaneous rate for cause $k$ among people still event-free, not a cause-specific probability over an interval.
Suitable joint-model dependence structure. The longitudinal and event processes must be connected through the specified latent quantities, with their uncertainty and dependence represented appropriately. A current-value link alone does not guarantee that all relevant dependence has been captured.

These are modeling assumptions, not facts established merely by fitting the equation. In particular, a strong association parameter does not show that changing the biomarker would change event risk. Unmeasured factors, model misspecification, or selection among those who remain event-free can affect the observed association.

Worked Interpretations

Worked Example: A Positive Association for One Cause

Suppose a hypothetical fitted model for cause 1 estimates $\hat{\alpha}_1=0.35$, where the biomarker is measured in units of 10 points. The cause-specific hazard has the form above, with the current value entering as $\alpha_1m_i(t)$. Calculate and interpret the one-unit hazard ratio.

The one-unit hazard ratio is $$ \exp(\hat{\alpha}_1)=\exp(0.35)=1.419. $$

Among subjects who are still event-free at the same time and have the same modeled covariates and latent effects, a 10-point higher latent current biomarker value corresponds to an estimated cause-1 hazard about 1.42 times as large, or approximately 41.9% higher. This is a conditional association in the fitted model, not a 41.9-percentage-point increase in event probability.

The units matter: the comparison is per 10 points because that is one unit on the specified biomarker scale. If the biomarker were instead entered in single-point units without changing the model otherwise, the numerical coefficient would rescale accordingly.

Worked Example: Different Directions Across Causes

Consider hypothetical estimates $\hat{\alpha}_1=0.20$ and $\hat{\alpha}_2=-0.30$, using the same biomarker scale. Calculate each cause-specific hazard ratio for a one-unit increase in latent current value.

For cause 1, $$ \exp(0.20)=1.221. $$ For cause 2, $$ \exp(-0.30)=0.7408. $$

Conditional on the same modeled covariates and other conditioning quantities, a one-unit higher current value is associated with an estimated cause-1 hazard about 1.22 times as large, and an estimated cause-2 hazard about 0.741 times as large. Equivalently, the estimated cause-2 hazard is about $100(1-0.7408)=25.9\%$ lower per unit.

These comparisons do not imply that the overall event risk must rise for cause 1 or fall for cause 2 by those percentages. The cumulative incidence for a cause depends on the hazards for all causes over time and on remaining event-free. The signs and magnitudes here describe the two conditional cause-specific links.

Worked Example: Comparing Two Current Values

Suppose a hypothetical cause-1 model has $\hat{\alpha}_1=0.25$. At a particular time, compare two otherwise equivalent at-risk subjects whose latent current values are 6 and 8 units. Find the modeled cause-1 hazard ratio for the second subject relative to the first.

The latent-value difference is $8-6=2$ units. Applying the model, $$ \frac{h_{2,1}(t)}{h_{1,1}(t)} = \exp\{0.25(8-6)\} = \exp(0.50) = 1.649. $$

Thus the subject with a current value 2 units higher has an estimated cause-1 hazard about 1.65 times as large, conditional on the comparison assumptions. The result also follows by squaring the one-unit ratio: $\exp(0.25)^2=1.284^2\approx1.649$. The two subjects must be compared at the same time and with the same modeled covariates for the baseline hazard and covariate terms to cancel.

Inference and a Short Derivation

If an estimated standard error $SE(\hat{\alpha}_k)$ is available, a Wald interval for $\alpha_k$ can be formed as $\hat{\alpha}_k\pm1.96SE(\hat{\alpha}_k)$, under the usual large-sample normal approximation. Exponentiating both endpoints gives an interval for the one-unit hazard ratio. For example, if $\hat{\alpha}_k=0.35$ and $SE(\hat{\alpha}_k)=0.10$, the coefficient interval is $$ 0.35\pm1.96(0.10)=(0.154,\,0.546), $$ and the corresponding hazard-ratio interval is $$ (\exp(0.154),\exp(0.546))=(1.166,\,1.726). $$

In this hypothetical example, the interval estimates the cause-specific hazard ratio for a one-unit higher latent current value to be between 1.17 and 1.73, conditional on the model. Because the interval excludes 1, it is inconsistent with a zero association parameter at the corresponding two-sided 5% Wald level under the approximation. This does not measure the probability that the parameter lies in this particular interval.

The algebra behind the ratio also shows why the interpretation is local to a cause and time. Taking the logarithm of the hazard gives $$ \log h_{ik}(t)=\log h_{0k}(t)+\gamma_k^{\mathsf T}w_i+\alpha_km_i(t). $$ Holding time and covariates fixed, the difference in log hazards for current values $m+\Delta$ and $m$ is $\alpha_k\Delta$. Exponentiating gives the hazard ratio $\exp(\alpha_k\Delta)$. Setting $\Delta=1$ yields the one-unit interpretation; for any other difference, the same formula scales the log-hazard contrast by that difference.

Diagnostics, Alternatives, and Limits

Diagnostics should assess both the longitudinal trajectory and the event association. Check whether fitted trajectories describe the observed biomarker patterns over time, whether residual patterns suggest an inadequate longitudinal mean or variance structure, and whether the cause-specific event model shows evidence against its proportional-hazards or log-linear current-value form. Cause-specific event counts and event times should be examined by cause; a small number of events for a cause can make its association estimate imprecise. If the data do not cover the relevant range of biomarker values or follow-up times well, the fitted hazard ratio may rely heavily on model extrapolation.

A useful failure signal is a systematic pattern in event-model residual diagnostics over time or across fitted current values. Such a pattern can indicate that a constant $\alpha_k$ is inadequate, that the baseline hazard or covariate structure is insufficient, or that the longitudinal trajectory has not captured the biomarker process well. These checks do not prove the model correct when no pattern is detected. A further concern is whether each cause has enough events to support its own association estimate; a large standard error or unstable estimate is a warning against strong interpretation.

The current-value association is one possible way to connect longitudinal and event processes. A model with cause-specific loadings on shared random effects instead relates event hazards to latent subject-specific trajectory features such as random intercepts or slopes. Those parameters answer a different question: they describe association with the chosen latent effects, not directly with the biomarker’s current level. The current-value model makes the per-unit current-level contrast explicit, while its interpretation depends on how the trajectory is scaled and modeled.

The association also differs from a subdistribution-hazard association: it is defined through the cause-specific hazard among subjects who remain event-free. It should not be interpreted as a direct model for the cumulative incidence. Nor should it be confused with a slope association: $\alpha_k$ multiplies $m_i(t)$, not the rate of change in $m_i(t)$. These choices should be guided by the scientific question and data, rather than selected solely because one fitted model gives a more appealing estimate.

Common pitfall: “The coefficient is 0.35, so a one-unit increase raises the hazard by 35%.” This is incorrect because the coefficient is on the log-hazard scale. The one-unit hazard ratio is $\exp(0.35)=1.419$, corresponding to about a 41.9% higher conditional hazard. A second error is to apply that ratio to the probability of the event; a cause-specific hazard ratio is not a risk ratio or an absolute probability difference.

The most defensible interpretation therefore names the cause, the one-unit biomarker scale, the conditional risk-set comparison, and the fact that the association is not necessarily causal. The cause-specific parameters can differ across causes because the model permits each $\alpha_k$ to describe its own event process. That flexibility comes with a cost: each cause-specific estimate needs adequate event information and a plausible cause-specific model.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

If $\hat{\alpha}_k=-0.20$, what is the estimated hazard ratio for a one-unit increase in latent current value, and how should its direction be described?
A biomarker is recorded with substantial measurement error. Why does the model use $m_i(t)$ rather than simply substituting the most recent observed value?
For two otherwise equivalent at-risk subjects whose latent values differ by 3 units, express their cause-$k$ hazard ratio when $\alpha_k=0.10$, and calculate it.
Does a cause-specific current-value hazard ratio directly give the relative cumulative incidence of that cause? Explain.
What does a cause-specific association parameter describe differently from a loading on a shared random effect?

Solutions. (1) $\exp(-0.20)=0.819$. The estimated cause-$k$ hazard is about 0.819 times as large, or 18.1% lower per unit, conditional on the model. (2) The observed measurement includes error and may not equal the underlying current level; $m_i(t)$ represents the latent trajectory value specified by the longitudinal model. (3) The ratio is $\exp\{0.10(3)\}=\exp(0.30)=1.350$, so the subject with the higher value has an estimated cause-$k$ hazard 1.35 times as large under the stated comparison. (4) No. The cause-specific hazard is conditional on remaining event-free at time $t$; cumulative incidence also depends on the hazards of other causes over time. (5) $\alpha_k$ describes association per unit of latent current biomarker value, whereas a shared-random-effect loading describes association with the selected latent subject-specific effects.