Sensitivity to Cause-Specific Association Structures

Why Association Structure Is a Sensitivity Question

A joint model connects a subject’s underlying longitudinal trajectory to the hazards of competing events. But “connected” can mean different things: the hazard may depend on the marker’s current level, its rate of change, or its accumulated history. These structures encode different scientific descriptions of how a trajectory relates to event risk. When the data do not clearly distinguish them, predicted cause-specific hazards and cumulative incidence may depend on which structure is used.

A sensitivity analysis fits and compares plausible association structures while keeping other modeling choices as comparable as possible. Its goal is not to treat one fit as automatically correct, nor to select a structure solely because its coefficient is largest. It is to determine whether substantive conclusions, especially predicted risks, are robust to reasonable choices about how the longitudinal process enters each cause-specific hazard.

Model Structures and Their Interpretation

Let \(Y_{ij}\) be the longitudinal measurement for subject \(i\) at time \(t_{ij}\), and let \(m_i(t)\) denote the subject’s latent, or underlying, marker trajectory. Write \(m_i'(t)\) for its instantaneous slope. For cause \(k\), let \(h_{ik}(t\mid b_i)\) be the cause-specific hazard conditional on the subject’s random effects \(b_i\). One common specification is

h_{ik}(t\mid b_i) = h_{0k}(t)\exp\left\{ \gamma_k^\top w_i +\alpha_k^{V}\bigl[m_i(t)-m_{\mathrm{ref}}\bigr] +\alpha_k^{S}m_i'(t) +\alpha_k^{A}A_i(t) \right\},

Here \(h_{0k}(t)\) is the cause-specific baseline hazard, \(w_i\) is a vector of event-model covariates, and \(m_{\mathrm{ref}}\) is a fixed, meaningful marker reference value. The centered cumulative exposure is \(A_i(t)=\int_0^t[m_i(u)-m_{\mathrm{ref}}]\,du\). In a current-value-only model, the slope and cumulative-exposure terms are omitted; in a slope-only or cumulative-exposure-only model, the other association terms are omitted. A model can also include more than one link when that structure is scientifically justified and estimable.

Definition: A current-value association links a cause-specific hazard to the latent marker level at time \(t\). A slope association links it to the instantaneous rate of marker change at \(t\). A cumulative-exposure association links it to the accumulated centered marker history through \(t\). The coefficient for each link is cause-specific when it is indexed by \(k\).

The coefficients have different units. If the marker is measured in mmol/L and time in years, then \(\alpha_k^V\) is measured per mmol/L, \(\alpha_k^S\) per mmol/L per year, and \(\alpha_k^A\) per mmol/L-year. Their numerical magnitudes should not be ranked as if they were on a common scale. Each coefficient describes the change in log hazard per one-unit increase in its own association quantity, holding the other modeled terms fixed.

A cause-specific hazard describes the instantaneous rate of cause \(k\) among subjects who have not yet experienced any event. A larger cause-specific hazard does not by itself determine the probability of experiencing that cause by a given time. The probability also depends on the hazards of all competing causes and on remaining event-free.

Worked Example: Interpreting Three Association Coefficients

Suppose three separately fitted models for cause 1 give the following association estimates: \(\alpha_1^V=0.12\) per mmol/L, \(\alpha_1^S=0.50\) per (mmol/L per year), and \(\alpha_1^A=0.04\) per (mmol/L-year). For a subject, suppose the current marker is 2 mmol/L above \(m_{\mathrm{ref}}\), the slope is 0.30 mmol/L per year, and cumulative exposure is 5 mmol/L-year. Calculate the association contribution to the hazard ratio in each model.

For the current-value model, the log hazard contribution is \(0.12(2)=0.24\), so the hazard ratio is \(\exp(0.24)=1.271\). For the slope model, it is \(0.50(0.30)=0.15\), giving \(\exp(0.15)=1.162\). For the cumulative-exposure model, it is \(0.04(5)=0.20\), giving \(\exp(0.20)=1.221\).

At this particular marker profile, the fitted current-value model implies a 27.1% higher cause 1 hazard relative to the reference profile, while the slope and cumulative-exposure models imply 16.2% and 22.1% higher hazards, respectively. These are conditional hazard comparisons, not comparisons of event probabilities. They also do not show that the current-value structure is superior: the coefficients refer to different quantities and came from different fitted models.

Making Comparisons Informative

A useful comparison changes the association structure without needlessly changing everything else. Use the same analysis sample, event definitions, longitudinal submodel, covariates, baseline-hazard strategy, random-effects distribution, and integration method when feasible. Refit each candidate model under its own likelihood; do not take a coefficient from one structure and insert it into another. This makes it easier to attribute prediction differences to the association link rather than to a changed baseline or covariate adjustment.

The joint likelihood integrates over the random effects rather than treating the subject-specific latent trajectories as known. Consequently, estimates and predictions depend on both the longitudinal and event components and on the assumed random-effects distribution. The usual modeling requirements still matter: the longitudinal mean and covariance structure should be adequate; the cause-specific baseline hazards and covariate effects should be sufficiently flexible; and censoring should be conditionally independent given the modeled information. Association terms should be identifiable from the observed longitudinal and event data. Sparse causes can make this especially difficult, as can highly correlated current-value, slope, and cumulative-exposure quantities.

For each fitted structure, inspect convergence, parameter uncertainty, and whether predictions are driven by a few subjects or extrapolated trajectories. Check the longitudinal fit over time and by relevant covariates, and compare cause-specific event predictions with observed event patterns. Large standard errors, estimates sensitive to starting values, boundary estimates, or unstable predictions are warnings that the data do not support a precise comparison. A converged optimizer alone does not establish that the association structure is adequate.

Worked Example: Comparing Hazards Along One Trajectory

Consider a hypothetical subject whose latent marker trajectory, in mmol/L, is \(m_i(t)=2+0.5t\), with \(t\) in years and \(m_{\mathrm{ref}}=2\). At year 4, the current excess is \(m_i(4)-2=2\) mmol/L, the slope is 0.5 mmol/L per year, and the cumulative exposure is

A_i(4)=\int_0^4 0.5u\,du =0.25(4^2) =4\ \text{mmol/L-year}.

Suppose the fitted current-value, slope, and cumulative-exposure models for cause 1 have association estimates 0.20 per mmol/L, 0.70 per (mmol/L per year), and 0.08 per mmol/L-year, respectively. Let the cause 1 baseline hazard at year 4 be 0.08 per year in each model, and suppose the covariate contribution is a common multiplier of 1.20. The hazard before applying the association term is \(0.08(1.20)=0.096\) per year.

The current-value model gives \(0.096\exp[0.20(2)]=0.096\exp(0.40)=0.1432\) per year. The slope model gives \(0.096\exp[0.70(0.5)]=0.096\exp(0.35)=0.1362\) per year. The cumulative-exposure model gives \(0.096\exp[0.08(4)]=0.096\exp(0.32)=0.1322\) per year.

These values illustrate how the structures can produce different fitted hazards for the same trajectory. For a real comparison, the baseline hazards and other parameters would generally be estimated separately in each model; they would not necessarily remain equal as stipulated here to isolate the link contribution. Even with a common baseline, the calculation is conditional on the stated trajectory and covariates, not a population-average risk.

From Cause-Specific Hazards to Cumulative Incidence

For two competing causes, conditional on random effects \(b_i\), define \(S_i(t\mid b_i)\) as the probability of remaining free of either event through time \(t\). If the cause-specific hazards are \(h_{i1}\) and \(h_{i2}\), then the total event hazard is their sum. Thus,

S_i(t\mid b_i) = \exp\left\{-\int_0^t \bigl[h_{i1}(u\mid b_i)+h_{i2}(u\mid b_i)\bigr]\,du\right\}, \qquad F_{ik}(t\mid b_i) = \int_0^t S_i(u\mid b_i)h_{ik}(u\mid b_i)\,du.

The second expression is the cause \(k\) cumulative incidence: at each time \(u\), the subject must have remained event-free until \(u\), then experience cause \(k\) at that time. Population predictions in a joint model average the conditional quantities over the random-effects distribution. In particular, one generally computes the expected conditional cumulative incidence over \(b_i\); evaluating the hazard at an average trajectory is not generally equivalent.

Worked Example: A Hazard Change and Its Cumulative Incidence

For a transparent calculation, suppose two cause-specific hazards are constant over a one-year interval. Under association structure A, let them be \(\lambda_1=0.10\) and \(\lambda_2=0.06\) per year. Under structure B, suppose the cause 1 hazard is 0.14 per year and the cause 2 hazard remains 0.06 per year. These are hypothetical fitted hazard predictions for a specified covariate and marker profile.

With constant hazards, \(F_1(T)=\frac{\lambda_1}{\lambda_1+\lambda_2}\{1-\exp[-(\lambda_1+\lambda_2)T]\}\). Under structure A at \(T=1\), this is \(\frac{0.10}{0.16}\{1-\exp(-0.16)\}=0.625(0.1479)=0.0924\). The cause 2 cumulative incidence is \(\frac{0.06}{0.16}(0.1479)=0.0554\).

Under structure B, the total hazard is 0.20 per year and \(1-\exp(-0.20)=0.1813\). Therefore \(F_1(1)=\frac{0.14}{0.20}(0.1813)=0.1269\), while \(F_2(1)=\frac{0.06}{0.20}(0.1813)=0.0544\). In this illustration, the one-year cause 1 incidence increases from 9.24% to 12.69%; cause 2 incidence changes slightly even though its own hazard is held fixed, because the probability of remaining event-free changes. The calculation is a constant-hazard illustration, not a general substitute for integrating time-varying fitted hazards.

A Practical Sensitivity Workflow

Specify plausible links.
Choose current value, slope, cumulative exposure, or a scientifically justified combination. State the reference value and units, especially for cumulative exposure.

Hold other choices steady.
Use comparable data, covariates, longitudinal specifications, baseline hazards, and numerical integration settings wherever possible.

Check estimation and fit.
Review convergence, uncertainty, trajectory fit, event calibration, and signs of weak identification before interpreting differences.

Compare quantities on a common scale.
Evaluate cause-specific hazards and cumulative incidences at the same times and covariate profiles, and report units and uncertainty.

Describe robustness, not certainty.
State which conclusions persist across plausible structures and which depend on the chosen link. A sensitivity analysis does not prove that any candidate structure is correct.

Comparing fitted likelihoods or information criteria can be useful when models are fitted to the same data and likelihood basis, but such summaries do not replace checking predictive behavior and scientific plausibility. Current-value, slope, and cumulative-exposure models are not always nested, and a small difference in a fit statistic may not resolve weak identification. A simpler, interpretable structure can be preferable when extra association terms are unstable.

Common Mistakes and Limitations

Comparing coefficient sizes across links. A slope coefficient and a current-value coefficient have different units. Compare estimated hazard contrasts over specified, meaningful changes in the marker quantities instead.
Changing several model components at once. If the baseline hazard, covariates, and association structure all change, the source of a prediction difference is unclear. Keep the comparison controlled where possible.
Equating a hazard ratio with a cumulative-incidence ratio. Cumulative incidence depends on all competing hazards and survival up to each time, not only on the hazard for the cause of interest.
Treating sensitivity as causal evidence. A fitted association describes a modeled relationship conditional on the model assumptions; it does not establish that changing the marker would change event risk.
Over-interpreting unstable estimates. A large association estimate with a wide interval or poor convergence is not strong evidence of a large effect. It may indicate inadequate information or model instability.

These comparisons are limited by the observed follow-up, measurement schedule, number of events for each cause, and assumptions about the latent trajectory and random effects. Predictions can be particularly uncertain beyond the time range with adequate data. Report uncertainty for predicted hazards or cumulative incidences when available, and avoid presenting one fitted structure as definitive when plausible alternatives lead to materially different predictions.

Check Your Understanding

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

A marker is measured in mg/L and time in years. What are the units of a current-value coefficient and a slope coefficient in the log-hazard model?
For a centered current marker value of 1.5 mg/L and association coefficient 0.20 per mg/L, calculate the hazard ratio contribution.
Why can a cause 2 cumulative incidence change when the cause 2 hazard is unchanged but the cause 1 hazard increases?
Name two modeling choices that should be held constant when comparing candidate association structures, if feasible.
What does a sensitivity analysis establish if predicted cumulative incidence differs substantially across plausible links?

Solutions. (1) The current-value coefficient is per mg/L. The slope has units mg/L per year, so its coefficient is per (mg/L per year). (2) The contribution is \(\exp[0.20(1.5)]=\exp(0.30)=1.350\), a 35.0% higher conditional hazard than at the reference marker value, holding other terms fixed. (3) A higher cause 1 hazard reduces the probability of remaining event-free, which changes the amount of time during which cause 2 can occur; cumulative incidence depends on both hazards. (4) Examples include the analysis sample, longitudinal submodel, event covariates, baseline-hazard specification, random-effects distribution, and integration method. (5) It establishes that the prediction is sensitive to the association assumption under the fitted candidate models. It does not, by itself, determine which link is true or imply a causal effect.