Identifiability of Cause-Specific Association Parameters

Why Identifiability Matters

A fitted joint model can return a numerical estimate for every cause-specific association parameter and still provide little evidence that those parameters have been uniquely determined by the data. A large standard error, strong dependence on modeling choices, or nearly equivalent likelihood values for different parameter combinations can signal that the data do not distinguish the proposed effects. This concern is especially important in competing-risks models: an individual experiences at most one first event, so the observed data contain limited direct information about how a longitudinal marker relates to each cause.

The key question is not merely whether an optimizer converges. It is whether different values of an association parameter imply measurably different distributions for the observed longitudinal measurements, event time, and event cause. Identifiability depends on the whole design and model: who remains at risk, how the marker varies and is measured, how many events of each cause occur, and how the random effects connect longitudinal and event processes.

Model and Identifiability Definitions

For clarity, consider a two-cause joint model with a current-value association. Let \(Y_{ij}\) be the longitudinal measurement for individual \(i\) at time \(t_{ij}\), and let \(m_i(t)\) be that individual's latent marker trajectory. A longitudinal model specifies the distribution of the measurements around this trajectory. Let \(T_i\) be the observed event or censoring time, and let \(\delta_{ik}=1\) if cause \(k\) occurs at \(T_i\), with \(\delta_{ik}=0\) otherwise. A person who is censored has \(\delta_{i1}=\delta_{i2}=0\).

Conditional on the relevant random effects and covariates \(w_i\), suppose the cause-specific hazard is

\lambda_{ik}(t\mid b_i)=\lambda_{0k}(t)\exp\{\gamma_k^\top w_i+\alpha_k m_i(t)\},\qquad k=1,2.

Here \(\lambda_{0k}(t)\) is the baseline hazard for cause \(k\), \(\gamma_k\) is its covariate coefficient vector, and \(\alpha_k\) is the cause-specific longitudinal association parameter. For example, \(\exp(\alpha_k)\) is the conditional hazard ratio for a one-unit increase in the latent marker at a fixed time, holding the covariates and other model quantities fixed. The unit of that marker must be stated when interpreting the effect. Random effects \(b_i\) may govern the latent trajectory and may also include correlated components for cause processes. The observed-data likelihood integrates over their distribution.

Definition: A model parameter is structurally identifiable if distinct values of that parameter (or parameter vector), with the remaining model structure specified, imply distinct probability distributions for the observable data. It is practically identifiable when the available design and sample provide enough information to estimate it with useful precision. A structurally identifiable parameter can be practically weakly identified in a finite sample.

These definitions distinguish an intrinsic ambiguity in the model from a shortage of information in a particular data set. An optimizer's finite estimate is not proof of structural identifiability; nor does a wide interval by itself prove structural nonidentifiability. The likelihood and the way its shape changes with the parameter are central to diagnosing both.

Where the Information About an Association Comes From

Write \(Y_i(t)=1\) while individual \(i\) remains at risk just before time \(t\), and zero after the event or censoring time. Conditional on the random effects, the event-process log likelihood includes the event term for the observed cause and the integrated hazards for all causes. If the marker trajectory does not itself depend on \(\alpha_k\), the part of its derivative involving \(\alpha_k\) is

\frac{\partial \ell_i}{\partial \alpha_k} = \delta_{ik}m_i(T_i) - \int_0^{T_i}Y_i(u)\lambda_{ik}(u\mid b_i)m_i(u)\,du.

This is a conditional, complete-data score: it treats the random effects and latent trajectory as known. Its first term reflects the marker value at an observed event of cause \(k\); its second term reflects the hazard contribution while the person was at risk. In the observed-data likelihood, the random effects are unobserved. The corresponding score averages the conditional score over their distribution given the observed data. Thus, uncertainty about the marker trajectory and random effects can reduce how sharply the data distinguish values of \(\alpha_k\).

The expression also clarifies why an event count alone does not determine identifiability. Events provide direct cause-specific information, but the hazard contributions for people who do not experience that cause also matter. What matters is whether the observed risk histories and marker trajectories produce distinguishable likelihood contributions as \(\alpha_k\) changes, after allowing the baseline hazard, covariate effects, and random-effects distribution to vary as specified.

Worked Example: Event Counts and Marker Contrast

Suppose a hypothetical screening calculation divides people into two groups with fixed marker values 0 and 1. Each group contributes 20 person-years of exposure to one cause, and the event counts are 2 in the marker-0 group and 6 in the marker-1 group. As a deliberately simplified approximation, assume independent Poisson event counts with rates \(\lambda_0\) and \(\lambda_0\exp(\alpha)\). This calculation ignores random effects and time-varying marker values; it illustrates how events across marker levels can inform an association, not how to fit the full joint model.

The estimated rates are \(2/20=0.1000\) events per person-year and \(6/20=0.3000\) events per person-year. Under the approximation, their rate ratio estimates \(\exp(\alpha)\), so

\hat{\alpha}=\log\left(\frac{6/20}{2/20}\right)=\log(3)=1.099

The approximate standard error is \(\sqrt{1/6+1/2}=0.8165\). A rough Wald interval is \(1.099\pm1.96(0.8165)\), or approximately \([-0.501,2.699]\) per one-unit increase in this marker. The interval is wide: even with eight events, the data give limited precision for this contrast. In the full joint model, repeated marker measurements, uncertain trajectories, nuisance parameters, and random effects also affect precision. If the marker had the same value for everyone at risk, these data would not supply this between-marker contrast for separating \(\alpha\) from the baseline level.

Marker Follow-Up, Baseline Hazards, and Random Effects

The marker needs to vary in informative ways over the times when people remain at risk. Repeated measurements can help estimate individual trajectories rather than treating noisy observations as exact marker values. The timing of those measurements matters: if nearly all marker information comes early but the association concerns risk later, the later latent values may be determined mainly by the assumed longitudinal model. People who experience an event also stop contributing post-event marker measurements. Consequently, weak or uneven follow-up can make association estimates sensitive to trajectory assumptions, especially when the observed measurements do not cover the risk period well.

A further issue is separation from the baseline hazard. If the marker takes a constant value for everyone at risk, the model contains \(\log\lambda_{0k}(t)+\alpha_k m\). With \(m\) constant, a change in \(\alpha_k\) can be offset by a change in the baseline log hazard. Whether the association can be separated then depends on other model structure or information; the data have no marker contrast at that time. Flexible baseline hazards can make this separation more demanding because they can absorb time patterns that a more restrictive baseline would otherwise constrain.

Random-effects variation is also relevant. If individuals' latent trajectories show little variation in the feature linked to the hazard, there may be little evidence for how risk changes across that feature. Conversely, large variation does not guarantee identification: the random-effects distribution and the association parameters may be entangled, particularly when marker measurements are sparse or noisy. Repeated measurements help distinguish within-person measurement error from between-person trajectory variation, but they do not remove the need to check whether the resulting likelihood has adequate curvature in the association parameters.

Worked Example: A Marker Constant Over Follow-Up

Consider a hypothetical cause-specific hazard with a constant marker value \(m=2\) for every person at risk and a constant baseline log hazard \(\gamma\) over the period. The log hazard is \(\gamma+2\alpha\). Suppose the data identify this log hazard as \(0.4\). They do not separately identify \(\gamma\) and \(\alpha\): for example, \((\gamma,\alpha)=(0.4,0)\) and \((0,0.2)\) both give \(0.4\). Infinitely many pairs on \(\gamma+2\alpha=0.4\) yield the same hazard.

Now suppose, purely to inspect the design geometry, that marker values 1 and 3 are both observed among people at risk, and the baseline log hazard is still a single constant. The corresponding linear predictor has design rows \((1,1)\) and \((1,3)\) for \((\gamma,\alpha)\). The cross-product matrix is

\begin{pmatrix}1&1\\1&3\end{pmatrix}^{\!\top} \begin{pmatrix}1&1\\1&3\end{pmatrix} = \begin{pmatrix}2&4\\4&10\end{pmatrix}, \qquad \det\begin{pmatrix}2&4\\4&10\end{pmatrix}=20-16=4>0.

The two columns are no longer perfectly collinear, so the intercept and association can in principle be separated in this simplified setup. This rank check is not a substitute for the joint-model likelihood: the amount and timing of data, uncertainty in \(m_i(t)\), and additional nuisance parameters still determine practical precision.

Correlation Across Causes and Parameter Separation

When random effects for the cause processes are correlated, the model uses the joint distribution of those effects to describe dependence between risks. Correlation does not automatically make the cause-specific association parameters unidentifiable. Cause labels and distinct event histories can still provide information about each association. But strong dependence can make the two association directions difficult to distinguish when the observed event patterns and marker information are similar across causes. The relevant issue is whether the likelihood contains separate information for changes in \(\alpha_1\) and \(\alpha_2\), after accounting for the random-effects covariance and other parameters.

Worked Example: A Weak Cause-Specific Contrast

Suppose a hypothetical local approximation to the information for \((\alpha_1,\alpha_2)\), after scaling both parameters to comparable units and accounting for nuisance parameters, is

I= \begin{pmatrix}1&0.98\\0.98&1\end{pmatrix}.

Its eigenvectors are proportional to \((1,1)\) and \((1,-1)\), with eigenvalues \(1.98\) and \(0.02\), respectively. The information is strong for a common movement in the two associations, but much weaker for a contrast in which one rises as the other falls. The eigenvalue ratio is \(1.98/0.02=99\), indicating severe local ill-conditioning in this scaled example.

This matrix is illustrative, not a rule that a random-effects correlation of 0.98 must produce this information matrix. Rather, correlation among random effects can contribute to weak separation when it is accompanied by similar cause-specific risk information and limited cause-specific events. The diagnostic target is near-collinearity in the likelihood's information after nuisance parameters are considered.

Assessing Identifiability in Practice

Identifiability should be assessed using the fitted likelihood and the data structure, not inferred from a single convergence message or standard error. Useful checks include the following:

Check support and variation. Summarize the number of events by cause and inspect marker values and follow-up times among people at risk. Determine whether there is meaningful marker variation over the relevant risk periods.
Inspect likelihood shape. Profile the likelihood over each association and, where relevant, over pairs of associations. A flat region, long ridge, or multiple similar maxima indicates weak separation or possible nonidentifiability.
Examine curvature and parameter dependence. Look for a nearly singular observed information matrix, large standard errors, extreme correlations among estimates, or estimates that change substantially with starting values.
Check sensitivity to modeling assumptions. Compare reasonable specifications of the longitudinal trajectory, random-effects covariance, and baseline hazards. Large changes in association estimates can reveal that the data do not strongly determine the parameter separately from these components.

These checks diagnose limitations; they do not mechanically prove that a model is invalid. A numerical integration error can also distort likelihood shape, so integration accuracy should be adequate before interpreting curvature or profiles. The earlier material on Numerical Integration for Competing-Risks Likelihoods is relevant to that computational check.

Separate Associations and a Shared Constraint

A model with separate \(\alpha_1\) and \(\alpha_2\) permits the marker association to differ by cause. A constrained model that sets \(\alpha_1=\alpha_2\) estimates a common association instead. That restriction can improve precision when the common-effect assumption is scientifically appropriate, but it does not create information that distinguishes two effects: it removes their contrast from the parameterization. A narrow interval under the constraint is therefore an interval for a different, restricted model parameter, not evidence that the unrestricted cause-specific associations were individually well identified.

The choice should be guided by the scientific question and the available information. If event histories or marker follow-up do not support separate associations, report the limitation and consider whether a scientifically defensible constraint is useful. Do not interpret an imposed equality as a discovery about the causes. Likewise, identifiability is conditional on the model: it does not establish that the association is causal, and it cannot compensate for incorrect assumptions about event processes, marker measurement, or random-effects distributions.

Common Pitfall

Common mistake: “The optimizer converged and returned two finite association estimates, so both cause-specific effects are identified.” Convergence only says that the numerical algorithm met its stopping criterion. The likelihood may still have a flat ridge, and the estimates may depend strongly on starting values or modeling choices. Check likelihood profiles, information-matrix conditioning, marker support, and cause-specific event histories before treating the estimates as data-determined.

Key takeaway: A cause-specific association is well identified only when the observed longitudinal and event data distinguish its effect from the baseline hazards, other cause associations, and random-effects structure. Event counts matter, but so do marker variation and follow-up, risk-set histories, and dependence across causes.

Check Your Understanding

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

In the simplified rate model, the marker-0 and marker-1 groups each contribute 30 person-years and have 3 and 9 events, respectively. What is the estimated log rate ratio per marker unit? Solution: The rates are \(3/30=0.1000\) and \(9/30=0.3000\) events per person-year. Thus \(\hat{\alpha}=\log(0.3000/0.1000)=\log(3)=1.099\) per marker unit.
Why can a constant marker value among all people at risk make the association difficult to separate from the baseline hazard? Solution: With a constant value \(m\), the log hazard contains \(\log\lambda_{0k}(t)+\alpha_k m\). Changing \(\alpha_k\) can be offset by changing the baseline log hazard, so the data lack a marker contrast to distinguish them.
An information matrix has eigenvalues 3 and 0.03. What does this suggest? Solution: One parameter combination has much less local information than the other. The eigenvalue ratio is \(3/0.03=100\), indicating substantial ill-conditioning and likely weak practical identification in the low-information direction, though the matrix alone does not identify the cause of the problem.
Does a high correlation between cause-specific random effects prove that \(\alpha_1\) and \(\alpha_2\) cannot be separately identified? Solution: No. It can contribute to weak separation when event histories and marker information are similar, but separate event causes can still provide distinct information. The likelihood's curvature and profile for the association parameters should be checked.
What is the difference between structural and practical identifiability? Solution: Structural identifiability concerns whether distinct parameter values imply distinct observable-data distributions under the model. Practical identifiability concerns whether a particular sample and design provide enough information for useful precision. A structurally identifiable parameter can be practically weakly identified.