Comparing Cause-Specific Association Models

Why Compare Association Structures?

A competing-risks joint model can allow a longitudinal marker or its latent trajectory to be associated with each cause-specific hazard. The scientific and statistical question is often not simply whether an association exists, but whether the structure of association should be shared across causes or allowed to differ. A common association parameter is more parsimonious; separate parameters can represent cause-specific differences. Other candidate structures may link risk to different summaries of the longitudinal process.

A comparison is meaningful only when it contrasts models for the same observed information. In particular, the likelihoods must account for the same subjects, longitudinal records, event times, event-cause information, and censoring. This requirement is especially important when event causes are uncertain or misclassified: changing how those records enter the likelihood can change the data being modeled, not merely the association structure.

Define the Candidate Structures and Likelihood

For clarity, consider \(K\) competing event causes. For subject \(i\), let \(T_i\) be the observed event or censoring time, let \(\delta_i=1\) indicate an observed event and \(\delta_i=0\) indicate right censoring, and let \(J_i\) denote the observed cause when \(\delta_i=1\). Let \(b_i\) represent the subject-specific random effects in the longitudinal and event model. A familiar cause-specific hazard form is

Here \(m_i(t;b_i)\) is the modeled longitudinal quantity entering the hazard, \(\alpha_k\) is its association parameter for cause \(k\), and \(\lambda_{0k}(t)\) is the cause-specific baseline hazard. The comparison in this tutorial concerns how the association parameters or association terms are structured; other components should be held fixed unless their variation is explicitly part of the candidate model.

For example, the common-association model imposes \(\alpha_1=\cdots=\alpha_K=\alpha\), while the separate-association model estimates \(\alpha_1,\ldots,\alpha_K\). The former is nested in the latter. By contrast, a model using current marker value and a model using marker slope may be nonnested: neither is generally obtained merely by fixing or equating parameters in the other.

With right-censored event times and known causes for observed events, a subject's event contribution conditional on \(b_i\) is

The observed-data likelihood combines this contribution with the longitudinal-data contribution and integrates over the random-effects distribution. Denote the resulting log likelihood by \(\ell\). If the event-cause information is uncertain, its likelihood contribution must represent that uncertainty consistently in every candidate model. The same principle applies to delayed entry or interval censoring: use the appropriate observed-data likelihood for the actual observation scheme in every model being compared.

Nested Models: Likelihood-Ratio Comparison

Let \(M_R\) be a restricted model and \(M_F\) a larger model containing it. Write their maximized log likelihoods as \(\ell_R\) and \(\ell_F\). Under the null hypothesis that the restrictions hold, a likelihood-ratio statistic is

Under standard regularity conditions, \(D\) is approximately distributed as a chi-square random variable with \(q\) degrees of freedom, where \(q\) is the number of independent restrictions. These conditions include correctly specified likelihoods, a true parameter in the interior of the larger model's parameter space, identifiable parameters, and sufficiently regular large-sample behavior. A test of equal association parameters across two causes usually imposes one restriction, \(\alpha_1-\alpha_2=0\), so \(q=1\).

The justification follows from the local quadratic approximation to the log likelihood around its maximum. Under the null, the \(q\) directions released by the larger model behave asymptotically like \(q\) independent standard-normal components after appropriate scaling. The increase in maximized log likelihood is approximately half the sum of their squares; doubling it gives a sum of \(q\) squared standard-normal variables, which has a \(\chi^2_q\) distribution. This is an asymptotic argument, not a guarantee for a small sample or a poorly identified model.

Nonnested Models: Information Criteria

A routine likelihood-ratio test is not justified for nonnested models. A likelihood difference alone does not have the chi-square reference distribution used for a nested-model test. A common likelihood-based comparison uses the Akaike information criterion (AIC) or Bayesian information criterion (BIC):

where \(p\) is the number of estimated parameters in the likelihood being compared and \(n\) is the number of independent subjects for the usual subject-level BIC convention. Smaller values indicate a better tradeoff, under that criterion, between fit and complexity. AIC targets expected relative predictive performance under its assumptions; BIC applies a stronger complexity penalty as sample size grows and has a large-sample model-selection interpretation under specific conditions. Neither criterion supplies a test that a model is true.

Uncertainty, Comparability, and Diagnostics

Model rankings and parameter uncertainty answer different questions. For an estimated association \(\hat{\alpha}_k\), report its standard error and, where appropriate, a confidence interval such as \(\hat{\alpha}_k\pm1.96\,SE(\hat{\alpha}_k)\). A confidence interval describes uncertainty under the fitted model and its regularity assumptions; it does not account automatically for searching across many association structures. For a cause contrast, report the contrast itself and its standard error, using the covariance between estimates:

Do not calculate this standard error by treating estimates from a joint model as independent unless their covariance is actually zero. If the likelihood is asymmetric or a variance estimate is near a boundary, a profile-likelihood interval or a carefully designed subject-level bootstrap may be more reliable than a Wald interval. Resampling should preserve the independent sampling unit and refit the models using the same fitting procedure.

Before interpreting comparisons, check the following:

• Same observed information. Confirm that subjects, longitudinal measurements, event times, censoring indicators, and event-cause treatment are the same. Differences in missing-data handling or cause classification can invalidate a direct comparison.
• Same likelihood target. Compare maximized observed-data likelihoods, not a marginal likelihood for one model with a conditional likelihood for another. Keep the censoring and entry-time treatment consistent.
• Convergence and integration. Review convergence messages, gradients, Hessian behavior, and sensitivity to numerical integration settings. A lower log likelihood can reflect failed optimization rather than a worse model.
• Identification and sparsity. Cause-specific parameters may be weakly identified when few events occur for a cause. Very large standard errors, unstable estimates across starting values, or nearly singular Hessians are warnings that an apparent ranking may be fragile.
• Model adequacy. AIC or BIC selects only among the candidates considered. Check whether the baseline hazards, covariate effects, event-cause assumptions, and longitudinal model are plausible; a preferred model can still fit poorly.

When fitting uses numerical integration over random effects, use settings accurate enough that small likelihood differences are not numerical noise. A practical check is to refit leading candidates with tighter integration settings and alternative reasonable starting values. If their rankings or estimates shift materially, report that instability rather than treating the smallest criterion value as decisive.

What the Comparison Can and Cannot Establish

The nested likelihood-ratio test asks whether the data are inconsistent with a specified set of restrictions, subject to its assumptions. AIC and BIC rank candidate models according to different fit-complexity criteria. Neither procedure establishes a causal effect, and a cause-specific association parameter remains a model-based association, not a causal claim. Likewise, a selected association structure does not establish that the underlying biological mechanism is shared or cause-specific.

Selection also adds uncertainty. If a structure is chosen after comparing many alternatives, the usual standard errors and confidence intervals computed as though that structure had been fixed in advance can understate total uncertainty. Present the candidate structures, criterion values, parameter estimates, and relevant uncertainty summaries together. If scientific conclusions change across plausible structures, that sensitivity is itself an important result.

Common Mistakes

• Using a likelihood-ratio test for nonnested models. A difference in maximized log likelihood does not by itself have a chi-square reference distribution. Use an appropriate comparison method, such as AIC or BIC, for nonnested candidates.
• Comparing likelihoods from different records. A larger likelihood can arise because the subjects or event-cause contributions differ. Refit using the same observed information before comparing.
• Calling the lower AIC a significant winner. AIC is a relative criterion, not a hypothesis test or probability that a model is true. Describe the size of the difference and the candidate set.
• Ignoring uncertainty in a cause contrast. Subtracting two estimates is not enough; its standard error depends on their covariance.
• Interpreting an association as causal. A fitted cause-specific association describes the model's conditional relationship; causal interpretation requires additional design and assumptions.