Testing Whether Additional Markers Improve Joint Fit

Why Compare Joint Models?

A marker can be measured repeatedly and still contribute little to explaining event risk once other markers are included. Conversely, a marker that appears weak on its own may improve the joint description when its association is considered alongside another marker. Model comparison gives a disciplined way to assess whether adding marker-related parameters improves fit enough to justify the added complexity.

The comparison must target a clearly defined question. For example, one may ask whether a second marker's current-value association is needed after accounting for the first marker, or whether adding a set of level and slope associations improves the model. This is not the same as asking whether the marker itself should be measured: a marker may be retained in the longitudinal model while its association with the event hazard is tested.

For subject \(i\), let \(Y_{ik}\) denote the repeated measurements of marker \(k\), \(T_i\) the observed event or censoring time, and \(\delta_i\) the event indicator. A general joint model has marker-specific longitudinal models conditional on latent random effects \(b_i\), together with an event hazard such as

where \(W_i\) contains covariates, \(m_{ik}(t;b_i)\) is the model-implied value of marker \(k\) at time \(t\), and \(\alpha_k\) is its current-value association parameter. Other association structures, including slope terms, are possible; the comparison logic is the same, provided the models are nested and the tested parameters are identified under the null.

The observed-data likelihood integrates over the random effects. Writing \(f_i(Y_i\mid b_i)\) for the longitudinal contribution, \(h_i\) for the conditional hazard, and \(S_i\) for the conditional event-free probability, a subject's likelihood contribution has the form

The integral is generally evaluated numerically or through an estimation algorithm. The model-comparison calculations use the maximized observed-data log likelihood \(\ell\), not a likelihood conditional on estimated random effects.

Likelihood-Ratio Tests

Let \(M_0\) be the restricted model and \(M_1\) the larger model. If \(M_0\) imposes \(q\) independent restrictions on parameters that are otherwise identifiable and are not on a boundary, the likelihood-ratio statistic is

Here \(\ell_0\) and \(\ell_1\) are the maximized log likelihoods, and \(q\) is the difference in the number of freely estimated parameters under the regular nested-model conditions. A large value of \(G^2\) is evidence against \(H_0\), because the larger model improves the maximized likelihood. The p-value is the upper-tail probability \(P(\chi^2_q\geq G^2)\), calculated under the null reference distribution.

For example, testing whether one additional current-value coefficient is zero gives \(q=1\), if all other model components are unchanged. Testing two added association coefficients gives \(q=2\). Do not count the number of markers alone: count the independent parameter constraints. If adding a marker also adds several association parameters, or changes a covariance structure, the parameter difference may be larger.

Information Criteria and Parameter Costs

Likelihood-ratio tests answer a hypothesis-testing question for nested models. Information criteria instead balance fit and complexity, and can also be used to compare a set of candidate models that are not all arranged as one simple null-versus-alternative test. For a model with \(p\) estimated parameters, the Akaike information criterion and Bayesian information criterion are

For BIC, \(n\) is the number of independent subjects in this setting, not the number of repeated measurements treated as if they were independent. Both criteria are on a relative scale: among comparable candidate models, lower values indicate a preferred fit-complexity trade-off under that criterion. AIC and BIC do not provide a p-value or prove that the selected model is true. BIC imposes a larger penalty per parameter than AIC when \(\log(n)>2\).

For a valid comparison, the candidates should use the same observed outcomes and likelihood basis. In particular, if the question concerns whether marker B's event association adds information, retain marker B's longitudinal submodel in both candidates. Omitting the marker's measurements from one likelihood changes the data being modeled and can make the likelihood and information-criterion comparison misleading.

Derivation and Conditions for the Test

The likelihood-ratio statistic compares the maximized likelihoods, or equivalently their log likelihoods. Since a nested larger model can reproduce the restricted model by setting the added parameters to their null values, its maximized log likelihood cannot be lower in an exact optimization: \(\ell_1\geq\ell_0\). Under regularity conditions, the local likelihood around the true parameter is approximately quadratic. The improvement from freeing \(q\) restrictions behaves asymptotically like the squared length of \(q\) independent standard-normal directions, which has a \(\chi^2_q\) distribution.

More explicitly, if \(\theta\) is the full parameter and the null sets \(q\) components to zero, a second-order expansion around the null estimate gives the likelihood gain as a quadratic form in the \(q\)-dimensional score, scaled by its information matrix. Under the null, the standardized score converges to a \(q\)-variate standard normal vector \(Z\). Hence the statistic has limiting form

This argument requires an interior null value, local identifiability, a correctly specified likelihood with sufficiently regular behavior, and adequate estimation. The independent units are subjects; repeated measurements within a subject are handled within that subject's joint likelihood contribution. The chi-square approximation can be poor in small samples, with weak identification, or when numerical integration and optimization are unreliable.

Practical Checks, Alternatives, and Limitations

Before interpreting a comparison, verify that both fits use the same subjects and outcomes, and that the restricted model can be obtained by imposing the stated constraints on the larger model. Check that maximization has converged from reasonable starting values and that the reported log likelihoods correspond to the same likelihood approximation. A tiny negative value of \(\ell_1-\ell_0\) can arise from numerical error; a substantial negative value signals a failed fit, a non-nested comparison, or inconsistent likelihood calculations and should be resolved before testing.

Also inspect whether the added parameters are estimable. Strong collinearity among marker trajectories can produce large standard errors, unstable estimates, or nearly singular information matrices. A small likelihood gain does not prove that a marker is scientifically irrelevant: the study may have limited event information, the added effect may be weakly identified, or the chosen association form may be inadequate. Conversely, a statistically significant likelihood gain does not guarantee useful prediction or a well-calibrated model.

The main alternatives serve different purposes. The likelihood-ratio test assesses a specified nested restriction and gives a calibrated p-value only when its reference distribution is appropriate. AIC and BIC compare fit against different complexity penalties and can help rank a prespecified candidate set, but do not test a single null hypothesis. Predictive validation can assess out-of-sample performance, but it answers a prediction question rather than the same parameter-restriction question. These approaches should not be treated as interchangeable.

When many markers or many association structures are tested, repeated ordinary p-values can overstate evidence if the search process is ignored. Specify the comparison of primary interest in advance when possible, describe exploratory searches as such, and consider multiplicity or validation when making broader claims. Model selection also adds uncertainty: estimates and p-values from the selected model do not automatically account for the selection step.