Joint Models Versus Time-Dependent Cox Models

Why Two Models Can Give Different Answers

Repeated omics measurements are noisy snapshots of biological processes that may change between visits. When the event risk depends on that evolving process, two common strategies are to put an observed, time-updated measurement into a Cox model or to model the longitudinal measurements and event time jointly. They do not simply offer two computational routes to the same quantity: they can use different representations of the marker and therefore estimate different associations.

A time-dependent Cox model relates the hazard to a specified observed covariate history. A shared-random-effects joint model instead represents a subject’s underlying marker trajectory as latent and links that trajectory to the hazard. The joint approach can account for measurement error and for the fact that the event process and marker measurements share subject-specific features, but its conclusions depend on stronger modeling assumptions. Neither approach automatically fixes a visit schedule that depends on unobserved or evolving health.

Here \(h_0(t)\) is the baseline hazard, \(W_i\) denotes other covariates, and \(b_i\) denotes subject-specific random effects. In the joint model, \(\alpha\) is an association parameter: conditional on the model and other covariates, it describes how the hazard changes with the latent marker level. It is not, by itself, a causal effect of changing the marker.

What Each Model Assumes

The time-dependent Cox model requires a covariate history that is defined at each event time and uses only information available up to that time. Such a history is called predictable. A common implementation carries the most recent observed marker value forward until the next measurement. This is a choice about the covariate history, not a consequence of the Cox model. It assumes that this carried-forward value is an adequate representation of the risk-relevant exposure during the interval. The proportional-hazards assumption is also required: the covariate’s log hazard ratio is constant over time unless time-varying effects are explicitly included. Standard inference further relies on suitable censoring assumptions and correct risk sets.

A time-dependent Cox model can be useful when the scientific target concerns the association of risk with the recorded, operationally defined covariate history. But a marker measured with error is not the latent current biology. If measurement error is substantial, the observed value can misrepresent the risk-relevant trajectory; the resulting coefficient may be attenuated or otherwise biased. With time-varying error and irregular visits, the bias need not be simple attenuation.

A shared-random-effects joint model commonly specifies a longitudinal submodel such as \(Y_{ij}=m_i(t_{ij})+\epsilon_{ij}\), with \(m_i(t)=x_i(t)^\top\beta+z_i(t)^\top b_i\), \(b_i\sim N(0,D)\), and \(\epsilon_{ij}\sim N(0,\sigma^2)\). It links this latent trajectory to the event hazard, for example through \(\alpha m_i(t)\). Typical assumptions include a correctly specified trajectory and hazard form, an appropriate random-effects distribution, conditionally independent measurement errors given the random effects and covariates, and conditional independence of the longitudinal measurement errors and event time given the modeled random effects and covariates. Censoring must also be noninformative conditional on the variables in the model, or be modeled appropriately.

In this standard formulation, event-related cessation of measurements is handled through the joint event and longitudinal likelihood: subjects who experience an event stop contributing later measurements, and their event time contributes information about the latent trajectory. This does not mean every missing measurement or visit pattern is automatically ignorable. If visit timing itself responds to health status beyond what the model captures, the observation process may need explicit treatment. The later material on informative omics visit processes addresses that issue in detail.

Likelihood and the Source of the Difference

For a joint model, the random effects are unobserved. The subject-specific likelihood therefore averages over their possible values rather than substituting a single estimated trajectory as if it were known. Under the assumptions above, a typical contribution is

Here \(f(Y_{ij}\mid b_i)\) is the longitudinal measurement density, \(S_i(t\mid b_i)\) is the conditional event-free survival probability, and \(f(b_i)\) is the random-effects density. The event term is the event density when \(\delta_i=1\), and the survival probability through the censoring time when \(\delta_i=0\). The integral combines the marker and event information while accounting for uncertainty about \(b_i\).

By contrast, a time-dependent Cox model uses the covariate value defined at each event time in a risk-set comparison. In schematic form, its partial likelihood is a product over observed events of the failing subject’s relative risk divided by the sum of relative risks among subjects still at risk. It does not, by itself, include a measurement model for a latent trajectory. Thus, replacing an observed value with a model-estimated trajectory is not merely a harmless preprocessing step: it changes how uncertainty and the marker history enter the analysis.

Worked Comparisons

Choosing, Checking, and Comparing the Approaches

Model choice should follow the scientific target. If the target is the association with the most recent assay result as used in practice, a time-dependent Cox model may be a natural starting point, provided the value history is defined without using future information. If the target concerns the association of event risk with an underlying evolving biological process, a joint model may be more aligned with that question. A joint model is not automatically superior: misspecifying the trajectory or random-effects distribution can produce misleading results, while the simpler Cox model may be more robust to those particular distributional assumptions.

Diagnostics should examine whether the longitudinal submodel describes marker patterns over time, whether residual variation and subject-specific trajectory patterns are plausible, and whether the survival model’s proportional-hazards form is adequate. Investigate whether visits cluster near clinical changes or are more frequent among subjects with worsening status. For a time-dependent Cox model, document exactly how measurements define \(X_i(t)\), check that it is predictable, and assess whether conclusions change under reasonable alternatives to carrying the last value forward. For either model, compare predictions on the same target population and information set; apparent performance differences can otherwise arise from different prediction times or available data.

The approaches can also be used as complementary sensitivity analyses, but their coefficients should not be treated as estimates of an identical parameter. Dynamic prediction from a joint model and prediction at a specified landmark answer prediction questions with different information sets; comparison should respect those definitions. Neither model removes confounding by unmeasured variables, establishes causation, nor guarantees transportability to a population with a different assay or visit schedule.

Common Pitfall