Centered and Noncentered Random-Effects Parameterizations in Joint Models

Why the Random-Effects Coordinates Matter

In a shared-random-effects joint model, a subject’s latent effects connect that subject’s longitudinal measurements to their event-time information. The probability model is unchanged if those effects are represented in different coordinates, but the posterior can look very different to a sampler. Since computation explores the posterior in its chosen coordinates, that difference can determine whether the sampler moves efficiently or spends long periods in difficult regions.

The choice is not a general rule that one parameterization is always better. It depends in part on how strongly each subject’s data determine their random effects, relative to the prior. Longitudinal measurements can be highly informative about some random-effect directions, while an individual’s survival contribution may add little information, or may add useful information through the shared-effects association. The resulting posterior geometry, rather than the model label alone, motivates a choice.

Model and Two Parameterizations

Let \(y_{ij}\) denote subject \(i\)’s longitudinal measurement at time \(t_{ij}\), and let \(b_i\) be a \(q\)-dimensional subject-specific random-effects vector. Write \(T_i\) for the observed event or censoring time and \(\delta_i\) for the event indicator. Let \(\theta\) collect the other model parameters, including longitudinal fixed effects, association parameters, and baseline-hazard parameters. A standard shared-effects model has a subject-level likelihood contribution \(p(y_i,T_i,\delta_i\mid b_i,\theta)\).

For example, a longitudinal mean can depend on \(b_i\), and a survival hazard can depend on a latent biomarker trajectory constructed using the same \(b_i\). The usual working assumptions include conditional independence of the longitudinal and event-time observations given the shared effects and model parameters, and independent random-effects vectors across subjects conditional on their population distribution. The event-time likelihood must also use an appropriate censoring assumption. These formulations concern how \(b_i\) is represented; they do not remove or replace the model’s assumptions.

In the centered form, a typical posterior density, up to proportionality, is

In the noncentered form, the corresponding target in \(z_{1:n}\) is

where \(\phi_q\) is the density of \(N_q(0,I_q)\). The noncentered formulation makes the standard-normal prior independent of \(D\); the data still connect \(z_i\) to \(D\) through \(Lz_i\). These are alternative coordinates for the same model, not different scientific assumptions.

Transformation and Posterior Geometry

For fixed \(D\), the change of variables \(b_i=Lz_i\) has absolute determinant \(|L|\) for each subject. The centered prior density contains the factor \(|D|^{-1/2}=|L|^{-1}\). Thus, when the centered density is transformed to \(z_i\), its Jacobian \(|L|\) cancels that prior normalizing factor, leaving the standard-normal density for \(z_i\). This is why a determinant should not be added again to the displayed noncentered target: the standard-normal prior already gives the transformed density.

The key geometry can be seen in a scalar random-intercept model. Suppose \(b\sim N(0,\tau^2)\), and the observed data reduce to a Gaussian summary \(\bar{y}\mid b\sim N(b,\sigma^2/n)\), with \(\tau,\sigma\) known. Define the dimensionless information ratio \(a=n\tau^2/\sigma^2\). Conjugate updating gives

With \(b=\tau z\), the posterior variance of \(z\) is \(1/(1+a)\). When \(a\) is small, the data barely update the standard-normal \(z\). In centered coordinates, by contrast, the scale of \(b\) remains tied to \(\tau\), which can create a narrow funnel when \(\tau\) is small. When \(a\) is large, the data constrain \(b\) tightly; in noncentered coordinates, that tight constraint acts on \(\tau z\), so the allowable range of \(z\) depends strongly on \(\tau\). This can instead make the noncentered geometry difficult.

How Survival Information Enters the Choice

A survival contribution can add information about \(b_i\) because the hazard depends on the shared effects. For an event at time \(T_i\), a conditional survival likelihood contribution has the form \(h_i(T_i\mid b_i)^{\delta_i}\exp\{-H_i(T_i\mid b_i)\}\), where \(H_i\) is the cumulative hazard. Its curvature in \(b_i\) depends on the hazard model, event time, covariates, and association parameters. It need not be equally informative for every subject, and its curvature can vary across the posterior.

The following calculation uses local curvature only. It is a diagnostic approximation, not a substitute for the full event-time likelihood. Suppose a scalar random intercept enters the hazard as \(h(t\mid b)=h_0(t)\exp(\alpha b)\). The log survival contribution is

Its negative second derivative, the local likelihood curvature, is \(H_0(T)\alpha^2\exp(\alpha b)\). This quantity helps compare local survival information with the prior precision \(\tau^{-2}\) and longitudinal likelihood curvature. It is only a local comparison: curvature alone does not capture skewness, multiple modes, or all dependence among parameters.

A Practical Choice and Its Diagnostics

A useful starting principle is to match coordinates to the information pattern. When subject-level data strongly identify \(b_i\), centered effects often express the narrow, data-determined posterior directly. When those data weakly identify \(b_i\), noncentered effects can reduce the direct dependence between the subject effects and their population scale or covariance. In a joint model, assess the information from both data components: a subject with few biomarker measurements may still receive some information from the event process, but sparse or weakly associated survival data may contribute little.

Check the sampling behavior in the coordinates actually used. In Hamiltonian Monte Carlo, repeated divergences, poor exploration of small covariance scales, or strong dependence between a random effect and a scale parameter can signal difficult geometry. Compare behavior across relevant parameters and subjects; a single well-mixing fixed effect does not establish that the random-effects posterior is explored well. Also verify that changing coordinates has not changed the implied prior or omitted the transformation in a density written on a different set of variables.

A parameterization is not a cure for a misspecified or weakly identified model. If both forms mix poorly, the cause may include limited information, near-nonidentifiability between association and random-effects parameters, or an overly restrictive covariance structure. The centered/noncentered comparison is about sampling coordinates, not about resolving the scientific adequacy of the model.

Common Mistakes and Limitations

• Assuming noncentered is always better. It often helps when individual data are weak, but when the data constrain effects tightly, dependence between \(z_i\) and covariance parameters can make it inefficient. Assess the information and posterior diagnostics.
• Interpreting a reparameterization as a new prior. Replacing \(b_i\) by \(z_i\) must preserve \(b_i\sim N_q(0,D)\). A standard-normal prior on \(z_i\) is correct only together with \(b_i=Lz_i\).
• Using the longitudinal sample size alone as the information measure. Measurement noise, design times, and the survival contribution all affect how well the shared effects are identified. In multiple dimensions, information can differ by direction.
• Treating local curvature as a global guarantee. A curvature calculation near one point does not rule out skewness, tails, or other difficult posterior features. Use it to motivate a parameterization, then examine sampling behavior.

The centered and noncentered forms are the main alternatives considered here. Intermediate or partially noncentered coordinates can be useful when information is neither clearly weak nor strong, but they require additional choices and do not eliminate the need to check the fitted posterior. The best representation can differ across random-effect components and datasets.