Centered and Noncentered Random-Effect Parameterizations

Why the Choice of Coordinates Matters

In a Bayesian NLME model, an individual parameter is linked to a population distribution. The likelihood may contain little information about a particular individual’s parameter, even when the model has many observations overall. In that setting, the random effect is largely governed by its population distribution. How that distribution is expressed in the sampled coordinates can strongly affect posterior geometry and computational efficiency.

The centered and noncentered formulations describe the same hierarchical model. Centering samples an individual parameter, or its random effect, directly. Noncentering instead samples a standardized random effect and constructs the individual quantity from the population parameters. Neither is universally preferable: the useful choice depends on how informative the individual data are relative to the between-subject variability.

For example, if the positive individual parameter \(\theta_i\) is modeled on the log scale, a centered formulation is \(\log(\theta_i)=\mu+\eta_i\), with \(\eta_i\mid\tau\sim N(0,\tau^2)\). The noncentered formulation is \(\log(\theta_i)=\mu+\tau z_i\), with \(z_i\sim N(0,1)\). Here \(\mu\) is the population mean on the log scale and \(\tau\) is the between-subject standard deviation on that scale. The observation model and the priors on population parameters are otherwise unchanged.

Equivalence and the Change of Variables

Consider a scalar random effect and fix \(\mu\) and \(\tau>0\). In centered coordinates, the conditional prior density of \(\eta_i\) is normal. Set \(z_i=\eta_i/\tau\), or equivalently \(\eta_i=\tau z_i\). The density transformation includes the Jacobian \(d\eta_i/dz_i=\tau\). This factor is essential: it shows why the standardized effect has a unit normal prior, and why merely substituting \(\tau z_i\) into a density without transforming the measure would be incorrect.

Here \(\phi\) is the standard normal density. The data likelihood in the noncentered coordinates is evaluated at the same individual effect, \(\eta_i=\tau z_i\). The resulting joint posterior is therefore the same probability distribution expressed in different coordinates, provided that the Jacobian is handled correctly. Posterior summaries for quantities such as \(\theta_i\) are unchanged after transforming draws back to those quantities; the coordinates explored by a sampler, however, can have very different geometry.

How Data Information Changes Posterior Geometry

The geometry is easiest to see in a simple normal observation model. Suppose an individual-level summary \(y_i\) has likelihood \(y_i\mid\eta_i\sim N(\mu+\eta_i,s_i^2)\), and the centered random effect has prior \(\eta_i\mid\tau\sim N(0,\tau^2)\). Treat \(\mu\), \(\tau\), and the observation standard deviation \(s_i\) as fixed for the moment. This conjugate example isolates the effect of the individual data; real NLME likelihoods may be nonlinear and need not be normal.

Completing the square gives a normal conditional posterior for \(\eta_i\). Its variance is determined by the sum of prior and likelihood precision, and its mean is the precision-weighted combination of the population mean and the observation.

When \(s_i\) is large compared with \(\tau\), the likelihood is relatively imprecise. The posterior for \(\eta_i\) remains close to its population prior: its standard deviation is near \(\tau\), and its mean is close to zero. In centered coordinates, the width of the random-effect distribution changes with \(\tau\). If \(\tau\) can approach zero, the centered posterior can become very narrow in \(\eta_i\) for small \(\tau\), while remaining much wider for larger \(\tau\). This changing width is the characteristic funnel-like geometry of a weakly informed hierarchical effect.

In noncentered coordinates, \(z_i\) has a standard normal prior whose scale does not change with \(\tau\). When the likelihood contributes little information about an individual, it also contributes little constraint linking \(z_i\) to \(\tau\). This often makes the posterior easier to explore. When an individual is strongly informed by data, however, the likelihood may constrain \(\mu+\tau z_i\) tightly. Changes in \(\tau\) may then require coordinated changes in \(z_i\), producing posterior dependence in the noncentered coordinates. In this regime, the centered effect can be easier to sample.

Vector Random Effects and Practical Choice

For a vector random effect \(\eta_i\) with covariance matrix \(\Omega\), the centered form is \(\eta_i\sim N(0,\Omega)\). A noncentered form is \(\eta_i=A z_i\), where \(z_i\sim N(0,I)\) and \(A\) is any factor satisfying \(AA^\mathsf{T}=\Omega\). The population mean component is added in the same way as in the scalar case. The particular factorization is a computational and modeling implementation detail; the essential distinction is whether the sampled effect is expressed on the population scale or on a standardized scale.

A useful qualitative comparison is the relative scale of between-subject variation and individual-level information. In the normal example, \(\tau^2/s_i^2\) compares prior variance with observation variance. A small ratio means the data are weak relative to the population distribution and often favors noncentering. A large ratio means the data are informative about the individual effect and may favor centering. This is a diagnostic heuristic, not a universal threshold: in nonlinear models, local curvature, parameter transformations, and dependence among effects also matter.

Diagnostics, Alternatives, and Limitations

A parameterization problem may appear as slow movement in traces, persistent dependence between a population scale and individual effects, or poor exploration across the range of plausible values for \(\tau\). In centered coordinates, inspect whether individual effects contract strongly toward zero as \(\tau\) becomes small, while having much wider conditional spread at larger \(\tau\). In noncentered coordinates, check whether the data force strong dependence between \(z_i\) and \(\tau\). These patterns are evidence about the geometry, not proof that the statistical model is wrong.

The main alternative to choosing one parameterization for all subjects is to use a partially noncentered formulation, which interpolates between centered and noncentered coordinates. Another alternative is to integrate out some or all random effects when constructing the target distribution, rather than sampling every individual effect directly. These approaches change the computational representation of the model and can be useful in different settings, but they do not remove the need to check that the likelihood, population distribution, and priors are specified appropriately.

A reparameterization cannot create information about poorly observed subjects, resolve a weakly identified population variance, or correct a misspecified likelihood. Nor does noncentering guarantee improved sampling: highly informative individual data can make it inefficient. If the random-effect distribution is not normal, the standardized construction must use the appropriate base distribution and transformation; the simple normal derivation above cannot be transferred unchanged. In every case, transformations must preserve the intended prior, including any required Jacobian.