Recovering Individual Parameters from EBEs

From an EBE to an Individual Parameter

A fitted NLME model often reports an empirical Bayes estimate (EBE) for each individual’s random effects. These estimates are on the random-effect scale, not necessarily on the scale of clearance, volume, or absorption. To obtain an individual parameter value, first calculate the individual’s covariate-conditioned typical value, then apply the random-effect relationship specified by the model. The conversion is simple algebraically, but the result depends on the parameterization and on what kind of EBE is being used.

The covariate-conditioned typical value is the model’s typical parameter for an individual with the observed covariates and random effects set to zero. For example, it may differ from the population reference value because of that individual’s body weight. An EBE then supplies the estimated random-effect contribution around that typical value.

Exponential and Additive Parameterizations

In an exponential parameterization, the random effect is additive on the log-parameter scale and multiplicative on the parameter scale. This is commonly used when a parameter must remain positive. The random effect is dimensionless: a value of \(0.10\) represents a multiplicative factor of \(\exp(0.10)\), not an increase of 0.10 units of clearance or volume.

The second equation defines the usual plug-in reconstruction from an EBE. A positive EBE gives a factor greater than one, and a negative EBE gives a factor between zero and one. The reconstructed parameter remains positive whenever \(P_{i,\mathrm{typ}}>0\).

In an additive parameterization, the random effect is measured in the same units as the parameter:

Thus an additive clearance EBE is in clearance units, while an additive \(k_a\) EBE is in inverse-time units. Unlike the exponential form, the additive form does not guarantee a positive parameter. The fitted model and the resulting individual values must be checked for that limitation.

Covariate Conditioning Comes Before EBE Conversion

The typical value in the conversion formula is not automatically the population reference value. It must include the fixed-effect covariate function used by the fitted model, evaluated at the individual’s covariates and the model’s coding conventions. For a categorical covariate, the relevant reference-cell or contrast coding determines the fixed-effect contribution. For a continuous covariate, use the model’s stated reference value, scaling, and functional form.

A practical calculation therefore has two distinct parts: evaluate the typical parameter from the fixed effects and individual covariates, then apply the EBE using the correct random-effect parameterization. Reversing those steps is harmless only when the chosen formula makes the operations equivalent; it should not be assumed for more complicated parameter relationships.

Correlated Random Effects

Random effects for clearance, volume, and absorption may be correlated. Write the individual random-effect vector as \(\boldsymbol{\eta}_i\), with population covariance matrix \(\Omega\). A nonzero off-diagonal element of \(\Omega\) means that the corresponding random effects vary together in the population model. It does not, by itself, add a cross-term to each parameter’s reconstruction equation.

If the model specifies \(CL_i=CL_{i,\mathrm{typ}}\exp(\eta_{i,CL})\) and \(V_i=V_{i,\mathrm{typ}}\exp(\eta_{i,V})\), each parameter is reconstructed with its own EBE. The estimated effects may be statistically dependent, and their joint estimation may reflect that dependence, but one does not add \(\widehat{\eta}_{i,V}\) to the clearance equation unless the model explicitly includes it there.

What the Reconstructed Value Does—and Does Not—Represent

An EBE is commonly a conditional mode or conditional mean of a random effect given an individual’s data and the fitted model. Those summaries are not interchangeable, and a transformed EBE should be described as a plug-in value unless the estimation target is specified more precisely. In particular, exponentiating a conditional mean on the eta scale does not generally produce the conditional mean of the parameter.

For example, if the conditional distribution of \(\eta_{i,P}\) given the data is normal with mean \(m\) and variance \(s^2\), then

These differ when \(s^2>0\). Thus, if the reported EBE is the conditional mean \(m\), the usual plug-in parameter is \(P_{i,\mathrm{typ}}\exp(m)\), while the conditional mean parameter under the stated normal conditional distribution includes the additional factor \(\exp(s^2/2)\). The latter requires conditional uncertainty information, not just the EBE. In nonlinear models the conditional distribution need not be normal, so that expression is not a general substitute for integrating the fitted conditional distribution.

Likewise, transforming a conditional mode on the eta scale gives the conventional plug-in value, but it need not equal the mode of the parameter-scale density: a nonlinear transformation changes density through its Jacobian. State which quantity is reported rather than calling every transformed EBE a posterior mean or mode.

Diagnostics, Alternatives, and Common Pitfalls

Before using reconstructed parameters, verify the model equation, covariate reference and coding, EBE-to-parameter mapping, and units. Check that exponential EBEs are treated as dimensionless log-scale quantities and additive EBEs have parameter units. Review whether additive reconstructions violate positivity or other scientifically required bounds. Also consider the amount of individual information: shrinkage can pull EBEs toward zero, making reconstructed values close to covariate-conditioned typical values even when an individual’s true parameter differs. A small EBE is not proof that the individual is typical.

The exponential and additive forms answer different modeling needs. Exponential effects enforce positivity and describe proportional deviations; additive effects describe absolute deviations and are easier to interpret in the parameter’s original units. Neither form is universally preferable. The additive form can produce invalid values, while the exponential form can produce large multiplicative changes for large positive EBEs. The appropriate form is the one specified and supported by the fitted model, not a choice made after seeing an inconvenient estimate.

• Wrong: “The clearance EBE is 0.12, so add 0.12 L/h.” Correction: Under an exponential model, \(0.12\) is dimensionless and multiplies the typical clearance by \(\exp(0.12)\).
• Wrong: “The typical clearance is the reference clearance for everyone.” Correction: First evaluate the covariate-conditioned typical value for the individual.
• Wrong: “Clearance and volume EBEs are correlated, so the volume EBE must be added to clearance.” Correction: Correlation is a feature of their joint distribution; each parameter follows its specified equation.
• Wrong: “Exponentiating the EBE gives the conditional mean clearance.” Correction: It gives a plug-in transformation; a conditional mean generally requires accounting for the full conditional distribution.

A reliable reconstruction is therefore explicit about both its formula and its target: covariate-conditioned typical value plus the model-defined random-effect contribution, evaluated at the reported EBE. This supplies individual parameter values on the model scale without implying that the values are directly observed or free of uncertainty.