Defining Parameter Uncertainty in Nonlinear Mixed-Effects Models

What Does a Parameter Confidence Interval Describe?

An NLME model can estimate several kinds of quantities at once. A confidence interval for a population typical clearance, a confidence interval for between-subject variability in clearance, and a prediction interval for one patient’s clearance do not answer the same question. Before interpreting any interval, identify the exact quantity it targets.

For subject \(i\) at observation \(j\), a general model can be written as

Here \(y_{ij}\) is the observation, \(f\) is the structural model, \(t_{ij}\) and \(D_{ij}\) represent the observation time and dosing information, \(\theta\) contains population fixed effects, and \(\eta_i\) contains subject-specific random effects. The covariance matrix \(\Omega\) describes between-subject variation and association among those random effects. The residual error \(\varepsilon_{ij}\) describes observation-level variation not accounted for by the structural and random-effects parts of the model. A complete fitted model may also have residual-error parameters.

In a frequentist interpretation, once the data have been observed, the parameter is treated as fixed and the computed interval is fixed. The 95% refers to the long-run coverage of the procedure, not to a 95% probability that this particular interval contains the parameter. The interval is also conditional on the chosen model, its assumptions, and the analysis design; it does not automatically account for uncertainty about whether the model form is correct.

Population Effects, Variability, and Individual Effects

A population fixed effect describes a feature of the population model. For example, \(\theta_{\mathrm{CL}}\) might be the typical clearance on the model’s parameter scale, or a fixed-effect coefficient might describe the change in log clearance associated with a covariate. The meaning depends on the structural model and parameterization.

By contrast, a random effect \(\eta_i\) represents a subject’s deviation from the population relationship. In the model above, the \(\eta_i\) values vary across subjects, while \(\Omega\) is a population-level covariance parameter. The fitted estimate of a subject’s random effect is not itself a population parameter. Nor is its uncertainty interval interchangeable with a confidence interval for a component of \(\Omega\).

Residual-error parameters are another target class. They describe the assumed observation-level error distribution, not between-subject differences in clearance or volume. Thus, an interval for a residual standard deviation and one for a random-effect standard deviation refer to different sources of variation, even if both are reported in similar units.

Variance Components, Standard Deviations, and Correlations

For a random effect with variance \(\omega^2\), the corresponding standard deviation is \(\omega=\sqrt{\omega^2}\). The variance is in squared units of the random effect; the standard deviation is in the original units. If random effects act on a log scale, those units are log-scale units, and the standard deviation is not directly a clearance standard deviation in L/h.

For two random effects with variances \(\omega_1^2\) and \(\omega_2^2\), and covariance \(\omega_{12}\), their correlation is

A variance, covariance, and correlation are distinct fitted-model quantities. Variances and covariances have scale-dependent units; correlation is dimensionless and constrained to \([-1,1]\). An interval for a variance does not automatically provide an interval for its standard deviation or for a correlation. Each requires its own target and appropriate treatment of the parameter’s constraints.

Derived Pharmacokinetic Quantities

Some quantities of pharmacokinetic interest are functions of fitted parameters rather than parameters estimated independently. Examples include a treatment-group clearance ratio, an individual arithmetic mean derived from a lognormal model, or a half-life calculated from clearance and volume. An interval for such a quantity targets the function itself. It is not necessarily equivalent to an interval for any one input parameter.

For example, suppose individual clearance follows a lognormal random-effects model:

Then \(\exp(\mu_{\mathrm{CL}})\) is the median individual clearance (and the typical value under this parameterization), whereas the arithmetic population mean depends on both \(\mu_{\mathrm{CL}}\) and \(\omega_{\mathrm{CL}}^2\). This distinction matters because a confidence interval for \(\mu_{\mathrm{CL}}\) does not, by itself, give an interval for the arithmetic mean individual clearance.

Why Derived Quantities Depend on Multiple Estimates

The dependence in the previous example follows directly from the lognormal model. If \(Z\sim N(0,\omega^2)\), its exponential moment is \(E[\exp(Z)]=\exp(\omega^2/2)\). Therefore,

This derivation shows why the interval target must be stated precisely: the arithmetic mean is a function of two population parameters, while the typical value is a function of one. In general, if a quantity of interest is \(g(\phi)\), where \(\phi\) is a vector of fitted population parameters, an interval for one component of \(\phi\) is not automatically an interval for \(g(\phi)\). The uncertainty calculation must correspond to the complete function and its inputs.

Assumptions, Diagnostics, and Limits

An interval’s interpretation requires that the fitted model and the interval procedure be appropriate for the target. At a minimum, the population structure, random-effects distribution, residual-error model, and independence assumptions must be defensible for the data and design. The target must also be identifiable: the observations need to contain enough information to distinguish it from competing parameters. These assumptions do not become true merely because an estimation program returns an interval.

Warning signs include estimates on or near a parameter boundary, extremely wide intervals, strong dependence between parameters, unstable estimates across reasonable model fits, and intervals that conflict with the parameter’s allowed range. A variance cannot be negative and a correlation cannot fall outside \([-1,1]\); an interval extending into impossible values is a reason to question the approximation or procedure, not to interpret those values literally. Sparse sampling, weak information about a random-effect distribution, or confounding between fixed effects can also make a target poorly determined.

The main alternatives in interpretation are not competing labels for the same interval. A confidence interval for a population quantity describes uncertainty in its estimation; a prediction interval describes a future or individual outcome under a specified model; and an interval for an individual random effect concerns a subject-specific latent quantity. These may be reported together, but each answers a different question. Likewise, an interval for a directly fitted parameter and one for a derived PK quantity differ by their target, even if they are based on the same fitted model.

A confidence interval is conditional on the model and does not quantify every source of scientific uncertainty. It does not, by itself, account for model-selection uncertainty, misspecification, or uncertainty about whether the selected population is relevant to a new setting. Reporting the target, scale, units, and population interpretation makes these limits visible.

Common Mistakes

• Calling a population interval an individual range. An interval for typical clearance is not the range expected to contain most patients’ clearances; that requires a distributional or predictive quantity.
• Confusing variability with uncertainty. A large random-effects variance indicates substantial population heterogeneity, not necessarily imprecise estimation of that variance. Conversely, an imprecise variance estimate can occur even when the estimated variability is modest.
• Reporting a derived quantity without naming its inputs. A mean individual clearance under a lognormal model depends on both the typical log clearance and its random-effect variance.
• Ignoring scale and units. A variance on a log scale is not a clearance variance in \((\mathrm{L/h})^2\), and a correlation has no units.
• Reading an interval as a probability statement about the fixed parameter. A 95% confidence procedure has long-run 95% coverage under its assumptions; it does not assign a 95% probability to the parameter being in the observed interval.