What Does the Interval Describe?
A numerical interval is interpretable only after its target has been named. In a pharmacometric simulation, an interval might describe outcomes for different individuals, uncertainty about a population-level prediction, or both. These are different quantities. A narrow interval for a population mean does not imply that individual outcomes are tightly clustered, and a wide range of simulated individual outcomes does not by itself show that the population mean is poorly estimated.
Consider a model in which an outcome \(Y\), such as concentration at a specified time, depends on population parameters \(\theta\), an individual's covariates \(x\), an individual random effect \(\eta\), and residual error \(\epsilon\). For a new individual with a specified dosing history and covariates, the model first predicts an outcome conditional on \(\theta\), \(\eta\), and the relevant error structure. The interval changes according to which of those quantities are allowed to vary.
These labels are not interchangeable. In particular, a population-distribution interval is often called a population prediction interval or prediction band, while an interval around a predicted population mean is often called an uncertainty or confidence interval for that mean. Because terminology varies, name the quantity being bounded rather than relying on the label alone.
Three Sources of Variation
For clarity, separate three sources. Interindividual variability is variation in \(\eta\) across individuals from the target population. Residual variability is observation-level variation represented by \(\epsilon\), conditional on the model and individual. Parameter uncertainty is uncertainty about \(\theta\) after fitting the model. It reflects finite information in the data and is not the same as variation between people.
A plug-in simulation fixes \(\theta\) at an estimate, often written \(\hat{\theta}\). Sampling many virtual individuals with different \(\eta\) values then describes a population distribution conditional on that fixed estimate. If residual error is also simulated, the distribution concerns observations; if it is not, it concerns latent model predictions for individuals. Either way, the plug-in simulation does not automatically include uncertainty in \(\hat{\theta}\).
An interval for a population prediction instead varies the parameter values according to an uncertainty procedure, then recalculates the target prediction. The target could be a population mean concentration at a time, a population quantile, or a population summary such as mean exposure. Whether individual and residual variation also enter depends on whether the target itself is a population summary or a future outcome.
Worked Example: Individual Outcomes Versus the Population Distribution
Suppose, hypothetically, that at one time point a concentration for a new individual is represented by \(Y=20+\eta+\epsilon\) mg/L. Assume \(\eta\sim N(0,4^2)\) mg/L and \(\epsilon\sim N(0,3^2)\) mg/L, independently. Treat the population parameters as known for this example.
For a new individual, both the individual's random effect and the observation-level residual error vary. Their independent normal contributions give \(Y\sim N(20,4^2+3^2)\), with variance \(25\) \((\mathrm{mg/L})^2\) and standard deviation \(5\) mg/L. A central 95% individual-outcome predictive interval is therefore
If the target is instead the distribution of latent individual predictions before residual error, the outcome is \(20+\eta\). Its standard deviation is \(4\) mg/L, so the central 95% population-distribution interval is
The second interval describes variation across individuals in their latent predictions; it does not include residual scatter in measured concentrations. Neither interval includes parameter uncertainty, because the example treats the parameters as known.
Intervals for Population Predictions
A population prediction is a specified function of the model parameters and target population. For example, let \(g(\theta)\) be the predicted mean concentration at a time, calculated for a defined covariate distribution and dosing regimen. An uncertainty interval for \(g(\theta)\) describes how uncertain that population-level prediction is because \(\theta\) is estimated. It is not an interval containing a stated proportion of individuals.
A common simulation approach is to obtain parameter draws \(\theta^{(1)},\ldots,\theta^{(B)}\) from a procedure that represents parameter uncertainty, and calculate \(g(\theta^{(b)})\) for each draw. The interval endpoints may be selected quantiles of these calculated values. In a Bayesian analysis, the draws may come from the posterior distribution. In a frequentist analysis, they may come from a suitable bootstrap or an approximation to the sampling distribution of the estimator. The interpretation depends on that procedure; the same computational quantiles do not have identical inferential meaning under every framework.
If \(g\) is a population mean, its parameter-uncertainty interval is an interval for that mean. If \(g\) is a population 90th percentile, the interval is an interval for the estimated 90th percentile. In neither case should it be described simply as “the range of individual values.” Conversely, a 5th-to-95th percentile range across simulated individuals at fixed \(\hat{\theta}\) is not an uncertainty interval for the population mean.
Worked Example: Uncertainty in a Population Mean Prediction
Suppose a hypothetical model predicts a population mean concentration of \(20.0\) mg/L at a specified time. A valid parameter-uncertainty calculation gives an estimated standard error of \(1.50\) mg/L for that mean prediction. Assume a normal approximation is appropriate and use a 95% interval.
The interval is the estimate plus or minus \(1.96\) estimated standard errors:
This interval concerns uncertainty in the population mean prediction. Under a frequentist interpretation, the 95% coverage statement concerns the performance of the interval procedure over repeated data sets under its assumptions; it does not mean there is a 95% probability that a fixed mean is inside this particular interval. The interval also says nothing directly about how widely individual concentrations vary around the mean.
Combining Parameter and Outcome Uncertainty
Sometimes the target is a future outcome while the parameters are also uncertain. Then a predictive distribution must account for both. A useful justification follows from the law of total variance. Let \(D\) denote the data used to fit the model. Conditional on a parameter value \(\theta\), variation in a future outcome \(Y\) has variance \(\operatorname{Var}(Y\mid\theta,D)\). The conditional mean, \(\operatorname{E}(Y\mid\theta,D)\), can itself vary across plausible parameter values. The total variance is
To see why, write \(Y-\operatorname{E}(Y\mid D)\) as the sum of \(Y-\operatorname{E}(Y\mid\theta,D)\) and \(\operatorname{E}(Y\mid\theta,D)-\operatorname{E}(Y\mid D)\). The first term has conditional mean zero given \(\theta,D\), so its cross-product with the second term has expectation zero. The expected square of the first term is the first variance component; the expected square of the second is the second component. This proves the decomposition.
In a hierarchical model, the first term can include interindividual and residual variability, while the second captures parameter-driven variation in the conditional prediction. Those components need not be independent in every model or for every target. The decomposition is a variance identity, not permission to add selected standard errors without checking the model structure. For nonlinear predictions, the resulting distribution may also be asymmetric, so quantiles from simulated predictions can be more appropriate than a normal interval.
Worked Example: A Predictive Interval Including Parameter Uncertainty
Continue the hypothetical concentration example, but now suppose the population mean \(\mu\) is uncertain, with an approximately normal uncertainty distribution centered at \(20.0\) mg/L and standard deviation \(1.50\) mg/L. Assume this parameter uncertainty is independent of the individual random effect and residual error, whose standard deviations remain \(4\) and \(3\) mg/L.
For a future individual observation, all three sources contribute. Under the stated normality and independence assumptions, the predictive variance is
Thus the predictive standard deviation is \(\sqrt{27.25}=5.220\) mg/L. The central 95% predictive interval is
This wider interval describes uncertainty about a future individual observation, combining parameter uncertainty, interindividual variability, and residual error. It is not an interval for the population mean. If the target were only the population mean, the individual and residual terms would not be added as sources of variation in that mean's parameter-uncertainty interval.
Choosing and Checking an Interval
Before calculating or reporting an interval, specify the prediction target and the sources allowed to vary. For a profile, state the time, dosing regimen, covariate target, and whether the profile is latent or includes observation error. For a summary such as an exposure metric, state whether the interval describes variation across individuals or uncertainty in the estimated population summary.
Identify whether the interval concerns one future outcome, a distribution across individuals, or a population-level estimand such as a mean or quantile.
Record whether the calculation varies individual random effects, residual error, parameter values, or some combination.
Use individual simulations for population distributions and parameter draws with recalculation of the target for parameter uncertainty. Use a combined predictive calculation when both are part of the target.
Review the assumed distributions, parameter covariance or resampling procedure, and Monte Carlo precision of the reported endpoints.
A practical diagnostic is to compare the interval's width and purpose with its components. If an interval for individual concentrations is unexpectedly narrow, check whether interindividual variability or residual error was omitted. If an interval for a population mean is as wide as the spread of individual outcomes, check whether the simulation has mistakenly treated between-person variation as uncertainty in the mean. Repeating the simulation with more replicates can assess Monte Carlo error in estimated quantiles, but it cannot repair omitted uncertainty or an incorrectly defined target.
| Interval target | Typical sources included | What it does not mean by itself |
|---|---|---|
| Future observation for one new individual | Individual variability, residual error; parameter uncertainty if propagated | A confidence interval for a population mean |
| Distribution across individuals at fixed parameters | Interindividual variability; residual error only if observations are simulated | Uncertainty in the estimated parameters |
| Population mean or other population prediction | Parameter uncertainty in the prediction | A range containing a specified fraction of individuals |
| Future outcome with parameter uncertainty | Outcome variation and parameter uncertainty | A single-source measure of variability |
Alternatives differ mainly in how they represent parameter uncertainty. A covariance-based approximation can be efficient when the estimator is approximately normal and the prediction is locally well behaved. Bootstrap refitting can represent nonlinearity and estimation behavior more directly, but can be computationally demanding and may fail in some resamples. Posterior draws provide a Bayesian uncertainty distribution when the model and prior are specified. None of these choices changes the target automatically: each must be paired with a calculation that recomputes the intended population prediction.
All approaches have limits. A normal approximation can produce implausible parameter values or poor intervals for strongly nonlinear predictions. A bootstrap may be unstable when the model is weakly identified or resampling frequently fails. Posterior intervals depend on the likelihood, prior, and computation. Simulation-based quantiles also have Monte Carlo error, especially in the tails. Report the method, target, conditioning, and included sources so readers can tell what the interval supports.
Common Mistakes
- Calling a population percentile band a confidence interval. The percentiles across virtual individuals describe population spread conditional on the model inputs; they do not quantify parameter uncertainty.
- Calling a mean-prediction interval an individual prediction interval. An interval for a population mean does not include the full spread of outcomes among individuals.
- Assuming every simulation includes parameter uncertainty. Simulating many individuals at fixed \(\hat{\theta}\) increases Monte Carlo precision but does not make \(\hat{\theta}\) uncertain.
- Adding variability components without checking the target. Residual error belongs in an interval for an observation, but not automatically in an interval for a latent individual prediction or population mean.
- Reporting only “95% prediction interval.” State what is predicted, for whom, under what conditions, and which sources of uncertainty are included.
Check Your Understanding
Answer each question by identifying the target and the uncertainty sources, not just the interval label.
- A simulation fixes the estimated parameters, samples random effects across virtual patients, and reports the 5th and 95th percentiles of latent concentration. What does the interval describe, and what source is omitted?
- A future measured concentration has independent residual standard deviation \(2\) mg/L and interindividual standard deviation \(3\) mg/L. With parameters treated as known and a normal model, calculate a central 95% interval around a predicted mean of \(10\) mg/L.
- An interval for a predicted population mean is reported as \([8,12]\) mg/L. Does this imply that 95% of individual concentrations are between \(8\) and \(12\) mg/L? Explain.
- Why can increasing the number of virtual patients narrow Monte Carlo uncertainty in a simulated percentile without reducing parameter uncertainty?
Solutions. (1) It describes the central 90% of latent individual predictions in the target population conditional on the fixed parameter estimates. It omits residual error and parameter uncertainty. (2) The predictive standard deviation is \(\sqrt{3^2+2^2}=\sqrt{13}=3.606\) mg/L. The interval is \(10\pm1.96(3.606)=10\pm7.068\), or approximately \([2.93,17.07]\) mg/L. (3) No. It is an interval for uncertainty in the population mean prediction; individual concentrations can vary beyond it. (4) More virtual patients reduce simulation error in estimating the percentile under the chosen model and fixed inputs. They do not change the parameter estimates or represent their uncertainty; that requires a parameter-uncertainty procedure.