Prediction Intervals Versus Confidence and Tolerance Intervals

Three Intervals, Three Questions

An interval is useful only when its target is clear. In pharmacokinetics, one question may concern a future concentration for a particular patient; another may concern the population mean concentration; a third may ask whether a dose keeps concentrations within a range for a large fraction of the target population. These are different inferential tasks. Their intervals generally have different widths and cannot be substituted for one another.

For an individual forecast, the conditioning information matters. It could include a planned dose and covariates, or observed concentrations for that individual. A prediction interval describes uncertainty about a future observation under those conditions. A confidence interval describes uncertainty in estimating a population quantity. A tolerance interval describes an interval intended to contain a stated fraction of a population, with a stated confidence level.

For example, “95% prediction interval” refers to the probability content of a procedure for future observations, under its model and conditioning conditions. “95% confidence interval” refers to repeated-sample coverage of a parameter. A “95%/95% tolerance interval” aims to contain at least 95% of the target population with 95% confidence. The two percentages in a tolerance statement have distinct roles.

Prediction and Confidence Intervals in a Normal Benchmark

A simple normal model makes the distinction in width transparent. Suppose \(X_1,\ldots,X_n\) are independent observations from \(N(\mu,\sigma^2)\), and \(X_{\text{new}}\) is an independent future observation from the same distribution. Assume normality and common variance, and estimate \(\mu\) and \(\sigma\) by \(\bar{x}\) and \(s\). Under these assumptions, the usual two-sided \(100(1-\alpha)\%\) confidence interval for \(\mu\) is \(\bar{x}\pm t_{1-\alpha/2,n-1}s/\sqrt{n}\). The corresponding prediction interval for \(X_{\text{new}}\) is \(\bar{x}\pm t_{1-\alpha/2,n-1}s\sqrt{1+1/n}\).

The prediction interval is wider because it must account for both uncertainty in the estimated mean and the future observation’s variation around that mean. The confidence interval concerns only the mean parameter.

These formulas are a benchmark, not a substitute for an NLME forecast. In an NLME model, the target might be a future concentration conditional on an individual’s data, dose, and covariates; the forecast distribution is obtained from the specified model and conditioning information. Its interval is still about a future observation, not automatically about a population parameter or the fraction of the population inside a range.

Tolerance Intervals and Population Coverage

A tolerance interval addresses a population-level coverage question. For example, a clinician or development team may ask whether a concentration range contains at least 95% of concentrations for a defined target population at a specified time and regimen. The target population must be stated: it may refer to individuals with specified covariates, or to a broader covariate distribution. Changing that target changes the population coverage being assessed.

In a fitted NLME model, the distribution relevant to tolerance coverage includes the model’s variation across individuals and the observation process if the target is measured concentrations. The exact distribution depends on the question. A range for latent individual typical concentrations is not necessarily a range for observed concentrations, which also reflect residual error. Likewise, a population-wide tolerance interval is not an interval for a particular patient’s next observation.

One practical model-based approach uses a parametric bootstrap. Repeatedly generate data under the fitted model, refit the model, and estimate the target population fraction inside a candidate interval for each refit. The lower tail of those estimated fractions quantifies uncertainty in coverage due to sampling and fitting. This is an approximate tolerance procedure: its repeated-sample interpretation depends on the model, bootstrap design, and accuracy of the coverage calculation.

A Confidence Interval for a Population Parameter

A confidence interval can target a parameter other than the mean, such as a fixed-effect coefficient or a variance parameter. Its interpretation remains parameter-focused: its repeated-sample coverage concerns that parameter. For an estimated population mean, the normal-theory interval above uses the sampling uncertainty in \(\bar{x}\); it does not describe the spread of individuals around the mean.

Choosing, Checking, and Interpreting the Interval

Before reporting an interval, state its target, conditioning information, units, and assumptions. For a prediction interval, check that the model represents the intended future setting and that the forecast distribution accounts for the variability relevant to the observation being predicted. For a confidence interval, check whether its sampling or model-based approximation is appropriate for the parameter and data. For a tolerance interval, check that the target population and coverage definition are unambiguous and that the fitted model reproduces important distributional features, including tails.

Useful diagnostics include comparing simulated and observed concentration distributions, checking residual patterns and extreme values, assessing whether the covariate distribution used for a population target is appropriate, and examining whether bootstrap fits are stable. A tolerance interval can be especially sensitive to misspecified tails: a model that fits the center well may still give unreliable estimates of the fraction outside the interval. A prediction interval may also be poorly calibrated if the individual’s conditioning information or future dosing conditions differ from those represented by the model.

The main alternatives answer different questions. A confidence interval is appropriate for uncertainty in a parameter, not individual spread. A prediction interval is appropriate for a future individual observation, not the fraction of an entire population captured. A tolerance interval directly targets population content, but usually requires more modelling or computation to account for uncertainty in that content. A credible interval for a Bayesian parameter has a posterior probability interpretation conditional on the model and prior; a posterior predictive interval instead concerns future observations. The label alone is not enough: always identify what random quantity the interval concerns.

A common mistake is to call a narrow confidence interval for the mean a “95% range for patients.” That reasoning confuses uncertainty in the estimated mean with variability among individuals. The correction is to choose a prediction interval for one future observation or a tolerance interval for a stated population fraction, depending on the scientific question. Neither interval eliminates model dependence: misspecification, unrepresentative covariates, or an incorrect residual model can make the reported coverage misleading.