Prediction Intervals, Confidence Intervals, and Credible Intervals

Three Intervals, Three Targets

A PBPK analysis may produce several intervals from the same set of simulations. Their numerical ranges can overlap, but their meanings depend on what is uncertain and what quantity the interval targets. An interval for a future patient’s concentration is not an interval for the mean concentration, and neither is automatically an interval for an uncertain model parameter.

This distinction matters when a result is used to assess a dose or communicate model uncertainty. For example, a wide range of simulated concentrations may reflect variability among individuals, uncertainty in model inputs, or both. A narrow interval around the simulated mean may instead describe only the Monte Carlo precision of that mean estimate.

The target should be named whenever an interval is reported. For a PBPK concentration example, possible targets include a future individual’s concentration at a specified time, the expected concentration under a specified input distribution, or a clearance parameter given observed data. A statement such as “the 95% interval is 2 to 8 mg/L” is incomplete unless it identifies which target the interval describes.

Prediction Intervals for Future Concentrations

Let \(C_{\mathrm{new}}(t)\) denote the concentration at time \(t\) for a future individual or exposure scenario drawn from the predictive setup. A central 90% prediction interval is formed from the 5th and 95th percentiles of the distribution of \(C_{\mathrm{new}}(t)\). Its endpoints depend on what sources of uncertainty and variability were included in that distribution.

For example, the simulated prediction distribution might propagate uncertain physiological inputs through the PBPK model, or it might additionally include between-subject variability. The interval represents only the components included in the predictive setup. It does not, by itself, guarantee that real future concentrations will fall within the interval if the model, input distributions, or target population do not adequately represent reality.

A prediction interval can be asymmetric, especially when concentration predictions are skewed or bounded below by zero. It is therefore generally defined by quantiles rather than by assuming that the prediction distribution is normal and adding and subtracting a fixed multiple of a standard deviation. The preceding material on constructing prediction intervals gives methods for estimating the relevant quantiles; the interpretive task here is to keep their future-outcome target clear.

Confidence Intervals for Estimated Summaries

A confidence interval for an estimated summary has a different target. For example, let \(X\) be a concentration drawn from a specified PBPK prediction distribution, and let \(\mu_C=E(X)\) be its expected concentration. If Monte Carlo draws are used to estimate \(\mu_C\), then the sample mean \(\bar{x}\) is an estimate of that expectation. The variability of \(\bar{x}\) across repeated simulation runs is described by its standard error, not by the spread of individual predicted concentrations.

For independent, identically distributed draws with finite variance, the estimated standard error of the sample mean is \(SE=s/\sqrt{n}\). When the draws are approximately normal, or when the sample is sufficiently large for a normal approximation to the sampling distribution of \(\bar{x}\), a two-sided 95% interval can be calculated as \(\bar{x}\pm 1.96SE\). This interval is about estimating the expectation, not predicting a new concentration.

The justification follows from the sampling distribution of the mean. If the draws have mean \(\mu_C\) and variance \(\sigma^2\), independence gives \(\operatorname{Var}(\bar{x})=\sigma^2/n\), so the standard deviation of \(\bar{x}\) is \(\sigma/\sqrt{n}\). Replacing the unknown \(\sigma\) with \(s\) estimates this standard error. When the standardized mean is approximately normal, about 95% of its values lie between \(-1.96\) and \(1.96\); rearranging that statement for \(\mu_C\) gives the interval formula above. If simulation draws are dependent, the independent-draw standard error formula need not apply.

Credible Intervals for Parameters

A Bayesian credible interval targets a parameter whose uncertainty is represented by a posterior distribution. Let \(CL\) denote clearance and \(D\) denote observed data. A 95% equal-tailed credible interval has 2.5% posterior probability below its lower endpoint and 2.5% above its upper endpoint. Its interpretation is conditional on the specified likelihood, prior distribution, and data.

The confidence and credible interval statements differ even when their endpoints happen to be numerically similar. A confidence interval’s 95% refers to the long-run coverage of its procedure for a fixed quantity. A Bayesian credible interval’s 95% is a probability assigned to parameter values by the posterior, conditional on the model and data. Neither interpretation should be transferred to a different interval merely because it is also labelled “95%.”

How the Targets Relate—and How They Do Not

A predictive distribution can depend on uncertain parameters and on variability in a future individual. Conceptually, a future concentration is generated by drawing parameter values under the relevant uncertainty representation, drawing any individual-specific inputs, and evaluating the PBPK model. A prediction interval summarizes the resulting distribution of future concentrations. A credible interval summarizes a posterior distribution for parameters. A confidence interval may summarize uncertainty in an estimated quantity such as the mean of the prediction distribution.

These intervals answer different questions, as the comparison below shows.

A confidence interval around a simulated mean can be far narrower than a prediction interval because increasing the number of Monte Carlo draws reduces the simulation error in \(\bar{x}\), while it does not remove the spread among possible future concentrations. Conversely, a credible interval for a parameter is not automatically a prediction interval: it has a parameter as its target. A posterior predictive distribution is needed to describe future outcomes when posterior parameter uncertainty is part of the prediction.

Diagnostics, Pitfalls, and Limitations

Before interpreting an interval, check that the computational procedure and the stated target agree. For a Monte Carlo confidence interval for a mean, verify that the draws support the assumed standard-error calculation. If draws are dependent, or if the simulation is too small for the normal approximation to be adequate, the simple formula may understate or misrepresent the uncertainty in the estimated mean. For a prediction interval, check that the simulations correspond to the intended future scenario and include the uncertainty and variability that the interpretation claims to cover. For a credible interval, check that the reported posterior is in fact the distribution used to calculate its endpoints.

Monte Carlo error is a further limitation. Estimated prediction quantiles and posterior quantiles can change across simulation runs, particularly when the number of draws is limited. A reported interval should not be treated as exact merely because its endpoints are printed to several decimal places. Also, all three interval types rely on assumptions: a prediction interval is conditional on a predictive model, a confidence interval on its sampling procedure, and a credible interval on its prior, likelihood, and posterior calculation.

The practical consequence is simple: report an interval together with its target, population or scenario, time point, and included uncertainty sources. The same numerical endpoints cannot be interpreted safely without this context. Choosing the right interval is not a matter of selecting the most familiar formula; it is a matter of matching the interval to the scientific question.