Constructing Central and One-Sided Prediction Intervals

Why interval direction matters

A PBPK simulation can produce a distribution of a scalar exposure outcome, such as AUC or Cmax, under specified input distributions, model assumptions, and a dosing scenario. An interval compresses that distribution into boundaries. The boundaries are useful only if their direction and tail allocation match the question: a central interval describes a middle portion of the distribution, whereas an upper or lower one-sided limit addresses a question about one tail.

For example, an upper limit may be relevant when asking how large an exposure could be while still covering most simulated outcomes. A lower limit may be relevant when asking whether a substantial fraction of simulated exposures exceed a threshold. The interval describes the output distribution induced by the chosen simulation setup; it does not correct inappropriate input distributions, model assumptions, or simulation errors.

Definition: Let \(Y\) be a scalar PBPK output under a specified scenario, and let \(F_Y(y)=P(Y\le y)\) be its predictive distribution. The \(p\)-quantile is \(q_p=\inf\{y:F_Y(y)\ge p\}\), for \(0<p<1\). A prediction interval formed from these quantiles reports a range intended to contain a stated proportion of outcomes from \(F_Y\). The quantiles have the units of \(Y\).

Here, “predictive distribution” means the distribution of the selected model output induced by the uncertain inputs represented in the simulation. The output and scenario must be stated: for instance, AUC over a specified dosing interval for a particular regimen. Quantiles from one output or scenario do not automatically describe another.

Equal-tail prediction intervals

For nominal coverage \(1-\alpha\), an equal-tail interval allocates \(\alpha/2\) probability to each tail. Its boundaries are the \(\alpha/2\) and \(1-\alpha/2\) quantiles. Thus a nominal 90% equal-tail interval uses the 5th and 95th percentiles; a nominal 95% interval uses the 2.5th and 97.5th percentiles.

I_{\mathrm{central}}= \left[q_{\alpha/2},q_{1-\alpha/2}\right]

For a continuous distribution, the probability of falling between these boundaries is \(1-\alpha\), with \(\alpha/2\) in each tail. This follows directly from the definition of a quantile: \(F_Y(q_p)=p\) when the distribution is continuous and strictly increasing at that quantile.

P(q_{\alpha/2}\le Y\le q_{1-\alpha/2}) =F_Y(q_{1-\alpha/2})-F_Y(q_{\alpha/2}) =(1-\alpha/2)-\alpha/2 =1-\alpha

This calculation also explains why the result requires care for discrete or tied outcomes. With the generalized quantile definition, \(F_Y(q_p)\) can be greater than \(p\). The interval’s actual probability can then exceed its nominal coverage, and the two tail probabilities need not equal \(\alpha/2\) exactly. Report the quantile convention and treat the stated coverage as nominal unless the distribution supports exact equality.

Worked Example: A Central Interval for AUC

Hypothetical scenario. A simulation propagates uncertain inputs through a PBPK model and produces \(M=2{,}000\) AUC values for one regimen. AUC is measured in mg·h/L. Suppose the empirical 5th and 95th percentiles are 11.8 and 24.6 mg·h/L, respectively. Construct and interpret a nominal 90% equal-tail interval.

For 90% nominal coverage, \(\alpha=1-0.90=0.10\), so each tail receives \(\alpha/2=0.05\). The interval is

[q_{0.05},q_{0.95}] =[11.8,\;24.6]\ \text{mg·h/L}

Under the simulated output distribution, this central interval runs from the 5th percentile to the 95th percentile. For a continuous target distribution, it contains 90% of outcomes, with 5% below and 5% above. The empirical quantiles are estimates from a finite simulation, so they are not exact population quantiles; their Monte Carlo precision should be checked, especially when the tails matter to a decision.

One-sided limits for directional questions

A one-sided interval puts the entire excluded probability in one tail. An upper \(100(1-\alpha)\%\) limit uses the \(1-\alpha\) quantile: the interval extends from negative infinity to that upper boundary. A lower \(100(1-\alpha)\%\) limit uses the \(\alpha\) quantile: the interval begins at that lower boundary and extends to positive infinity. These forms express which side of the output distribution is being bounded.

I_{\mathrm{upper}}=(-\infty,q_{1-\alpha}], \qquad I_{\mathrm{lower}}=[q_{\alpha},\infty)

For a continuous distribution, \(P(Y\le q_{1-\alpha})=1-\alpha\), and \(P(Y\ge q_\alpha)=1-\alpha\). If the output is known to have support on nonnegative values, it can be convenient to report the upper interval as \([0,q_{1-\alpha}]\); this uses the known support and has the same coverage when \(P(Y<0)=0\). The quantile boundary, not the choice to write a support-based endpoint, determines the tail allocation.

Match the tail to the question before calculating. “What upper value covers 95% of outcomes?” calls for \(q_{0.95}\). “What lower value is exceeded by 90% of outcomes?” calls for \(q_{0.10}\). The latter is a lower 90% limit because \(1-\alpha=0.90\) implies \(\alpha=0.10\). A common source of error is to use the same percentile for both directions: upper and lower limits at the same coverage generally use different quantiles.

Worked Example: An Upper Limit for Cmax

Hypothetical scenario. For a fixed dosing regimen, \(M=4{,}000\) simulated Cmax values are recorded in mg/L. Their estimated 95th percentile is 172 mg/L. A reviewer asks for a one-sided limit intended to place 95% of the simulated Cmax distribution at or below the limit. The relevant tail fraction is \(\alpha=1-0.95=0.05\), so the upper boundary is \(q_{1-\alpha}=q_{0.95}\).

q_{1-\alpha}=q_{1-0.05}=q_{0.95} =172\ \text{mg/L}

The upper 95% prediction limit is 172 mg/L. For a continuous target distribution, 95% of outcomes lie at or below this value and 5% lie above it. If a specified exposure ceiling is 180 mg/L, the estimated upper limit is below that ceiling: \(172\ \text{mg/L}<180\ \text{mg/L}\). This supports the statement that the simulated 95th-percentile boundary is below the ceiling, but it does not establish that no simulated or possible outcome exceeds 180 mg/L.

Worked Example: A Lower Limit for an Efficacy-Related Exposure Question

Hypothetical scenario. A team examines simulated AUC values in mg·h/L and asks for a lower boundary that 90% of outcomes meet or exceed. The estimated 10th percentile is 8.6 mg·h/L. Since the desired lower-tail exclusion is 10%, \(\alpha=0.10\), and the lower 90% limit is \(q_\alpha=q_{0.10}\).

I_{\mathrm{lower}}=[q_{0.10},\infty) =[8.6\ \text{mg·h/L},\infty)

For a continuous distribution, 90% of outcomes are at or above 8.6 mg·h/L. If the exposure threshold of interest is 8.0 mg·h/L, the boundary is greater than the threshold: \(8.6\ \text{mg·h/L}>8.0\ \text{mg·h/L}\). In the continuous case, this indicates that at least 90% of the distribution exceeds the threshold. With ties or a point mass at the boundary, distinguish “at or above” from “strictly above” and inspect the simulated threshold probability directly.

From an exposure question to boundaries

A quantile boundary can summarize a threshold question, but it is not the only way to answer one. If the question is specifically “What is the probability that \(Y\) exceeds \(c\)?”, the direct quantity is \(P(Y>c)=1-F_Y(c)\). In a simulation with \(M\) equally weighted outputs, estimate it by counting outputs strictly greater than \(c\) and dividing by \(M\). Comparing \(c\) with a quantile is often useful for a target probability, while counting exceedances gives the estimated probability at a particular threshold.

For example, if a question asks for a boundary such that no more than 5% of simulated Cmax values exceed it, use the 95th percentile as the upper limit. If instead the question fixes a ceiling \(c\) and asks how often it is exceeded, calculate the fraction of outputs above \(c\). These questions are related but not identical: a quantile answers a probability-targeted boundary question; an exceedance fraction answers a threshold-targeted probability question.

Practical rule: First name the output, scenario, and direction of the question. Then choose the quantile: central \(1-\alpha\) coverage uses \(q_{\alpha/2}\) and \(q_{1-\alpha/2}\); an upper \(1-\alpha\) limit uses \(q_{1-\alpha}\); a lower \(1-\alpha\) limit uses \(q_\alpha\).

Assumptions, diagnostics, and alternatives

An interval is meaningful only relative to the distribution it summarizes. The simulated draws must represent the intended uncertain inputs and their dependence, and each PBPK run must correspond to the stated scenario and produce a valid output. The model’s structural and physiological assumptions remain part of the interpretation. A quantile interval does not establish that these assumptions are true or that the input distributions represent every relevant source of uncertainty.

Simulation design also matters. Monte Carlo draws may be independent, while Latin hypercube or other stratified designs deliberately organize the inputs. Either may be used to estimate output quantiles, but the interpretation and diagnostics must respect the design. Check that the requested tail quantiles are stable as the simulation is enlarged or repeated, and examine numerical failures and extreme outputs rather than automatically treating them as valid tail behavior. The earlier material on quantile Monte Carlo error and convergence is especially relevant for tail boundaries.

Too few effective tail observations. A 99th percentile estimated from a small run depends heavily on very few outputs. Increase simulation effort and assess quantile stability before reporting a precise-looking limit.
Wrong tail for the question. Using \(q_{0.05}\) as an upper 95% limit reverses the intended direction. Write the desired probability statement first, then identify the percentile.
Ambiguous ties. Rounded or discrete outputs may place substantial probability at a boundary. Report the quantile convention and use a direct exceedance count when strict threshold crossing matters.
Overinterpreting coverage. A 95% interval does not mean every output lies inside it, nor does it guarantee the result for a different regimen or model setup. It summarizes the specified predictive distribution.

Equal-tail intervals are easy to calculate and communicate, but they need not be the narrowest interval with the stated coverage. A skewed distribution can yield unequal distances from its median to the two endpoints; equal tail probabilities do not imply equal interval widths. A one-sided limit is preferable when the question genuinely concerns one direction, but it omits information about the other tail. If the full shape or the probability at a particular threshold is important, report the distribution or the corresponding exceedance estimate alongside the interval rather than relying on a boundary alone.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A continuous PBPK output has \(q_{0.025}=3.1\) and \(q_{0.975}=9.7\) mg/L. What is the nominal coverage and tail allocation of the equal-tail interval?
Which quantile gives a one-sided upper 90% prediction limit? Which gives a one-sided lower 90% limit?
A simulated Cmax threshold is \(c=200\) mg/L. What calculation estimates the probability of exceeding this fixed threshold?
Why can an empirical 99th percentile be unstable even when the simulation’s mean appears stable?

Solutions. (1) The interval is \([3.1,9.7]\) mg/L, with nominal coverage \(1-0.05=0.95\), or 95%. Each tail has probability 0.025, or 2.5%, for a continuous distribution. (2) For an upper 90% limit, \(\alpha=0.10\), so use \(q_{1-\alpha}=q_{0.90}\). For a lower 90% limit, use \(q_\alpha=q_{0.10}\). (3) Count simulated values strictly greater than 200 mg/L and divide by the total number of equally weighted outputs; this estimates \(P(Y>200\ \text{mg/L})\). (4) The 99th percentile is determined by the extreme upper tail, where relatively few draws provide information. A mean can stabilize while that tail quantile remains sensitive to individual outputs or additional simulation draws.