Conditional Prediction Intervals Across Patient Subgroups

Why Condition on Patient Subgroups?

A PBPK prediction for a patient with impaired renal function need not have the same distribution as a prediction for a patient with normal renal function. The subgroup may differ in physiology, covariates, or the way uncertain inputs affect the model output. A single interval for a population combining both groups can conceal these differences. Conditional prediction intervals make the target explicit: they describe predictions for patients who meet a stated subgroup definition, under a stated dose and prediction context.

The calculation uses the same basic simulation outputs as other prediction intervals, but changes which predictive distribution is summarized. The key is to condition the distribution on subgroup membership, rather than to treat all simulated patients as one undifferentiated sample. For example, the target might be the concentration at a specified time for a new patient in a defined renal-function group. The interval is then about that patient-level prediction distribution, not an interval for a group mean.

The distribution being summarized must match the prediction question. It may include patient-to-patient variability and parameter uncertainty, if both are part of the predictive target. If simulations use nested sampling, the subgroup condition applies to the patient-level covariates and outputs, while the parameter draws represent the uncertainty included in the target prediction. The model, dose, endpoint, time point, and any other fixed conditions should be held consistent across subgroup comparisons.

Constructing a Conditional Interval

For a categorical subgroup, such as a prespecified renal-function category, generate or select predictive draws whose patient covariates satisfy that category. Then calculate the desired quantiles of the corresponding outputs. For a continuous covariate, the condition must be defined precisely: for example, an interval of renal-function values, or a prediction at a specified value. These are different targets. A category interval averages over the covariate values represented within that category; a prediction at a particular value conditions on that value or uses a model that predicts at it.

In Monte Carlo work, suppose \(M_g\) predictive outputs belong to subgroup \(g\), and write them in increasing order as \(Y_{(1),g},\ldots,Y_{(M_g),g}\). One simple empirical quantile convention is the inverse empirical CDF: \(\hat q_{p\mid g}\) is the smallest observed value for which the empirical proportion at or below that value is at least \(p\). With this convention, the estimated central interval is formed from the empirical quantiles at probabilities \(\alpha/2\) and \(1-\alpha/2\). Other software may interpolate between ordered values, so the quantile convention should be reported.

Here \(\widehat F_g\) is the empirical CDF of outputs for the target subgroup. The quantiles are not calculated from the subgroup mean and standard deviation unless a particular predictive distribution is justified and those summaries correctly determine its quantiles. PBPK outputs can be skewed or otherwise non-normal; simulation quantiles avoid imposing a normal shape, though they still depend on adequate simulation and a suitable model.

Conditioning Must Preserve the Relevant Covariates

A valid conditional distribution reflects how covariates are distributed within the target subgroup and how those covariates depend on one another. For instance, if age and renal function are associated in the target population, sampling age independently of renal-function stratum can create an unrealistic subgroup. Similarly, conditioning on a renal-function category should not be replaced by setting every patient’s renal function to the category midpoint unless that is specifically the prediction target and the model assumptions justify it.

There are two common ways to obtain subgroup predictions. One is to generate patient covariates from a joint distribution and retain simulated patients whose covariates meet the subgroup definition. The other is to sample directly from the covariate distribution conditional on subgroup membership. Either can be appropriate. In both cases, preserve relevant dependencies and ensure that the retained or generated patients represent the intended target population. If a subgroup is rare, simple filtering can leave too few outputs for stable tail quantiles; direct conditional sampling may use simulation effort more efficiently.

For continuous covariates, conditioning can be more demanding than assigning a category. A prediction at exactly one continuous value may have probability zero under a continuous covariate distribution, so it is usually defined through the model evaluated at that value or through a conditional model. A prediction for a range instead averages over the covariate values within that range. The choice changes the predictive distribution and should be stated, not left implicit.

Why a Marginal Interval Is Not a Subgroup Interval

If a population contains several subgroups, its marginal predictive distribution is a mixture of the conditional distributions. If \(w_g=P(G=g)\) is the subgroup proportion in the target population, then the marginal CDF is the weighted sum of subgroup CDFs. Its quantiles generally cannot be obtained by averaging subgroup quantiles, and a marginal interval does not automatically provide the desired interval for any one subgroup.

The reason is that a quantile is defined by where a CDF reaches a probability threshold. Adding CDFs with subgroup weights changes the shape and position of that combined CDF. There is no general algebraic rule that makes its threshold equal to a weighted average of the subgroup thresholds.

Diagnostics, Comparisons, and Limitations

Conditional intervals are useful when decisions concern specific patient groups, but they can be less stable than marginal intervals because fewer simulations contribute to each subgroup’s tails. Check the number of outputs in each stratum, inspect the simulated covariate values and their dependencies, and assess whether additional draws materially change the endpoint quantiles. Earlier material on Monte Carlo error in prediction quantiles provides relevant methods for assessing endpoint stability. A large total simulation count does not guarantee precise conditional tails if only a small fraction of draws belongs to the subgroup.

Also check that subgroup membership is applied to the intended patient covariates and that model predictions are evaluated under consistent dose, time, and other conditions. Implausible covariate combinations, abrupt changes at category boundaries, or large endpoint shifts from small changes in the stratum definition can signal sparse support or model sensitivity. If a subgroup is absent or poorly represented in the input data, simulation does not create empirical support for it; predictions may depend heavily on model extrapolation.

A marginal interval is appropriate when the target is a new patient drawn from a specified overall population, including its subgroup proportions. A conditional interval is appropriate when the target is a new patient known to belong to a specified subgroup. A model-based conditional prediction evaluated at a fixed covariate value is another option when the question is about that particular profile. These approaches answer different questions and should not be presented as interchangeable.

Finally, subgroup intervals describe model-based predictive distributions, not guarantees for individual patients. Their interpretation depends on the model, input distributions, subgroup definition, and uncertainty sources included in the simulation. Wide intervals may reflect real within-group variability as well as uncertainty represented in the model; narrow intervals can be misleading if important sources of variation were omitted. Report the subgroup definition, endpoint and time, dose or scenario, interval level, quantile method, and units so the target is clear.