Why Uncertainty Belongs in a Label Claim
A model-derived estimate can be useful for prescribing information only if its precision is communicated honestly. A predicted exposure ratio of 1.30, for example, may be a central estimate, but it does not tell a reader how uncertain the population estimate is, how much exposure may vary among patients, or how results change under alternative assumptions. Those are distinct questions and call for distinct summaries.
The aim is not to place every model output into label text. It is to choose wording that preserves the intended claim while making its evidential limits clear. A confidence interval can describe uncertainty in an estimated population quantity; a prediction interval can describe the range of outcomes expected for a new patient under a specified model; and scenario results can show sensitivity to chosen assumptions. None of these should be presented as if it answered the other two questions.
Match the Interval to the Claim
First identify the quantity the proposed wording is meant to describe. It might be a population geometric mean exposure ratio, the probability of a clinical outcome in a defined population, or an individual patient's future exposure. The interval must correspond to that quantity. A narrow interval for a population mean does not imply that individual patients will have outcomes close to that mean.
For a model estimate \( \hat{\theta} \) with standard error \( SE \), an approximate two-sided 95% confidence interval is often calculated as \( \hat{\theta} \pm 1.96SE \) when the estimator's sampling distribution is approximately normal and the standard error is appropriate. For a positive ratio, a log-scale interval is often more suitable: calculate the interval for the log ratio, then exponentiate both endpoints. This respects the positive scale and produces an interval that is generally asymmetric around the ratio estimate.
For a new individual outcome, a prediction interval must also account for the outcome's variability around the population prediction. In a simple normal model, if that variability is independent of estimation error, the predictive variance is approximately the sum of the variance of the estimated mean and the residual variance. For nonlinear population models, intervals are commonly obtained by propagating parameter uncertainty and simulating new outcomes under the specified model. The method and assumptions matter: a confidence interval for the mean is not a substitute for this predictive calculation.
Worked Example: Uncertainty in a Population Exposure Ratio
Suppose a hypothetical pharmacokinetic model estimates the geometric mean AUC ratio for a defined organ-function group relative to a reference group as 1.30. The standard error of the estimated log ratio is 0.12. Assume the log-ratio estimator is approximately normal, the standard error reflects the relevant estimation uncertainty, and the model and comparison are specified as intended.
Calculate the interval. The estimated log ratio is \( \log(1.30)=0.2624 \). The approximate 95% interval on the log scale is:
Exponentiating the endpoints gives \( (\exp(0.0272),\exp(0.4976))=(1.028,1.645) \), or about 1.03 to 1.64. The model-based point estimate is a 1.30 geometric mean ratio; the interval describes uncertainty about the population ratio under the stated estimation assumptions. It does not mean there is a 95% probability that the fixed ratio lies within these particular endpoints.
Possible communication. “The estimated geometric mean AUC ratio was 1.30; the approximate 95% confidence interval was 1.03 to 1.64.” Whether that full sentence belongs in a label depends on the purpose and available space, but the interval should not be silently replaced by a more precise-sounding point claim.
Prediction Intervals Describe New Outcomes
A prediction interval answers a different question: how widely might a future individual outcome vary under the model? It includes uncertainty about the population prediction and the expected spread of individual outcomes around it. It is conditional on the model, covariates, regimen, and population definition used for the prediction. It does not automatically cover every source of uncertainty, such as a misspecified model or a population outside the intended setting.
In the next calculation, suppose the log AUC ratio for an individual has a population mean estimated as \( \log(1.30)=0.2624 \). Assume the estimated mean has standard error 0.12, the individual residual standard deviation on the log scale is 0.25, and residual variability is independent of estimation error. These assumptions permit a simple approximate predictive calculation. In a full model, parameter covariance and the model's predictive distribution may require simulation rather than this shortcut.
Worked Example: A Predictive Range for an Individual
Using the assumptions above, the approximate predictive standard deviation on the log scale is:
An approximate 95% predictive interval for the log individual ratio is:
Exponentiating gives approximately \( (\exp(-0.2811),\exp(0.8059))=(0.755,\ 2.239) \). Under this illustrative model, a new individual ratio may vary much more widely than the estimated population ratio's confidence interval. The range is a prediction interval under the stated assumptions, not a 95% confidence interval for the population geometric mean and not a guarantee about every patient.
Possible communication. If individual variability is material to the prescribing decision, the label claim should not suggest that all patients will experience the population-average change. A predictive range may help reviewers assess that distinction, even when the label ultimately uses a concise population estimate.
Scenario Results Are Not Probability Intervals
Some uncertainties are represented by alternative plausible assumptions rather than by sampling variability or individual residual variation. For example, model predictions may be calculated under several assumptions about an unmeasured covariate or an uncertain mechanistic contribution. Reporting the minimum and maximum across those cases can be useful, but the endpoints do not have a 95% coverage interpretation unless a probability model and interval procedure justify one.
Scenario results should name what changes and what stays fixed. A range without that context can be mistaken for a confidence interval, a prediction interval, or an observed range. It can also imply that the selected scenarios exhaust all plausible possibilities when they do not. A clear statement identifies the scenarios as conditional predictions and avoids describing them as “the 95% range” unless that is what was actually computed.
Worked Example: Wording Across Dose Scenarios
A hypothetical model predicts a geometric mean AUC ratio of 1.15 under a lower-dose scenario, 1.35 under the base scenario, and 1.60 under an alternative scenario. The scenarios differ in an assumed covariate value; no probabilities are assigned to the alternatives. Assume each prediction uses the same population and outcome definition.
The numerical span is \(1.60-1.15=0.45\), and the ratio of the highest to lowest prediction is \(1.60/1.15=1.391\), or about 1.39. Those calculations describe the spread among the three selected scenario predictions. They do not establish how likely any scenario is or the chance that a future patient's ratio will fall between 1.15 and 1.60.
Possible communication. “Predicted geometric mean AUC ratios were 1.15, 1.35, and 1.60 under the lower-dose, base, and alternative covariate scenarios, respectively.” This wording identifies the conditional nature of the results. Calling 1.15 to 1.60 a “95% interval” would be unsupported because no probability distribution or 95% interval procedure was used.
From Statistical Summary to Label Language
The interval alone does not determine the label wording. Consider what quantity is relevant to the prescribing decision, which population and regimen the claim concerns, and whether uncertainty could change the practical interpretation. State a central estimate when useful, but avoid extra decimal places that suggest greater accuracy than the model supports. Use units and identify ratios, percentages, time points, or population groups explicitly.
A confidence interval describes uncertainty in an estimated population quantity under the estimation procedure. A prediction interval includes individual outcome variability as well. Scenario comparisons describe conditional changes under specified alternatives. The distinctions can be summarized as follows:
| Summary | Question answered | Appropriate qualification |
|---|---|---|
| Confidence interval | How uncertain is the estimated population quantity? | Identify the estimate, confidence level, and quantity. |
| Prediction interval | What outcomes may occur for a new individual under the model? | Identify the target population, outcome, and predictive assumptions. |
| Scenario range | How do predictions differ across specified assumptions? | Name the scenarios; do not imply probability coverage without justification. |
These summaries are not interchangeable with other useful displays. A point estimate alone is concise but hides uncertainty. A standard error quantifies estimation precision on a particular scale but may be difficult to interpret directly in label language. A range across scenarios can make sensitivity transparent but does not represent sampling or patient-level variability. The appropriate choice depends on the claim and decision, not on which display looks most definitive.
Assumptions, Diagnostics, and Limits
Before presenting an interval, check that its construction matches the model output and target quantity. For a normal approximation, assess whether the estimator's sampling distribution is plausibly close to normal and whether the standard error is reliable. For a log-scale interval, confirm that the target quantity is positive and that the transformation and back-transformation are correctly applied. For a prediction interval, include the relevant outcome variability rather than only uncertainty in the estimated mean. For scenario results, describe the assumptions defining each scenario.
Common warning signs include an interval that is symmetric on a ratio scale despite being calculated from a log-scale model, a prediction interval that is no wider than the confidence interval despite meaningful individual variability, or a scenario range presented with a confidence level that was never calculated. Strong skew, boundary estimates, sparse data, or uncertainty in model parameters may also make a simple normal approximation unreliable. Simulation or other appropriate interval procedures may be needed; even then, intervals are conditional on the model and the quantities included in the calculation.
A model-based interval does not automatically include structural uncertainty, uncertainty about whether the model applies in a new setting, or every data limitation. If those concerns are not quantified, wording should not imply that the interval covers them. This tutorial's intervals communicate uncertainty for a specified claim; they do not by themselves establish that the claim is suitable for a particular population or that the model's assumptions are adequate.
A Practical Communication Sequence
Specify the population quantity or individual outcome the claim is intended to describe, including units, comparator, and relevant regimen or time point.
Use a confidence interval for uncertainty in an estimated population quantity, a prediction interval for a new outcome, or explicitly identified scenarios for conditional alternatives.
Verify the scale, standard error or predictive variability, assumptions, and back-transformation. Do not attach a confidence level to a range that has no coverage procedure.
Use appropriately rounded values and wording that preserves the distinction between an estimated population effect, patient-to-patient variability, and scenario sensitivity.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A model estimates a population exposure ratio of 1.20 with a 95% confidence interval from 1.05 to 1.38. What does the interval describe, and what does it not establish about individual patients?
- A predictive calculation has estimation variance 0.04 and individual-level variance 0.09 on the same scale, with independence assumed. What is the predictive variance?
- Three predictions under selected assumptions span 0.90 to 1.40, with no probabilities assigned to the assumptions. Can this be called a 95% interval?
- Why can an interval calculated on the log scale be asymmetric after it is expressed as a ratio?
Solutions. (1) It describes uncertainty in the estimated population exposure ratio under the interval procedure and its assumptions. It does not state that 95% of individuals have ratios from 1.05 to 1.38; that requires an individual prediction interval. (2) The predictive variance is \(0.04+0.09=0.13\), so the predictive standard deviation is \(\sqrt{0.13}=0.3606\) on that scale. (3) No. It is a scenario range; without a probability model and interval procedure, it has no 95% coverage interpretation. (4) Exponentiation is nonlinear: equal distances around the log-scale estimate become multiplicative changes on the ratio scale, so the endpoints are not generally equally distant from the ratio estimate.