Three Different Questions a Model Can Answer
A model can summarize what was observed, predict an outcome for a new case under conditions represented by the data, or estimate an outcome beyond those conditions. These uses may involve the same model equation, but they answer different questions and rely on different assumptions. A close match between model and observed data is evidence about description; by itself, it does not establish that predictions for new patients will be accurate or that extrapolations to a different population will be credible.
In model-informed drug development (MIDD), the distinction matters because a model output may inform a decision about a regimen, population, or development step. The intended use and decision rule establish what output matters; the next question is whether the evidence supports that output for the conditions in which it will be used.
These are uses, not mutually exclusive properties of a model. One model may describe existing data and also be used to predict outcomes for new patients. Whether those predictions are justified depends on the target conditions and assumptions. Extrapolation can also be a legitimate modeling objective, but it requires a defensible account of why the relevant relationships should continue in the new conditions.
Description: What the Observations Show
A descriptive result is tied to a defined set of observations. For example, the mean exposure among patients in a hypothetical study describes that study’s observed sample. A model can also describe an observed pattern, such as how exposure varied with dose in that study. The result does not automatically apply to all patients with the disease, to patients treated with another regimen, or to outcomes that were not measured.
Description requires clarity about what was measured, which records were included, and the conditions under which they were collected. It also requires care with dependence and sampling: repeated measurements from one patient are not automatically independent observations from different patients. A summary that ignores such structure may still calculate correctly while giving a misleading picture of the evidence.
Worked Example: Describing Observed Exposure
In a hypothetical study, five patients receiving the same regimen have observed steady-state exposure values of 7, 8, 9, 10, and 11 mg·h/L. Calculate the sample mean and describe what it says.
Mean. For observations \(x_1,\ldots,x_n\), the sample mean is \(\bar{x}=\sum_{i=1}^{n}x_i/n\). Substituting the five observed values gives
The mean exposure in these five hypothetical patients is 9 mg·h/L. This is a description of the observed sample under the stated regimen. It is not, without further assumptions and evidence, a prediction for a different dose or a claim about every patient in the broader treatment population.
A descriptive model may smooth noisy observations or express a pattern compactly. For example, a curve fitted to observed dose and exposure values can describe their relationship in the analyzed data. Fit alone, however, does not tell us whether that curve will predict accurately for another patient. A model can track the observed points yet be too sensitive to noise, omit a relevant feature, or represent a relationship that changes outside the observed conditions.
Prediction: A New Observation Under Represented Conditions
A prediction has a specified target: an outcome for a new observation under conditions similar in relevant respects to those represented by the data. The target might be the expected outcome for patients receiving a particular regimen, or the outcome for one new patient. These are different targets. An expected outcome describes an average conditional on the stated conditions; a prediction for an individual also has to account for variation among individuals and measurement or residual variation.
A simple model makes the distinction explicit. Let \(Y\) be an outcome and \(X\) the conditions or predictors. A model can be written as \(Y=m(X)+\varepsilon\), where \(m(X)\) is the model’s expected outcome at \(X\), and \(\varepsilon\) represents variation not explained by that expected value. Predicting a new outcome requires both a suitable expected-value relationship and assumptions about the variation around it.
Worked Example: Predicting Within the Represented Dose Range
Suppose a hypothetical study represents patients receiving doses from 50 to 150 mg. A model gives an expected exposure of 8 mg·h/L at 100 mg, with residual standard deviation 2 mg·h/L for individual patients under comparable conditions. Assume, for illustration, normally distributed residual variation with mean zero and that the model and variability apply at 100 mg. Find an approximate interval containing 95% of individual outcomes, ignoring uncertainty in estimated model parameters.
Under these assumptions, an individual outcome at 100 mg is modeled as \(Y=8+\varepsilon\), where \(\varepsilon\sim N(0,2^2)\). About 95% of a normal distribution lies within 1.96 standard deviations of its mean, so the interval is
Thus, under the illustrative model and assumptions, approximately 95% of individual exposures at 100 mg are expected to lie between 4.08 and 11.92 mg·h/L. This is not a 95% interval for the model’s expected exposure: that would address uncertainty in the estimated mean, which this calculation explicitly ignores. Nor does the calculation establish that the model’s assumptions hold in practice. The dose is within the represented range, but the prediction still depends on comparability of patients and adequacy of the model.
The prediction target should be stated precisely. “Predict exposure at 100 mg” may mean the expected exposure for a comparable patient group or the exposure for one new patient. It may also leave unspecified whether the outcome is a single measurement, an average over time, or a summary such as steady-state exposure. Ambiguity about the target makes it difficult to judge whether the model’s uncertainty and assumptions are appropriate.
Extrapolation: Extending Beyond Represented Conditions
Extrapolation occurs when the target conditions extend beyond the evidence represented in the data. This may be a numerical extension, such as predicting at a dose above the studied range, or a change in context, such as applying an adult relationship to children. A target can also be outside the evidence even when each variable separately looks familiar—for example, if the new population has a combination of covariates not represented in the data.
The central issue is that observed data cannot directly show whether the modeled relationship continues in the new conditions. Extrapolation therefore requires additional assumptions about what remains stable and what may change. Those assumptions should be made explicit, and the result should be communicated as conditional on them rather than as a direct description of observed outcomes.
Worked Example: Extending an Age Relationship Beyond the Data
A hypothetical adult dataset includes ages from 18 to 60 years. A fitted relationship for a clearance summary is \(CL=5+0.08(\text{age}-18)\), with clearance measured in L/h and age in years. Use the equation to calculate its prediction at age 12, then identify why interpreting it requires extrapolation assumptions.
Substitution gives
The equation returns 4.52 L/h at age 12. But the observed ages began at 18 years, so this is not a prediction within the represented age range. It assumes that the adult linear relationship continues below age 18 and that no relevant age-related change alters the relationship. The arithmetic is valid; the biological and statistical warrant is not supplied by the equation itself. The value should therefore be reported as an extrapolation conditional on those assumptions, not as an observed or directly supported estimate for 12-year-olds.
Extrapolation is not inherently invalid. Development decisions sometimes require predictions before direct data are available in the target setting. The important distinction is between an output calculated by extending a model and evidence that supports the extension. Mechanistic knowledge, relevant external evidence, or a justified account of how the system behaves may support an extrapolation. If the new setting changes important processes, however, a relationship that was adequate in the original setting may no longer be adequate.
A Short Justification: Why Predictions Include More Variation Than Means
The difference between predicting an average and predicting one new observation follows from the model \(Y=m(X)+\varepsilon\). Conditional on \(X\), suppose the residual has mean zero and variance \(\sigma^2\), and suppose the estimated mean \(m(X)\) has variance \(V_m\) due to uncertainty in its estimation. For a new observation, the uncertainty comes from both the estimated mean and the new residual. If those components are independent, then
The equality uses independence, so the covariance term is zero. The \(\sigma^2\) term is present because a new individual can differ from the conditional mean even if that mean were known exactly. By contrast, uncertainty about the mean concerns estimation of \(m(X)\), not the extra residual for a particular new individual. This is why an interval for individual outcomes is generally wider than an interval for the expected outcome. If parameter uncertainty, model uncertainty, or other sources of variability are omitted, the resulting prediction may be too narrow.
How to Tell When the Use Is Going Wrong
A practical check is to state the target observation and its conditions, then compare those conditions with the evidence supporting the model. Warning signs include:
- A descriptive statistic is presented as though it applied to a wider population, without addressing how the observed cases relate to that population.
- A prediction is called “for a new patient,” but the relevant patient characteristics or treatment conditions differ from those represented in the data.
- An extrapolated value is reported without naming the relationship assumed to continue beyond the observed conditions.
- An interval for an expected outcome is interpreted as covering most individual outcomes, or individual variability is omitted from a patient-level prediction.
- The model’s output changes sharply under small, plausible changes to an extrapolating assumption, but the reported conclusion hides that dependence.
These signs do not by themselves prove that the model is unusable. They indicate that the stated use needs qualification or additional support. Description, prediction, and extrapolation also are not a ranking from “bad” to “good”: description may be exactly the right use for summarizing a completed study, while extrapolation may be essential for a development question. The appropriate standard depends on the decision and the consequences of being wrong.
Comparing the Three Uses
| Use | Target | Key question | Main limitation |
|---|---|---|---|
| Description | Observed records and conditions | What pattern or summary is present in these observations? | Does not, by itself, establish behavior in new cases or settings |
| Prediction | A new observation under represented conditions | How well does the model represent outcomes for comparable new cases? | Depends on comparability, model adequacy, and appropriate uncertainty |
| Extrapolation | An observation under conditions beyond those represented | What must remain true for the relationship to extend to this target? | Depends on additional, often less directly testable assumptions |
A common mistake is to reason, “The model fits the observed data, so it predicts the new population.” This moves directly from description to extrapolation without establishing that the new population is represented or that the relationship transports to it. The correction is to name the target conditions, classify the use, and state what evidence and assumptions support that specific use. A good fit may be relevant evidence, but it is not a substitute for this reasoning.
Check Your Understanding
For each question, identify the relevant use and explain the reasoning. Short solutions follow the questions.
- A study reports the mean concentration among 40 patients given a specified regimen. Is this necessarily a prediction for the next patient? Why or why not?
- A model is used to predict an outcome at a dose inside the studied range, for patients similar to those studied. What type of use is this, and name two assumptions it requires.
- A clearance model fitted only to adults is used to estimate clearance in children. What type of use is this? Give one assumption that must be considered.
- Why is an interval for a new individual’s outcome generally wider than an interval for the model’s expected outcome?
- A model matches observed outcomes closely, but the proposed target population has characteristics not represented in the data. Does close fit alone justify the proposed use? Explain.
Solutions. (1) No. The reported mean describes the 40 observed patients; prediction for the next patient requires assumptions about comparability and individual variability. (2) It is within-condition prediction. It requires, among other things, that the new patients and treatment conditions are sufficiently comparable to those represented and that the model relationship and uncertainty are appropriate. (3) It is extrapolation because the target population differs from the adult data. One must consider whether the modeled clearance relationship remains appropriate in children. (4) An individual prediction includes residual variation among new individuals as well as uncertainty in the estimated expected outcome; the expected-outcome interval does not include that new-individual residual. (5) No. Close fit describes performance on the observed data, but does not establish that the relationship applies to the unrepresented population; assumptions supporting that extension are needed.