Selecting Doses From Sparse Dose-Ranging Data

Why Sparse Dose Levels Need Careful Interpretation

A Phase 2 study may include many participants at each dose but only a few distinct dose levels. Those participants can estimate responses at the doses studied with useful precision; they do not, by themselves, reveal how response behaves at every untested dose. Dose selection therefore requires two separate judgments: what the data say at tested doses, and how much additional structure a model must assume to speak about doses between or beyond them.

The key distinction is between interpolation—estimating a response at a dose inside the observed range—and extrapolation—predicting a response outside that range. Interpolation is often more defensible because nearby tested doses constrain the estimate, but it is not assumption-free. Extrapolation relies more heavily on the chosen dose-response shape and can be especially fragile when the study has not tested doses near the proposed dose.

The comparison must use a consistent response definition: the same endpoint, analysis time, population, and treatment-effect scale. The dose scale also matters. Interpolation on the dose scale assumes local linearity per dose unit; interpolation on the log-dose scale assumes local linearity per log-dose unit. The scale should be selected for pharmacological and clinical reasons, not chosen after seeing which gives the preferred recommendation.

Local Interpolation and Its Assumptions

Suppose the estimated response changes at adjacent tested doses \(D_1\) and \(D_2\) are \(Y_1\) and \(Y_2\), with \(D_1<D_2\). For a dose \(D\) between them, linear interpolation assigns weights to the two neighboring estimates according to the relative distance from each dose.

This formula follows from assuming a straight line between the two observed dose-response points. At \(D=D_1\), \(w=0\) and the estimate is \(Y_1\); at \(D=D_2\), \(w=1\) and it is \(Y_2\). For doses strictly between them, the estimate is a weighted average. That property is one reason interpolation is more constrained than extrapolation: the estimate cannot exceed both endpoint responses or fall below both when the weights are between zero and one.

The calculation is defensible only to the extent that a straight line is an adequate local approximation and the two estimates are comparable. It does not establish that the true response is linear between the doses. A sharp bend, delayed response, or change in variability between dose levels could make the estimate misleading. Sparse dose levels make such features difficult to detect, even when model diagnostics at the observed levels look acceptable.

Why a Model Can Change the Answer

Linear interpolation is not the only way to use sparse dose-ranging data. A pharmacologically motivated model can represent curvature and can make predictions at untested doses. But if only a few dose levels are available, several plausible models may describe those observed responses similarly while predicting different responses between them. Agreement at the tested doses is not proof that the models agree where no data were collected.

For illustration, the Emax model for a dose-related response above baseline is \(E(D)=E_{\max}D/(ED_{50}+D)\), where \(E_{\max}\) is the limiting response increment and \(ED_{50}\) is the dose associated with half that increment. The model assumes a particular saturating shape. Using it to interpolate may be reasonable when its form is pharmacologically and empirically defensible; it does not make sparse data more informative than they are.

The example is deliberately simple: the curve’s parameters are specified to illustrate the consequence of shape, not claimed to be uniquely estimated or validated by the two observations. In an actual analysis, model adequacy, uncertainty in parameters, the response scale, and the full evidence matrix matter. Earlier material on model choice, Emax interpretation, target-dose uncertainty, posterior target probabilities, model averaging, and model misspecification provides tools for those evaluations. Here the practical lesson is narrower: a model-based dose recommendation remains conditional on its shape assumptions, even when the proposed dose is within the observed range.

Uncertainty in an Interpolated Response

An interpolated point estimate should not be presented as if it were known precisely. If \(Y_1\) and \(Y_2\) are independent estimates with standard errors \(SE_1\) and \(SE_2\), and the interpolation weights are treated as fixed, then the variance of their weighted estimate is the sum of the weighted variances. Independence is important: if the dose contrasts share a placebo group or otherwise have correlated errors, their covariance must also be included.

This formula follows because the variance of a sum of independent random variables is the sum of their variances, and multiplying an estimate by a constant multiplies its variance by the square of that constant. It quantifies sampling uncertainty conditional on the interpolation rule; it does not account for uncertainty that the linear segment is the right shape.

Diagnostics, Alternatives, and Limits

Sparse dose selection is most credible when the recommendation is anchored by observed data and when reasonable analytical choices lead to similar conclusions. Before making a recommendation, check whether the proposed dose is inside the tested range, whether a response target is bracketed by observed responses, and whether the interpolation scale and local shape are scientifically plausible. Examine uncertainty at the proposed dose and sensitivity to reasonable alternative models. If the dose lies beyond the range, explicitly label the result as extrapolation and show how much it depends on model assumptions.

Warning signs include a target response that is not bracketed by the observed responses, a recommendation that shifts substantially under plausible dose-response models, an estimate driven mainly by one sparse dose level, or a very wide interval around the predicted response. Such findings do not automatically prohibit a dose recommendation. They indicate that the evidence supports a less precise claim, or that additional dose-ranging evidence may be needed before making a high-confidence choice.

These approaches are not interchangeable guarantees of truth. A tested dose is directly represented in the study, but its estimated response still has uncertainty. Interpolation uses neighboring data but imposes a local shape. A parametric model can borrow structure across dose levels but can mislead if its form is inappropriate. The choice should be tied to the clinical objective and the evidence, with the strength of the recommendation matched to the strength of the support.