Locating Minimum Effective and Plateau Doses

Why Locate These Doses?

A fitted dose-response curve can answer two useful but different questions. First, what is the lowest dose predicted to meet a clinically meaningful efficacy criterion? Second, at what dose does increasing the dose provide so little additional benefit that the response can reasonably be called near its plateau? Neither question has a universal numerical answer: the first depends on the response threshold, and the second depends on how much additional benefit is considered negligible and over what dose range.

These are model-based dose locations, not automatically recommended doses. Their interpretation depends on the endpoint, dose range, fitted model, and uncertainty. In particular, a dose at which the estimated mean response crosses a threshold is not necessarily a dose for which the evidence establishes that the threshold is met.

For a monotone increasing benefit curve, the remaining gain from dose \(d\) to the top of the range is \(B(D_{\max})-B(d)\). The plateau criterion can therefore be written \(B(D_{\max})-B(d)\leq\epsilon\). This definition describes a practical plateau over the specified range; it does not require the curve to reach its mathematical asymptote. If benefit is not monotone, the greatest additional benefit over higher doses must be examined rather than assuming the endpoint value represents the maximum.

Model, Assumptions, and Thresholds

A common model for an increasing, saturating treatment effect is the Emax model:

Here \(E_{\max}>0\) is the asymptotic benefit, and \(ED_{50}>0\) is the dose producing half of that benefit. The curve is increasing and approaches \(E_{\max}\), since its derivative is \(E_{\max}ED_{50}/(ED_{50}+d)^2>0\). The model is a useful illustration, not a default truth: it assumes a saturating monotone response of this form over the dose range of interest.

Dose locations are meaningful only if the response scale and direction are clear. If lower values on the measured endpoint represent improvement, define \(B(d)\) so that positive values still mean benefit, for example by taking the decrease from the reference response. Use the clinically meaningful threshold \(\Delta\) on that same benefit scale. The threshold should come from clinical interpretation, not from whichever value makes a fitted curve look persuasive.

The analysis also assumes that the fitted curve adequately represents the relevant population response and that the doses considered are within a range supported by the data. Model uncertainty, sparse observations near a crossing, or misspecification can make the estimated dose location unstable. These assumptions should be checked before interpreting either dose as a useful decision point.

Locating the Minimum Effective Dose

For a monotone increasing curve, the plug-in minimum effective dose solves \(\hat B(d)=\Delta\), where \(\hat B\) is the fitted response curve. In the Emax model, when \(0<\Delta<E_{\max}\), rearranging the threshold inequality gives:

If \(\Delta\geq E_{\max}\), the fitted curve does not meet the threshold at any finite dose when \(\Delta=E_{\max}\), or at any dose when \(\Delta>E_{\max}\). A computed root outside the studied or permitted dose range is not a supported selection: it indicates that the model does not establish the criterion within that range.

Uncertainty changes the question. A point estimate crossing \(\Delta\) describes the fitted curve, but does not show how precisely the crossing is known. One approach is to form a confidence interval for the dose crossing by repeatedly refitting the model to bootstrap samples and recalculating the crossing. Another is to use a lower confidence band for the response curve: the first dose at which that band reaches \(\Delta\) is a conservative, uncertainty-qualified crossing under the chosen band procedure. If doses across a continuum are scanned, a simultaneous band is preferable to separate pointwise intervals when the claim concerns the whole curve; pointwise intervals do not generally control uncertainty across all doses searched.

Locating a Plateau Dose

A plateau criterion must specify both the dose range and what counts as negligible extra benefit. For a monotone curve over \(0\) to \(D_{\max}\), define the remaining gain at dose \(d\) as \(B(D_{\max})-B(d)\). The plateau dose is the smallest \(d\) for which that remaining gain is at most \(\epsilon\), where \(\epsilon\) is expressed in response units. This operational definition avoids calling a curve “flat” without saying how flat is sufficient.

For the Emax model, the criterion is:

This is a practical, range-specific definition. A different \(D_{\max}\) can produce a different plateau dose, even with the same fitted curve. A slope-based criterion is another possible approach, but it needs a tolerance in benefit per dose unit and can behave differently from a criterion based on total remaining benefit.

Uncertainty, Diagnostics, and Alternatives

The plateau location is estimated from the fitted curve just as the minimum effective dose is. For a bootstrap assessment, each resample can produce a new fitted curve and a new plateau root; the resulting distribution describes uncertainty in that dose-location estimate. A broad interval signals that the data do not pinpoint a narrow plateau dose. It may be especially broad when few observations approach saturation, because \(E_{\max}\) and \(ED_{50}\) can be difficult to distinguish from a curve that is still rising.

Useful diagnostics include plotting observed responses and the fitted curve together, checking residual patterns, examining uncertainty near the proposed crossing, and confirming that the crossing lies within the studied dose range. Look for systematic residual trends, influential observations, implausible extrapolation, or an apparent response pattern that is nonmonotone despite a monotone model. If the curve never reaches the efficacy threshold within the allowed range, report that result rather than extrapolating a dose as though it were supported. If the curve is still rising substantially at the highest observed dose, a plateau estimate may be driven mainly by model assumptions.

The main alternatives answer different questions. A statistical significance test against no effect asks whether the data are inconsistent with zero effect under its assumptions; it does not establish that the effect meets the clinical threshold. A fixed fraction of the fitted maximum, such as the dose reaching 90% of \(E_{\max}\), provides a model-relative location, but the chosen fraction is not necessarily a clinically negligible-gain criterion. A slope threshold instead asks whether the instantaneous rate of response change is small, and its tolerance has units of response per dose. These approaches can yield different dose locations, so state the criterion rather than using “effective” or “plateau” without qualification.

A population mean curve also does not describe the proportion of individuals who meet a response target. Likewise, locating a dose from a fitted curve does not by itself account for safety, preferences, or other decision objectives. Those limitations matter when translating a descriptive dose location into a dosing decision.

Common Pitfalls

• Calling any statistically significant dose effective. Statistical evidence against zero benefit does not show that the clinically meaningful threshold is met. Compare the fitted benefit and its uncertainty with \(\Delta\).
• Treating the fitted crossing as certain. The crossing is estimated. Report uncertainty in the response curve or dose root, and distinguish a point-estimate crossing from an uncertainty-qualified crossing.
• Calling the maximum tested dose the plateau. Being the largest dose studied does not show that further benefit is negligible. Apply a stated tolerance over a stated range and assess whether the data support the curve near its upper end.
• Reporting a plateau without its definition. State the dose range and the response-scale tolerance \(\epsilon\); otherwise the reported location cannot be interpreted or reproduced.