From a Flat-Looking Curve to a Supported Plateau
A fitted dose-response curve can look nearly horizontal at its upper end without establishing that additional dose produces little or no response. The curve may be flat because the response has reached a plateau, but it may also be flat-looking because estimates are imprecise, the tested doses are too close together, or the curve’s apparent leveling depends on model assumptions. In particular, the highest tested dose has no higher tested dose against which to assess whether the response has stopped increasing.
The practical question is therefore not simply whether the fitted curve looks flat. It is whether the response differences between a proposed plateau dose and higher observed doses are small enough, with sufficient precision, to count as negligible for the dose decision. The introductory material on interpreting flat dose-response curves motivates this distinction; here the focus is an explicit contrast-and-interval rule.
If the endpoint is naturally oriented so that smaller values are better, reverse its sign before applying this definition, or define the contrast consistently in the benefit direction. The margin \(\delta\) must be chosen before inspecting the contrasts and justified for the response scale and decision context. It is not a universal constant, and it is not estimated by looking for the margin that makes the data appear to plateau.
For a candidate dose below the maximum tested dose, comparing it with every higher tested dose is important. A small contrast between adjacent doses does not rule out a meaningful accumulated difference from the candidate dose to a much higher dose. A simultaneous set of intervals is preferable when several contrasts are used to make one plateau claim. If the candidate dose is chosen after examining many doses, that selection should also be considered when assessing uncertainty.
What the Interval Must Show
A confidence interval that includes zero does not by itself show that two doses have equivalent responses. It may include zero because the estimated contrast is small, or because its standard error is large enough to leave substantial positive and negative differences plausible. To support a plateau under the rule above, the interval must be narrow enough to exclude differences as large as the prespecified margin in either direction.
For example, if a model-based contrast estimate is \(\widehat{\Delta}\) with standard error \(SE\), a large-sample two-sided 90% interval is approximately \(\widehat{\Delta}\pm1.645SE\). When the normal approximation is justified, requiring this entire interval to lie inside \((-\delta,\delta)\) corresponds to the usual two one-sided equivalence tests at level 0.05. A different interval procedure may be needed for small samples, non-normal estimates, or strongly nonlinear model parameters; the essential requirement remains that the uncertainty interval be contained within the prespecified negligible-difference range.
The interval should reflect uncertainty in the fitted response predictions, including covariance between predictions at different doses. Treating \(\mu(d_j)\) and \(\mu(d^*)\) as independent when they are estimated from the same fitted model can give an incorrect standard error for their difference. Profile-likelihood or simulation-based intervals can be useful when a simple normal approximation is unreliable, provided their construction matches the fitted model and target contrast.
Worked Examples
Worked Example: Evidence for a Plateau at an Observed Dose
Suppose a hypothetical trial’s fitted mean benefit is reported in points, with larger values indicating greater benefit. Doses of 50, 100, 150, and 200 mg were tested. A prespecified negligible-difference margin is \(\delta=1.5\) points. For a candidate plateau dose of 100 mg, model-based fitted contrasts and standard errors are: 100 to 150 mg, 0.40 points with \(SE=0.25\); and 100 to 200 mg, 0.55 points with \(SE=0.30\). Assume the model-based normal approximation is adequate and these are the simultaneous 90% intervals for the contrasts being assessed.
For 150 mg versus 100 mg, the interval is $$ 0.40\pm1.645(0.25)=0.40\pm0.411 $$ so the interval is \((-0.011,\,0.811)\) points. It lies wholly within \((-1.5,1.5)\).
For 200 mg versus 100 mg, the interval is $$ 0.55\pm1.645(0.30)=0.55\pm0.494 $$ so the interval is \((0.056,\,1.044)\) points. This also lies wholly within \((-1.5,1.5)\). Both higher-dose contrasts meet the equivalence criterion. Under the stated model, interval assumptions, and margin, the results support 100 mg as a plateau dose across the tested range from 100 to 200 mg. They do not establish what would happen at doses above 200 mg.
Worked Example: The Maximum Tested Dose Is Not an Observed Plateau
In a second hypothetical trial, doses of 50, 100, 150, and 200 mg were tested. The fitted benefit estimates at 150 and 200 mg are close, and the fitted contrast from 150 to 200 mg is 0.50 points with \(SE=0.60\). Let the prespecified margin be \(\delta=1.0\) point. Using a 90% normal interval, $$ 0.50\pm1.645(0.60)=0.50\pm0.987, $$ which gives \((-0.487,\,1.487)\) points.
The interval crosses the upper equivalence boundary of 1.0 point. Although it includes zero and the point estimate is only 0.50 points, it does not fit wholly inside \((-1.0,1.0)\). Therefore, these data do not establish equivalence between 150 and 200 mg. The contrast is also only between the two highest tested doses: since 200 mg is the maximum tested dose, there is no observed higher-dose contrast showing that response remains negligible beyond 200 mg. Calling 200 mg a plateau dose would go beyond the evidence; it is the maximum tested dose.
Worked Example: A Flat Point Estimate With Inadequate Precision
A third hypothetical analysis compares a candidate dose of 75 mg with higher tested doses of 100 and 150 mg. The prespecified margin is \(\delta=2.0\) units of benefit. The fitted contrasts are 0.20 units with \(SE=0.90\) for 100 versus 75 mg, and 0.30 units with \(SE=1.00\) for 150 versus 75 mg. Assume the reported intervals are simultaneous 90% normal intervals.
For 100 versus 75 mg, $$ 0.20\pm1.645(0.90)=0.20\pm1.481, $$ giving \((-1.281,\,1.681)\) units. This interval is contained within \((-2.0,2.0)\).
For 150 versus 75 mg, $$ 0.30\pm1.645(1.00)=0.30\pm1.645, $$ giving \((-1.345,\,1.945)\) units. This interval is also contained within \((-2.0,2.0)\), narrowly. Thus, under the stated assumptions, both contrasts meet the prespecified equivalence rule and support a plateau at 75 mg over the observed higher doses. However, this conclusion depends on the chosen margin: if the decision required a tighter margin of 1.5 units, the second interval would not be contained within \((-1.5,1.5)\), and equivalence would not be established under that stricter criterion.
Why the Contrast-and-Interval Rule Works
The margin defines the range of differences treated as negligible, while the interval represents the values of the contrast that remain compatible with the data under the specified procedure. Requiring the entire interval to lie within the margin ensures that the data are sufficiently precise not only to make a small difference plausible, but also to rule out differences larger than the chosen threshold.
Choose \(\delta\) in advance, in the units of the fitted response, and state why differences smaller than this threshold count as negligible for this comparison.
For each proposed plateau dose, subtract its fitted response from the fitted response at each higher tested dose, using one consistent benefit-oriented scale.
Use the fitted model’s uncertainty, including covariance, and account for multiple contrasts or data-driven candidate selection where relevant.
Claim support for a plateau over the observed higher-dose range only if each required interval is wholly within \((-\delta,\delta)\). Otherwise report that equivalence was not established.
This rule is a decision criterion about response contrasts; it is not a test that the curve has an exactly horizontal segment. For a normal contrast estimate with standard error \(SE\), the two one-sided tests ask whether the true contrast is greater than \(-\delta\) and less than \(\delta\). At one-sided level 0.05, each condition must be supported. These conditions hold together when the 90% two-sided interval is contained within the equivalence bounds. Thus, the interval rule makes explicit why “not statistically distinguishable from zero” is not enough: a wide interval can contain zero while also allowing a meaningful benefit increase.
Diagnostics, Alternatives, and Limits
Before interpreting a plateau claim, check that the contrast and its interval actually correspond to the intended fitted estimand and dose comparisons. Confirm the response direction and units, verify that the margin was specified independently of the observed results, and inspect whether the model adequately represents the data over the tested range. Residual patterns, influential observations, sparse dose groups, or strong sensitivity to plausible model choices can undermine confidence in model-based predictions. The uncertainty calculation should also be checked: intervals that omit shared-parameter covariance can misstate the uncertainty in a difference.
When multiple candidate doses or several higher-dose contrasts are examined, a collection of unadjusted intervals may overstate the evidence for finding at least one apparent plateau. Prefer a prespecified candidate and comparison set, or use an approach that provides appropriate simultaneous uncertainty. If no tested dose has higher tested doses for comparison, the contrast rule cannot support a plateau there. A fitted asymptotic model may imply that the curve will level off at a larger dose, but that is model-based extrapolation, not direct evidence from an observed higher-dose comparison.
| Approach | What it can establish | Main caution |
|---|---|---|
| Visual inspection of a fitted curve | Whether the displayed fit appears to level off in the observed range | Does not quantify whether higher-dose differences are negligible or how uncertain they are |
| Test of a zero contrast | Whether the data provide evidence of a nonzero difference under that test | Failure to reject zero is not evidence that the difference is within a negligible range |
| Model-based asymptote or target-dose estimate | Where the chosen model predicts its limiting response or a specified response level | May depend strongly on model form and extrapolation beyond observed doses |
| Contrast intervals against a prespecified margin | Whether observed higher-dose comparisons are sufficiently precise to support negligible differences | Conclusion is limited to the selected margin, comparisons, model, and tested range |
No one approach removes the need to examine model adequacy and uncertainty. A plateau supported by fitted contrasts is not automatically a dose recommendation: it does not, by itself, address other aspects of dose selection. Nor does an equivalence conclusion imply that the true contrasts are exactly zero. It means that the uncertainty intervals meet a specified practical-equivalence criterion under the analysis assumptions.
Common Mistake
Wrong reasoning: “The 150 mg and 200 mg estimates are not significantly different, so 150 mg is the plateau dose.” This treats absence of evidence for a difference as evidence that the difference is negligible. It also ignores whether contrasts from 150 mg to every higher tested dose are sufficiently precise and whether the maximum tested dose has been mistaken for an established asymptote.
Correction: Specify a negligible-difference margin, calculate uncertainty intervals for the relevant fitted contrasts, and require those intervals to lie inside the margin. If a contrast interval is too wide or extends beyond the margin, report that a plateau was not established. If the candidate is the maximum tested dose, state that it is the maximum tested dose; a plateau beyond it remains unverified by observed higher-dose comparisons.
Check Your Understanding
Answer each question using the contrast-and-interval rule. Short solutions follow the questions.
- A candidate dose has a contrast estimate of 0.1 units with a 90% interval of \((-1.8,2.0)\). If \(\delta=1.5\), has equivalence been established?
- Why is a nonsignificant test of zero contrast not enough to establish a plateau?
- A candidate dose is equivalent to the next higher dose, but no contrast has been assessed against a still higher tested dose. What remains to be checked?
- Can an uncertainty interval for a contrast between two tested doses establish that the response has plateaued at doses above the maximum tested dose?
Solutions: 1. No. The interval is not wholly within \((-1.5,1.5)\), because its upper limit is 2.0 units. 2. A nonsignificant result may arise from a wide interval that still allows differences larger than the negligible-difference margin; equivalence requires interval containment within that margin. 3. Compare the candidate dose with every higher tested dose required by the plateau definition, using suitable uncertainty intervals. 4. No. Such a contrast concerns only the observed dose range; behavior beyond the maximum tested dose is extrapolation, not an observed higher-dose comparison.