From Tested Doses to a Dose Between Them
Dose selection often begins with a finite set of doses that were actually administered. A fitted dose-response model can compare their predicted responses while accounting for the fact that predictions from the same model are correlated. In some situations, the decision may then be to use a dose between two tested doses. That step is not merely a finer reading of a dose-response curve: it relies on the model adequately describing the response in the interval and on the dose being clinically usable.
The first decision is whether the available evidence supports choosing among the tested doses. Interpolation is a separate decision. It is most defensible when the proposed dose lies between tested doses that provide informative coverage of the relevant response range, and when uncertainty and clinical constraints are acceptable. A dose outside the tested range is extrapolation, not interpolation.
Comparing the Tested Doses
Suppose fitted model predictions at tested doses are collected in a vector \(\hat{\boldsymbol m}\), with estimated covariance matrix \(V\). For a contrast defined by a coefficient vector \(c\), the estimated contrast is \(c^\mathsf{T}\hat{\boldsymbol m}\), and its estimated variance is \(c^\mathsf{T}Vc\). This calculation matters because predictions at different doses are generally not independent: they share the same fitted model parameters and data.
The model and covariance calculation presume that the dose-response model is an adequate representation for the intended use, that the parameter estimates and their covariance are reliable in the dose range being considered, and that the response scale and direction are defined consistently. An interval based on an approximate normal distribution also relies on that approximation being reasonable. These assumptions should be assessed before using a contrast as a decision input.
Worked Example: Contrasting Two Tested Doses
Consider a hypothetical study with tested doses of 10, 20, and 40 mg. A fitted model predicts mean efficacy responses of 2.0, 4.2, and 6.0 response units, respectively. For the predictions at 20 and 40 mg, suppose the estimated variances are 0.16 and 0.25 response-units squared, and their covariance is 0.08 response-units squared. Assume an approximate normal distribution for these fitted predictions.
The model-based contrast for 40 mg minus 20 mg is \(6.0-4.2=1.8\) response units. Its variance accounts for the shared model fit:
Thus the standard error is \(\sqrt{0.25}=0.50\) response units. An approximate 95% confidence interval for the contrast is \(1.8\pm1.96(0.50)=1.8\pm0.98\), or 0.82 to 2.78 response units. Under the model and approximation, the estimated mean response at 40 mg exceeds that at 20 mg; the interval quantifies uncertainty in that difference. It does not by itself establish which dose should be selected.
For illustration, suppose the decision criterion already specified for this use is a mean response of at least 5 response units. Approximate marginal 95% intervals are \(4.2\pm1.96(0.4)=(3.416,4.984)\) at 20 mg and \(6.0\pm1.96(0.5)=(5.020,6.980)\) at 40 mg. These intervals are just on opposite sides of the target. Subject to the model assumptions and other clinical constraints, 40 mg is therefore better supported than 20 mg for this criterion. The calculation does not justify a dose below 40 mg; that requires considering interpolation explicitly.
A contrast is useful for comparing doses, but the selection criterion should remain explicit. A larger estimated response need not outweigh greater safety risk, treatment burden, or uncertainty. Those elements belong in the decision framework established for the intended use; a contrast is evidence for that decision, not a decision rule by itself.
When Local Linear Interpolation Is Defensible
If a clinically relevant target response \(T\) lies between the predicted responses at two tested doses \(d_1\) and \(d_2\), a simple local approximation is to treat the response as linear between them. Write \(\hat m_1=\hat m(d_1)\) and \(\hat m_2=\hat m(d_2)\). When \(\hat m_1<T<\hat m_2\), the interpolated dose is
The ratio is the fraction of the predicted response change from \(\hat m_1\) to \(\hat m_2\) needed to reach \(T\). It lies between zero and one when the target is bracketed, so the estimated dose lies between the tested doses. This formula assumes that a straight line is a useful local approximation across the interval; bracketing alone does not prove that assumption.
Worked Example: Interpolating to a Target Response
Use the hypothetical predictions of 4.2 response units at 20 mg and 6.0 response units at 40 mg from the preceding example. Suppose the target is 5.0 response units and the local-linear approximation is considered adequate over this interval.
The target is bracketed because \(4.2<5.0<6.0\). Substituting into the formula gives
Thus the point estimate is approximately 28.9 mg. This is an estimate, not an administered dose or a guarantee that the target will be achieved. Any proposed practical dose must also conform to available formulations, dose increments, and clinical constraints.
Uncertainty in the two fitted responses produces uncertainty in \(\hat d_T\). For this illustration, suppose their variances are 0.16 and 0.25 response-units squared and their covariance is 0.08 response-units squared. The gradient of the interpolation function with respect to \((\hat m_1,\hat m_2)\), evaluated at the fitted responses, is
Using the first-order delta method, the approximate variance is
The approximate standard error is \(\sqrt{17.07057}=4.13166\) mg. A normal-approximation 95% interval is \(28.8889\pm1.96(4.13166)\), or approximately 20.79 to 36.99 mg. The interval remains within the tested-dose bracket, but its width shows that the target dose is not known precisely. This approximation treats the target as fixed and relies on a smooth interpolation function and a sufficiently accurate joint normal approximation for the two predictions.
Model Shape and Dose Coverage
A local straight line is not the only way to interpolate. If a scientifically suitable nonlinear model has been fitted, a target dose can instead be obtained by solving \(m(d;\hat\theta)=T\). The model’s shape can matter: for a saturating response, a straight line between two doses may give a different target-dose estimate from inversion of the fitted curve. The choice should follow evidence about response shape and the intended range, not convenience.
Worked Example: Target-Dose Inversion Under an Emax Model
Suppose a hypothetical efficacy model has the Emax form \(m(d)=E_{\max}d/(ED_{50}+d)\), with fitted \(E_{\max}=10\) response units and \(ED_{50}=20\) mg. Assume tested doses of 20 and 40 mg support the interval of interest. Their model-predicted responses are \(10(20)/(20+20)=5.0\) and \(10(40)/(20+40)=6.667\) response units. For a target of 6 response units, solve the model equation:
The model-based estimate is 30 mg, within the tested interval. Linear interpolation between the two predicted responses would instead give \(20+20(6-5)/(6.667-5)\approx32.0\) mg. The difference arises because the Emax curve is not linear over the interval; the interpolation method is consequential even when the target is bracketed.
Uncertainty in the parameters also affects the target dose. Rearranging the model gives \(d=ED_{50}T/(E_{\max}-T)\). Suppose, hypothetically, that the fitted parameter variances are 4 mg squared for \(ED_{50}\) and 0.36 response-units squared for \(E_{\max}\), with covariance 0.20 mg-response units. At \(ED_{50}=20\), \(E_{\max}=10\), and \(T=6\), the gradient with respect to \((ED_{50},E_{\max})\) is \((1.5,-7.5)\). The delta-method variance is
The approximate standard error is \(\sqrt{24.75}=4.97494\) mg, giving an approximate normal 95% interval of \(30\pm1.96(4.97494)\), or 20.25 to 39.75 mg. This interval reflects the supplied parameter uncertainty under the assumed model; it does not include uncertainty that the Emax form itself is wrong.
Diagnostics, Alternatives, and Limitations
Before interpolating, inspect whether the tested doses actually cover the target region. A target response should be bracketed by supported predictions, with adequate dose spacing and data that inform the curve locally. Check fitted responses and residual patterns for evidence that the model misses curvature, a plateau, or an unexpected reversal. Examine uncertainty in the predictions and the target-dose estimate, not only the point estimate. Also verify that the proposed dose is usable and compatible with clinical constraints.
Several warning signs call for caution: a target outside the range of fitted responses; a very wide dose interval between supporting arms; an estimated target dose highly sensitive to small changes in model parameters; or uncertainty intervals that extend beyond the studied range. A nearly flat response curve is particularly problematic for inversion: a small response change can correspond to a large dose change. In the linear formula, this appears as a small denominator \(\hat m_2-\hat m_1\), which amplifies uncertainty.
| Approach | Useful when | Main limitation |
|---|---|---|
| Choose among tested doses | The decision can be made from the studied options and their model-based predictions. | May not identify a dose between tested options that meets the decision criterion. |
| Local linear interpolation | The target is bracketed and response is plausibly close to linear over a sufficiently narrow interval. | Can be misleading when curvature is important or the interval is wide. |
| Invert a fitted nonlinear model | A supported model shape is appropriate over the bracket and the target has a unique solution. | Target-dose estimates depend on model form and parameter uncertainty. |
Interpolation is not automatically superior to choosing a tested dose. The tested-dose option is more directly anchored to observed treatment arms, whereas interpolation can offer a more tailored dose at the cost of additional model dependence. Neither approach removes the need to consider safety, feasibility, and the intended dose-selection criterion. If a dose is outside the adequately supported range, the appropriate description is extrapolation, and the evidence required is stronger than for interpolation.
Common Pitfall
A common error is to compute a target dose between two tested doses and treat the result as if it were directly tested. That reasoning ignores uncertainty in the fitted responses, possible curvature, and clinical constraints. The correction is to label the result as a model-based estimate, check that the target is bracketed, quantify uncertainty using the joint covariance or parameter covariance, and verify that the model is credible across the interval. A point estimate alone is not adequate evidence for a dose decision.
Check Your Understanding
Answer each question, then compare with the solutions.
- Two fitted responses have estimates 3.0 and 5.0 units, variances 0.36 and 0.64 units squared, and covariance 0.12 units squared. What is the standard error of the second-minus-first contrast?
- Tested doses are 10 and 30 mg, with fitted responses of 2 and 6 units. What dose does local linear interpolation estimate for a target of 5 units?
- Why is solving for a target dose outside the tested response range extrapolation, even if the model returns a numerical answer?
- Name one reason a target-dose estimate can be unstable when the target is bracketed.
Solutions. 1. The contrast variance is \(0.64+0.36-2(0.12)=0.76\) units squared, so its standard error is \(\sqrt{0.76}=0.8718\) units. 2. The target fraction is \((5-2)/(6-2)=0.75\), so the dose is \(10+(30-10)(0.75)=25\) mg. 3. The estimate lies beyond the interval supported by the tested doses, so the result depends on extending the model beyond observed dose coverage rather than estimating between supported doses. 4. For example, if the two fitted responses are very close, the interpolation denominator is small and uncertainty in the responses can produce a large change in the estimated dose.