Worked Dose-Selection Analysis from Trial Data

From Trial Results to a Dose Recommendation

A dose-ranging trial can show that several doses have activity without resolving which dose should advance. The analysis must connect observed trial results to a decision: what clinical benefit is sufficient, what safety risk is acceptable, and how should uncertainty about the dose-response relationship affect the choice? This worked analysis follows those steps using hypothetical data. The numerical decision rule is specified before examining the model predictions, so the recommendation follows a stated criterion rather than a threshold chosen to favor a particular dose.

Suppose a randomized Phase II trial assigns 200 participants to placebo or each of four once-daily doses. The efficacy endpoint is improvement on a clinical scale, in points, with larger values better. The values below are estimated mean improvements relative to placebo; for example, 2.4 points at 10 mg/day means an estimated improvement 2.4 points greater than the placebo mean. Safety is the proportion of participants with at least one specified adverse event during the trial.

For this illustration, assume that randomization was implemented as planned, endpoint definitions and follow-up were consistent across arms, and the reported mean effects are suitable for the target population. The candidate efficacy fits use the four active-dose means, with equal weight because the arm sizes and precision are assumed comparable. These assumptions make a transparent worked analysis possible; a real analysis should use participant-level data, the prespecified estimand, and the actual uncertainty of each estimate.

Fit and Compare Candidate Efficacy Curves

Consider a simple Emax relationship with a fixed \(ED_{50}\), the dose associated with one-half of the model’s maximum effect:

Here \(d\) is dose in mg/day, \(E(d)\) is the placebo-adjusted mean improvement in points, and \(E_{\max}\) is the model’s maximum effect in points. The curve assumes a nonnegative, saturating effect, with no downturn at high doses. For a fixed \(ED_{50}\), only \(E_{\max}\) is estimated. We examine \(ED_{50}=20\) and \(ED_{50}=40\) mg/day as alternative candidate shapes; these values are illustrative, not claimed to be known biological constants.

For observations \(y_i\) at doses \(d_i\), write \(x_i=d_i/(ED_{50}+d_i)\). Least squares chooses \(E_{\max}\) to minimize the sum of squared residuals, \(\sum_i(y_i-E_{\max}x_i)^2\). Differentiating with respect to \(E_{\max}\), setting the derivative to zero, and solving gives:

This derivation presumes that the model’s fixed shape is appropriate and that the equal-weight least-squares criterion is a reasonable description of the precision of the means. If arm estimates have materially different variances, weighted fitting is more appropriate. Least squares also minimizes squared error; it does not account for all uncertainty in choosing a model or selecting a dose.

For \(ED_{50}=20\), the \(x_i\) values are \(1/3\), \(3/5\), \(3/4\), and \(5/6\). Applying the same formula gives \(\widehat{E}_{\max,20}\approx7.315\) points. Its predicted mean at 30 mg/day is \(7.315(30/(20+30))=4.389\) points. The two fits can be compared by their residual sums of squares, \(SSE=\sum_i(y_i-\widehat{E}(d_i))^2\).

Estimate Safety Risk and Apply a Prespecified Rule

For a compact safety summary, fit a straight line to the five observed event proportions, including placebo, as a function of dose. This is a local descriptive approximation across the tested range, not a general probability model. Let \(\hat r(d)=a+bd\) denote predicted risk, where risk is expressed as a proportion. The line is fitted by ordinary least squares to the five arm-level proportions. It may be used here only because the predicted values over the dose range remain between zero and one; a linear probability model can produce impossible risks outside its fitted range.

Now specify a rule for this hypothetical decision: a dose must have predicted efficacy of at least 4.0 points under both efficacy candidates, and its predicted safety risk under the fitted line must be no more than 12.0%. Among tested doses satisfying both criteria, select the lowest. As an additional safety sensitivity check, increase the fitted risk by 2 percentage points at each dose and repeat the assessment. That increase is a stress test, not a confidence bound or a probability statement.

Deriving and Documenting the Recommendation

The recommendation follows from the intersection of two conditions, not from the efficacy curve alone. If \(\mathcal{D}\) is the set of tested doses, the rule selects the smallest \(d\in\mathcal{D}\) for which the minimum efficacy prediction across the two candidate models is at least 4.0 points and the stress-tested safety prediction is at most 12.0%. At 30 mg/day the minimum efficacy prediction is 3.922 points, so the condition fails. At 60 mg/day the minimum is 5.486 points and the stressed risk is 11.788%, so both conditions hold. Therefore 60 mg/day is the lowest tested dose that satisfies the stated rule.

The use of a safety stress test makes the choice less dependent on the fitted line’s point prediction, but does not turn the result into a guarantee. For instance, the observed 60 mg/day event risk is based on 18 events among 200 participants, so it has sampling uncertainty. A real protocol might instead specify an upper confidence or posterior bound for safety risk. That would require an explicit interval method, model, and threshold chosen before reviewing the results.

Diagnostics, Alternatives, and Limitations

Before accepting the illustrative result, inspect whether the efficacy residuals show a systematic pattern. A consistent underprediction at middle doses and overprediction at the ends, for example, would suggest that a candidate curve misses the observed shape. Check that the safety trend is plausible, that predicted risks remain within zero and one, and that no single arm dominates the fit. With only four active-dose means, a low residual sum of squares can be unstable and does not establish that a saturating response is correct. The observed efficacy means also rise by smaller amounts across successive dose increments, which is compatible with saturation but not sufficient to rule out other shapes.

The analysis makes several consequential simplifications. It treats the arm-level efficacy means as equally precise, uses fixed \(ED_{50}\) candidates rather than estimating shape uncertainty, and summarizes safety with a linear trend instead of a binomial model. It uses point predictions in the main rule and does not account for uncertainty in the efficacy parameters, safety risks, or thresholds. It selects only among tested doses; a recommendation for an untested dose would add interpolation or extrapolation assumptions. The result is a transparent demonstration of a decision process, not a clinical recommendation.

Other approaches address some of these limitations. MCP-Mod can formally test dose-response shapes and estimate effects under selected models; Bayesian adaptive dose-ranging can incorporate prior information and update during a trial. Model-averaging approaches can reflect uncertainty across candidate efficacy curves, while a binomial likelihood is more natural than a linear probability model for adverse-event counts. Direct dose-arm contrasts can avoid imposing a smooth curve but may be less precise and do not directly describe response between tested doses. These alternatives are not interchangeable: the choice depends on the trial design, estimand, available data, and prespecified decision objective.

A useful decision record states the recommendation and its conditions: here, advance 60 mg/day under the specified 4.0-point efficacy target, 12.0% safety threshold, two-candidate efficacy check, and 2-percentage-point safety stress test. It should also report the competing predictions, observed event counts, assumptions, and unresolved uncertainty. This makes it possible to distinguish a dose selected by a rule from a dose proven to be optimal.