Worked Sensitivity Analysis for a Dose-Selection Decision

From uncertain inputs to a dose decision

A dose-selection analysis is not just a ranking of predicted efficacy. The decision also depends on how much value is assigned to benefit relative to harm, which outcomes are included, and how uncertain the model-based predictions are. A useful analysis makes these ingredients explicit so that a reviewer can see both the preferred dose under the stated assumptions and whether that preference changes when the assumptions change.

Consider a hypothetical comparison of doses of 50, 100, and 150 mg. For illustration, suppose a model provides the probability of meeting a prespecified efficacy target and the probability of an adverse event at each dose. These probabilities are hypothetical decision inputs, not results from a real study. The analysis treats the target and adverse event as decision-relevant outcomes and uses a simple utility scale to combine them.

The coefficient 10 sets the scale and relative value of efficacy-target attainment; \(w\) expresses the decision-maker's concern about adverse events on that same scale. Neither is a statistical estimate. They are preference inputs that must be justified for the decision at hand. The formula also assumes that the two outcome contributions can be added, that the probabilities refer to a common target population and time horizon, and that other relevant consequences do not alter the ranking. Those are substantive assumptions, not consequences of the model predictions.

For the reference analysis, set \(w=10\). The calculations are direct substitutions into the stated utility rule. No claim is made that this weight is uniquely correct; sensitivity to it is part of the analysis.

Pairwise differences and the tipping point

A pairwise utility difference shows which inputs are responsible for a ranking and where that ranking changes. For 100 mg versus 50 mg, subtract the two utilities:

The efficacy contribution is \(10(0.16)=1.6\) utility points, and the adverse-event contribution to the difference is \(0.07w\) points. The two doses tie when \(1.6-0.07w=0\), giving \(w=1.6/0.07=22.86\). Thus, for \(w<22.86\), 100 mg has greater utility than 50 mg; for \(w>22.86\), 50 mg has greater utility than 100 mg. At the tie, these two options have equal utility, though another candidate dose could still be preferred.

This threshold is a tipping point for the pairwise comparison: it identifies the value of the stated preference weight at which the comparison reverses while the outcome probabilities are held fixed. It does not say that the weight has a probability distribution, and it does not by itself settle the overall decision among all available doses.

Scenario analysis: make assumptions visible

A scenario analysis recalculates the decision under a set of clearly described alternative inputs. It is useful when inputs may change together or when a decision team wants to understand specific plausible combinations. Scenarios should be internally coherent: for example, a scenario with weaker efficacy predictions may also represent a different model assumption, rather than a collection of unrelated worst-case values. Scenario results are comparisons, not probabilities, unless probabilities have separately been assigned to the scenarios.

The reference case above gives 100 mg the largest utility. Consider two additional hypothetical scenarios. In the first, the estimated efficacy gain at 100 mg is smaller and its adverse-event probability is higher than in the reference case. In the second, the efficacy predictions are more favorable and adverse-event probabilities are lower. Keep the 50 mg estimates fixed in both scenarios for clarity.

Probabilistic analysis: summarize uncertainty over inputs

When a joint uncertainty distribution for the inputs is available, a probabilistic analysis evaluates the decision across draws or states from that distribution. A discrete distribution is sufficient to illustrate the calculation: assign probabilities to a set of joint input states, calculate the utility of each dose in each state, and then summarize expected utility and how often each dose is preferred. In an applied analysis, the distribution and dependence among inputs should come from an appropriate uncertainty analysis, such as joint model-based uncertainty or documented expert elicitation; they should not be chosen merely to produce a desired ranking.

The following three states reuse the scenarios above and assign illustrative probabilities of 0.50, 0.30, and 0.20. These probabilities are hypothetical and are included only to demonstrate the method. They sum to 1.00. Because the inputs are grouped into joint states, each state has a coherent combination of efficacy estimates, adverse-event estimates, and weight.

How to report stability, alternatives, and limitations

Expected utility and probability of being preferred answer different questions. Expected utility averages the utility values over the specified uncertainty distribution and supports a choice when the decision rule is to maximize average utility. Probability of being preferred counts how often an option ranks first. A dose can have the highest expected utility without being preferred in every state; that is exactly what happens to 100 mg here. If decision-makers use a different criterion, such as avoiding a dose that performs poorly in any plausible case, that criterion should be stated and evaluated directly rather than silently substituted for expected utility.

Several checks help identify an analysis that may be misleading. Confirm that the target outcome, adverse event, time horizon, and population are defined consistently across doses. Check that probabilities lie between 0 and 1, that the signs in the utility rule match the intended preferences, and that every calculated ranking follows from the displayed arithmetic. For probabilistic work, verify that state probabilities sum to 1 and that the joint states represent plausible combinations. If a small change in a weight or prediction reverses the recommendation, report that fragility clearly; if only one extreme combination changes the ranking, describe that condition specifically rather than hiding it in an overall stability label.

A common mistake is to say, “100 mg is robust because it has the greatest expected utility.” That conclusion confuses an average over assumed uncertainty with stability across assumptions. Here 100 mg has the greatest expected utility, but it is not preferred in the less favorable state. A correction is to report both findings: “100 mg maximizes expected utility under the illustrative three-state distribution, while 50 mg is preferred in the less favorable state.” This is more informative and does not overstate what the analysis establishes.

The simple utility rule is intentionally limited. It includes only one efficacy target and one adverse event, treats their contributions as additive, and uses a single weight for adverse events. A real decision may need multiple outcomes, severity distinctions, treatment burden, uncertainty about the utility weights, or constraints such as a maximum acceptable safety risk. Model uncertainty and uncertainty in preference values should also be distinguished: uncertainty in predicted probabilities is not the same as disagreement about how outcomes should be valued. The analysis supports structured discussion; it does not replace clinical, regulatory, or patient input.