Threshold Plots for Decision-Relevant Parameters

What a Threshold Plot Shows

A sensitivity plot becomes especially useful when it answers a direct decision question: at what value of an uncertain input would the preferred action change? In dose selection, the actions might be two dose regimens; in development planning, they might be proceed and stop. A threshold plot displays the decision value of each action as one decision-relevant parameter varies, while the other inputs are held at specified reference values. A change in which action has the highest value marks a decision threshold.

The plot is conditional on the model, the definition of decision value, the reference values of other inputs, and the parameter range selected. It does not show how likely the parameter is to fall on either side of a threshold. Nor does a crossing by itself establish that one dose causes a better clinical outcome: it compares model-based decision values under the stated assumptions.

Decision value must put the options on a comparable scale. It may be an expected utility, a net benefit, or another explicitly defined objective that incorporates relevant benefits and costs. Raw response rates and treatment costs, for example, cannot be compared by simply subtracting one from the other because they have different units. If the decision value is expressed in utility points, then the difference between two actions is also in utility points. The parameter \(x\) retains its own units, such as percentage points of response.

Constructing and Reading the Plot

Start by naming the action being compared, the parameter being varied, the plausible range, and the values held fixed. Calculate \(V(a;x)\) for each action over that range. Plot either the values themselves or the incremental value \(\Delta(x)\). On a plot of action values, the preferred action is the one with the higher value at each \(x\); on a plot of \(\Delta(x)\), positive values favor \(a_1\), negative values favor \(a_0\), and zero indicates a tie.

A useful plot labels units on both axes, identifies the reference assumptions, and marks any crossing within the displayed range. The range should be justified by plausible values for the parameter, rather than chosen merely to make a crossing appear. If a crossing is outside that range, report that the preference does not change over the range examined; do not present an extrapolated crossing as decision-relevant without additional justification.

Why the Crossing Marks a Switch

The two-action result follows directly from the definition of incremental value. If \(\Delta(x)>0\), then \(V(a_1;x)>V(a_0;x)\), so \(a_1\) is preferred between those two actions. If \(\Delta(x)<0\), the inequality reverses and \(a_0\) is preferred. A zero is therefore a tie. A zero is a preference switch only when the sign differs on its two sides; a curve that merely touches zero and returns to the same sign has a tie but no switch.

For a linear difference, the threshold can be calculated exactly. Suppose \(\Delta(x)=\beta_0+\beta_1x\). If \(\beta_1\ne 0\), solving the equality gives the following result. A switch occurs within the analyzed range only if this value lies in the range and the sign changes there.

If \(\beta_1=0\), the difference is constant: there is no threshold if it is nonzero, and the two actions tie throughout the range if it is zero. For a nonlinear difference, there may be no root, one root, or several. A plotted grid helps locate candidate crossings, but a grid alone can miss a crossing between grid points or conceal multiple crossings in a wide interval. Refine the calculation around each apparent crossing and check the signs on both sides.

Multiple Actions and Nonlinear Comparisons

With more than two actions, a pairwise crossing is not necessarily a threshold for the overall decision. The chosen action is the one with the highest value among all available actions. Two actions can cross while a third remains better than both. Plotting all action values, or checking the upper envelope of those values, shows whether a pairwise crossing changes the actual preferred action.

Nonlinear action values can also produce more than one switch. The following constructed example shows why checking only the range endpoints is not enough.

Diagnostics, Alternatives, and Limitations

Before interpreting a threshold, check that the decision values are defined consistently across actions and parameter values. Confirm that the parameter’s units and direction are clear, that other inputs are fixed at stated values, and that the analyzed range is plausible. Check the model calculations near every crossing, including signs just below and above it. For a numerical curve, increase resolution around apparent roots and verify the result using the underlying calculations rather than relying only on the visual intersection.

A plot of raw outcome values can be misleading if larger values are not always better, or if benefits and harms are shown on incompatible scales. A decision value should encode the stated objective; if the objective changes, the threshold can change too. Also inspect other available actions: a crossing between two options matters to the decision only where those options are competitive with the alternatives.

A threshold plot complements, rather than replaces, other sensitivity summaries. One-way sensitivity analysis shows how an output changes as one input varies, while a threshold plot emphasizes where that variation changes the preferred action. A tornado diagram ranks changes across selected inputs but does not, by itself, locate a decision switch. Scenario analysis compares specified combinations of assumptions, rather than tracing a continuous range for one input. Threshold plots are easy to communicate, but their one-input-at-a-time construction can conceal dependence among inputs or changes in preference when several assumptions vary together.

Most importantly, a threshold is not a probability statement. If a threshold is 8 percentage points, that does not mean there is an 8 percent chance of preferring one dose, nor does it tell how likely the uncertain parameter is to exceed 8. It identifies the boundary in the chosen conditional comparison. The model, decision objective, and fixed assumptions determine the boundary; uncertainty about the parameter determines how relevant each side may be, but requires information beyond the threshold alone.