From a Single Threshold to a Boundary
A one-parameter sensitivity analysis varies one assumption and asks where a chosen conclusion changes. With two sensitivity parameters, there is generally no single value at which the conclusion changes: different combinations of the parameters can produce the same decision, and the change may occur along a curve. A useful summary is therefore the tipping boundary, the frontier separating parameter combinations that lead to different conclusions.
For example, two delta shifts might represent assumptions about missing outcomes in two treatment arms. The exact meanings and units of the parameters depend on the analysis. What matters here is that the conclusion is evaluated using a specified rule for each pair of values. A boundary is meaningful only relative to that rule and the parameter range under consideration.
The domain \(\mathcal{D}\) might be a rectangle of clinically plausible values, but it must be stated or otherwise made clear. A mathematical boundary outside that domain is not a tipping boundary encountered in the sensitivity analysis. Conversely, the boundary may enter the domain only over part of its range, or may not intersect it at all.
The boundary is not itself a conclusion that the result is “almost significant” or clinically unimportant. It identifies combinations at which the specified decision changes. The interpretation of those combinations depends on the sensitivity parameters, the decision rule, and the substantive context.
Decision Scores and the Boundary Equation
A convenient way to define the frontier is to express the decision rule through a score. For instance, if a positive treatment contrast is the desired conclusion, the score could be the estimated contrast after applying the sensitivity assumptions. The decision might be positive when that score exceeds a prespecified cutoff. Subtracting the cutoff gives a score \(g\) whose sign encodes the decision.
This equation requires care. It is appropriate when the score is continuous and the two conclusions are determined by which side of zero it lies on. At points where \(g=0\), the decision rule must specify what happens exactly at equality; the boundary can still be reported as the equality set, while noting that equality is a separate decision state if the rule defines one. For a more general rule that assigns conclusions directly without a continuous score, define the frontier as the boundary between the resulting conclusion regions rather than assuming that an equation \(g=0\) exists.
The boundary may be a curve that cannot be written as one value of \(\delta_2\) for each \(\delta_1\). Even when it can be written that way locally, it may turn back, have multiple branches, or intersect a fixed-\(\delta_1\) slice more than once. Thus a boundary is a set of parameter combinations, not automatically a function or a unique threshold.
Worked Example: A Straight Boundary for Two Arm-Specific Shifts
Suppose a hypothetical analysis uses \(\delta_T\) and \(\delta_C\), measured in outcome points, for sensitivity shifts in the treatment and control arms. Its adjusted treatment contrast is \(\theta(\delta_T,\delta_C)=1.2-0.4\delta_T-0.2\delta_C\) outcome points. The prespecified conclusion is favorable when \(\theta>0\). Consider the domain \(0\leq\delta_T\leq4\), \(0\leq\delta_C\leq4\). Find the boundary and identify the favorable side.
The decision changes where the adjusted contrast equals zero:
Within the stated square, the line meets the upper edge \(\delta_C=4\) when \(4=6-2\delta_T\), so \(\delta_T=1\). It meets the lower edge \(\delta_C=0\) when \(0=6-2\delta_T\), so \(\delta_T=3\). Thus only the segment from \((1,4)\) to \((3,0)\) lies in the domain.
To identify the conclusion on each side, use the original score rather than relying on a sketch. At \((\delta_T,\delta_C)=(1,1)\), \(\theta=1.2-0.4(1)-0.2(1)=0.6\) outcome points, so the conclusion is favorable. At \((2,3)\), \(\theta=1.2-0.8-0.6=-0.2\) outcome points, so it is not favorable. The favorable combinations satisfy \(\delta_C<6-2\delta_T\); the unfavorable combinations satisfy \(\delta_C>6-2\delta_T\), within the stated domain.
Why the Equality Set Separates Conclusions
The connection between a zero-score equation and a boundary follows from continuity. Suppose \(g\) is continuous, and the two conclusion regions are \(R_+=\{(\delta_1,\delta_2):g>0\}\) and \(R_-=\{(\delta_1,\delta_2):g<0\}\). At a point where \(g=0\), arbitrarily small changes can lead to different conclusions only if positive and negative values occur nearby. Some zero-score points can instead lie along a flat region or border a region with no sign change, so the equality set should not automatically be treated as a frontier separating both conclusions.
For a point that is genuinely on the frontier between \(R_+\) and \(R_-\), continuity implies \(g=0\). If \(g\) were positive there, continuity would keep it positive in a sufficiently small neighborhood; if it were negative, the same argument would keep it negative. Either case would contradict the point being a frontier between positive and negative regions. This justifies using \(g=0\) to locate a sign-changing boundary, while checking that the two sides actually have different conclusions.
For a path through the parameter domain, the familiar one-dimensional threshold is recovered by intersecting that path with the boundary. For example, fixing \(\delta_1\) and varying \(\delta_2\) can produce no intersection, one intersection, or several. Consequently, a single threshold describes a particular path or slice; it does not generally summarize the full two-parameter result.
Worked Example: A Curved Boundary and a Restricted Domain
In a second hypothetical analysis, let the decision score be \(g(\delta_1,\delta_2)=0.9-0.3\delta_1-0.2\delta_2-0.05\delta_1\delta_2\), with conclusion A when \(g>0\) and conclusion B when \(g<0\). Both sensitivity parameters are dimensionless and restricted to \(0\leq\delta_1,\delta_2\leq3\). Find the boundary in the domain.
Setting the score equal to zero and collecting the terms involving \(\delta_2\) gives \(0.9-0.3\delta_1=\delta_2(0.2+0.05\delta_1)\). Since \(0.2+0.05\delta_1>0\) throughout the domain,
At \(\delta_1=1\), this gives \(\delta_2=0.6/0.25=2.4\). At \(\delta_1=2\), it gives \(\delta_2=0.3/0.3=1\). At \(\delta_1=3\), it gives \(\delta_2=0\). The curve first enters the domain through \(\delta_2=3\): solving \(3=(0.9-0.3\delta_1)/(0.2+0.05\delta_1)\) gives \(0.6+0.15\delta_1=0.9-0.3\delta_1\), hence \(\delta_1=2/3\). The in-domain boundary therefore runs from \((2/3,3)\) to \((3,0)\).
At \((1,1)\), \(g=0.9-0.3-0.2-0.05=0.35\), so the conclusion is A. At \((2,2)\), \(g=0.9-0.6-0.4-0.2=-0.3\), so the conclusion is B. These checks confirm the change of conclusion across the boundary; the equation alone would not tell us which side has which label without checking the score.
Worked Example: A Boundary That Is Not One Threshold
Consider a hypothetical score \(g(\delta_1,\delta_2)=0.5-(\delta_1-1)^2-(\delta_2-1)^2\) on the domain \(0\leq\delta_1,\delta_2\leq2\). Conclusion A applies when \(g>0\), and conclusion B when \(g<0\). Describe the boundary and examine the slice \(\delta_2=1\).
The boundary satisfies \((\delta_1-1)^2+(\delta_2-1)^2=0.5\). It is a circle centered at \((1,1)\) with radius \(\sqrt{0.5}\), entirely contained in the domain. The score is positive inside the circle and negative outside it. For instance, at the center \(g=0.5\), whereas at \((0,0)\), \(g=0.5-1-1=-1.5\).
On the slice \(\delta_2=1\), the boundary equation becomes \((\delta_1-1)^2=0.5\). Thus \(\delta_1=1\pm\sqrt{0.5}\), or approximately \(0.2929\) and \(1.707\). Conclusion A holds between these two values on the slice, while conclusion B holds outside them. There are two tipping values along this particular path, not one. Reporting one as “the” threshold would conceal the other change and the region where conclusion A holds.
Checks, Alternatives, and Limitations
Before interpreting a proposed boundary, verify that the sensitivity parameters have clear meanings and units, that the decision rule is fixed, and that the parameter domain is explicit. Evaluate the decision score at points on opposite sides of the proposed frontier. If both sides produce the same conclusion, the candidate equation may include a non-separating zero set, or the calculation may be wrong. Check as well whether the boundary actually intersects the domain and whether it has multiple branches or repeated intersections along a chosen slice.
A two-parameter boundary is richer than a single threshold, but it can be harder to communicate. A one-at-a-time analysis gives a threshold along each selected path, which can be easy to explain but may miss interactions or changes elsewhere in the domain. A table of selected scenarios is transparent about the evaluated combinations, but it can miss changes between those combinations. A boundary summarizes where a fixed decision changes; it does not, by itself, say which parameter values are plausible or whether the assumptions are credible.
A practical diagnostic is to compare the fitted or computed score with the stated rule at several parameter combinations, including combinations well away from the proposed boundary. If the score is not continuous, if the decision has more than two categories, or if results are computed with numerical procedures that introduce instability, a simple equality curve may not fully describe the frontier. In those cases, define the conclusion regions directly and describe their separating edge with appropriate qualifications.
- Do not call the whole boundary a threshold. A threshold is a value along a specified one-dimensional path; a boundary is a set of combinations.
- Do not assume the boundary is monotone. Curvature, turns, or multiple branches can create more than one change along a slice.
- Do not extrapolate beyond the domain. A mathematical continuation may involve sensitivity assumptions that were not included in the analysis.
- Do not confuse a boundary with robustness. Whether the result is robust depends on how the boundary relates to plausible assumptions, not just on its equation.
Check Your Understanding
Use the definitions and decision rules in each question. Solutions follow the questions.
- If \(g(\delta_1,\delta_2)=2-\delta_1-\delta_2\), with conclusion A for \(g>0\), what is the boundary and which side has conclusion A?
- Why is a boundary not generally a single tipping threshold?
- A candidate equation \(g=0\) is found, but small moves to either side give the same conclusion. What should be checked?
- What does it mean if the boundary does not intersect the stated sensitivity-parameter domain?
Solutions. (1) The boundary is \(2-\delta_1-\delta_2=0\), or \(\delta_2=2-\delta_1\). Conclusion A holds when \(\delta_1+\delta_2<2\), and conclusion B when \(\delta_1+\delta_2>2\), within the domain. (2) A threshold is an intersection along one specified path; a two-dimensional boundary can curve, branch, or intersect a path more than once. (3) Check whether the equality set is a genuine frontier between conclusion regions, whether the decision rule was applied correctly, and whether the score changes sign nearby. (4) Under the stated rule and domain, there is no parameter combination in the domain at which the conclusion changes across that boundary; this does not establish robustness outside the domain or under a different rule.