Tipping Surfaces with Multiple Sensitivity Parameters

From a Contour to a Higher-Dimensional Boundary

A two-parameter tipping-point contour can show how two sensitivity parameters jointly change a conclusion. With three or more delta parameters, the same question lives in a higher-dimensional space: which combinations of assumptions change the conclusion? A single curve or two-dimensional plot cannot show that whole space without fixing, omitting, or summarizing some coordinates.

Let \(\boldsymbol{\delta}=(\delta_1,\ldots,\delta_q)\) denote the vector of sensitivity parameters, with \(q\geq 3\). These parameters describe the chosen departures from the reference imputation assumptions. Let \(\mathcal{D}\) be the specified range of values considered for \(\boldsymbol{\delta}\). At each point in \(\mathcal{D}\), the same imputation and analysis procedure is used to evaluate the same estimand and decision rule. For example, the rule might classify a result according to whether a pooled confidence interval excludes a null value, or whether an estimate exceeds a prespecified threshold.

Write \(C(\boldsymbol{\delta})\) for the resulting conclusion category. The tipping region is the part of \(\mathcal{D}\) assigned a specified category, such as “the interval excludes the null.” Its tipping boundary separates points with different categories. The boundary is relative to the chosen domain \(\mathcal{D}\): a plot only describes the delta combinations in that domain, not every conceivable assumption.

When a scalar signed margin \(h(\boldsymbol{\delta})\) defines the decision—for example, a confidence-limit endpoint minus a threshold—one can label points by the sign of \(h\). If \(h>0\) indicates one conclusion and \(h<0\) the other, the equality set \(h=0\) is a candidate boundary. It is the actual tipping boundary only where the conclusion changes across it. A zero at which the sign merely touches zero and returns to the same side need not be a boundary between categories.

Why Slices and Projections Are Different

A slice fixes selected coordinates at specified values and displays the boundary as the remaining coordinates vary. For instance, a slice of a three-parameter space might fix \(\delta_3\) at a value judged relevant and show the two-dimensional contour in \((\delta_1,\delta_2)\). Changing the fixed value can change the contour substantially; reporting one slice without its fixed value is incomplete.

A projection suppresses one or more coordinates to summarize where a region or boundary occurs in the coordinates retained. For example, projecting a tipped region onto the \(\delta_1\)-axis asks whether there is any combination of the omitted parameters that tips the conclusion at each displayed \(\delta_1\). That is an existence statement. It does not imply that the conclusion tips for every combination of omitted parameters. Projecting the boundary itself and projecting the tipped region are also different operations, so a plot must say which set is being projected.

A full surface display can preserve more of the joint structure, but is difficult to read when there are more than three parameters. Slices are usually easier to interpret, while projections can compactly summarize a region. Neither automatically conveys all the information in the original \(q\)-dimensional space.

Geometry and Local Surface Conditions

For a scalar margin, an equation \(h(\boldsymbol{\delta})=0\) is often a convenient way to draw a boundary. But the equation alone does not guarantee a regular, well-behaved surface. A standard local geometric justification comes from the implicit function theorem. At a boundary point, if \(h\) is continuously differentiable in a neighborhood and its gradient is nonzero there, then at least one partial derivative is nonzero. Locally, the coordinate corresponding to that partial derivative can be expressed as a continuously differentiable function of the other coordinates.

For \(q\) parameters, that local representation has \(q-1\) free coordinates. In three dimensions it looks locally like a two-dimensional surface; in four dimensions it is a three-dimensional boundary and cannot be directly drawn as an ordinary surface in three-dimensional space. If the gradient is zero, or if the response is not continuously differentiable, this theorem does not provide that local representation. The boundary may still exist, but it may have a singular point, a sharp feature, or a more complicated shape.

The condition matters because a smooth-looking fitted surface can conceal a region where the actual response is irregular. The theorem is a local geometric result, not a guarantee that an estimated surface is correct, that the boundary is globally connected, or that a numerical plot has located every piece of it.

Reading Projections Without Overclaiming

This distinction also applies to projections onto two coordinates. A shaded point in a projected region may correspond to a tipping combination at one omitted-coordinate value and a non-tipping combination at another. If the scientific question concerns which assumptions jointly produce a change, a projection alone is not enough: show slices, a range over the omitted coordinates, or a higher-dimensional representation that preserves the relevant combinations.

Practical Construction and Failure Modes

A responsible display begins with a clearly specified domain, a consistent conclusion rule, and a clear description of what each plotted coordinate means. The delta ranges should be stated in their chosen units, and fixed slice values should be labelled. A grid or fitted surface can help visualize the response, but the displayed boundary should be checked against the underlying analyses at points on both sides. In particular, a finely drawn interpolated curve is not evidence that the original calculations were equally fine or that all relevant boundary components were found.

Common failures include treating one slice as representative of all values of the fixed parameters; interpreting a projection as if it gave a conclusion for every omitted combination; and drawing a smooth, connected surface when the evaluated results show gaps, discontinuities, or separate pieces. A coarse grid can miss a narrow tipped region or a small disconnected component. Conversely, interpolation may suggest a boundary between evaluated points where the response has not actually been checked. An empty boundary in the displayed domain means only that no boundary was found there under the chosen rule; it does not establish that no tipping is possible outside the domain.

A plot also cannot establish that the chosen delta ranges are plausible, that the imputation assumptions are correct, or that the conclusion is robust to every unexamined modeling choice. It describes how the specified analysis behaves under the specified sensitivity parameters and domain. Interpret it alongside the assumptions that define those parameters and the numerical evidence used to construct the display.

The main alternatives serve different purposes. A collection of labelled slices is usually clearest for showing how one or two parameters behave while the others are held at interpretable values. A projection is compact when the question is whether tipping is possible somewhere among omitted values, but it loses combination-specific information. A full boundary representation preserves more joint structure but becomes difficult to visualize as the number of parameters increases. No one display removes the need to state what was fixed, omitted, or summarized.