Why Vary Two Assumptions Together?
A decision can depend on more than one uncertain assumption. For example, a preferred dose may change with both the assumed probability of a clinically important safety event and the assumed penalty assigned to exposure-related harm. A one-way analysis varies one input while fixing the other; that is useful for seeing a conditional effect, but it may miss combinations under which the preferred action changes. A two-way sensitivity analysis evaluates decisions across combinations of two inputs.
The purpose is to show which action is preferred under each specified combination, and where that preference changes. A grid does not, by itself, say how likely any combination is. In particular, its cells should not be read as probabilities or as a summary of uncertainty unless an additional probability model has been defined.
The grid is meaningful only if its inputs, ranges, and decision rule are clear. Here, each point represents a deliberate combination of assumptions. These combinations need not be equally plausible, and varying two inputs over ranges does not establish that they are independent. The assumption register, scenario analysis, and plausible-bounds reasoning developed earlier help determine which values and combinations merit examination.
Constructing and Reading a Grid
Consider a hypothetical decision among low, medium, and high doses, denoted \(L\), \(M\), and \(H\). Let \(x\) be an assumed probability of a clinically important safety event, ranging from 0 to 0.4, and let \(y\) be a dimensionless multiplier for an exposure-related penalty, ranging from 0.5 to 1.5. Suppose the development team uses the following hypothetical decision scores, in decision-score units:
These formulas are stipulated for illustration, not estimated from a real study. In this example, a higher score is preferred, and all three doses are evaluated using the same input values. The coefficients specify how each score changes as the assumptions change; they are not probabilities or dose effects in clinical units. Before interpreting a grid, check that the score definition and direction of preference are consistent with the decision problem.
A practical construction is to select values for \(x\) and \(y\), evaluate all candidate scores at each pair, and record the maximum. The grid can display the winning dose in each cell, or show the scores as well so that close decisions and ties remain visible. A small difference between the top scores may matter: it indicates that the nominal winner has little advantage under that combination, even if the winning label is unambiguous.
Worked Example: Filling a Two-Way Grid
Use the three hypothetical score functions above at \(x=0,\ 0.2,\ 0.4\) and \(y=0.5,\ 1.0,\ 1.5\). For the first cell, \(x=0,\ y=0.5\):
Thus \(H\) is preferred at this pair because \(13.5\) is the largest score. Repeating the same calculations for the other pairs gives the following scores and winners. Scores are shown in the order \(V_L,\ V_M,\ V_H\).
| \(y\) | \(x\) | \(V_L,\ V_M,\ V_H\) | Preferred dose |
|---|---|---|---|
| 0.5 | 0 | 9, 12.5, 13.5 | H |
| 0.5 | 0.2 | 7, 8.5, 7.5 | M |
| 0.5 | 0.4 | 5, 4.5, 1.5 | L |
| 1.0 | 0 | 8, 10, 9 | M |
| 1.0 | 0.2 | 6, 6, 3 | L and M tie |
| 1.0 | 0.4 | 4, 2, -3 | L |
| 1.5 | 0 | 7, 7.5, 4.5 | M |
| 1.5 | 0.2 | 5, 3.5, -1.5 | L |
| 1.5 | 0.4 | 3, -0.5, -7.5 | L |
This grid shows that the preferred dose depends on the combination of assumptions. At \(y=0.5\), the winner changes from \(H\) to \(M\) to \(L\) as \(x\) increases across the chosen values. At \(y=1.0\), \(L\) and \(M\) tie at \(x=0.2\). These are results at the specified grid points; they do not imply that a particular cell or assumption combination is probable.
Pairwise Boundaries and Preferred Regions
A boundary between two options occurs where their decision scores are equal. Solving for this equality shows where a pairwise preference changes, but a tie does not automatically mean either option is preferred overall: a third action could have a higher score. Therefore, after finding a pairwise boundary, compare the tied scores with every other candidate at that point.
For the example, subtract the medium-dose score from the low-dose score:
The pairwise tie occurs when \(V_L-V_M=0\), so \(10x+3y-5=0\). At a fixed \(y\), this gives \(x=(5-3y)/10\). A positive difference means \(L\) scores higher than \(M\); a negative difference means \(M\) scores higher than \(L\). This direction follows directly from the definition of the difference.
Worked Example: Checking a Dose-Preference Boundary
At \(y=1\), the low- and medium-dose boundary is
At this point, evaluate the scores, rather than relying only on the boundary equation:
The tied scores are \(6\), and the high-dose score is \(3\), so \(L\) and \(M\) are tied for the best score at this point. For \(y=0.5\), the same pairwise equation gives \(x=(5-1.5)/10=0.35\). Substitution gives \(V_L=8-10(0.35)-2(-0.5)=5.5\), \(V_M=10-20(0.35)-5(-0.5)=5.5\), and \(V_H=9-30(0.35)-9(-0.5)=3\). Again, the tie is for the best score, not merely a tie below another option.
Interpreting the Grid in Context
A preferred region is the set of input combinations at which one action has the highest score among those considered. The region depends on the chosen ranges, resolution, score functions, and candidate actions. A plotted boundary is not a confidence limit, and its location does not quantify the probability that an assumption will fall on either side.
For a grid to support a development decision, ask whether its displayed combinations are plausible and whether they are described transparently. If two assumptions cannot reasonably take extreme values together, a rectangular grid that includes that corner may show a mathematically valid but decision-irrelevant scenario. Conversely, excluding a combination requires a defensible rationale; do not remove a consequential combination merely because it changes the preferred action.
It is also useful to show score differences, not only winner labels. A region where one dose wins by a small margin may call for a different discussion from a region where it wins by a large margin. The scale and meaning of the score difference must be explained; it should not be described as a clinical effect unless the score has that interpretation.
Worked Example: Finding a Change Within a Plausible Range
Suppose the team considers \(x\) between 0.08 and 0.30 and \(y\) between 0.9 and 1.2 plausible for this hypothetical decision. At the upper multiplier \(y=1.2\), the \(L\)-versus-\(M\) tie occurs at
At \(x=0.10,\ y=1.2\), the scores are \(V_L=8-10(0.10)-2(0.2)=6.6\), \(V_M=10-20(0.10)-5(0.2)=7.0\), and \(V_H=9-30(0.10)-9(0.2)=4.2\). Thus \(M\) is preferred. At \(x=0.30,\ y=1.2\), they are \(V_L=8-3-0.4=4.6\), \(V_M=10-6-1=3.0\), and \(V_H=9-9-1.8=-1.8\), so \(L\) is preferred. The boundary lies within the stated range, and the calculated scores confirm a change in the winner between these two points. This identifies a decision-relevant region to examine; it does not establish how likely either dose is to be optimal.
Assumptions, Diagnostics, and Limitations
The calculations above require that the decision scores be defined consistently, that the same candidate actions be compared at every grid point, and that other inputs not shown on the grid be fixed or explicitly specified. The selected ranges and combinations must also be interpretable for the decision. If a score is constructed from model predictions, its dependence on the two assumptions must be implemented correctly; an algebraic grid cannot repair an inappropriate model or an unsuitable decision criterion.
Useful checks include recomputing a few cells independently, confirming that the winner in each cell really has the largest score, and checking tie points by evaluating all candidate scores. Look for abrupt changes that may signal a coding, unit, or formula error, and check whether the reported boundary lies inside the plotted range. A grid can also be too coarse: if the winner differs at adjacent cells, add points near the change and calculate the boundary where possible. A tie between grid points may otherwise go unnoticed.
Two-way sensitivity analysis complements, rather than replaces, one-way sensitivity analysis, threshold plots, and scenario analysis. A one-way analysis isolates change in one assumption conditional on a fixed value of the other. A threshold plot can focus on where a decision changes as an input varies. A scenario analysis compares selected coherent sets of assumptions. The two-way grid makes joint variation visible at specified combinations, but it does not assign probabilities to them, integrate uncertainty, or establish that the two inputs vary independently.
Finally, a region map is a summary of the decision rule under the assumptions supplied. It does not prove that one dose is clinically superior, nor does it resolve uncertainty about model structure, the score itself, or the plausibility of the input ranges. State the assumptions and score direction alongside the grid, report ties or small score margins, and avoid turning a conditional preference into an unconditional recommendation.
Common Pitfalls
- Reading cells as probabilities. A grid cell is a specified input combination, not a probability mass. The fraction of cells favoring an action depends on the grid design and is not a probability of that action being optimal.
- Reporting a pairwise tie as the overall decision. Two options can tie while a third scores higher. Evaluate every candidate at the boundary.
- Using implausible combinations without explanation. A wide rectangular range can include pairs that cannot reasonably co-occur. Justify the combinations or clearly identify them as stress tests.
- Showing only the winning label. Winner maps can conceal near-ties. Include scores or differences where they affect interpretation.
- Claiming more precision than the grid supports. A boundary drawn between widely spaced grid values is not precisely located unless it is calculated or refined.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In a two-way sensitivity grid, what does a cell represent, and what does it not represent?
- For the hypothetical score functions in this tutorial, calculate \(V_L\), \(V_M\), and \(V_H\) at \(x=0.1,\ y=1.0\). Which dose is preferred?
- At \(y=1.0\), the \(L\)-versus-\(M\) boundary is \(x=0.2\). Why must \(V_H\) also be checked there?
- What should be done if adjacent grid points have different winners but the location of the decision change matters?
- Why is the proportion of grid cells favoring a dose not generally the probability that the dose is optimal?
Solutions. (1) A cell represents one specified pair of input values, evaluated using the stated decision rule; it is not an assigned probability for that pair. (2) \(V_L=8-10(0.1)-2(0)=7\), \(V_M=10-20(0.1)-5(0)=8\), and \(V_H=9-30(0.1)-9(0)=6\). Therefore \(M\) is preferred, with a score of 8 decision-score units. (3) The tied pair might both score below the third dose, so all candidate scores must be compared to establish an overall preference. (4) Add grid points near the change or solve and check the relevant boundary using the score functions. (5) The fraction depends on how many points are chosen and where they are placed; without a probability distribution and appropriate weighting, it has no general probabilistic interpretation.