Why Map Two Delta Parameters?
A one-parameter sensitivity analysis varies one delta adjustment while holding the others fixed. When two delta parameters may jointly matter, a single curve can hide how their combinations change the conclusion. A two-parameter tipping-point contour displays the combinations on a two-dimensional delta plane at which a prespecified threshold is reached. One side of the contour satisfies the decision criterion; the other does not.
The contour is conditional on choices: the imputation model, the two delta definitions and ranges, any other parameters held fixed, and the rule defining a “tip.” It is not a data-estimated boundary that is free of assumptions. Its purpose is to show clearly how a stated conclusion responds to specified departures from the imputation assumptions.
The delta parameters need specified meanings and units. For example, they might shift imputed values for two different missing-data groups by amounts on the outcome scale, or by amounts on a prespecified transformed scale. The sign and interpretation of each adjustment must follow its definition. A positive value should not automatically be described as increasing the outcome: it may increase or decrease a particular contrast depending on which group is shifted and how the estimand is coded.
For each pair \((\delta_1,\delta_2)\), the analysis must apply both adjustments as defined, complete the imputation-and-analysis procedure, and calculate the pooled quantity used in the criterion. A response function written as \(\Psi(\delta_1,\delta_2)\) may be computed directly, evaluated on a grid and interpolated, or approximated for display. It must represent the analysis actually being studied; a convenient equation is not a substitute for recalculating the estimand under the specified adjustments.
Constructing and Reading a Contour
First select the threshold and the parameter ranges. For a practical threshold, the boundary may be \(\Psi=T\), such as a risk difference reaching zero. For a statistical-significance rule, specify the exact inferential criterion. For example, if significance in a positive direction means that a two-sided 95% confidence interval lies entirely above zero, the boundary is where the lower endpoint equals zero. These two definitions answer different questions: crossing a clinical threshold is not necessarily the same as losing statistical significance.
Next evaluate the criterion over combinations of the two deltas. On a grid, label each point according to whether the criterion is met. Then locate adjacent points with different labels and refine the boundary by further calculations or interpolation. The contour consists of the boundary points, not all the grid points and not merely the single most extreme scenario. Plotting conventions should label each axis with its delta definition, units, and displayed range; the threshold and the side satisfying it should also be stated.
Name the pooled estimand, the threshold or decision rule, and which side counts as meeting the criterion.
State what \(\delta_1\) and \(\delta_2\) change, their units, their plausible search ranges, and which other assumptions remain fixed.
Calculate the pooled result over the plane, identify where the criterion changes, and refine those boundary locations.
Verify representative points on both sides, report the contour’s limitations, and avoid implying that it selects the plausible delta pair.
A short calculus argument explains why a contour’s shape is informative. Suppose the boundary is defined by a differentiable function \(F(\delta_1,\delta_2)=0\), with a nonzero partial derivative with respect to \(\delta_2\) locally. Along a smooth part of the boundary, a small change in \(\delta_1\) must be accompanied by a change in \(\delta_2\) that keeps \(F\) at zero. Differentiating the boundary equation gives:
Thus the local slope reflects how strongly the boundary criterion responds to each delta near that point. This slope is not a causal effect or evidence about how likely either delta is. If the denominator is zero, this representation does not give a finite local slope; the contour may have a vertical tangent or require a different local description.
Worked Example: A Contour for an Effect Threshold
Consider a hypothetical continuous-outcome contrast, measured in outcome points. In a specified range, suppose recalculation under two delta adjustments gives the illustrative response \( \Psi(\delta_1,\delta_2)=1.20-0.30\delta_1-0.20\delta_2 \). Here each delta is defined in the model’s stated adjustment units, and the coefficients describe the response of this particular contrast in this hypothetical analysis. Suppose the threshold is zero contrast, and inspect nonnegative delta values.
The boundary satisfies \(\Psi=0\). Substitution gives:
At \(\delta_1=0\), the crossing occurs at \(\delta_2=1.20/0.20=6.00\). At \(\delta_2=0\), it occurs at \(\delta_1=1.20/0.30=4.00\). At \(\delta_1=2.00\), the boundary value is \(\delta_2=(1.20-0.30(2.00))/0.20=3.00\). These are points on a straight contour, provided they fall within the ranges the analyst chose to examine.
At \((\delta_1,\delta_2)=(2.00,2.00)\), the contrast is \(1.20-0.60-0.40=0.20\) outcome points, above the threshold. At \((3.00,2.00)\), it is \(1.20-0.90-0.40=-0.10\) outcome points, below it. The contour separates these two results. It does not say that either pair is plausible; plausibility must come from the substantive justification for the delta ranges.
Contours for Inferential Decisions
A significance contour must use the inferential quantity in the decision rule, not just the pooled point estimate. Consider a hypothetical analysis with pooled estimate and standard error \( \bar{Q}(\delta_1,\delta_2)=0.80-0.25\delta_1-0.15\delta_2 \) and \(SE=0.20\) throughout the displayed region. For illustration, assume the appropriate 95% interval uses critical value 1.96 throughout. This simplification is an assumption of this example; in a real MICE analysis, the pooled standard error and degrees of freedom should be recalculated at each delta pair if they vary.
Worked Example: Loss of a Positive 95% Confidence Interval
The decision rule is that the two-sided 95% confidence interval lies entirely above zero. Under the stated constant-standard-error approximation, its lower endpoint is \(\bar{Q}-1.96SE\). The boundary occurs when that endpoint equals zero:
The right-hand side is \(0.80-0.392=0.408\). If \(\delta_1=0\), the boundary is \(\delta_2=0.408/0.15=2.72\). If \(\delta_2=0\), it is \(\delta_1=0.408/0.25=1.632\). At \((1.00,1.00)\), the estimate is \(0.80-0.25-0.15=0.40\), and the lower endpoint is \(0.40-0.392=0.008\); the interval remains just above zero. At \((1.20,1.20)\), the estimate is \(0.32\) and the lower endpoint is \(0.32-0.392=-0.072\); the interval includes zero. The contour marks loss of the stated positive-significance criterion, not a zero point estimate.
In an actual analysis, the interval would ordinarily be formed using the appropriate pooled estimate, standard error, and degrees of freedom for each delta pair. If those change across the plane, the boundary need not be a straight line, even if the point estimate responds linearly.
Nonlinear Boundaries and Grid Checks
When delta effects interact or the estimand is nonlinear, a contour may curve, bend sharply, or have more than one crossing along a line. A grid is a practical way to identify boundary regions, but a coarse grid can miss a narrow feature or place the apparent crossing inaccurately. Refine the grid where adjacent points switch classification, and check that the contour is stable to finer spacing. If values are interpolated, describe the interpolation and verify selected boundary points by recalculating the analysis itself.
Worked Example: A Curved Clinical-Threshold Contour
Suppose a hypothetical risk-difference response, in percentage-point units, is \( \Psi(\delta_1,\delta_2)=0.10-0.04\delta_1-0.03\delta_2-0.01\delta_1\delta_2 \). Both delta parameters are examined from 0 to 2 in their specified adjustment units. The clinical threshold is a risk difference of zero.
A coarse grid gives the following values, all in percentage points:
| \(\delta_1\) | \(\delta_2=0\) | \(\delta_2=1\) | \(\delta_2=2\) |
|---|---|---|---|
| 0 | 0.10 | 0.07 | 0.04 |
| 1 | 0.06 | 0.02 | -0.02 |
| 2 | 0.02 | -0.03 | -0.08 |
The grid suggests crossings between \(\delta_2=1\) and 2 when \(\delta_1=1\), and between \(\delta_2=0\) and 1 when \(\delta_1=2\). Solving the threshold equation gives:
At \(\delta_1=1\), this gives \(\delta_2=0.06/0.04=1.50\); substitution yields \(0.10-0.04-0.045-0.015=0\). At \(\delta_1=2\), it gives \(\delta_2=0.02/0.05=0.40\); substitution yields \(0.10-0.08-0.012-0.008=0\). At \(\delta_1=0\), the crossing would require \(\delta_2=0.10/0.03=3.33\), outside the displayed range. Thus the contour enters the displayed rectangle between its upper and right edges; it does not cross the left edge within the examined range.
Diagnostics, Alternatives, and Limitations
Several checks help distinguish a meaningful contour from a plotting artifact. Verify that the delta definitions, signs, units, and parameter ranges are explicit. Recompute the pooled result at points on both sides of the contour and at selected boundary points. Check that the decision labels agree with the numerical rule. If the imputation procedure has Monte Carlo variation, assess whether that variation is small relative to the distance from the threshold; an apparent jagged or shifting boundary may reflect simulation noise rather than a substantive feature. A contour near the edge of the search region should be described as unresolved beyond that region, not extrapolated as if its shape were known.
A one-parameter tipping-point search or root-finding method answers a narrower question: it finds a crossing along a specified path, such as fixing \(\delta_2\) and varying \(\delta_1\). That is useful when the second parameter is fixed by design or substantive judgment, but it cannot show how the crossing changes when both vary. A two-dimensional grid makes that joint dependence visible, at the cost of more calculations and potential interpolation error. Compared with reporting a few selected scenarios, a contour can show the boundary more completely, but it still depends on the chosen plane, range, and threshold.
A contour also does not assign probabilities to delta values, determine which combinations are credible, or establish robustness outside the displayed region. It can be difficult to interpret if the boundary folds, if there are multiple disconnected regions satisfying the rule, or if one delta has little effect across the examined range. In these cases, report the relevant regions and crossings directly, and use a sufficiently detailed grid to avoid suggesting a single simple boundary where none exists.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- For \(\Psi=0.60-0.20\delta_1-0.10\delta_2\), find the contour where \(\Psi=0\), including both axis intercepts.
- If a contour is defined by the lower 95% confidence limit equaling zero, does it necessarily mark where the pooled point estimate equals zero? Explain.
- For the curved response in the third worked example, calculate the threshold value of \(\delta_2\) when \(\delta_1=1.50\).
- Name two checks that can reveal a misleading contour drawn from a coarse or noisy grid.
Solutions. (1) Set \(0.60-0.20\delta_1-0.10\delta_2=0\), so \(0.20\delta_1+0.10\delta_2=0.60\). The intercepts are \(\delta_1=3.00\) at \(\delta_2=0\), and \(\delta_2=6.00\) at \(\delta_1=0\). (2) No. It marks where the lower interval endpoint reaches zero; the estimate can remain positive because the interval also depends on its standard error and critical value. (3) Substitute into the derived contour formula: \(\delta_2=(0.10-0.04(1.50))/(0.03+0.01(1.50))=0.04/0.045=0.889\) adjustment units, to three significant figures. (4) Recalculate selected boundary and opposite-side points to check their classifications, and refine the grid to see whether the contour location is stable. When simulation noise is possible, assess whether it is small relative to the threshold distance.