Asymmetric Plausibility Ranges Across Arms

Why the Two Arms Need Not Share a Plausibility Range

A sensitivity analysis asks how a treatment-effect conclusion changes as assumptions change. It is tempting to give the treatment and control arms identical ranges around zero. That choice may be convenient, but it is not automatically scientifically justified. The arms can differ in missingness, clinical context, or the plausible direction and size of departures from the reference assumptions. Assigning each arm its own range makes those differences explicit.

The central consequence is that asymmetry can do more than widen uncertainty. It can move the entire range of treatment-effect estimates up or down. A range that is wider in the direction that reduces the treatment contrast may undermine a favorable conclusion more than an equally wide range in the opposite direction.

Definition: Let \(\hat{\tau}_0\) be a treatment-effect estimate under specified reference assumptions. Let \(\delta_T\) and \(\delta_C\) denote the departures from those assumptions for the treatment and control arms, measured here as changes in their respective arm-level estimands on the outcome scale. In the additive model, \(\hat{\tau}(\delta_T,\delta_C)=\hat{\tau}_0+\delta_T-\delta_C\). A pair of arm-specific plausibility ranges is \(L_T\leq\delta_T\leq U_T\) and \(L_C\leq\delta_C\leq U_C\). If every combination in the resulting rectangle is considered plausible, this is a rectangular plausible set.

This parameterization treats each \(\delta\) as an arm-level change in the estimand, not necessarily as a raw shift applied only to missing outcomes. If a model instead parameterizes a raw missing-outcome shift, its effect on the arm-level estimand may be multiplied by a weight, such as the proportion of outcomes missing in that arm. In that case, the treatment-effect formula must include those weights. The examples below use arm-level departures, so the coefficients are one.

The formula also fixes the contrast direction: the treatment-arm estimand minus the control-arm estimand. A positive treatment-arm departure raises the contrast; a positive control-arm departure lowers it. If the outcome is undesirable, such as symptom severity, the sign of a beneficial contrast may be reversed, but the arithmetic must still follow the declared contrast convention.

From Arm-Specific Ranges to a Treatment-Effect Range

For the additive model and rectangular set, the smallest treatment effect occurs when the treatment-arm departure is as low as allowed and the control-arm departure is as high as allowed. The largest effect occurs at the opposite pair of endpoints. Thus,

\hat{\tau}_{\min}=\hat{\tau}_0+L_T-U_C, \qquad \hat{\tau}_{\max}=\hat{\tau}_0+U_T-L_C.

The opposite directions for the two arms follow from subtracting the control-arm departure. An increase in the treatment-arm mean increases the treatment-minus-control contrast, whereas an increase in the control-arm mean decreases it.

Result: Over a rectangular plausible set, the treatment-effect range is \([\hat{\tau}_0+L_T-U_C,\ \hat{\tau}_0+U_T-L_C]\). Its width is \((U_T-L_T)+(U_C-L_C)\). Consequently, each arm's range contributes its full width to the total effect range, while the locations of the arm-specific ranges determine where that total range is centered.

To see the location-and-width distinction, write the midpoint and half-width of each arm's range as \(m_T=(L_T+U_T)/2\), \(h_T=(U_T-L_T)/2\), and similarly \(m_C\) and \(h_C\). The treatment-effect range is centered at \(\hat{\tau}_0+m_T-m_C\), with half-width \(h_T+h_C\). An off-zero midpoint in either arm therefore shifts the effect range; the widths determine how far it extends on either side. If the ranges are asymmetric around zero, their midpoint need not be zero.

Worked Example: A Favorable Estimate With Unequal Arm Ranges

Suppose a hypothetical trial has a reference treatment-effect estimate of 2.4 points, with higher scores indicating better health. The treatment-arm departure is considered plausible from −0.5 to 1.0 points, while the control-arm departure is considered plausible from −0.2 to 0.6 points. These are assumed to be arm-level changes, and every combination in the rectangle is judged plausible.

The additive model gives \(\hat{\tau}(\delta_T,\delta_C)=2.4+\delta_T-\delta_C\). The minimum is attained at \(\delta_T=-0.5\) and \(\delta_C=0.6\):

\hat{\tau}_{\min}=2.4+(-0.5)-0.6=1.3\text{ points}.

The maximum is attained at \(\delta_T=1.0\) and \(\delta_C=-0.2\):

\hat{\tau}_{\max}=2.4+1.0-(-0.2)=3.6\text{ points}.

The plausible treatment-effect range is therefore 1.3 to 3.6 points. The treatment-arm range has midpoint 0.25 and half-width 0.75; the control-arm range has midpoint 0.20 and half-width 0.40. The effect range has midpoint \(2.4+0.25-0.20=2.45\) and half-width \(0.75+0.40=1.15\), reproducing endpoints \(2.45\pm1.15\), or 1.3 and 3.6. This arithmetic checks that the midpoint shift and width are both consistent with the stated ranges.

Uncertainty Around the Assumption-Indexed Estimates

A treatment-effect range summarizes variation in the point estimate across assumptions. It is not, by itself, a confidence interval: it does not include sampling uncertainty. A confidence interval conditional on one pair \((\delta_T,\delta_C)\) describes sampling uncertainty under that pair of assumptions. A confidence envelope over the plausible set covers the union of those conditional intervals.

If the standard error is constant over the set and a two-sided confidence interval uses critical value \(q\), then each conditional interval is \(\hat{\tau}(\delta_T,\delta_C)\pm qSE\). The smallest lower endpoint is the minimum estimate minus \(qSE\), and the largest upper endpoint is the maximum estimate plus \(qSE\). Hence the envelope is:

[\hat{\tau}_{\min}-qSE,\ \hat{\tau}_{\max}+qSE].

This result depends on the stated constant-standard-error condition. If the standard error or critical value changes with the assumptions, calculate the conditional interval endpoints over the plausible set rather than adding one common margin. For a rectangular set, the point-estimate extrema are at the corners under the additive model, but extrema of interval endpoints need not follow from the same corner rule if their standard errors vary.

Worked Example: Checking Whether a Favorable Sign Is Robust

Consider a hypothetical trial with \(\hat{\tau}_0=1.1\) points on a higher-is-better outcome. Suppose \(\delta_T\) ranges from −0.8 to 0.3 and \(\delta_C\) from −0.1 to 0.7 points. Treat these as arm-level departures, assume every combination in the rectangle is plausible, and suppose the conditional standard error is constant at 0.35 points.

The effect range is \(\hat{\tau}_{\min}=1.1-0.8-0.7=-0.4\) and \(\hat{\tau}_{\max}=1.1+0.3-(-0.1)=1.5\) points. Thus, plausible assumptions include estimates below zero as well as above zero.

For an illustrative 95% interval using \(q=1.96\), the common margin is \(1.96(0.35)=0.686\) points. The confidence envelope is \([-0.4-0.686,\ 1.5+0.686]=[-1.086,\ 2.186]\) points, or approximately −1.09 to 2.19 points.

Because the envelope includes zero, a positive treatment effect is not robust to all the specified assumptions when judged by whether the conditional 95% intervals exclude zero. This does not mean every assumption produces a nonsignificant result: some conditional intervals may exclude zero. The envelope summarizes the full set, not a claim about each scenario individually.

Clinical Thresholds and the Cost of Symmetric Defaults

Robustness can concern a clinical threshold as well as the sign of an effect. If a benefit of at least \(c\) points is required, uniform threshold robustness over the plausible set requires \(\hat{\tau}_{\min}\geq c\). If the purpose is to determine whether any plausible assumption reaches that threshold, the relevant condition is \(\hat{\tau}_{\max}\geq c\). These statements concern point estimates; a threshold conclusion that accounts for sampling uncertainty must also specify how the conditional intervals are being used.

Worked Example: Asymmetry Can Shift a Threshold Conclusion

Suppose a hypothetical trial estimates a 1.8-point benefit at the reference assumptions, and a 1-point benefit is the prespecified clinical threshold. Let the plausible treatment-arm departure range be −1.2 to 0.2 points and the control-arm range be −0.3 to 0.4 points. Assume the ranges are arm-level changes and all pairs in their rectangle are plausible.

The minimum is \(1.8-1.2-0.4=0.2\) points, and the maximum is \(1.8+0.2-(-0.3)=2.3\) points. The range crosses the 1-point threshold: some plausible assumptions produce an estimate below it, while others produce an estimate above it. It is therefore not uniformly above the clinical threshold.

The treatment range has midpoint −0.5 and half-width 0.7; the control range has midpoint 0.05 and half-width 0.35. The treatment-effect range is centered at \(1.8-0.5-0.05=1.25\) points with half-width \(0.7+0.35=1.05\), giving 0.2 to 2.3 points.

For comparison, retaining these widths but replacing both ranges with ranges centered at zero would give effect endpoints \(1.8-0.7-0.35=0.75\) and \(1.8+0.7+0.35=2.85\) points. That symmetric-around-zero substitute has the same total width but a different location. Its lower endpoint, 0.75, still falls below the threshold, but it understates how far below the threshold the lower endpoint lies in the stated asymmetric analysis. A symmetric default is not a neutral simplification when the scientific ranges have nonzero midpoints.

Assumptions, Diagnostics, and Alternatives

The endpoint calculation is exact under a small set of conditions: the effect is additive in the two arm-level departures; the contrast is treatment minus control; the plausible set contains all combinations in the specified ranges; and those ranges have a defensible interpretation on the outcome scale. If arm departures enter with known coefficients, for example \(\hat{\tau}=\hat{\tau}_0+w_T\delta_T-w_C\delta_C\) with nonnegative \(w_T,w_C\), the endpoints become \(\hat{\tau}_0+w_TL_T-w_CU_C\) and \(\hat{\tau}_0+w_TU_T-w_CL_C\). The coefficients must be included rather than silently treating raw parameter changes as full arm-level changes.

Before interpreting a range, check that the arm-specific assumptions are anchored to the same reference analysis, use compatible units and outcome scales, and have plausible directions and magnitudes for their respective arms. Check whether the corners that determine the extrema are meaningful combinations. A rectangular set asserts that every pair of values in it is plausible, including the most adverse cross-arm combination. If clinical knowledge rules out some pairs, the rectangle may be too broad; if important values lie outside the ranges, it may be too narrow.

Alternative summaries answer different questions. A shared-departure model restricts the arms to move together and can produce a narrower range, but it imposes a relationship that separate ranges do not. Symmetric ranges around zero simplify calculation, but can misstate the location of the range when plausible departures are directionally unbalanced. A nonrectangular joint set can encode dependence or exclude implausible pairs, but its treatment-effect extrema must be found over that set rather than by mechanically combining all four arm endpoints.

The simple endpoint formulas also rely on an additive effect function. If the effect is nonlinear in the assumptions, the minimum and maximum need not occur at the four corners. In that case, evaluate or optimize the conditional effect over the specified plausible set. Finally, plausible ranges are assumption statements, not confidence limits or estimates of how likely each assumption is. The analysis cannot establish that its chosen bounds are correct; it shows what follows if they are adopted.

Common pitfall: Treating “asymmetric” as only a difference in range widths. The midpoints matter too: \(\delta_T\) shifts the treatment-minus-control effect upward, while \(\delta_C\) shifts it downward. Calculate both the range’s midpoint and width, and verify the endpoint signs from the declared contrast.

Define the departures.
State whether each parameter is an arm-level estimand change or a raw model parameter requiring a coefficient.

State the plausible set.
Give separate bounds for treatment and control, and say whether all combinations are considered plausible.

Compute the effect range.
For the additive treatment-minus-control contrast, combine the low treatment bound with the high control bound for the minimum, and vice versa for the maximum.

Assess the intended conclusion.
Compare the range with zero or a clinical threshold, and include conditional sampling uncertainty when the question concerns interval-based inference.

Check Your Understanding

Use the additive treatment-minus-control model. Each departure below is an arm-level change, and each rectangular set includes every combination. Short solutions follow the questions.

If \(\hat{\tau}_0=3\), \(\delta_T\in[-1,0.5]\), and \(\delta_C\in[0,1]\), what is the treatment-effect range?
For the ranges in Question 1, calculate the midpoint and half-width of the effect range.
Suppose the treatment-arm departure is always zero while the control-arm departure ranges from −0.4 to 0.6. Does the control range shift or widen the effect range, and in which direction?
Why is it incorrect to call the range of point estimates across assumptions a confidence interval?
If the plausible pairs are restricted to \(\delta_T=\delta_C\), can the rectangular endpoint formula be used without checking the restriction?

Solutions. (1) The minimum is \(3-1-1=1\), and the maximum is \(3+0.5-0=3.5\), so the range is 1 to 3.5. (2) The midpoint is \((1+3.5)/2=2.25\), and the half-width is \((3.5-1)/2=1.25\). (3) The control departure is subtracted: the minimum is \(3-0.6=2.4\), and the maximum is \(3-(-0.4)=3.4\). It widens the range by 1 point and centers it at 2.9, a downward shift of 0.1 from 3. (4) It varies assumptions in point estimates and does not, by itself, quantify sampling uncertainty. (5) No. The restriction excludes most pairs in the rectangle; optimize the effect over the restricted set instead. Here \(\delta_T=\delta_C\) makes their contributions cancel, so the effect is exactly 3 throughout the restricted set.