Treatment-Effect Surfaces Across Two Sensitivity Parameters

Why View Two Sensitivity Parameters Together?

When outcomes are missing in both trial arms, a treatment-effect estimate can depend on assumptions about the unobserved outcomes in each arm. Treating one arm’s assumption as fixed while varying the other can hide their joint influence. A treatment-effect surface displays the estimate across pairs of arm-specific sensitivity parameters, making it possible to see where a conclusion changes as both assumptions vary.

The surface is an assumption-indexed collection of estimates, not a probability distribution over assumptions. Its purpose is to show the consequences of specified assumptions and to identify regions that matter for interpretation. It does not determine which assumptions are plausible; that depends on the sensitivity ranges and substantive reasoning used to define them.

Defining the Surface

Let $a=1$ denote the treatment arm and $a=0$ the control arm. For a continuous outcome, define $\delta_a$ as the difference, in outcome units, between the mean among participants with missing outcomes in arm $a$ and the corresponding reference-model mean for those participants. Thus $\delta_a=0$ represents the reference assumption, while a positive or negative value shifts the missing-outcome mean upward or downward. The exact reference model is part of the analysis specification.

Let $q_a$ be the fraction of participants in arm $a$ whose outcomes are missing, and let $\hat\mu_a(0)$ be that arm’s estimated mean under the reference assumption. Suppose the sensitivity analysis shifts the estimated missing-outcome mean by the same $\delta_a$ for all participants with missing outcomes in that arm, leaving the observed-outcome contribution unchanged. The arm-specific estimate then shifts by $q_a\delta_a$.

Definition: Under this additive pattern-mixture mean-shift specification, the treatment-effect surface for the treatment-minus-control contrast is $$ \hat\tau(\delta_1,\delta_0) = \hat\tau(0,0)+q_1\delta_1-q_0\delta_0, \qquad \hat\tau(0,0)=\hat\mu_1(0)-\hat\mu_0(0). $$ The surface gives a treatment-effect estimate for each pair $(\delta_1,\delta_0)$. Its units are the units of the outcome.

The minus sign before $q_0\delta_0$ follows from defining the effect as treatment minus control: raising the control-arm mean reduces that contrast. The formula assumes that the target estimand and outcome scale are held fixed across the sensitivity analysis, and that $q_a$ and the reference estimates are treated as fixed inputs when constructing this point-estimate surface. If the analysis uses another sensitivity parameterization, a nonlinear link, or a different estimand, the surface must be derived from that model rather than assumed to have this form.

Why the Surface Is Planar in This Model

The form can be derived directly from the arm means. Write the reference estimate in arm $a$ as the weighted combination of its observed-outcome mean and the reference-model mean for participants with missing outcomes. Denote those means by $\hat\mu_{a,\mathrm{obs}}$ and $\hat\mu_{a,\mathrm{mis}}(0)$, respectively. The arm-specific estimate at departure $\delta_a$ is

\hat\mu_a(\delta_a) = (1-q_a)\hat\mu_{a,\mathrm{obs}} + q_a\{\hat\mu_{a,\mathrm{mis}}(0)+\delta_a\}.

At $\delta_a=0$, the right-hand side is $\hat\mu_a(0)$. Subtracting that reference estimate from the expression at $\delta_a$ leaves $q_a\delta_a$. Taking the treatment-arm mean minus the control-arm mean therefore gives

\begin{aligned} \hat\tau(\delta_1,\delta_0) &=\hat\mu_1(\delta_1)-\hat\mu_0(\delta_0)\\ &=\hat\mu_1(0)+q_1\delta_1-\{\hat\mu_0(0)+q_0\delta_0\}\\ &=\hat\tau(0,0)+q_1\delta_1-q_0\delta_0. \end{aligned}

This derivation shows why the missing-outcome fractions determine how strongly each sensitivity parameter affects the contrast. For example, if $q_1$ is small, even a sizeable change in $\delta_1$ has a relatively small effect on the treatment estimate. The partial changes per unit of sensitivity parameter are $q_1$ for $\delta_1$ and $-q_0$ for $\delta_0$.

Worked Example: Evaluating the Surface at a Pair of Assumptions

Consider a hypothetical trial with outcomes measured in score points. Suppose the reference estimates are $\hat\mu_1(0)=52$ and $\hat\mu_0(0)=48$, and the missing-outcome fractions are $q_1=0.20$ and $q_0=0.30$. Evaluate the estimated treatment effect when $\delta_1=-5$ points and $\delta_0=2$ points.

First, the reference treatment effect is $52-48=4$ points. Apply the surface formula:

\begin{aligned} \hat\tau(-5,2) &=4+(0.20)(-5)-(0.30)(2)\\ &=4-1-0.6\\ &=2.4\text{ points}. \end{aligned}

The treatment-arm departure lowers the contrast by $1$ point, while the positive control-arm departure lowers it by $0.6$ points. Under this pair of assumptions, the estimated treatment-minus-control difference is $2.4$ points. This is a conditional estimate: it describes the result under the specified sensitivity values, not an average over possible values of $\delta_1$ and $\delta_0$.

Threshold Boundaries and Regions

A surface is especially useful when a conclusion is expressed through a threshold. Let $c$ be a treatment-effect threshold, such as zero for the sign of the effect or a prespecified minimum clinically important benefit. The boundary where the estimate equals $c$ satisfies

\hat\tau(0,0)+q_1\delta_1-q_0\delta_0=c.

If $q_0>0$, this can be rearranged as

\delta_0 = \frac{\hat\tau(0,0)-c+q_1\delta_1}{q_0}.

On one side of this line, the estimate exceeds $c$; on the other, it is below $c$. For the stated treatment-minus-control contrast and positive $q_0$, the estimate exceeds $c$ when $\delta_0$ is less than the boundary value at that $\delta_1$. A region of materially different conclusions is therefore defined by the chosen threshold and the sensitivity values considered plausible, not by the surface alone.

Worked Example: Finding Where a Clinical Threshold Is Crossed

Use the hypothetical surface $\hat\tau(\delta_1,\delta_0)=4+0.20\delta_1-0.30\delta_0$, with all effects and departures in score points. Suppose a benefit of at least $2$ points is considered clinically important, and consider $-10\leq\delta_1\leq5$ and $-2\leq\delta_0\leq10$. Find the boundary and assess two locations.

Set the estimate equal to the clinical threshold:

\begin{aligned} 4+0.20\delta_1-0.30\delta_0&=2,\\ \delta_0&=\frac{2+0.20\delta_1}{0.30} =6.667+0.6667\delta_1. \end{aligned}

The estimate is at least $2$ points when $\delta_0\leq6.667+0.6667\delta_1$. At $(\delta_1,\delta_0)=(-10,10)$, it is $4+(0.20)(-10)-(0.30)(10)=4-2-3=-1$ point, below zero and below the clinical threshold. At $(5,-2)$, it is $4+(0.20)(5)-(0.30)(-2)=4+1+0.6=5.6$ points, above the threshold.

These calculations show that the specified range includes assumptions producing materially different point-estimate conclusions. They do not show that either location is more plausible. That judgment requires substantive justification for the sensitivity ranges.

Point Estimates and Conditional Uncertainty

The surface above describes point estimates. Sampling uncertainty is separate: a confidence interval can be calculated conditional on each pair $(\delta_1,\delta_0)$, using the standard error appropriate to that assumption. Do not infer from a point-estimate contour that statistical significance changes at the same location. A significance boundary is determined by the conditional interval, and its location can differ from the boundary where the point estimate crosses zero or a clinical threshold.

The next example uses a deliberately simplified setting in which the conditional standard error is assumed to be constant. In an actual analysis, this must be checked or obtained from the fitted model; it cannot generally be presumed constant across sensitivity values.

Worked Example: Separating a Point-Estimate Change From an Interval

Suppose the surface is $\hat\tau(\delta_1,\delta_0)=4+0.20\delta_1-0.30\delta_0$ points, and, for illustration, the conditional standard error is $1.20$ points at every pair under consideration. Use a normal-approximation 95% confidence interval at $(-5,2)$ and at $(-8,8)$.

At $(-5,2)$, the point estimate is $2.4$ points. The interval is estimate $\pm1.96SE$:

2.4\pm1.96(1.20) =2.4\pm2.352 =[0.048,\ 4.752]\text{ points}.

Under the stated normal approximation and conditional standard error, this interval lies just above zero. At $(-8,8)$, the point estimate is $4+(0.20)(-8)-(0.30)(8)=4-1.6-2.4=0$ points, giving

0\pm1.96(1.20)=[-2.352,\ 2.352]\text{ points}.

This second interval includes zero. The comparison is conditional on each sensitivity pair; it does not assign probabilities to those pairs. Nor does the first interval establish clinical importance: its lower limit is above zero but below the $2$-point clinical threshold used in the preceding example.

Reading, Checking, and Comparing Surfaces

A useful interpretation starts by stating the contrast direction, outcome units, sensitivity-parameter meanings, and the range over which the surface is examined. Then identify the relevant boundary or boundaries and describe which parts of the range lie on each side. If a clinical threshold is relevant, report it separately from a zero-effect boundary. If inferential claims are made, use conditional intervals rather than treating the point-estimate surface as an uncertainty display.

Several checks can reveal a mis-specified or misread surface:

Check the contrast signs. With treatment minus control, raising the treatment-arm missing-outcome mean raises the contrast, while raising the control-arm mean lowers it. Reversing the contrast reverses these signs.
Check the units. A departure is in outcome units, but its impact on the arm estimate is multiplied by the missing fraction. Mixing a percentage with a proportion, or using a departure on the wrong scale, changes the result.
Check the reference point. At $(0,0)$, the surface must equal the stated reference estimate. If not, the intercept or parameter definitions are inconsistent.
Check whether the model really implies a plane. A constant additive shift to missing-outcome means gives the formula above. If sensitivity parameters affect variances, weights, nonlinear transformations, or other parts of the model, the resulting surface may be curved and must be calculated from that model.
Check the considered ranges. A boundary outside the defined sensitivity set does not imply a crossing within that set. Conversely, a boundary that cuts through the set identifies assumptions where the point-estimate classification changes.

A two-dimensional surface gives more direct information about joint variation than a one-dimensional analysis that fixes one arm’s parameter. A scenario table can report selected pairs, but a surface also shows where a threshold boundary lies between those selected points. Neither representation replaces the assumptions used to define the sensitivity parameters and ranges. An envelope of conditional confidence intervals answers a different question: it summarizes sampling uncertainty across assumptions rather than displaying only how the point estimate varies.

Limitations and a Common Mistake

The surface is only as informative as its parameterization and considered range. A single shift per arm may be too simple if departures differ across participant groups or depend on observed outcomes. The planar formula also relies on an additive outcome scale; conclusions can change when the treatment effect is expressed on another scale. When the model or estimand is nonlinear, the analyst should evaluate the treatment effect over the parameter pairs using that model instead of forcing a planar representation.

Common mistake: “The surface crosses zero, so the treatment effect is equally likely to be positive or negative.” This is incorrect. A crossing shows that different specified assumptions yield estimates on different sides of zero. It says nothing about how likely those assumptions are, and it does not by itself describe sampling uncertainty. The correction is to report the crossing within the justified sensitivity set, explain the meaning of the assumptions on each side, and present conditional uncertainty separately when making inferential statements.

Key takeaway: A treatment-effect surface makes the joint consequences of two arm-specific assumptions explicit. Its threshold regions describe where conclusions differ under those assumptions; they are not probabilities, and they do not replace conditional uncertainty intervals or substantive justification of the sensitivity ranges.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

For $\hat\tau(\delta_1,\delta_0)=3+0.25\delta_1-0.10\delta_0$, calculate the estimate at $(4,-5)$.
For that same surface, find the boundary where the estimate is zero.
If $q_1=0.10$ and $q_0=0.40$, which arm-specific departure has the larger absolute effect on the contrast per one-unit increase?
What does it mean if the surface crosses a clinical threshold within the sensitivity range?
Why is a point-estimate surface not a confidence surface?

Solutions. (1) $3+(0.25)(4)-(0.10)(-5)=3+1+0.5=4.5$, in the outcome’s units. (2) Set $3+0.25\delta_1-0.10\delta_0=0$, giving $\delta_0=30+2.5\delta_1$; estimates are positive below that line and negative above it. (3) The control-arm departure: its coefficient has absolute value $0.40$, compared with $0.10$ for the treatment-arm departure. (4) Some considered assumption pairs yield point estimates on opposite sides of the clinical threshold; this is not a probability statement. (5) The surface shows conditional point estimates only. A confidence surface or envelope also requires conditional standard errors and an uncertainty calculation at the relevant assumption values.