Treatment-Arm-Specific Delta Parameters

Why Use Separate Delta Parameters by Treatment Arm?

A sensitivity analysis can allow missing outcomes in the treatment arm and control arm to depart differently from their respective MAR reference distributions. This matters when the reasons for missingness, the treatment context, or the clinical interpretation of an unobserved outcome could differ across arms. A single shared shift rules out that possibility by assumption; separate shifts let the analyst examine it directly.

The notation here is for a continuous outcome with an additive shift on its original scale. Let \(A=1\) denote the treatment arm and \(A=0\) the control arm. Let \(R=1\) indicate that the outcome is observed and \(R=0\) that it is missing. For each arm, a reference model supplies a MAR distribution for the missing outcomes. A delta parameter specifies how the missing-outcome distribution is shifted from that reference. The earlier material on delta adjustment establishes this reference-and-departure framework; the focus here is allowing the departure to vary by arm.

This definition makes a specific modeling assumption: within each arm, the same additive shift applies to all missing outcomes, relative to the chosen MAR reference. It does not require the treatment and control shifts to be equal. A positive \(\delta_a\) moves missing outcomes upward and a negative \(\delta_a\) moves them downward. These statements concern the outcome scale: if higher values represent worse health, a negative shift may be favorable rather than unfavorable.

The shifts are sensitivity parameters, not quantities identified by the observed outcomes alone. Observed data can help fit the reference model, but they cannot reveal the unobserved outcomes or determine which delta values are correct. The analyst therefore evaluates conclusions across clinically interpretable choices of \((\delta_1,\delta_0)\).

How the Two Shifts Affect the Arm Contrast

Let \(q_a=P(R=0\mid A=a)\) be the proportion of outcomes missing in arm \(a\). Let \(\mu_a^{\text{MAR}}\) be the arm mean under the MAR reference, combining observed outcomes with the reference predictions for missing outcomes. The arm-specific additive model changes only the missing-outcome component. Consequently, the adjusted arm mean changes by the missing fraction times the arm's delta:

To see why, write the reference arm mean as the weighted mean of its observed and missing components. The missing component receives weight \(q_a\). Adding \(\delta_a\) to that component therefore adds \(q_a\delta_a\) to the overall mean; the observed component is unchanged. This identity is useful for understanding the direction and size of the sensitivity effect. It is not a claim that the delta parameters or reference means are estimated without uncertainty.

Define the treatment contrast as treatment minus control, \(\tau=\mu_1-\mu_0\), so a positive contrast means a higher mean outcome in the treatment arm. The adjusted contrast is

This equation separates the assumptions from their consequences. The pair \((\delta_1,\delta_0)\) specifies departures in the two arms; the missing fractions determine how strongly those departures affect the overall contrast. In particular, equal shifts do not necessarily cancel. If \(\delta_1=\delta_0=\delta\), the contrast changes by \((q_1-q_0)\delta\), which is zero only when the missing fractions are equal or the common shift is zero.

Interpreting Unequal and Oppositely Signed Shifts

Unequal shifts express different assumptions about the missing outcomes in the two arms. For example, a more negative treatment-arm delta than control-arm delta says that treatment-arm missing outcomes are assumed to be worse relative to their own MAR reference. It does not, by itself, say that their absolute outcomes are worse than control outcomes: the two arms can have different MAR reference distributions.

Oppositely signed shifts make a stronger directional contrast between the assumptions. If \(\delta_1<0\) and \(\delta_0>0\), missing treatment outcomes are shifted downward from their reference and missing control outcomes upward. If higher outcomes are better, this specification makes the treatment arm less favorable relative to control than the MAR reference does. The interpretation reverses if lower outcomes are better. Always state the direction in terms of both the reference and the outcome scale.

Choosing and Checking an Arm-Specific Sensitivity Analysis

A useful analysis begins by stating what each delta means in context. Specify the outcome scale, identify which arm each parameter applies to, and describe a positive or negative shift relative to that arm's reference. Then choose a range of pairs \((\delta_1,\delta_0)\) that includes the MAR reference \((0,0)\) and scenarios justified for sensitivity assessment. Reporting only the numerical pair is insufficient if readers cannot tell whether a shift represents improvement or worsening.

The same parameter value can have different implications across arms because the missing fractions can differ. More generally, the contrast's sensitivity to \(\delta_1\) is \(q_1\), and its sensitivity to \(\delta_0\) is \(-q_0\). For example, increasing \(\delta_1\) by one outcome unit raises the adjusted contrast by \(q_1\) units; increasing \(\delta_0\) by one unit lowers it by \(q_0\) units. These are algebraic consequences, not evidence about which shifts are plausible.

A common alternative is a single shared delta, \(\delta_1=\delta_0\). It is simpler and imposes the assumption that both arms depart from their MAR references by the same amount on the selected scale. Arm-specific deltas relax that restriction. Reference-based methods such as J2R and Copy-Reference encode their own reference-distribution assumptions; they are not interchangeable with arbitrary arm-specific additive shifts. The choice depends on the scientific question and the assumptions investigators wish to examine.

Several checks help identify a poorly specified analysis:

• Check the direction and units. A delta measured in score points must not be interpreted as a percentage or standardized effect. Confirm whether higher outcomes indicate benefit or harm.
• Check the arm labels. Swapping \(\delta_1\) and \(\delta_0\) changes the scenario. Write out which parameter applies to treatment and which to control.
• Check the missing fractions. A shared delta cancels from the contrast only if \(q_1=q_0\) (or the shared delta is zero). Recalculate the weighted effects rather than reasoning from the word “shared.”
• Check the reference distributions. A delta is a departure from a reference, not an absolute mean for missing participants. A shift that seems reasonable in one arm may not be reasonable in the other if their reference means differ.
• Check plausibility across the chosen range. Extreme shifts can imply implausible outcome values or conclusions. Such results may reveal that a scenario is not meaningful on the chosen scale; they do not establish the correct delta.

Arm-specific deltas do not solve every missing-data problem. They do not make the unobserved outcomes empirically identifiable, and a constant shift within each arm may be too restrictive if departures vary across people or other characteristics. The reference model also matters: a sensitivity shift cannot repair a reference distribution that is poorly specified. Results should therefore be presented as conditional on both the reference model and the chosen delta pair, with enough scenarios to show how conclusions respond to those assumptions.