Delta Adjustments for Interaction Contrasts

Why an Interaction Needs a Contrast-Specific Sensitivity Analysis

A delta adjustment can change an estimated treatment effect, but its effect on an interaction is not necessarily the same. An interaction asks whether a treatment contrast differs between subgroups. Consequently, the change depends on the shifts in all four treatment-by-subgroup cells and on how many outcomes were missing in each cell. A shift that moves both subgroup-specific treatment contrasts equally can leave the interaction unchanged; unequal missingness can prevent that cancellation even when the same shift is used in both subgroups.

The purpose of this analysis is to quantify that dependence for a prespecified sensitivity scenario. It does not identify the unobserved outcomes or establish that any chosen delta is correct. Rather, it shows how the interaction estimate would change if missing outcomes departed from their MAR imputations by specified amounts in specified cells.

Definition: Let $A$ denote treatment ($A=1$ or $A=0$), $G$ denote subgroup ($G=1$ or $G=0$), and $Y$ be the outcome. Write $\mu_{ag}$ for the completed-data mean outcome in treatment cell $a$, subgroup cell $g$. On the additive outcome scale, the interaction contrast is $$ \theta=(\mu_{11}-\mu_{01})-(\mu_{10}-\mu_{00}) =\mu_{11}-\mu_{01}-\mu_{10}+\mu_{00}. $$ It is the treatment contrast in subgroup 1 minus the treatment contrast in subgroup 0. Its units are the units of $Y$.

How Cell-Specific Deltas Change the Contrast

Suppose a completed-data analysis under MAR gives cell mean $\bar{Y}^{\,MAR}_{ag}$. Let $n_{ag}$ be the number of individuals in that cell and $m_{ag}$ the number whose outcome is missing. Define the missing fraction $q_{ag}=m_{ag}/n_{ag}$. For an additive delta scenario, add $\delta_{ag}$ to each MAR-imputed missing outcome in cell $(a,g)$, leaving observed outcomes unchanged. Here $\delta_{ag}$ is measured in the units of $Y$.

Because only the fraction $q_{ag}$ of the cell receives the shift, the adjusted cell mean is the MAR completed-data mean plus $q_{ag}\delta_{ag}$. Substituting those four adjusted means into the interaction contrast gives the central calculation:

\theta^{\delta} =\theta^{MAR} +q_{11}\delta_{11}-q_{01}\delta_{01} -q_{10}\delta_{10}+q_{00}\delta_{00}.

The signs come from the definition of the interaction contrast, not from whether a delta is positive or negative. A positive shift in cell $(1,1)$ increases the contrast; the same shift in $(0,1)$ decreases it. A shift in cell $(0,0)$ increases it because that cell has a positive sign in the contrast.

Assumptions for this calculation: The target is the stated additive interaction in the cell means. The baseline imputation model is appropriate under its MAR assumption conditional on included observed information; the delta scenario then specifies the departure for missing outcomes. Within each cell, the additive shift is applied to missing outcomes as stated, and observed outcomes are not shifted. Cell definitions, outcome scale, and delta values are prespecified and interpreted consistently. The calculation describes that scenario; it does not prove MAR or the delta assumptions.

The formula also clarifies how simpler delta schemes behave. If deltas depend only on treatment, so that $\delta_{ag}=\delta_a$, the change is

\theta^{\delta}-\theta^{MAR} =\delta_1(q_{11}-q_{10})-\delta_0(q_{01}-q_{00}).

Thus, an arm-specific shift does not automatically cancel from the interaction. It cancels for each arm when that arm's missing fraction is equal across the two subgroups. If all four cells have the same missing fraction and the delta is common to every cell, the four signed contributions cancel. But a common delta can still change the contrast when the missing fractions differ across cells.

Worked Cell-Mean Calculations

Worked Example: Treatment-Arm Deltas with Unequal Missingness

Hypothetical data. Consider a continuous outcome measured in points, with 100 people in each treatment-by-subgroup cell. Suppose the MAR completed-data cell means and missing counts are as follows: subgroup 0 has mean 50 for control and 55 for treatment, with 20 and 30 missing outcomes, respectively; subgroup 1 has mean 52 for control and 62 for treatment, with 10 and 40 missing outcomes, respectively. These completed-data means are the starting point for the sensitivity calculation.

The MAR interaction is the treatment difference in subgroup 1 minus that in subgroup 0:

\theta^{MAR}=(62-52)-(55-50)=10-5=5\text{ points}.

Now suppose the sensitivity scenario adds 4 points to missing treated outcomes in both subgroups and adds 0 points to missing control outcomes. The missing fractions are $q_{00}=20/100=0.20$, $q_{10}=30/100=0.30$, $q_{01}=10/100=0.10$, and $q_{11}=40/100=0.40$. The arm-specific formula gives

\theta^{\delta}-\theta^{MAR} =4(0.40-0.30)-0(0.10-0.20) =4(0.10)=0.40\text{ points}.

Therefore, $\theta^{\delta}=5+0.40=5.40$ points. Directly, the adjusted means are 50, $55+0.30(4)=56.2$, 52, and $62+0.40(4)=63.6$, in cells $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. The adjusted contrast is $63.6-52-56.2+50=5.4$ points. The common 4-point shift in the treatment arm does not cancel because the treated subgroup missing fractions differ.

Worked Example: Subgroup-Specific Deltas

Hypothetical scenario. Use the same MAR means and missing fractions as above. Suppose missing outcomes in subgroup 1 receive a $+2$-point delta in both treatment arms, while missing outcomes in subgroup 0 receive a $-1$-point delta in both arms. The change in the interaction is

\begin{aligned} \theta^{\delta}-\theta^{MAR} &=0.40(2)-0.10(2)-0.30(-1)+0.20(-1)\\ &=0.80-0.20+0.30-0.20\\ &=0.70\text{ points}. \end{aligned}

Thus the adjusted interaction is $5+0.70=5.70$ points. The cell-mean check gives adjusted means $50+0.20(-1)=49.8$, $55+0.30(-1)=54.7$, $52+0.10(2)=52.2$, and $62+0.40(2)=62.8$. The resulting contrast is $62.8-52.2-54.7+49.8=5.7$ points.

A subgroup-specific delta is applied to both treatment arms in a subgroup, but its effect on the treatment contrast can differ across subgroups because the missing fractions differ. Here the adjustment increases the interaction by 0.70 points; the arithmetic, not the label “subgroup-specific,” determines the result.

From the Formula to an Imputation Analysis

The cell-mean expression is a useful check and, when the analysis is a saturated comparison of cell means, directly gives the adjusted contrast. In a general MICE analysis, however, the analysis may include covariates, additional terms, or a nonlinear model. The sensitivity calculation should then be implemented on the missing outcomes in each imputed data set, followed by the prespecified complete-data analysis. Estimate the interaction contrast and its variance separately in each adjusted imputed data set, then pool the estimates and variances using the usual multiple-imputation combining procedure. Apply the same fixed delta scenario to each imputation; do not shift observed outcomes.

Specify the target.
Define the treatment-by-subgroup interaction and its scale, such as a difference of mean outcomes or a difference of risk differences.

Set the sensitivity scenario.
Specify the delta for each relevant treatment-by-subgroup cell and state the units or transformed scale. Identify which variable's missing values receive the adjustment.

Adjust and analyze each imputed data set.
Apply each cell's delta only to its imputed missing outcomes, then fit the same substantive model and calculate the target contrast.

Pool and inspect.
Pool the contrast and its uncertainty across imputations. Check that the result agrees with a cell-mean calculation when that calculation applies, and examine how conclusions vary across prespecified scenarios.

The imputation and analysis models must represent the interaction relevant to the scientific question. For example, if subgroup-specific treatment effects are the target, an imputation model that omits information needed to preserve the treatment-by-subgroup relationship can distort the MAR baseline before any delta is applied. The delta calculation cannot repair a misspecified imputation model.

Worked Example: A Marginal Risk Interaction

Hypothetical binary-outcome data. Suppose the MAR completed-data risks are 0.10, 0.16, 0.20, and 0.32 in cells $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$, respectively. The corresponding missing fractions are 0.20, 0.30, 0.10, and 0.40. Define the interaction on the probability scale as the treatment risk difference in subgroup 1 minus that in subgroup 0. Its MAR value is

\theta^{MAR}=(0.32-0.20)-(0.16-0.10) =0.12-0.06=0.06,

or 6 percentage points. Consider an explicitly probability-scale scenario in which the deltas for missing outcomes in cells $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$ are 0, 0.02, 0.01, and 0.05, respectively. The weighted change is

\begin{aligned} \theta^{\delta}-\theta^{MAR} &=0.40(0.05)-0.10(0.01)-0.30(0.02)+0.20(0)\\ &=0.0200-0.0010-0.0060\\ &=0.0130. \end{aligned}

The adjusted interaction is $0.060+0.013=0.073$, or 7.3 percentage points. This calculation assumes an additive probability-scale shift for missing outcomes; the resulting probabilities must remain between 0 and 1. It is not the interaction coefficient from a logistic regression, which is on the log-odds scale and is a different estimand.

Diagnostics, Alternatives, and Limitations

Before interpreting the adjusted contrast, inspect the quantities that drive it. Report or tabulate missing fractions by treatment and subgroup, because large differences can amplify or prevent cancellation. Check that each cell has enough observed information to support the MAR imputation and that the imputation model retains the relationships needed for the analysis. Verify the actual adjusted values, especially for outcomes with natural bounds. A probability-scale shift that creates values below 0 or above 1 is not a coherent probability scenario; use a suitable scale and ensure the implied values are valid.

For nonlinear analyses, distinguish the model coefficient from the desired interaction contrast. A product term in a logit model is an interaction on the log-odds scale; it is not generally equal to a difference of subgroup-specific risk differences. If the scientific target is a marginal risk interaction, calculate the four marginal risks under the specified model and form the probability-scale contrast. Apply deltas and assess validity on the scale that matches the stated sensitivity assumption.

A common alternative is a single delta applied to every missing outcome. It is simpler, but it encodes a different assumption from treatment- or subgroup-dependent deltas. Another alternative is to specify deltas by treatment arm only or subgroup only. Those schemes may be useful when they reflect the substantive concern, but the formula shows that their effects still depend on the cell-specific missing fractions. None is automatically preferable; the choice should reflect a defensible scenario about missing outcomes.

Delta sensitivity analysis has important limits. Delta values are not estimated from the observed data without additional assumptions or external information. A fixed delta scenario also treats the chosen values as given; uncertainty about plausible delta values is not automatically included in the pooled standard error. Further, an additive shift may be implausible for some outcome distributions, and a constant shift may not represent departures that vary within cells. These limitations motivate examining multiple defensible scenarios rather than treating one adjusted estimate as definitive.

Common pitfall: “The same treatment-arm delta is used in both subgroups, so it must cancel from the interaction.” This is only true for that arm's contribution when its missing fraction is equal in both subgroups. In general the contribution is $\delta_a(q_{a1}-q_{a0})$, so unequal missingness leaves a nonzero change.

Check Your Understanding

For each question, calculate the requested result before reading its solution.

In the additive outcome setting, state the signs of the four cell adjustments in the interaction contrast.
A common delta of 3 points is applied to missing outcomes in every cell. The missing fractions $q_{11},q_{01},q_{10},q_{00}$ are 0.30, 0.20, 0.10, and 0.20. By how much does the interaction change?
An arm-specific delta $\delta_1=2$ points is applied to treated missing outcomes, while $\delta_0=0$. The treated missing fractions are $q_{11}=0.25$ and $q_{10}=0.15$. What is the interaction change?
Why should a logistic-regression treatment-by-subgroup coefficient not automatically be interpreted as a probability-scale interaction?

Solutions. 1. The contrast is $\mu_{11}-\mu_{01}-\mu_{10}+\mu_{00}$, so the signs are $+,-,-,+$. 2. The change is $3(0.30-0.20-0.10+0.20)=3(0.20)=0.60$ points. 3. The change is $2(q_{11}-q_{10})=2(0.25-0.15)=0.20$ points. 4. The logistic coefficient describes a product term on the log-odds scale; a probability-scale interaction is a difference of risk differences and must be calculated from probabilities. These are distinct contrasts.