Why the Reference Group Matters
When sensitivity assumptions vary by group, a relative delta contrast describes how the shift for one group differs from the shift for a chosen reference group. The reference therefore determines the interpretation of the coordinates: a contrast of 2 means that one group’s shift is 2 scale units greater than the reference group’s shift. It does not, by itself, say what either group’s shift is relative to the original imputation model.
This distinction matters when changing references. Re-expressing a fixed set of group-specific shifts using a different reference changes the contrast values but need not change any imputations. However, if the new reference group is simply assigned a shift of zero and the other shifts are left at their old numerical values, the shifts have changed in absolute terms. That is a different sensitivity scenario.
The scale is part of the definition. For a continuous outcome, \(\eta_g\) might be an additive shift in outcome units. For a binary outcome, it might be a shift in log odds. These are different parameterizations and interpretations; contrasts must be calculated on the scale on which the delta adjustment is specified. The observed-data model, the groups, and the chosen scale are held fixed when comparing reference groups.
Changing Coordinates Without Changing the Scenario
Suppose a fixed sensitivity scenario has absolute shifts \(\eta_1,\ldots,\eta_G\). With reference group \(r\), report the anchor \(\eta_r\) and contrasts \(\theta_g^{(r)}=\eta_g-\eta_r\). If the reference changes to group \(s\), then
The anchor also changes coordinates: \(\eta_s=\eta_r+\theta_s^{(r)}\). Thus, the absolute shift for each group remains \(\eta_g=\eta_r+\theta_g^{(r)}=\eta_s+\theta_g^{(s)}\). This is a re-expression of one vector of shifts, not a refitting of a different sensitivity assumption.
The derivation is simple but important. A relative contrast subtracts the reference shift from each group shift. Changing the reference from \(r\) to \(s\) means subtracting \(\eta_s\) instead. Since \(\eta_s=\eta_r+\theta_s^{(r)}\), the new contrast is the old contrast minus \(\theta_s^{(r)}\). Adding the new anchor back recovers the original absolute shift for every group. The common shift is not contained in the contrasts: adding the same constant to all \(\eta_g\) leaves every relative contrast unchanged.
Worked Example: Re-referencing Continuous-Outcome Shifts
Suppose hypothetical missing continuous outcomes in groups A, B, and C have baseline imputation-model conditional means of 10, 12, and 8 units, respectively. Consider the fixed absolute shifts \(\eta_A=0\), \(\eta_B=2\), and \(\eta_C=-1\) outcome units. For this illustrative location-shift calculation, the shifted conditional means are the baseline means plus the shifts.
Using A as the reference. The anchor is \(\eta_A=0\). The relative contrasts are \(\theta_A^{(A)}=0-0=0\), \(\theta_B^{(A)}=2-0=2\), and \(\theta_C^{(A)}=-1-0=-1\). The resulting means are \(10+0=10\) units in A, \(12+2=14\) units in B, and \(8-1=7\) units in C.
Using B as the reference. The anchor is now \(\eta_B=2\). The contrasts are \(\theta_A^{(B)}=0-2=-2\), \(\theta_B^{(B)}=2-2=0\), and \(\theta_C^{(B)}=-1-2=-3\). Adding the anchor to each contrast recovers the same absolute shifts: \(2+(-2)=0\) for A, \(2+0=2\) for B, and \(2+(-3)=-1\) for C. The shifted means remain 10, 14, and 7 units.
If one instead sets B’s absolute shift to zero and applies the other new-reference contrasts \(-2\) and \(-3\) directly, the shifts become \(-2,0,-3\). The means would then be \(10-2=8\), \(12+0=12\), and \(8-3=5\) units. This is a common shift of \(-2\) units from the original scenario, not merely a change of reference.
Scale-Specific Interpretation
The reference transformation is algebraic and applies on the scale of the delta parameter. What changes with the scale is what a contrast represents and how a specified absolute shift modifies an imputation. For instance, a log-odds contrast is a difference between log-odds shifts, not a difference between probabilities. The following example keeps the same absolute log-odds shifts while changing the reference.
Worked Example: Re-referencing Log-Odds Shifts
Suppose hypothetical baseline imputation probabilities are \(p_{0,A}=0.20\), \(p_{0,B}=0.30\), and \(p_{0,C}=0.40\). Set the absolute log-odds shifts to \(\eta_A=0\), \(\eta_B=\log(2)\), and \(\eta_C=-\log(2)\). The adjusted probability for group \(g\) is \(p_g=\operatorname{expit}\{\operatorname{logit}(p_{0,g})+\eta_g\}\), where \(\operatorname{expit}(x)=1/(1+\exp(-x))\).
For A, the baseline odds are \(0.20/0.80=0.25\). The shift is zero, so the adjusted odds remain \(0.25\), giving \(p_A=0.25/(1+0.25)=0.20\).
For B, the baseline odds are \(0.30/0.70=0.4286\). Adding \(\log(2)\) to the log odds multiplies the odds by 2, giving adjusted odds \(0.4286(2)=0.8571\) and \(p_B=0.8571/(1+0.8571)=0.4615\).
For C, the baseline odds are \(0.40/0.60=0.6667\). Adding \(-\log(2)\) multiplies the odds by \(1/2\), giving adjusted odds \(0.6667(0.5)=0.3333\) and \(p_C=0.3333/(1+0.3333)=0.2500\).
With B as the reference, the contrasts are \(\theta_A^{(B)}=-\log(2)\), \(\theta_B^{(B)}=0\), and \(\theta_C^{(B)}=-2\log(2)=\log(0.25)\). The anchor remains \(\eta_B=\log(2)\). For example, B’s anchor plus C’s contrast is \(\log(2)+\log(0.25)=\log(0.5)=\eta_C\). Thus, the three adjusted probabilities stay 0.20, 0.4615, and 0.2500. The contrasts are differences in log-odds shifts; they are not differences in adjusted probabilities.
Choosing and Reporting a Reference
A reference is a choice of coordinates, not a scientific finding. A useful reference often has a substantive role, such as the usual-care group, or makes the contrasts easy to communicate. It can also be selected for a practical reason, such as having a well-understood delta anchor. That choice should not be confused with the reference category in an analysis model: the two choices may coincide, but they serve different purposes.
For a specification based only on relative contrasts, explicitly state how the common shift is anchored. For example, one may set the reference group’s absolute shift to zero, or specify a separate common shift and report contrasts around it. These choices answer different questions. A zero shift for the reference group means its missing outcomes receive no delta adjustment relative to the specified baseline imputation model. A zero contrast for the reference group means only that it is being compared with itself; it does not imply that its absolute shift is zero.
A complete report should identify the groups, the delta scale, the reference, the contrasts, and the anchor. If a scenario is translated to a new reference, report the transformation or enough information to reconstruct the same \(\eta_g\) values. This lets readers distinguish a change in wording from a change in assumptions.
| What is reported | What it tells the reader |
|---|---|
| Scale and units | How a delta contrast is measured and how it enters the imputation model |
| Reference group | Which group’s shift is subtracted to form each relative contrast |
| Relative contrasts | How each group’s shift differs from the reference shift |
| Common-shift anchor | The absolute shift needed, together with the contrasts, to recover the scenario |
Diagnostics and Common Failure Modes
Before interpreting results, check that the reference transformation preserves the intended absolute shifts. For each group, recover \(\eta_g\) by adding the anchor to its contrast. Then verify that the imputation procedure applies that shift on the stated scale and to the intended group. In chained-equations implementations, inspect the group-specific imputed values or probabilities after adjustment; the labels and contrasts alone do not guarantee that the programmed shift was applied as intended.
- Resetting the new reference to zero without translating the anchor. This subtracts the old reference contrast from every absolute shift and produces a different scenario, unless that contrast is already zero.
- Comparing contrast numbers across references as if they were absolute shifts. A contrast changes when its reference changes. Recover the absolute shifts before deciding whether scenarios differ.
- Mixing scales. A contrast specified in log odds cannot be interpreted as a probability difference. Check that the transformation and interpretation use the same scale.
- Leaving out a group or using an inconsistent group definition. Every group whose imputed values receive a distinct shift must be included in the mapping, with the same membership definition before and after re-referencing.
These checks rely on the stated delta model being suitable for the intended sensitivity analysis: the scale and group-specific shift are defined in advance, the same baseline imputation model is used, and the adjustment is applied consistently to the relevant missing values. The shift parameterization does not establish that a particular delta is scientifically plausible. Nor can relative contrasts alone determine the common level of all the shifts. For bounded outcomes, transformations and range constraints may also affect the realized values, so verify the actual imputed distribution rather than assuming that every raw additive shift remains possible without modification.
A practical alternative is to report absolute shifts \(\eta_g\) directly, avoiding reference-dependent coordinates. That is often clearest for a small number of groups. Relative contrasts are useful when comparisons to a substantively meaningful group are the main aim, or when many groups make pairwise comparisons useful. Either representation can describe the same scenario if all required information is retained. Neither is inherently more valid; the important point is to avoid treating a coordinate change as a changed or unchanged scenario without checking the anchor.
Common Mistake
Wrong reasoning: “I changed the reference from A to B, so I set B’s delta to zero and kept the other delta values. The sensitivity scenario is unchanged because only the reference label changed.” This is not generally correct. Setting B’s absolute shift to zero changes the common level of the shifts. In the continuous example, the shifts changed from \((0,2,-1)\) to \((-2,0,-3)\), so all shifted means decreased by 2 units.
Correction: First decide whether the aim is to re-express an existing scenario or define a new one. To re-express it, transform every contrast using \(\theta_g^{(s)}=\theta_g^{(r)}-\theta_s^{(r)}\) and carry forward the new anchor \(\eta_s=\eta_r+\theta_s^{(r)}\). To define a scenario with zero absolute shift in the new reference group, say so: that is a new sensitivity assumption unless the old scenario already had zero shift there.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- For three groups, the absolute shifts are \((\eta_A,\eta_B,\eta_C)=(1,4,2)\) units. What are the contrasts with B as reference, and what is the anchor?
- Using the contrasts from question 1, translate to C as reference. What are the new contrasts and anchor?
- If relative contrasts are known but no anchor is specified, can the absolute shift for each group be recovered? Explain.
- For a binary outcome, a delta contrast is 0.5 on the log-odds scale. Is this a 0.5 probability-point difference? Explain.
- When would changing the reference while setting the new reference’s absolute shift to zero leave the scenario unchanged?
Solutions. (1) The contrasts are \(\theta_A^{(B)}=1-4=-3\), \(\theta_B^{(B)}=0\), and \(\theta_C^{(B)}=2-4=-2\) units; the anchor is \(\eta_B=4\) units. (2) Subtract \(\theta_C^{(B)}=-2\) from each old contrast: \(\theta_A^{(C)}=-3-(-2)=-1\), \(\theta_B^{(C)}=0-(-2)=2\), and \(\theta_C^{(C)}=0\). The anchor is \(\eta_C=4+(-2)=2\) units. Recovering the shifts gives \(2+(-1)=1\), \(2+2=4\), and \(2+0=2\), as before. (3) No. Adding the same constant to every absolute shift leaves all contrasts unchanged, so an anchor is required. (4) No. It is a difference between log-odds shifts; its probability-scale consequence depends on the baseline probability. (5) It leaves the scenario unchanged if the new reference group’s absolute shift was already zero under the original scenario; otherwise resetting it to zero changes the shifts.