Why Shift a Binary Reference Risk?
In a pattern-mixture sensitivity analysis, a reference distribution supplies a starting point for outcomes that are missing. For a binary outcome, that starting point can be expressed as a probability: for example, the reference risk of an event among participants with missing outcomes. A risk-difference delta changes that probability by a specified additive amount. It therefore describes departure from the reference distribution on the probability scale, in percentage points.
This scale has a direct interpretation, but it imposes a strict constraint: probabilities cannot be below zero or above one. A proposed delta that violates this constraint is not a valid additive risk-difference shift for that reference risk. The adjustment must be checked against the reference probability, not just applied mechanically.
The interpretation depends on how the event is defined. If \(Y=1\) means a harmful event, a negative delta means a lower event probability than under the reference distribution, while a positive delta means a higher probability. If \(Y=1\) means a beneficial response, the clinical direction reverses. State what \(Y=1\) represents before describing whether a shift is favorable or unfavorable.
The shift is applied to the reference risk for the missing outcomes under consideration. It does not alter observed outcomes. When a model supplies a different reference risk for different participants or covariate profiles, the same proposed delta must be checked at each risk to which it is applied.
Probability Bounds and the Feasible Delta Range
A valid adjusted probability must lie between zero and one, inclusive. Starting with \(p_{\delta}=p_{\mathrm{ref}}+\delta\), require \(0\leq p_{\delta}\leq1\). Subtracting \(p_{\mathrm{ref}}\) from all parts gives the allowable range:
The limits depend on the reference risk. For example, when \(p_{\mathrm{ref}}=0.20\), a delta cannot be less than \(-0.20\), because that would produce a negative risk. Nor can it exceed \(0.80\), because that would produce a risk above one. At the boundary \(p_{\mathrm{ref}}=0\), no negative delta is feasible; at \(p_{\mathrm{ref}}=1\), no positive delta is feasible.
This is a feasibility condition on the model, not a statistical test. It does not determine which feasible delta is scientifically plausible. Rather, substantive judgment may select a sensitivity value, after which the probability bounds reveal whether that value can represent the stated additive shift for the relevant reference risks.
Worked Example: Interpreting a Feasible Risk-Difference Shift
Suppose the reference event risk for a hypothetical group with missing outcomes is \(p_{\mathrm{ref}}=0.35\). Consider an additive delta of \(\delta=-0.10\). Assume the event is an adverse outcome.
Check feasibility. The allowable range is \(-p_{\mathrm{ref}}\leq\delta\leq1-p_{\mathrm{ref}}\), or \(-0.35\leq\delta\leq0.65\). Since \(-0.10\) is in this range, the shift is feasible.
Calculate the adjusted risk. \(p_{\delta}=p_{\mathrm{ref}}+\delta=0.35+(-0.10)=0.25\). Thus, the adjusted event probability is 0.25, or 25%.
Interpret the result. The assumed event risk for these missing outcomes is 10 percentage points lower than the reference risk: \(25\%-35\%=-10\) percentage points. This is not a 10% relative reduction. The relative reduction would be \(0.10/0.35\), approximately 28.6%, which is a different comparison.
From Adjusted Probabilities to Binary Imputations
A probability is not itself a binary outcome. If the sensitivity analysis uses stochastic imputation, an imputed outcome can be drawn from a Bernoulli distribution with the adjusted probability: \(Y_{\mathrm{mis}}\sim\operatorname{Bernoulli}(p_{\delta})\). This means the imputed value is 1 with probability \(p_{\delta}\) and 0 with probability \(1-p_{\delta}\). If the analysis instead uses expected outcomes, the adjusted probability is the expected value of the binary outcome because \(E(Y_{\mathrm{mis}})=p_{\delta}\).
The delta specifies a change in the probability used for missing outcomes; it does not require every imputed value to change by the same amount. A binary draw remains either 0 or 1. Consequently, the realized number of imputed events can differ from its expectation even when the adjustment is correctly applied.
Worked Example: Expected Events and Stochastic Imputation
In a hypothetical analysis, 40 participants have missing binary outcomes. Suppose the reference risk for each is 0.35 and the chosen risk-difference delta is \(-0.10\). Assume the same risk applies to all 40 participants and the adjusted outcomes are drawn independently conditional on the specified probability.
The adjusted probability is \(0.35-0.10=0.25\), which is within the feasible range. The expected number of events among the 40 missing outcomes is \(40(0.25)=10\). Under the reference risk, the corresponding expected number would be \(40(0.35)=14\). The difference in expected event counts is \(10-14=-4\), matching a reduction of \(40(0.10)=4\) events in expectation.
With stochastic imputation, the imputed number of events need not equal 10; it is a random count. The calculation gives its expectation, not a promise that exactly 10 events will be imputed. The model’s probability shift is 10 percentage points for each participant in this example.
Checking Bounds Across Reference Risks
A delta may be feasible for one reference risk and infeasible for another. If the analysis applies a common delta across participants whose reference risks vary, the proposed value must satisfy the probability bounds for every applicable risk. In particular, a negative delta that is valid at a moderate risk may produce a negative adjusted probability at a low reference risk.
Worked Example: A Common Delta at Two Reference Risks
Consider two hypothetical covariate profiles with reference event risks of 0.12 and 0.58. A common risk-difference delta of \(-0.10\) is proposed for missing outcomes in both profiles. Assume the same event definition and the same additive shift are intended in each profile.
For the first profile, \(p_{\delta}=0.12+(-0.10)=0.02\), which is valid. Its allowable delta range is \([-0.12,0.88]\), so \(-0.10\) is feasible. For the second profile, \(p_{\delta}=0.58+(-0.10)=0.48\), also valid; its allowable range is \([-0.58,0.42]\).
The adjusted risks are 2% and 48%, respectively. Each is 10 percentage points below its own reference risk. If instead the proposed common delta were \(-0.15\), the first profile would give \(0.12-0.15=-0.03\), which is impossible, even though the second profile would give \(0.58-0.15=0.43\). A common additive delta of \(-0.15\) therefore cannot be applied to both profiles as stated.
A useful diagnostic is to calculate the lower and upper feasible limits for the smallest and largest reference risks, or directly check every risk when there are many. For a common negative delta, the smallest reference risk is often the limiting one; for a common positive delta, the largest reference risk can be limiting. If reference risks are estimated rather than known, the analyst should also consider whether uncertainty in those estimates could make the proposed adjustment infeasible in relevant parts of the analysis.
Derivation and Model Interpretation
The risk-difference scale is defined by subtracting the reference risk from the adjusted risk. Write the difference as \(\delta=p_{\delta}-p_{\mathrm{ref}}\). Rearranging gives \(p_{\delta}=p_{\mathrm{ref}}+\delta\). Since a Bernoulli probability must satisfy \(0\leq p_{\delta}\leq1\), substitution yields \(0\leq p_{\mathrm{ref}}+\delta\leq1\). Subtracting \(p_{\mathrm{ref}}\) proves the stated bounds, \(-p_{\mathrm{ref}}\leq\delta\leq1-p_{\mathrm{ref}}\). Thus, the limits follow directly from the definition of a probability, not from an approximation.
The model also makes a substantive assumption: the chosen number \(\delta\) represents the intended departure from the reference risk on the absolute probability scale for the missing outcomes to which it is applied. That assumption is a sensitivity specification, not something established merely by observing the data. The reference risk and the definition of the missing-outcome group must be clearly stated, and the delta must be chosen and reported in a way that readers can interpret.
State what \(Y=1\) means and identify the reference probability for the missing outcomes under analysis.
Interpret its sign and magnitude as an absolute change in event probability, expressed in percentage points when reporting percentages.
Calculate \(p_{\delta}=p_{\mathrm{ref}}+\delta\) and verify that it lies in \([0,1]\) for each reference risk to which the delta applies.
For stochastic imputation, draw binary outcomes using that probability; do not alter observed outcomes or treat the expected event count as a realized count.
Comparisons, Diagnostics, and Limitations
A risk-difference delta is one possible scale for expressing departure from a reference distribution. With \(\delta=0\), the adjusted risk equals the reference risk. A log-odds delta instead adds a shift on the log-odds scale, as covered in the earlier material on log-odds delta adjustments for binary outcomes. Because odds are nonlinear functions of risk, a fixed log-odds shift generally does not produce a constant risk difference across different reference risks. Conversely, a fixed risk-difference shift does not represent a constant odds ratio. The scales answer different sensitivity questions; one is not automatically preferable in every application.
| Approach | What is held fixed | Key check or interpretation |
|---|---|---|
| No departure from reference | The adjusted risk equals \(p_{\mathrm{ref}}\) | Equivalent to a zero risk-difference delta |
| Risk-difference delta | An absolute probability difference | Interpret in percentage points; verify every adjusted risk is between 0 and 1 |
| Log-odds delta | A shift on the log-odds scale | Risk differences depend on the starting risk; interpret on the odds scale |
A common error is to compute a negative adjusted probability and then replace it by zero, or to replace a value above one by one, while still reporting the originally specified delta. For example, if \(p_{\mathrm{ref}}=0.08\) and \(\delta=-0.12\), the additive formula gives \(-0.04\), which is not a probability. Replacing it with zero changes the realized difference to \(-0.08\), not \(-0.12\). This clipping rule is a different adjustment and should not be presented as the original constant risk-difference shift. If a bounded alternative is intended, it must be specified and justified as a different model rather than silently substituted.
Other warning signs include interpreting a risk difference as a relative percentage change, failing to identify whether the event is beneficial or harmful, applying the shift to observed outcomes, or checking bounds at only one risk when the adjustment is used across multiple risks. A valid probability calculation does not by itself demonstrate that the selected delta is clinically plausible. Sensitivity results should therefore report the reference risk, the delta, the adjusted risk, the event definition, and any constraints that limit the range of values considered.
The additive risk-difference model is transparent but has a boundary-dependent range: the same delta cannot be feasible for every possible reference risk. It also imposes the same absolute change wherever a common delta is used, which may or may not be a credible assumption. The choice of scale and sensitivity values remains substantive. The bounds check ensures mathematical coherence; it does not resolve uncertainty about the unobserved outcomes.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A reference event risk is 0.42 and \(\delta=0.06\). What is the adjusted risk, and how should the shift be interpreted?
- For \(p_{\mathrm{ref}}=0.07\), is \(\delta=-0.09\) feasible? If not, what is the feasible range?
- Two groups have reference risks 0.15 and 0.70. Is a common delta of \(-0.12\) feasible for both? Give both adjusted risks or identify the failure.
- Why is replacing an adjusted risk of \(-0.04\) by zero not equivalent to preserving the originally specified risk-difference delta?
Solutions. (1) \(p_{\delta}=0.42+0.06=0.48\). The event risk is 6 percentage points higher than the reference risk. (2) The feasible range is \([-0.07,0.93]\), so \(-0.09\) is infeasible and would give \(0.07-0.09=-0.02\). (3) The first adjusted risk is \(0.15-0.12=0.03\); the second is \(0.70-0.12=0.58\). Both lie in \([0,1]\), so the common delta is feasible for these risks. (4) If the reference risk is 0.08 and the proposed delta is \(-0.12\), the stated additive formula gives \(-0.04\). Replacing that value by zero yields an actual difference of \(0-0.08=-0.08\), not \(-0.12\); clipping therefore changes the model.