Why Express a Delta in Two Scales?
An additive delta specifies how far the assumed distribution of missing outcomes is shifted from its MAR reference distribution. Its outcome-unit form is often easiest to explain clinically: for example, “the missing outcomes are assumed to be 4 points lower.” A standardized form expresses the same shift relative to residual variability: “the shift is one-half of a residual standard deviation.” Each scale answers a different question. Clinical units communicate the size of the assumed departure on the endpoint; residual-SD units help describe its magnitude relative to variability.
These are two parameterizations of a shift, not two different missing-data mechanisms. The conversion is straightforward, but it is only interpretable when the denominator is specified. In particular, the residual standard deviation is not automatically the overall sample standard deviation, and standardization does not by itself make a sensitivity assumption clinically plausible.
Here \(\delta_y\) is a signed numerical shift: it is added to the MAR reference outcome for a missing value. A positive value therefore raises the numerical outcome; whether that is clinically favorable or unfavorable depends on how the outcome is defined. For an endpoint where lower values are better, a positive shift may represent a worse outcome. For an endpoint where higher values are better, the clinical interpretation may be the opposite. The sign convention for the outcome must remain explicit when the parameter is reported.
For a simple location shift, the adjusted MAR reference mean for missing outcomes is increased or decreased by \(\delta_y\), while the standardization itself does not alter that mean. The residual standard deviation is the scale used to express the shift; it is not the standard error of an estimated mean. In a model-based analysis, it should correspond to the residual variability used for the relevant MAR reference distribution. The model, endpoint, population and method for obtaining this scale should be stated.
Converting and Interpreting the Parameters
To convert from outcome units to residual-SD units, divide by the chosen residual standard deviation. To convert back, multiply by that standard deviation. Units provide a useful check: \(\delta_y\) and \(\sigma_{\mathrm{res}}\) have the same units, so their ratio has no units; \(d_{\Delta}\sigma_{\mathrm{res}}\) has the outcome’s units.
A value such as \(d_{\Delta}=0.5\) says that the specified shift is half of the chosen residual standard deviation. It does not mean half of the observed outcome, half of a clinically important difference, or half of the standard error. Nor does a standardized value identify a unique clinical shift without its denominator: if the residual SD is 8 points, \(d_{\Delta}=0.5\) corresponds to 4 points; if it is 16 points, it corresponds to 8 points.
Worked Example: From HbA1c Units to Residual-SD Units
Suppose a hypothetical continuous HbA1c endpoint is reported in percentage points, with lower values clinically preferable. For the MAR reference distribution of missing outcomes, the specified residual SD is \(0.80\) percentage points. Consider an additive delta of \(+0.40\) percentage points.
The conversion is $$ d_{\Delta}=\frac{\delta_y}{\sigma_{\mathrm{res}}} =\frac{+0.40\ \text{percentage points}}{0.80\ \text{percentage points}} =+0.50. $$
Thus the assumed shift is \(0.50\) residual SD upward. If the MAR reference mean for these missing outcomes were \(7.50\%\), applying this location shift would give \(7.50+0.40=7.90\%\). Because higher HbA1c is less favorable in this hypothetical setting, this positive numerical shift represents a less favorable assumption for the missing outcomes. The interpretation depends both on the sign and on the clinical direction of the endpoint.
Worked Example: From a Clinical Shift to Standardized Units
Consider a hypothetical function score from 0 to 100, where higher scores indicate better function. Suppose the residual SD for the relevant MAR reference distribution is 12 score points. A sensitivity assumption says that missing outcomes are 9 points lower than their MAR reference, so \(\delta_y=-9\) points.
Standardize the shift: $$ d_{\Delta}=\frac{-9\ \text{points}}{12\ \text{points}}=-0.75. $$
The assumption is a downward shift of \(0.75\) residual SD. If a 5-point change is regarded as a clinically important difference for this hypothetical score, the assumed shift has magnitude \(9/5=1.8\) such differences. The standardized value and the clinical comparison are complementary: \(-0.75\) describes the shift relative to residual variation, while \(-9\) points describes it directly on the score scale. The clinical benchmark is not used in the conversion.
Why the Conversion Works
The conversion follows from the definition of a standard deviation as a scale for numerical deviations. Dividing an outcome-unit deviation by an SD measured in the same units expresses that deviation as a number of SDs. Multiplication reverses the operation. No distributional result, such as normality, is required for this arithmetic identity. A model may make additional assumptions about the outcome distribution, but those assumptions are separate from the conversion.
The scale behavior can also be checked by changing the units of the outcome. Suppose a new outcome scale is \(Y^*=a+bY\), where \(b>0\). An additive shift becomes \(\delta_y^*=b\delta_y\), and the residual SD becomes \(\sigma_{\mathrm{res}}^*=b\sigma_{\mathrm{res}}\). Therefore,
So changing from, for example, one positive linear unit to another changes the numerical value of the raw shift and SD together, but leaves the standardized delta unchanged. If the scale is reversed (\(b<0\)), the standardized delta changes sign because the numerical direction of the outcome has been reversed.
Worked Example: Interpreting the Same Standardized Delta Across Scales
Two hypothetical studies assess systolic blood pressure in mmHg and use the same signed standardized sensitivity value, \(d_{\Delta}=-0.60\). The relevant residual SD is 8 mmHg in Study A and 12 mmHg in Study B. The corresponding outcome-unit shifts are obtained by multiplication.
For Study A: $$ \delta_y=d_{\Delta}\sigma_{\mathrm{res}} =(-0.60)(8\ \text{mmHg})=-4.8\ \text{mmHg}. $$ For Study B: $$ \delta_y=d_{\Delta}\sigma_{\mathrm{res}} =(-0.60)(12\ \text{mmHg})=-7.2\ \text{mmHg}. $$
Both assumptions are shifts downward by \(0.60\) of the specified residual SD, but their raw magnitudes differ by \(2.4\) mmHg. Whether a downward shift is favorable or unfavorable must be interpreted for the particular analysis and endpoint; the arithmetic alone does not assign clinical desirability. The example shows why a standardized value should not be reported without the SD used to translate it to clinical units.
Choosing and Reporting the Standardizing Scale
The residual SD should be selected to match the question the sensitivity analysis is meant to answer. If the aim is to describe a shift in units of the variability of the MAR reference distribution, use the residual SD associated with that reference. State its source—for example, a specified fitted outcome model or a prespecified estimate—and use the same definition consistently when comparing values of \(d_{\Delta}\). A different denominator changes the standardized parameter even if the assumed clinical shift is unchanged.
Do not substitute a convenient SD without explaining why it is relevant. The overall outcome SD may include variation explained by predictors, whereas a residual SD describes variation remaining after the model’s predictors are accounted for. A standard error describes uncertainty in an estimate and generally depends on sample size; it is not the denominator for this standardization. Using either by mistake can make the delta appear much larger or smaller than intended.
When communicating a sensitivity analysis, report enough information to recover the clinical assumption. A useful report gives the signed \(\delta_y\), its outcome units, the value and source of \(\sigma_{\mathrm{res}}\), and the resulting \(d_{\Delta}\). If the delta was chosen clinically, explain the clinical rationale as well. If it was chosen in standardized units, translate it back into outcome units before judging its plausibility.
| Parameterization | What it communicates | Key caution |
|---|---|---|
| Outcome-unit delta, \(\delta_y\) | The numerical shift on the endpoint scale | Values are difficult to compare across endpoints with different units or scales |
| Standardized delta, \(d_{\Delta}\) | The shift relative to a specified residual SD | Its clinical meaning depends on the denominator and the endpoint scale |
| Clinical benchmark comparison | How the shift relates to a meaningful outcome difference | A benchmark is not a substitute for defining the residual SD |
Diagnostics, Pitfalls, and Limitations
A useful diagnostic is to perform the conversion in both directions for a few candidate values. If multiplying \(d_{\Delta}\) by the stated residual SD does not recover the intended outcome-unit shift, the scale or arithmetic has been mishandled. Also inspect whether the resulting shifts are plausible across the full range of the outcome. Standardizing cannot make an implausible shift plausible, and it does not remove constraints imposed by a bounded or otherwise restricted endpoint.
The sensitivity parameter is an assumption about missing outcomes relative to a MAR reference, not a quantity identified from observed data alone. The data may help estimate the reference distribution and its residual SD, but they do not determine which delta is clinically credible. Consequently, a standardized delta is not an estimate of how far the true missing outcomes differ from MAR, and its magnitude should not be described as evidence for such a departure.
- Using the total outcome SD automatically. This can standardize against variation that includes explained differences rather than the residual variation of the reference model. State why the selected SD is appropriate.
- Confusing SD and standard error. The SD measures outcome variability; the standard error measures uncertainty in an estimate. Dividing by an SE does not express the shift in residual-SD units.
- Ignoring the sign. A positive delta raises the numerical outcome under the stated additive definition, but the clinical direction depends on whether higher values are better or worse.
- Comparing standardized deltas with different denominators as if they were identical clinical assumptions. The same \(d_{\Delta}\) can imply different shifts in outcome units when residual SDs differ.
The main alternatives answer different communication needs. A delta in outcome units is preferable when the clinical scale and a plausible difference are central. A standardized delta is useful for expressing relative magnitude or comparing shifts across settings with different units, provided the residual SDs are defined comparably. A clinically anchored description can help assess plausibility, but it does not replace the model-based scale needed for conversion. None of these choices resolves uncertainty about the missing outcomes; they make the sensitivity assumption more transparent.
Check Your Understanding
Answer each question, then check the solutions below.
- A hypothetical outcome has residual SD 0.50 units. What standardized delta corresponds to an additive shift of \(+0.15\) units?
- A specified standardized delta is \(-0.40\), and the relevant residual SD is 10 score points. What is the outcome-unit shift?
- For a shift of 4 units, why is dividing by a total outcome SD of 14 potentially inappropriate if the relevant residual SD is 8?
- Two studies use \(d_{\Delta}=0.50\), but their relevant residual SDs are 6 and 9 units. What are the implied raw shifts, and are they equal?
Solutions. (1) \(d_{\Delta}=0.15/0.50=0.30\). (2) \(\delta_y=(-0.40)(10)=-4\) score points. (3) The required denominator is the specified residual SD of 8, so the standardized shift is \(4/8=0.50\), not \(4/14\approx0.286\). The total SD may include variation not represented by the residual scale. (4) The shifts are \((0.50)(6)=3\) units and \((0.50)(9)=4.5\) units. They are not equal; the same standardized delta implies different raw shifts when the residual SDs differ.