Standardizing Delta Parameters Across Outcomes

Why Standardize Delta Parameters?

A delta adjustment is expressed on the scale of the outcome it changes. A shift of 2 points may be modest for a test scored out of 100 but large for an outcome whose standard deviation is 1 point. Similarly, a shift of 5 mmHg and a shift of 5 kg do not have comparable meanings merely because their numerical values match. When sensitivity analyses involve outcomes in different units, expressing additive deltas relative to a clearly chosen standard deviation can make their magnitudes easier to compare.

Standardization is a change of scale, not a way to determine which delta is scientifically plausible. The delta values still encode assumptions about the unobserved outcomes. The standard deviation used as a divisor must therefore be stated, and the resulting standardized values must be interpreted with care.

Definition: For an outcome $Y_j$ with an additive delta adjustment $\delta_j$, its standardized delta is $$ \delta_j^{*}=\frac{\delta_j}{s_j}, $$ where $s_j>0$ is a prespecified reference standard deviation for outcome $j$. Equivalently, $\delta_j=\delta_j^{*}s_j$. The raw delta $\delta_j$ has the units of $Y_j$; the standardized delta $\delta_j^{*}$ is dimensionless and describes the shift as a number of reference standard deviations.

What the Standardized Delta Represents

For a continuous outcome, suppose a chained-equations imputation model gives a predictive distribution for a missing value under a missing-at-random (MAR) reference. An additive delta adjustment shifts that predictive distribution on the outcome scale. In a simple constant-shift formulation, for a record with covariates $X$,

Y_j^{\mathrm{MNAR}} \mid X, R_j=0 \ \overset{d}{=}\ Y_j^{\mathrm{MAR}} \mid X, R_j=0+\delta_j, \qquad \delta_j=\delta_j^{*}s_j.

Here $R_j=1$ denotes that outcome $j$ is observed and $R_j=0$ that it is missing; $\overset{d}{=}$ indicates a distributional shift. In this formulation, $\delta_j$ is constant across the records to which this adjustment applies. A positive delta shifts the missing-outcome predictive distribution upward; a negative delta shifts it downward. Standardizing changes the units used to describe the shift, not its direction.

This interpretation presumes that an additive shift on the outcome scale is appropriate for the sensitivity question. The imputation model, its predictors, and the delta formulation must be specified before interpreting a standardized value. If the shift is meant to differ by subgroup, treatment arm, or another feature, then the relevant group-specific shift and reference scale must be defined separately. A single standardized number does not remove the assumptions in the model.

Choosing the Reference Standard Deviation

A usable comparison depends on choosing $s_j$ deliberately. For continuous outcomes, a practical choice is the standard deviation of observed values in a stated reference population, calculated before applying the delta shifts. For example, the reference could be all participants with an observed value of that outcome at baseline. Another possibility is a prespecified residual standard deviation from a particular model. These choices answer different questions: the marginal observed standard deviation scales the shift against overall variation in the outcome, whereas a residual standard deviation scales it against variation left after accounting for specified predictors.

Whichever scale is chosen, hold it fixed across the delta scenarios being compared. If the denominator changes across scenarios, a change in $\delta_j^{*}$ can reflect a changed scale as well as a changed shift. Report the outcome, the reference population or model, the value of $s_j$, the raw delta, and the standardized delta. If an estimated standard deviation is used, say how it was estimated and whether it is based only on observed values.

There is no universal rule that the marginal observed standard deviation is always the best scale. It is often transparent and easy to reproduce, but it can be unrepresentative if observed values come from a narrow subgroup or if the outcome has a markedly different spread across groups. The chosen reference should match the comparison being made and should not be selected after seeing which scale makes a sensitivity result look most dramatic.

Worked Example: Converting Shifts for Two Outcomes

Consider two hypothetical continuous outcomes. Observed systolic blood pressure has a reference standard deviation of 16 mmHg, and observed pain score has a reference standard deviation of 2.5 points. Suppose the analyst considers an additive delta of $-8$ mmHg for missing blood pressure values and $+1.0$ point for missing pain scores.

For blood pressure,

\delta_{\mathrm{BP}}^{*}=\frac{-8\ \mathrm{mmHg}}{16\ \mathrm{mmHg}}=-0.500.

Thus the assumed shift is one-half of a reference standard deviation downward. For pain score,

\delta_{\mathrm{pain}}^{*}=\frac{1.0\ \mathrm{point}}{2.5\ \mathrm{points}}=0.400.

The two adjustments are $-0.500$ and $+0.400$ standard deviations, respectively. This comparison communicates their magnitudes relative to the selected spreads, and it preserves their opposite directions. It does not establish that the two assumptions have equal clinical importance or equal plausibility.

Derivation and Unit Invariance

The conversion follows from expressing the raw shift in units of the chosen reference standard deviation. Because a standard deviation has the same units as its outcome, their ratio has no units. The reverse conversion is obtained by multiplying by the scale:

\delta_j^{*}=\frac{\delta_j}{s_j} \quad\Longrightarrow\quad \delta_j=s_j\delta_j^{*}.

This also explains why a correctly calculated standardized delta is unchanged by a simple unit conversion. If an outcome is changed from one unit to another by multiplying by a positive constant $c$, then both its delta and its standard deviation are multiplied by $c$. The ratio is unchanged:

\frac{c\delta_j}{cs_j}=\frac{\delta_j}{s_j}.

This invariance applies to multiplicative changes of units, such as converting centimetres to metres. It does not imply invariance to nonlinear transformations, such as taking logarithms, nor does it make an additive shift on one scale equivalent to an additive shift on another. If the imputation model defines delta on a transformed or link scale, first interpret that delta on its stated scale; dividing it by an outcome standard deviation on another scale would not be the definition above.

Worked Example: Verifying a Unit Conversion

Suppose a hypothetical outcome has observed standard deviation $12$ cm and the proposed missing-value shift is $6$ cm. Its standardized delta is

\delta^{*}=\frac{6\ \mathrm{cm}}{12\ \mathrm{cm}}=0.500.

In metres, the same standard deviation is $0.12$ m and the same shift is $0.06$ m. Recomputing gives

\delta^{*}=\frac{0.06\ \mathrm{m}}{0.12\ \mathrm{m}}=0.500.

The standardized delta is unchanged, as it should be. If the second calculation produced a different value, the raw shift and standard deviation were not converted consistently.

Standardized Values Still Need Context

A common aim is to compare sensitivity assumptions across outcomes, not to claim that the outcomes themselves are interchangeable. A standardized delta of $0.5$ means a shift of one-half of the chosen reference standard deviation for that outcome. It does not automatically mean a clinically small or large change. A clinically important difference can be quite different across outcomes even when each is expressed in standard-deviation units.

Standardization can also obscure meaning if the denominator is poorly matched to the question. For example, a marginal standard deviation may be large because of substantial differences between groups. A shift that is a small fraction of that marginal spread could still be consequential within a group. Conversely, a small residual standard deviation can make a modest raw shift appear large when expressed in residual standard-deviation units. Always retain the raw-unit delta alongside the standardized value.

Worked Example: How the Choice of Scale Changes the Comparison

Suppose a hypothetical continuous outcome has a missing-value shift of $3$ units. The observed marginal standard deviation is $15$ units, while a prespecified model's residual standard deviation is $6$ units. Using the marginal scale gives

\delta_{\mathrm{marginal}}^{*}=\frac{3}{15}=0.200.

Using the residual scale instead gives

\delta_{\mathrm{residual}}^{*}=\frac{3}{6}=0.500.

Both calculations are arithmetically correct, but they answer different scaling questions. The first describes the shift relative to overall observed spread; the second describes it relative to the residual spread defined by that model. Reporting “a delta of 0.5 SD” without identifying the residual scale would invite an incorrect comparison with a value standardized using the marginal SD.

Assumptions, Diagnostics, and Alternatives

A standardized additive delta is most useful when the outcome has a meaningful continuous scale, the delta is defined as an additive shift on that same scale, and the reference standard deviation is positive, relevant, and fixed across the scenarios being compared. It is also necessary to state which missing records the shift applies to. These conditions make the conversion interpretable; they do not validate the missing-data assumption or establish that a selected delta is plausible.

Before reporting a comparison, check that the raw delta and $s_j$ use the same units, that $s_j>0$, and that the sign agrees with the intended direction of the shift. Check that the same reference scale is used throughout the sensitivity scenarios. If outcomes are recorded on bounded scales, or if an additive shift would conflict with their scale, standardization does not fix the underlying modeling issue. The shift specification must itself be appropriate; transformations or constrained approaches require interpretation on their own terms.

Alternative ways to compare adjustments include reporting raw deltas in outcome-specific units, expressing them relative to a clinically meaningful difference, or using a common external benchmark when one is substantively justified. Raw-unit deltas are often the clearest for clinical interpretation but are awkward to compare numerically across unlike outcomes. Clinical thresholds can be more meaningful, but may not exist or may differ in evidence quality. Standard-deviation units are convenient for relative-scale comparison, but depend on the reference spread. These approaches are complementary: a careful report can give the raw delta, its standardized value, and a clinically relevant interpretation where one is available.

Do not apply this formula mechanically to every delta parameter. For example, a delta defined as a log-rate adjustment is on a logarithmic scale, not in the original count units. Dividing that parameter by the count outcome's standard deviation would mix scales. The standardization must respect the scale on which the sensitivity parameter was defined.

Common pitfall: “Both standardized deltas equal $0.5$, so the assumed missing-outcome shifts are equally important.” This conclusion does not follow. The values say only that each shift equals one-half of its stated reference standard deviation. The outcomes may have different clinical meanings, and the reference standard deviations may have been defined differently. Report the raw-unit shifts and the reference scales before making a substantive comparison.

Reporting a Cross-Outcome Comparison

A concise report should make the calculation reproducible. For each outcome, state the additive shift applied to missing values, its direction, the reference standard deviation and how it was obtained, and the resulting dimensionless ratio. Use the same scale definition across scenarios for that outcome. If different outcomes use different reference populations or scale types, make that difference explicit rather than presenting the standardized values as if they shared a single universal benchmark.

Standardization is a translation aid: it helps readers see whether a raw shift is small or large relative to a declared spread. It is not a replacement for external calibration, subject-matter judgment, or a careful explanation of what a shift means in the outcome's original units.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A hypothetical outcome has reference standard deviation $4$ units and additive delta $-2$ units. What is the standardized delta, and what is its direction?
An outcome is converted from millimetres to centimetres. What must happen to both the raw delta and its reference standard deviation for the standardized delta to remain unchanged?
Two outcomes each have standardized delta $0.3$. Does this prove that the assumptions have equal clinical importance? Explain.
Why should the reference standard deviation be held fixed across sensitivity scenarios for a given outcome?

Solutions. (1) $\delta^{*}=-2/4=-0.500$, a downward shift equal to one-half of the reference standard deviation. (2) Both quantities must be divided by 10; their ratio is unchanged. (3) No. Each shift is $0.3$ of its own stated reference standard deviation, but the outcomes and scales can differ in clinical meaning. (4) Otherwise, changes in the standardized delta could reflect changes in the denominator as well as changes in the raw shift, making the sensitivity scenarios difficult to compare.