Why the Scale of Delta Matters
A delta adjustment describes a departure from a reference distribution for outcomes that are missing. The reference distribution is often the distribution predicted under a missing-at-random (MAR) assumption, conditional on the observed information available for a missing case. A sensitivity analysis then changes that distribution to represent departures that the observed data alone cannot determine. The earlier material on delta adjustment establishes the role of this sensitivity parameter; here, the central question is which quantity should be shifted?
A shift of two units in the original outcome has a direct interpretation when those units are clinically meaningful. The same numerical shift after a logarithm has a different interpretation: it changes the outcome multiplicatively. A shift in a model’s linear predictor changes a model parameter, such as a conditional mean, and the residual distribution is then governed by the model specification. These choices can produce different missing-outcome distributions even when the numerical value of delta is the same.
The reference distribution must be specified before applying the adjustment. In a pattern-mixture analysis, it is a conditional distribution for outcomes in the relevant missingness pattern, based on the observed data and the chosen reference assumptions. A delta does not make this distribution empirically identified: it indexes alternative assumptions about missing outcomes. The scale determines how a fixed delta changes location, spread, and support.
Three Distinct Adjustment Rules
The following rules are not interchangeable. In each case, the reference distribution is conditional on the same observed history \(H\), so the difference comes from the adjustment itself.
| Scale adjusted | Rule | What it directly changes |
|---|---|---|
| Raw outcome | \(Y_{\delta}=Y_0+\delta\) | Every reference outcome is translated by a fixed number of outcome units. |
| Transformed outcome | \(g(Y_{\delta})=g(Y_0)+\delta_g\) | Every reference outcome is shifted after transformation, then mapped back to the outcome scale. |
| Model link | \(h(\mu_{\delta})=h(\mu_0)+\delta_{\eta}\) | A conditional model parameter, commonly the mean, changes; the model’s residual specification determines the rest of the distribution. |
Here \(Y_0\) denotes a draw from the reference distribution, \(\mu_0=E(Y_0\mid H)\), and \(h\) is a link function for the conditional mean. For a transformed-outcome rule, \(g\) must be defined on the relevant outcome support, and the inverse transformation must return valid outcomes. The delta values have different units across the three rows: outcome units, transformed units, and link units. Equal numerical values therefore do not imply equal departures.
Deriving the Implied Distributions
The distributional consequences follow directly from the adjustment rules. For a raw shift, if \(Y_0\) has cumulative distribution function \(F_0\), then \(P(Y_{\delta}\leq y\mid H)=P(Y_0\leq y-\delta\mid H)=F_0(y-\delta\mid H)\). Thus the entire distribution translates by \(\delta\), leaving its variance unchanged. Its support also translates, which can be problematic for bounded outcomes.
For a strictly increasing transformation \(g\), define \(Y_{\delta}=g^{-1}\{g(Y_0)+\delta_g\}\). Applying \(g\) to both sides gives the event equivalence \(Y_{\delta}\leq y\) if and only if \(Y_0\leq g^{-1}\{g(y)-\delta_g\}\). Therefore,
This is generally not a constant translation on the raw scale. For example, if \(g(y)=\log(y)\), then \(Y_{\delta}=\exp\{\log(Y_0)+\delta_g\}=Y_0\exp(\delta_g)\). Each draw is multiplied by the same positive factor, so the absolute change is larger for larger starting values.
For a link-scale mean adjustment, invert the link:
This equation determines the adjusted mean. To obtain a predictive distribution, one must additionally specify how outcomes vary around that mean. For example, a model that holds a shape parameter fixed as the mean changes implies a different variance change from a model that holds the variance fixed. The model link alone does not settle that choice.
Worked Example: Translating a Normal Reference Distribution
Suppose, for a hypothetical group of missing patients with comparable observed histories, the MAR reference distribution for a symptom score is \(Y_0\mid H\sim N(10,4)\). The score is measured in points and, for this illustration, is treated as unbounded. Consider a raw-scale delta of \(2\) points.
The rule is \(Y_{\delta}=Y_0+2\). Adding a constant shifts the mean and leaves the variance unchanged:
The adjusted mean is 12 points, and the standard deviation remains \(\sqrt{4}=2\) points. For example, a reference draw of 8 points becomes \(8+2=10\) points, while a draw of 12 becomes 14 points. This assumption is defensible only if a fixed two-point increase across the relevant range is meaningful and the resulting values are plausible. In an actually bounded score, translating the distribution could assign probability to impossible values; the unbounded illustration avoids that particular problem but does not establish clinical plausibility.
Worked Example: Shifting a Log-Transformed Outcome
Consider a hypothetical positive biomarker whose reference distribution satisfies \(\log(Y_0)\mid H\sim N(\log(10),0.25)\), where the variance on the log scale is \(0.25\). Apply a transformed-scale delta of \(0.2\).
The adjustment is \(\log(Y_{\delta})=\log(Y_0)+0.2\), so \(Y_{\delta}=Y_0e^{0.2}\). The multiplier is \(e^{0.2}\approx1.221\). The adjusted log-outcome distribution is \(N(\log(10)+0.2,0.25)\). Its median changes from \(10\) to \(10e^{0.2}\approx12.21\) biomarker units.
The reference mean is not 10, because a lognormal distribution is right-skewed. For \(X\sim N(m,v)\), the mean of \(e^X\) is \(e^{m+v/2}\). Hence the reference mean is \(10e^{0.25/2}=10e^{0.125}\approx11.33\), while the adjusted mean is \(10e^{0.2+0.125}=10e^{0.325}\approx13.84\) biomarker units. The mean ratio is \(e^{0.2}\approx1.221\). The shift preserves the log-scale variance, but it scales the raw-scale standard deviation and mean along with the outcome. This rule is suitable only if the positive support and relative, rather than fixed-unit, departure match the scientific assumption.
Worked Example: Shifting a Mean on a Log Link
Suppose a hypothetical Gamma model describes a positive outcome with reference conditional mean \(\mu_0=10\) units and fixed shape parameter \(k=4\). For a Gamma distribution parameterized by mean \(\mu\) and shape \(k\), the variance is \(\mu^2/k\). Use a log link and a link-scale delta of \(0.2\).
The adjusted mean is \(\mu_{\delta}=\exp\{\log(10)+0.2\}=10e^{0.2}\approx12.21\) units. Under the stated fixed-shape assumption, the adjusted variance is \(12.21^2/4\approx37.30\) square units, compared with the reference variance \(10^2/4=25.00\) square units. The adjusted standard deviation is approximately \(\sqrt{37.30}=6.107\) units, compared with \(5\) units at the reference mean.
This example shifts the model’s conditional mean and retains a particular Gamma shape rule. The mean increases by about 22.1%, and the variance increases because the shape is held fixed. If instead a model kept the variance fixed at 25, it would imply a different adjusted distribution. Thus the link-scale delta is not a complete distributional assumption by itself; both the mean rule and the dispersion rule must be reported.
Assumptions, Diagnostics, and Failure Modes
A scale choice is defensible only in relation to the outcome, the reference model, and the missingness pattern. Before interpreting adjusted estimates, check the following conditions:
- Reference predictions are adequate. The conditional reference distribution should be reasonable for the observed histories represented among missing cases. Poor prediction or extrapolation can make the adjusted distribution unreliable regardless of delta.
- The adjustment has a coherent interpretation. A raw delta should correspond to a fixed unit difference; a transformed delta should describe a plausible change on that transformation; a link delta should describe a plausible parameter shift.
- The support is respected. A raw shift can create impossible negative values or move outcomes beyond a known ceiling. A transformed rule may preserve support only if the transformation and inverse transformation are appropriate. A model-link rule must also respect the model’s parameter constraints.
- Distributional spread is checked. Plot or summarize reference and adjusted predictive distributions, not just their means. Nonlinear transformations and model dispersion assumptions can change tail behavior or raw-scale variability.
- The delta range is substantively justified. Observed data generally do not identify the departure for missing outcomes. Use clinically plausible values or scenarios, and report how conclusions vary across them rather than presenting one value as learned from the observed outcomes.
A common failure is to label all three approaches “adding delta” and then use the same number without checking its units. A value of \(0.2\) means 0.2 outcome units on the raw scale, a 0.2-unit displacement of \(g(Y)\) on a transformed scale, or a 0.2-unit displacement of the predictor on a link scale. These can imply radically different outcome changes. Another failure is to inspect only the adjusted mean: a mean may be plausible even when the predictive distribution has implausible support or spread.
Choosing Among Scales
The main alternative is not simply a different formula: it is a different substantive statement about how missing outcomes depart from the reference distribution. A raw shift is often easiest to explain when a fixed difference in clinical units is meaningful. A transformed shift is useful when a relative change or a scale with more stable variation is scientifically motivated. A model-link shift is natural when departures are formulated through a regression parameter, but it requires an explicit distributional model to determine the resulting spread.
No scale is universally preferable. A transformed or link scale can encode a multiplicative change on the outcome scale, but those interpretations should not be conflated: a transformed-outcome shift changes the distribution of transformed draws, whereas a link-scale mean shift changes a model parameter and relies on a separate residual specification. One practical strategy is to express several plausible assumptions in outcome units, examine their predictive distributions, and retain scenarios that have a clear scientific interpretation. This comparison does not make the sensitivity parameter identifiable; it makes its consequences visible.
When results are sensitive to scale, report the reference distribution, the adjusted quantity, the delta and its units, and the assumptions about residual variation. A conclusion that changes under reasonable scale choices is evidence that inference depends on unobserved outcomes, not evidence that one scale has been established by the observed data.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- A raw-scale reference outcome is 15 units and the raw delta is \(-3\). What is the adjusted value, and what interpretation does the rule impose?
- For a log-transformed outcome, the delta is \(\log(1.1)\). By what factor does each reference draw change on the raw scale?
- A model uses a log link and has reference conditional mean 20 units. A link delta of \(-0.1\) is applied. Calculate the adjusted mean to three significant figures.
- Why is a link-scale mean adjustment insufficient to specify a complete predictive distribution?
- Give one diagnostic concern when applying a raw additive delta to a bounded outcome.
Solutions. (1) \(15+(-3)=12\) units; every reference draw is translated downward by three outcome units. (2) Each draw is multiplied by \(\exp\{\log(1.1)\}=1.1\), a 10% increase. (3) \(\mu_{\delta}=20e^{-0.1}\approx18.1\) units. (4) It determines the adjusted conditional mean, but not the residual variability or distributional shape; those require a model assumption. (5) Translation may place positive probability below the lower bound or above the upper bound, producing impossible outcomes.