Covariate-Conditional Delta Adjustments

Why Let the Delta Vary With Covariates?

A single delta assigns the same assumed departure from a reference distribution to every person whose outcome is missing. That can be too restrictive when the departure is expected to vary with measured characteristics. For example, missing outcomes among people with more severe baseline symptoms might plausibly be farther from their reference predictions than missing outcomes among people with milder symptoms. A covariate-conditional delta represents that variation explicitly.

The purpose is not to discover the missing outcomes’ departures from observed data. Those departures are not identified by the observed outcomes alone. Instead, the analyst specifies a function that describes how the assumed departure changes with observed covariates, then assesses conclusions under that sensitivity assumption.

Definition: Let $Y$ be an outcome, $R=1$ indicate that it is observed and $R=0$ that it is missing, $A$ denote treatment arm, and $X$ denote observed covariates. Let $m_{\mathrm{ref},a}(x)$ be the conditional mean under a specified reference distribution for arm $a$. An additive covariate-conditional delta model specifies $$ E(Y\mid R=0,A=a,X=x)=m_{\mathrm{ref},a}(x)+\delta_a(x), $$ where $\delta_a(x)$ is the assumed departure from the reference conditional mean at covariate profile $x$, measured in units of $Y$. The same definition may be written relative to another explicitly chosen reference conditional mean.

The conditioning matters: $\delta_a(x)$ compares missing outcomes with the reference distribution at the same covariate profile. It is not a treatment effect, and it is not automatically the difference between missing and observed outcomes. Its value and sign are interpreted on the outcome scale. If larger $Y$ is worse, a positive delta says that the assumed missing-outcome mean is worse than the reference mean at that profile.

Specifying and Interpreting the Function

A useful starting point is to choose a covariate, specify how its values enter the delta function, and check the implied shifts at clinically interpretable profiles. A linear function for a continuous covariate might be $$ \delta_a(x)=\gamma_{0a}+\gamma_{1a}(x-c), $$ where $c$ is a centering value. The intercept $\gamma_{0a}$ is the shift at $x=c$; the slope $\gamma_{1a}$ is the change in the shift for a one-unit increase in $x$. The slope’s units are outcome units per covariate unit. Centering changes the intercept’s interpretation but not the shifts produced by the same fitted function.

The functional form should reflect the sensitivity question, not a claim that observed data have established how missing outcomes behave. A linear function is convenient when a steady change in the assumed departure is meaningful. Other prespecified functions can represent curvature. Whatever the form, report the resulting deltas at relevant covariate values; coefficients alone can obscure what assumptions the model actually imposes.

Worked Example: A Linear Shift Across Baseline Severity

Hypothetical setting. The outcome is a week-12 symptom score, in points, with larger scores indicating worse symptoms. Let $x$ be the baseline symptom score, also in points. Suppose the chosen reference model gives conditional means of 8, 12, and 17 outcome points at baseline scores of 20, 40, and 60, respectively. Consider the sensitivity function $$ \delta(x)=1+0.05(x-40). $$

At $x=20$, the assumed shift is $$ \delta(20)=1+0.05(20-40)=1-1=0\text{ points}. $$ The adjusted conditional mean is $8+0=8$ points.

At $x=40$, the shift is $1+0.05(0)=1$ point, so the adjusted conditional mean is $12+1=13$ points.

At $x=60$, the shift is $1+0.05(20)=2$ points, so the adjusted conditional mean is $17+2=19$ points. The slope, $0.05$, means that the assumed departure increases by 0.05 outcome points for each one-point increase in baseline score. These are conditional sensitivity assumptions, not estimates of the unobserved outcomes’ true departures.

A coefficient on a standardized covariate has a different unit interpretation. If $Z=(X-c)/s_X$, where $s_X$ is a chosen covariate scale such as a standard deviation, a slope on $Z$ describes the change in delta per one such scale unit. Stating the definition of $Z$ prevents ambiguity about what “one unit” means.

Worked Example: Reading a Standardized-Covariate Function

Hypothetical setting. Suppose a symptom outcome is measured in points and the standardized baseline severity is $Z=(X-30)/10$, where $X$ is baseline severity in points. The specified conditional delta is $$ \delta(Z)=1.5+0.8Z. $$

For a person with $X=20$, $Z=(20-30)/10=-1$, so $$ \delta(-1)=1.5+0.8(-1)=0.7\text{ points}. $$ For $X=30$, $Z=0$, and the shift is $1.5$ points. For $X=40$, $Z=1$, and the shift is $1.5+0.8=2.3$ points.

Thus the assumed shift increases by 0.8 outcome points for each 10-point increase in baseline severity. It does not increase by 0.8 points for each one-point increase in $X$; expressed per baseline point, the slope is $0.8/10=0.08$ outcome points per baseline point. Since the outcome is worse at higher values, all three positive shifts place the missing-outcome means above their reference means.

Why the Conditional-Mean Formula Holds

The additive equation is a definition of the mean-scale sensitivity model. One common implementation translates the entire reference distribution by the profile-specific delta. For continuous outcomes on a scale where translation is meaningful, write its conditional density as $f_{\mathrm{ref},a}(y-\delta_a(x)\mid x)$. A change of variable verifies the implied mean:

\begin{aligned} E(Y\mid R=0,A=a,X=x) &=\int y f_{\mathrm{ref},a}(y-\delta_a(x)\mid x)\,dy\\ &=\int (u+\delta_a(x))f_{\mathrm{ref},a}(u\mid x)\,du\\ &=m_{\mathrm{ref},a}(x)+\delta_a(x), \end{aligned}

In the second line, $u=y-\delta_a(x)$, so $y=u+\delta_a(x)$; the reference density integrates to one. This justification requires a valid reference distribution and a translation that is meaningful for the outcome scale. The conditional-mean equation can also be used directly as a mean assumption without asserting that every part of the distribution is translated in this way.

Worked Example: A Curved Conditional Shift

Hypothetical setting. Suppose $X$ is baseline age in years, and the outcome is a continuous functional score in points, with larger values indicating better function. A reference model gives conditional means of 52, 50, and 53 points at ages 40, 50, and 60. An analyst considers the prespecified sensitivity function $$ \delta(x)=0.04(x-50)^2. $$

At age 40, $\delta(40)=0.04(-10)^2=4$ points, giving an adjusted conditional mean of $52+4=56$ points. At age 50, $\delta(50)=0.04(0)^2=0$ points, giving $50+0=50$ points. At age 60, $\delta(60)=0.04(10)^2=4$ points, giving $53+4=57$ points.

This function assumes a larger positive departure at ages farther from 50, on either side. Its coefficient has units of outcome points per squared year. The arithmetic is straightforward, but the curved pattern is not justified merely because it can be written down: it must be a defensible sensitivity scenario. The positive shifts indicate higher, and therefore better, missing-outcome means than the reference means for ages 40 and 60.

Assumptions, Diagnostics, and Failure Modes

A covariate-conditional delta analysis depends on both the reference model and the sensitivity function. Important conditions include:

Covariate availability. The covariates used in $\delta_a(X)$ must be observed or otherwise available for the people whose outcomes are missing. If a required covariate is unavailable, its profile-specific delta cannot be applied as specified without additional assumptions.
Reference-model adequacy. The conditional reference mean $m_{\mathrm{ref},a}(x)$ must be a reasonable model for the chosen reference distribution over the profiles being analyzed. A delta cannot repair an unsuitable reference model simply by varying with $x$.
Support and overlap. Profiles among people with missing outcomes should be represented adequately by the data used to construct the reference conditional distribution. Predictions far beyond the covariate range with observed outcomes rely heavily on extrapolation.
Meaningful scale and range. The specified shifts must make sense for the outcome. For bounded outcomes, an additive shift may produce impossible values or means; a different scale or distributional construction may be needed.
Prespecified interpretation. The direction, units, covariate scale, and functional form should be explicit. A sensitivity function should not be selected after inspecting results merely to obtain a preferred conclusion.

Observed data can help assess covariate distributions and the reference model, but cannot determine $\delta_a(x)$ for outcomes that are missing. This is a central diagnostic boundary: good fit among observed outcomes is not evidence that the chosen missing-outcome delta is correct. Instead, examine the implied shifts across the covariate range, including the low and high profiles actually represented among people with missing outcomes. Look for implausible sign reversals, extreme values, and predictions driven by extrapolation.

How This Choice Compares With Alternatives

A constant delta is the special case $\delta_a(x)=\delta_a$: it is simpler to communicate and useful when a common shift is plausible, but it cannot express systematic variation by covariate profile. A covariate-conditional delta adds flexibility at the cost of more assumptions about the function’s shape and scale. Treatment-arm-specific deltas allow departures to differ by arm; a covariate-conditional function describes how they vary across profiles. These choices address distinct dimensions of the sensitivity assumption and should be described separately when both are used.

Reference-based imputation specifies a reference distribution for missing outcomes, while a delta adjustment specifies a departure from a chosen reference. A selection model represents missingness through a model for the observation process; it is a different way to express assumptions about unobserved outcomes. None of these approaches makes the missing outcomes’ distribution identifiable from the observed data alone. The appropriate choice depends on the scientific question, the available covariates, and which assumptions can be explained and defended.

Common Mistakes and Limitations

Calling the delta a covariate effect. It is the assumed departure from a reference conditional mean, not the effect of changing the covariate on the outcome.
Ignoring units. A delta is on the outcome scale; a slope in a delta function has outcome units per covariate unit (or per specified standardized unit).
Assuming the function is learned from observed outcomes. Observed data may inform the reference model, but the conditional departure for missing outcomes is a sensitivity assumption.
Extrapolating without noticing. A mathematical function returns values outside the observed covariate range, but those values may have little empirical support and may be clinically implausible.
Confusing a conditional statement with a marginal one. A delta at a particular profile describes the assumed shift at that profile. It does not, by itself, state the average shift across all missing outcomes.

The main limitation is that added flexibility does not add identification: a more elaborate $\delta_a(x)$ can represent richer assumptions, but observed data cannot verify its behavior for outcomes that were not observed. A useful analysis therefore keeps the function interpretable, checks its implied values across relevant profiles, and reports conclusions as conditional on the chosen sensitivity assumptions.

Check Your Understanding

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

A score has larger values for worse symptoms. If $\delta(x)=-2+0.1(x-30)$, calculate the delta at $x=20$ and $x=50$, and interpret both signs.
In $\delta(x)=\gamma_0+\gamma_1(x-c)$, what are the units of $\gamma_1$, and what does $\gamma_0$ represent?
A covariate is unavailable for people with missing outcomes, but the proposed delta depends on that covariate. Is the function directly applicable as written? Explain.
At a covariate profile, the reference conditional mean is 18 points and the specified delta is $-1.5$ points. Find the adjusted conditional mean and interpret it if higher outcomes are better.

Solutions. (1) At $x=20$, $\delta(20)=-2+0.1(-10)=-3$ points; at $x=50$, $\delta(50)=-2+0.1(20)=0$ points. The first assumes a missing-outcome mean 3 points lower (better) than its reference mean; the second assumes no departure. (2) $\gamma_1$ has outcome units per covariate unit. $\gamma_0$ is the delta at $x=c$, because substituting $x=c$ makes the slope term zero. (3) No. The profile-specific delta cannot be evaluated as specified without that covariate; applying it would require an additional assumption or another available covariate specification. (4) The adjusted conditional mean is $18+(-1.5)=16.5$ points. It is 1.5 points below the reference conditional mean, so the assumed missing-outcome mean is worse when higher scores are better.