The MAR Reference Distribution and the Delta Departure

Why a MAR Reference Distribution Is Needed

A missing outcome is not made observable by fitting a model. What an outcome model can provide is a reference distribution: the conditional distribution the model predicts for that outcome given the information that is observed. Under a missing-at-random assumption, this distribution is a principled reference for missing outcomes. A delta departure then asks how conclusions would change under a specified alternative to that reference.

The distinction matters. The observed data can inform the MAR reference distribution, subject to modeling and support conditions. They cannot, by themselves, reveal how the unobserved outcomes differ from that reference. The earlier material on Delta Adjustment as a Pattern-Mixture Sensitivity Parameter introduced delta as a sensitivity parameter; here the focus is the distribution to which that departure is applied and the extra assumption the departure entails.

Definition: Let $Y_{\mathrm{mis}}$ denote an outcome that is missing, $Y_{\mathrm{obs}}$ the relevant observed outcomes, $X$ fully observed covariates, and $R$ the missingness pattern. The MAR reference distribution for the missing outcome is the conditional distribution $$ F_{\mathrm{MAR}}(y\mid Y_{\mathrm{obs}},X) = P(Y_{\mathrm{mis}}\leq y\mid Y_{\mathrm{obs}},X,R), $$ as represented by an outcome model under MAR. Under the MAR assumption, this conditional distribution does not depend further on $R$ once the observed information is conditioned on. A delta model replaces it with a specified family $F_{\delta}(y\mid Y_{\mathrm{obs}},X,R)$, with the reference parameterization conventionally satisfying $F_{0}=F_{\mathrm{MAR}}$.

The reference distribution is conditional on the information available for the particular missing outcome. In longitudinal data, for example, this may include baseline covariates and the subject’s observed outcome history. It is not necessarily the marginal distribution of all observed outcomes, nor is it a single fitted mean. A conditional distribution describes both the range and relative plausibility of possible missing values, as well as their uncertainty.

What MAR Contributes—and What It Does Not

In this setting, MAR means that, conditional on the observed information, the probability of the missingness pattern does not depend on the unobserved outcome. For a missing outcome and a pattern $R=r$, the relevant statement is $$ P(R=r\mid Y_{\mathrm{obs}},Y_{\mathrm{mis}},X) = P(R=r\mid Y_{\mathrm{obs}},X). $$ Thus, among units with the same observed information, missingness does not additionally select units according to their missing values. MAR does not say that missingness is unrelated to observed information, and it does not guarantee that a particular outcome model is correct.

To construct the reference distribution, an analyst specifies and fits a model for the outcome given observed predictors and history. Under MAR, the conditional distribution for a missing value is taken to be the same as the model’s conditional distribution for a value that is observed, at the same observed information. This use of the model requires adequate data to estimate the relevant conditional distribution and a model that represents it sufficiently well.

Conditions for using the MAR reference distribution: The missingness mechanism must satisfy MAR for the information being conditioned on; the outcome model must be correctly specified well enough for the target prediction; and the observed data must provide support for the covariate and outcome-history values of the units with missing outcomes. If a needed history or covariate combination is absent, the model must extrapolate there, and the resulting reference distribution depends on additional modeling choices.

A short likelihood argument explains the role of MAR. Write $f(y_{\mathrm{obs}},y_{\mathrm{mis}}\mid x)$ for the outcome distribution and $\pi(r\mid y_{\mathrm{obs}},y_{\mathrm{mis}},x)$ for the probability of pattern $r$. The contribution to the observed-data likelihood is obtained by integrating over the missing outcome:

\begin{aligned} L_{\mathrm{obs}} &= \int f(y_{\mathrm{obs}},y_{\mathrm{mis}}\mid x) \pi(r\mid y_{\mathrm{obs}},y_{\mathrm{mis}},x)\,dy_{\mathrm{mis}}\\ &= \pi(r\mid y_{\mathrm{obs}},x) \int f(y_{\mathrm{obs}},y_{\mathrm{mis}}\mid x)\,dy_{\mathrm{mis}}\\ &= \pi(r\mid y_{\mathrm{obs}},x)f_{\mathrm{obs}}(y_{\mathrm{obs}}\mid x). \end{aligned}

The second line follows from MAR: the missingness probability no longer depends on $y_{\mathrm{mis}}$, so it can be taken outside the integral. The observed outcomes can therefore inform the outcome model without modeling a dependence of missingness on the unobserved values. This factorization does not create observations of missing outcomes. It supports the reference distribution only where the outcome model can be learned from observed data, under its modeling and support conditions.

Worked Example: A Conditional Normal Reference Distribution

Suppose, hypothetically, that a continuous outcome $Y$, measured in points, is modeled as $Y\mid X\sim N(10+2X,9)$, where $X$ is a fully observed baseline score on a specified scale. A participant has a missing outcome and $X=1$. Assume for this example that the model is correctly specified, that the relevant $X$ values are represented among observed outcomes, and that missingness is MAR conditional on $X$.

Substitute $X=1$ into the fitted conditional mean: $$ 10+2(1)=12\text{ points}. $$

The conditional variance is 9 points squared, so the conditional standard deviation is $\sqrt{9}=3$ points. The MAR reference distribution for this participant is therefore $N(12,9)$: it has mean 12 points and standard deviation 3 points.

For example, the model-implied probability that this participant’s missing outcome lies between 9 and 15 points is $$ P(9<Y<15\mid X=1) = P\left(\frac{9-12}{3}<Z<\frac{15-12}{3}\right) = P(-1<Z<1) \approx 0.6827, $$ where $Z\sim N(0,1)$. This is a probability under the stated conditional model, not evidence that this individual’s value is known. The calculation is justified only to the extent that the MAR, model, and support assumptions are credible.

The Extra Assumption in a Delta Departure

A delta departure does more than acknowledge uncertainty in the fitted MAR distribution. It specifies how the conditional distribution for missing outcomes differs from that reference. In general, write the alternative as $F_{\delta}(y\mid Y_{\mathrm{obs}},X,R)$, with $F_0=F_{\mathrm{MAR}}$. The choice of how delta indexes the alternative is part of the sensitivity model; there is no single transformation that applies to every outcome type and analysis.

For nonzero delta, the additional assumption is a claim about the unobserved conditional distribution: given the same observed information and missingness pattern, the missing outcomes follow the specified alternative distribution rather than the MAR reference. Since those outcomes are not observed, the data cannot determine whether that departure is correct or estimate its value without further assumptions. Delta is therefore a sensitivity parameter, not a parameter identified by the observed outcomes alone.

The departure must also say what happens to the distribution, not just name a direction of concern. Depending on its specification, it may change probabilities across possible outcomes or alter other aspects of the distribution. If a method changes only one feature while holding others fixed, that too is an assumption. A delta-based analysis should describe the family of distributions being considered and explain which aspects are allowed to change.

Worked Example: A Discrete Delta Departure

Consider a hypothetical missing outcome that can be 0, 5, or 10 points. Suppose the MAR reference model gives probabilities 0.20, 0.50, and 0.30, respectively. For illustration, define a particular sensitivity family by $$ \begin{aligned} P_{\delta}(Y=0)&=0.20+0.20\delta,\\ P_{\delta}(Y=5)&=0.50-0.10\delta,\\ P_{\delta}(Y=10)&=0.30-0.10\delta, \end{aligned} \qquad 0\leq\delta\leq 1. $$ This is one illustrative parameterization, not a universal definition of delta.

At $\delta=0$, the probabilities are the MAR reference probabilities. At the hypothetical sensitivity value $\delta=1$, they become 0.40, 0.40, and 0.20. They remain valid probabilities because each is nonnegative and $$ 0.40+0.40+0.20=1. $$

The reference mean is $$ 0(0.20)+5(0.50)+10(0.30)=0+2.50+3.00=5.50\text{ points}. $$ Under $\delta=1$, the mean is $$ 0(0.40)+5(0.40)+10(0.20)=0+2.00+2.00=4.00\text{ points}. $$ Thus this specified departure lowers the model-implied mean by 1.50 points and changes the probabilities at all three outcome values. The calculation does not show that the missing outcomes actually follow the $\delta=1$ distribution. It shows the consequence of the additional distributional assumption if that scenario is used.

Worked Example: Combining Reference Distributions Across Observed Histories

Suppose a hypothetical study has two observed-history strata, A and B. Under a fitted MAR model, the missing outcome has conditional means 8 points in stratum A and 12 points in stratum B. Three outcomes are missing: two participants have observed history A and one has history B. Assume the fitted conditional means are appropriate in both strata and the study has support for these histories.

The model-implied expected sum of the three missing outcomes is the sum of their conditional means: $$ 2(8)+1(12)=16+12=28\text{ points}. $$ The corresponding mean across the three missing outcomes is $$ \frac{28}{3}=9.333\text{ points}. $$

This calculation uses the observed history of each participant to select the appropriate MAR reference distribution. It does not imply that each missing value equals its conditional mean: the predictive distributions may have substantial variation around those means. If a delta departure is applied, the analyst must additionally specify how the distributions for the relevant histories depart from their MAR references. A single pooled departure cannot be presumed appropriate for both histories without an explicit assumption.

Diagnostics, Alternatives, and Limits

The MAR assumption concerns unobserved values and cannot be verified from the observed data alone. Diagnostics can still reveal problems with the reference model or its use. Check whether missing cases have covariate and observed-history values outside the range represented among observed outcomes; inspect fitted distributions for implausible values or poor fit; and assess whether predictions change sharply under reasonable model choices. When feasible, compare predictions with observed outcomes in similar histories or use a masking exercise that temporarily hides observed outcomes. Such checks can assess predictive performance in observed data, but they do not test whether the missing outcomes themselves follow MAR.

The MAR reference approach and a delta departure answer different questions from other missing-data approaches. A reference-based imputation method uses a specified reference trajectory or arm as its predictive basis; the MAR reference instead uses the outcome model conditional on observed information under MAR. A selection model represents the dependence of missingness on outcomes through a model for the missingness process. These approaches encode different assumptions. A delta sensitivity analysis is useful for examining departures from a chosen MAR reference, but it does not automatically validate that reference or replace a selection model.

Several limitations deserve emphasis:

Model error: A reference distribution can be wrong even when MAR is a reasonable missingness assumption. Incorrect functional forms, omitted predictors, or poorly modeled variability can distort predictions.
Weak support: If observed data do not cover the histories or covariates of missing cases, the reference relies on extrapolation. A delta departure built on that reference inherits this uncertainty.
Unidentified departure: The observed data do not select the correct nonzero delta distribution. Conclusions across plausible alternatives should be presented as sensitivity results, not as estimates of the true missing-outcome distribution.
Unclear scope: Applying a departure without stating which patterns, histories, or distributional features it affects leaves the sensitivity assumption ambiguous.

A useful report separates what is supported by data from what is assumed: describe the outcome model and information conditioned on, explain why MAR is used for the reference, state the delta family and chosen values, and show how conclusions vary across those choices. The earlier tutorial material on reference-based imputation can provide alternative predictive references, but none removes the need to state the assumptions behind predictions for unobserved outcomes.

Common pitfall: “The fitted outcome model predicts missing values, so its predictions are the actual missing-outcome distribution.” A fitted model supplies a conditional reference distribution only under its modeling, MAR, and support assumptions. A nonzero delta adds a further assumption about how that unobserved distribution differs from the reference; the observed data alone cannot confirm that departure.

Check Your Understanding

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

Under MAR, what conditioning information makes the missingness pattern irrelevant to the conditional distribution of a missing outcome?
A fitted MAR reference distribution is $N(20,16)$ for a missing outcome measured in milligrams. What are its conditional mean and standard deviation, with units?
In a delta family, $F_0=F_{\mathrm{MAR}}$. Does this imply that the observed data prove MAR is true?
Why is a conditional reference distribution for a history absent among observed outcomes especially uncertain?
In the discrete example above, calculate the mean under $\delta=0.5$ and explain what the result does and does not establish.

Solutions. (1) Conditional on the relevant observed outcomes and fully observed covariates or history, MAR states that missingness does not additionally depend on the missing value. (2) The conditional mean is 20 milligrams. The variance is 16 milligrams squared, so the standard deviation is $\sqrt{16}=4$ milligrams. (3) No. $F_0=F_{\mathrm{MAR}}$ defines the reference point in the sensitivity family; MAR remains an assumption that cannot be established from observed data alone. (4) The outcome model must extrapolate beyond histories represented by observed outcomes, so its predictions depend more strongly on the model form and other unverified assumptions. (5) At $\delta=0.5$, the probabilities are $0.30, 0.45,$ and $0.25$. The mean is $0(0.30)+5(0.45)+10(0.25)=0+2.25+2.50=4.75$ points. This is the mean under that specified sensitivity distribution, not evidence that the missing outcomes truly have that distribution.

The MAR Reference Distribution and the Delta Departure

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