Why Build a Purposeful Set?
An MNAR analysis can produce many candidate specifications simply by changing a link, an outcome history, or a sensitivity value. A long list is not necessarily informative: candidates that differ only by arbitrary parameter increments can obscure the scientific assumptions that matter. The goal here is a compact, purposeful set in which each model represents a distinct, interpretable account of how the probability of observation may depend on an unobserved outcome.
The earlier material on aligning candidate models on shared observed-data components is important: when the scientific question concerns missingness assumptions, avoid changing the outcome model or covariate structure at the same time without a reason. Then differences in the implied missing outcomes can be interpreted in light of the changed MNAR assumption. This does not mean the data determine every MNAR parameter. Such parameters are often weakly identified, or not identified without restrictive parametric assumptions, so a candidate set is primarily a structured sensitivity analysis.
A Selection-Model Specification
Let \(Y\) be an outcome, \(R=1\) indicate that it is observed, and \(R=0\) indicate that it is missing. Let \(X\) denote observed covariates and, where relevant, observed outcome history. A selection model factors the joint distribution into an outcome model and a response model:
For a continuous outcome, one useful sensitivity specification is
Here \(m(X)\) and \(s(X)>0\) define the outcome scale used for standardization; for example, they may be a conditional mean and standard deviation from the shared outcome model. The function \(a(X)\) describes response odds at \(z=0\). The parameter \(\gamma\) is the sensitivity parameter: holding \(X\) fixed, a one-unit increase in \(z\), or one conditional standard deviation in outcome, multiplies the odds of observation by \(\exp(\gamma)\). Thus \(\gamma<0\) says that larger outcomes have lower observation odds, while \(\gamma>0\) says they have higher observation odds.
The interpretation depends on the outcome scale and on the assumed linear relationship between \(z\) and log odds. The specification also assumes that the chosen \(X\) captures the observed information being conditioned on, that the response probabilities are well defined, and that the shared outcome model is adequate for the outcomes being extrapolated. In particular, \(P(R=1\mid Y,X)\) must lie strictly between zero and one in the relevant range. Setting \(\gamma=0\) removes direct dependence on the unobserved outcome; under the stated conditioning on \(X\), this is the MAR benchmark, not an MNAR candidate.
A useful derivation shows why \(\gamma\) changes the missing-outcome distribution. Write \(\pi(y,x)=P(R=1\mid Y=y,X=x)\). By the conditional probability rule,
Thus outcomes are reweighted by their probability of being missing. A negative \(\gamma\) gives relatively more weight to larger \(z\) values among missing outcomes; a positive \(\gamma\) gives relatively more weight to smaller values. This follows from the response model, not from an assumption that the observed outcomes themselves are biased in a particular direction.
Worked Example: Calculating a Missing-Outcome Mean
Suppose, hypothetically, that \(z\) takes the three values \(-1,0,1\), each with population probability \(1/3\). Let \(a(X)=\alpha=\log 4\), so the observation odds at \(z=0\) are \(4\), giving \(P(R=1\mid z=0)=0.8000\). Consider \(\gamma=-0.25\). The assumed model is logistic, the three possible \(z\) values are specified, and the probabilities are calculated from that model; no claim is made that these hypothetical values describe an actual study.
At \(z=-1,0,1\), the linear predictors are respectively \(\log 4+0.25\), \(\log 4\), and \(\log 4-0.25\). Applying the inverse-logit gives observation probabilities \(0.8370, 0.8000,\) and \(0.7570\). The corresponding missingness probabilities are \(0.1630, 0.2000,\) and \(0.2430\).
Because each \(z\) value has population probability \(1/3\), that common factor cancels when forming the conditional mean among missing outcomes:
The model therefore gives a missing-outcome mean of approximately \(0.1320\) standard-deviation units above the center \(z=0\). The positive value is consistent with the negative \(\gamma\): higher outcomes have lower observation odds.
For comparison, if \(\gamma=-0.50\), the observation probabilities are \(0.8683,0.8000,\) and \(0.7081\), and the missingness probabilities are \(0.1317,0.2000,\) and \(0.2919\). The implied mean is
The stronger negative dependence produces a larger upward shift in the missing-outcome mean. These values are model-based implications, not estimates of the true missing-outcome mean from the observed data alone.
Choosing Candidate Assumptions
Choose candidates by asking what unobserved-outcome differences could plausibly affect observation after conditioning on \(X\). Clinical knowledge, the study’s follow-up process, and prior evidence may suggest a direction or a range of response-odds changes. Translate those claims into the model’s parameter scale before fitting or calculating consequences. For the specification above, a statement about a one-standard-deviation increase translates directly to \(\exp(\gamma)\), the conditional observation-odds ratio.
A useful set usually includes a benchmark and a small number of scientifically distinct MNAR departures. The benchmark \(\gamma=0\) helps show what follows under MAR. MNAR candidates should represent plausible departures, not just convenient decimal values. If the direction is genuinely uncertain, include candidates on both sides of zero only when both directions have defensible rationales. A candidate that represents an extreme stress test can be useful, but label it as such rather than presenting it as equally plausible.
Worked Example: Translating Expert Judgments Into Candidates
Imagine a hypothetical follow-up study with a standardized symptom outcome, where higher scores mean worse symptoms. Investigators believe that people with worse symptoms may be less likely to complete follow-up, but they cannot say how strong this dependence is. They judge that a one-standard-deviation increase might multiply observation odds by about \(0.7\) in a moderate scenario or \(0.4\) in a stronger scenario.
The corresponding sensitivity values are \(\gamma=\log(0.7)\approx-0.3567\) and \(\gamma=\log(0.4)\approx-0.9163\). These are not selected because the numbers are convenient: they encode the two stated scientific judgments. The first candidate says that, holding \(X\) fixed, each standard-deviation increase multiplies the odds of observation by \(0.7\); the second says it multiplies them by \(0.4\).
For illustration, suppose the baseline observation odds at \(z=0\) are \(3\), so the baseline response probability is \(3/(1+3)=0.7500\). Under the moderate candidate, the odds at \(z=1\) are \(3(0.7)=2.1\), giving response probability \(2.1/(1+2.1)=0.6774\). At \(z=-1\), the odds are \(3/0.7=4.286\), giving probability \(4.286/(1+4.286)=0.8108\). Under the stronger candidate, the odds at \(z=1\) are \(3(0.4)=1.2\), giving probability \(1.2/(1+1.2)=0.5455\); at \(z=-1\), they are \(3/0.4=7.5\), giving probability \(7.5/(1+7.5)=0.8824\).
These calculations check the interpretation and direction of both candidates. They also expose a modeling assumption: the odds ratio applies per standard deviation across the specified outcome range. Before adopting either model, investigators should consider whether that scale and constant log-odds relationship remain credible for very high or low symptom scores.
Worked Example: Writing a Compact Candidate Set and Rationale
For a hypothetical trial with a continuous, higher-is-worse outcome, suppose the shared outcome model and observed covariates have already been specified. The scientific team considers worse unobserved outcomes more likely to be associated with dropout, but regards the strength as uncertain. It proposes this compact set:
| Candidate | Assumption | Scientific rationale |
|---|---|---|
| Benchmark: \(\gamma=0\) | Response does not depend directly on the unobserved outcome after conditioning on \(X\). | Reference case for assessing departures; not asserted as the sole plausible mechanism. |
| Moderate MNAR: \(\gamma=\log(0.7)\) | Each standard-deviation increase in outcome multiplies observation odds by 0.7. | Represents a meaningful but not severe tendency for participants with worse outcomes to be less likely to complete follow-up. |
| Stronger MNAR: \(\gamma=\log(0.4)\) | Each standard-deviation increase multiplies observation odds by 0.4. | Stress-tests a stronger version of the same clinically motivated dropout mechanism. |
This set varies one interpretable assumption while keeping the stated outcome scale and observed-data structure fixed. The two MNAR candidates share a direction and differ in strength, which matches the team’s scientific judgment. If the team had credible evidence for the opposite direction, it could add a positive-\(\gamma\) candidate with its own rationale; it should not add one merely to make the set look symmetric.
The written record should also state what “one standard deviation” means, how \(a(X)\) and the response model are handled, what outcome range the logit-linear dependence is intended to cover, and why the chosen values are plausible. That documentation lets a reader distinguish a reasoned sensitivity range from a post hoc selection of favorable results.
Diagnostics, Alternatives, and Limitations
A candidate set is going wrong if its mathematical implications conflict with its stated rationale or if its probabilities become implausible in regions that matter. Before interpreting results, calculate response probabilities over the observed and extrapolated outcome range, check that they stay away from numerical extremes where relevant, and confirm that increasing \(z\) moves the probabilities in the intended direction. Also check that the standardization is stable and meaningful; a poorly defined or near-zero \(s(X)\) makes the per-standard-deviation interpretation unreliable.
Other warning signs include a large unexplained change to the shared outcome model across candidates, a parameter range chosen only after seeing which treatment effect is preferred, and many near-duplicate values that add no substantive scenario. Observed-data fit alone cannot generally validate an MNAR dependence assumption: different assumptions about unobserved outcomes can be compatible with similar observed information. A strong fit to observed data is therefore not evidence that a particular \(\gamma\) is scientifically correct.
Selection models express the assumption through \(P(R\mid Y,X)\). Pattern-mixture models express it through distributions of outcomes within missingness patterns. These are alternative parameterizations of the missing-data problem, and translating between them requires specifying enough of the joint model; do not assume that a parameter value in one representation has a direct numerical counterpart in the other. The focus here is selecting and documenting candidates within a selection-model framework, not declaring one parameterization universally preferable.
Finally, the candidate set is not a probability distribution over mechanisms and its values are not confidence limits. It does not remove uncertainty about model form, guarantee that the true mechanism is included, or repair misspecified outcome or response models. Its value is transparency: the conclusions can be examined under a short, defensible range of departures from the benchmark.
Specify the outcome model, observed covariates, and outcome scale to be held common across candidates.
Describe why an unobserved outcome might affect observation and whether its direction is known or uncertain.
For this logit specification, convert a per-standard-deviation response-odds ratio \(q\) to \(\gamma=\log(q)\).
Calculate response probabilities, verify the direction and plausible range, and record why each candidate belongs in the set.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In the stated selection model, what does \(\exp(\gamma)=0.8\) mean for observation odds?
- For a one-standard-deviation odds ratio of \(0.5\), calculate \(\gamma\) and state the direction of the dependence.
- Why is \(\gamma=0\) useful even when the analysis focuses on MNAR candidates?
- Name two checks that could reveal a poorly specified or poorly documented candidate set.
Solutions. (1) Holding \(X\) fixed, a one-standard-deviation increase in outcome multiplies the odds of observation by \(0.8\), a 20% reduction in odds. (2) \(\gamma=\log(0.5)\approx-0.6931\); higher outcomes have lower observation odds. (3) It supplies a MAR benchmark against which the consequences of the MNAR departures can be described; it does not establish that MAR is true. (4) For example, calculate probabilities over the relevant outcome range and check that their direction matches the rationale; also verify the standardization and document a scientific basis for each sensitivity value.