Concept Architecture
Concept
Theoretically, Missing Not at Random (MNAR) is a missing-data mechanism in which the probability that an observation is missing depends on the unobserved value itself, even after conditioning on all observed data. Within the Rubin framework for missing-data theory, MNAR represents a non-ignorable missing-data mechanism because the process generating missingness contains information about the missing values. It exists to describe situations in which standard likelihood-based methods and multiple imputation under Missing at Random assumptions may produce biased estimates unless the missing-data process is explicitly modelled.
Mathematically, Missing Not at Random is defined by a conditional probability model in which the probability of missingness depends directly on the unobserved values. Since the missing-data mechanism cannot be ignored, estimation requires simultaneous modelling of both the outcome process and the missingness process or the use of sensitivity analyses that explore alternative assumptions regarding the missing values.
In practice, MNAR cannot be confirmed from the observed data alone because the missing values are unavailable. Its plausibility is evaluated using subject-matter knowledge, study design, external evidence and sensitivity analyses. In health economics, MNAR commonly arises when patients with poor health outcomes are less likely to complete quality-of-life questionnaires or when individuals with very high healthcare costs discontinue follow-up. Analyses under MNAR frequently employ selection models, pattern-mixture models or shared-parameter models to assess the robustness of cost-effectiveness conclusions.
Purpose
Used to characterise non-ignorable missing-data mechanisms in which missingness depends on unobserved values, supporting appropriate modelling strategies and sensitivity analyses in health economic evaluations.
Mathematical Formulae
Primary Formula
P(R | Yobs, Ymis, ?) = P(R | Yobs, Ymis, ?)
where:
- R = missingness indicator
- Yobs = observed data
- Ymis = missing data
- ? = parameters governing the missingness process
Under MNAR:
P(R | Yobs, Ymis, ?) ? P(R | Yobs, ?)
Supporting Formulae
Joint likelihood:
P(Y, R | �, ?) = f(Y | �) ? P(R | Yobs, Ymis, ?)
Observed-data likelihood:
L(�, ?) = ? f(Yobs, Ymis | �) ? P(R | Yobs, Ymis, ?)dYmis
Related Mathematical Methods
- Selection Models
- Pattern-Mixture Models
- Shared-Parameter Models
- Multiple Imputation
- Missing at Random
- Missing Completely at Random
- Sensitivity Analysis
Example
A health economist evaluates EQ-5D utility scores in a cancer trial.
Patients experiencing severe deterioration in health are substantially less likely to complete follow-up quality-of-life questionnaires. Even after accounting for age, treatment allocation and previous EQ-5D scores, the probability of missingness remains related to the patients' unobserved current utility values.
The missing-data mechanism is therefore considered Missing Not at Random.
Rather than relying solely on standard multiple imputation, the analyst performs a pattern-mixture sensitivity analysis by assuming that missing utility values are systematically lower than predicted under the Missing at Random assumption. The resulting incremental cost-effectiveness ratio is compared across alternative assumptions to evaluate the robustness of the economic conclusions.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| IF | =IF(ISBLANK(B2),1,0) | Create a missingness indicator variable. |
| ISBLANK | =ISBLANK(B2) | Identify missing observations. |
| COUNTIF | =COUNTIF(C2:C501,1) | Summarise missing-data patterns. |
| Data Table | Evaluate alternative assumptions for missing values. | Perform deterministic sensitivity analyses under MNAR scenarios. |
| Solver | Estimate simplified selection-model parameters under assumed MNAR mechanisms. | Illustrate sensitivity analyses involving non-ignorable missingness. |
VBA (Optional)
A VBA routine can generate alternative MNAR sensitivity scenarios, apply user-defined adjustments to missing values and summarise their impact on economic evaluation results.
Sources
- Rubin DB. Inference and Missing Data. Biometrika. 1976;63(3):581?592.
- Little RJA, Rubin DB. Statistical Analysis with Missing Data. Wiley.
- Carpenter JR, Kenward MG. Multiple Imputation and Its Application. Wiley.
- Molenberghs G, Kenward MG. Missing Data in Clinical Studies. Wiley.
- National Research Council. The Prevention and Treatment of Missing Data in Clinical Trials. National Academies Press.
- Faria R, Gomes M, Epstein D, White IR. A Guide to Handling Missing Data in Cost-Effectiveness Analysis Conducted within Randomised Controlled Trials. Pharmacoeconomics. 2014;32(12):1157?1170.
Related Concepts (3)
Library
Publications
1
Bayesian Methods in Health Economics — Gianluca Baio, 1st Edition ed., 2012 (Chapman & Hall / CRC Press)
An overview of Bayesian statistical methods for the analysis of health economic data, covering economic evaluation concepts, statistical cost-effectiveness analysis, Bayesian computation and MCMC, and applied health economic evaluation.
BookView source →
Frequently Asked Questions (6)
What does missing not at random mean?
A missing data classification indicating the probability of missingness depends on the unobserved value itself, even after accounting for other variables.
Source: Rubin 1976
What makes data missing not at random especially troublesome?
Data are missing not at random when the chance a value is absent depends on that very value, even after accounting for everything else observed, as when the sickest patients are the ones who fail to report their symptoms. This is especially troublesome because the observed data give no reliable way to fill the gaps, and standard methods that assume the gaps are explained by what was seen will be biased. Coping with it requires explicit assumptions about the unseen values and sensitivity analysis around them. Missingness tied to the hidden value is its nature. Little and Rubin (2002) describe this.
Source: Little & Rubin 2002
How does missing not at random differ from missing at random?
Missing not at random means the probability of missingness depends on the unobserved values themselves, while missing at random means it depends only on observed variables, not on the missing values once those are accounted for. Under missing at random the observed data can correct the missingness; under missing not at random they cannot fully. So the two differ in whether missingness relates to the unobserved values, with missing not at random being more problematic because the standard methods valid under missing at random can be biased, since the reason for missingness is tied to the missing values, which the observed data do not capture.
Source: Rubin 1976
Why is missing not at random problematic?
Missing not at random is problematic because the missingness depends on the unobserved values, so the observed data do not contain the information needed to correct for it, and standard methods valid under missing at random, such as multiple imputation as usually applied, can give biased results. So missing not at random poses a serious challenge, since no method can fully recover unbiased estimates from the observed data alone, which is why analyses under this mechanism rely on assumptions about the missingness that cannot be verified, and why sensitivity analyses and specialised models are used to explore the possible impact rather than to claim a definitive correction.
Source: Rubin 1976
How is missing not at random handled?
Missing not at random is handled by specialised approaches that make explicit assumptions about how missingness relates to the unobserved values, such as selection models and pattern-mixture models, and importantly by sensitivity analyses that examine how conclusions change under different assumptions about the missingness. So missing not at random is handled through methods and analyses that acknowledge the untestable dependence of missingness on the missing values, since no approach can eliminate the uncertainty, which is why the emphasis is on transparently stating assumptions and testing the robustness of conclusions to them, rather than on a single corrective method that would falsely imply the bias had been removed.
Source: Rubin 1976
Can missing not at random be distinguished from missing at random?
Missing not at random cannot be distinguished from missing at random using the observed data alone, because doing so would require knowing the missing values, which are unavailable, so the two mechanisms can fit the observed data equally well. The distinction must be argued from subject-matter knowledge. So whether data are missing at random or not at random is untestable from the data, which is a fundamental difficulty, and it is why reasoning about the likely reasons for missingness and conducting sensitivity analyses across assumptions are central to handling incomplete data, since the choice between assuming missing at random and missing not at random rests on judgement that the data cannot settle.
Source: Rubin 1976
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 18 Dec 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-ES-SA-123
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