Concept Architecture
Concept
Theoretically, Missing at Random (MAR) is a missing-data mechanism in which the probability that an observation is missing depends only on observed variables and not on the unobserved value itself, conditional on the observed data. Introduced within the Rubin framework for missing-data theory, MAR provides the assumptions under which likelihood-based methods and multiple imputation can produce valid statistical inference. It exists to distinguish situations in which missingness can be appropriately modelled using available information from those requiring stronger assumptions.
Mathematically, Missing at Random is defined through the conditional probability distribution governing the missing-data indicator. Under MAR, the probability of missingness may depend on observed variables but is conditionally independent of the missing values themselves. This assumption permits the missing-data mechanism to be treated as ignorable for likelihood-based inference when model parameters governing the data and missingness processes are distinct.
In practice, MAR cannot be verified directly because the missing values are unobserved. Instead, its plausibility is assessed using substantive knowledge, patterns of missingness and relationships between missingness indicators and observed variables. In health economics, MAR commonly underpins the use of multiple imputation, maximum likelihood estimation and mixed-effects models for analysing incomplete data on healthcare costs, quality-adjusted life years, utility scores and clinical outcomes collected in trials and observational studies.
Purpose
Used to characterise a missing-data mechanism under which likelihood-based estimation and multiple imputation provide valid statistical inference by conditioning on observed information.
Mathematical Formulae
Primary Formula
P(R | Yobs, Ymis, ?) = P(R | Yobs, ?)
where:
- R = missingness indicator
- Yobs = observed data
- Ymis = missing data
- ? = parameters governing the missingness process
Supporting Formulae
Observed-data likelihood:
L(�) = ? f(Yobs, Ymis | �)dYmis
Ignorable likelihood factorisation:
P(Y, R | �, ?) = f(Y | �) ? P(R | Yobs, ?)
provided that MAR and parameter distinctness hold.
Related Mathematical Methods
- Multiple Imputation
- Maximum Likelihood Estimation
- Expectation-Maximisation Algorithm
- Mixed-Effects Models
- Missing Completely at Random
- Missing Not at Random
- Sensitivity Analysis
Example
A health economist analyses EQ-5D utility scores collected at baseline, 6 months and 12 months in a randomised controlled trial.
Older participants are more likely to miss the 12-month questionnaire, but after accounting for age, treatment group and previous EQ-5D scores, the probability of missingness is unrelated to the unobserved 12-month utility value itself.
Under this assumption, the missing-data mechanism is considered Missing at Random. Multiple imputation is therefore performed using age, treatment allocation and previous utility scores to estimate the missing values before conducting the cost-effectiveness analysis.
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) | Count missing observations for diagnostic summaries. |
| AVERAGEIF | =AVERAGEIF(C2:C501,0,B2:B501) | Compare observed values between complete and incomplete groups. |
| Solver | Estimate parameters under likelihood-based models using complete observed data. | Support simplified MAR-based estimation workflows. |
VBA (Optional)
A VBA routine can identify missing-data patterns, generate missingness indicators and prepare datasets for multiple imputation or likelihood-based analyses under the MAR assumption.
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.
- 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.
- NICE. Health Technology Evaluation Manual.
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 at random mean?
A missing data classification indicating the probability of a value being missing depends on other observed variables, not the missing value itself.
Source: Rubin 1976
What does missing at random say about why data are absent?
Missing at random says that whether a value is absent depends on other information that was observed, but not on the missing value itself once those observed factors are taken into account. For example, if older patients are more likely to skip a measurement, the missingness is explained by age, which is recorded. This matters because principled methods such as multiple imputation give unbiased results under this assumption, using the observed data to fill the gaps. Missingness explained by what was seen is its meaning. Little and Rubin (2002) describe this.
Source: Little & Rubin 2002
How does missing at random differ from missing completely at random?
Missing at random means the probability of missingness depends on observed variables but not on the missing values themselves, while missing completely at random means it depends on neither observed nor unobserved variables, so missingness is entirely unrelated to the data. Missing completely at random is the stronger, more restrictive assumption. So the two differ in whether missingness can depend on observed data, with missing at random allowing such dependence and missing completely at random not, which means missing at random is more often plausible, and methods valid under missing at random, such as multiple imputation, are more broadly applicable than those requiring missing completely at random.
Source: Rubin 1976
Why does missing at random matter for analysis?
Missing at random matters because it is the assumption under which many principled missing data methods, such as multiple imputation and likelihood-based approaches, give valid results by using the observed variables to account for the missingness. Under this assumption, the missing data can be handled without bias if done properly. So missing at random matters as the condition that justifies the common methods for missing data, since under it the observed data contain the information needed to correct for the missingness, which is why establishing the plausibility of missing at random, and using methods valid under it, is central to handling incomplete data appropriately.
Source: Rubin 1976
Can the missing at random assumption be tested?
The missing at random assumption cannot be fully tested from the observed data alone, because distinguishing it from missing not at random would require knowing the missing values, which are unavailable. Its plausibility must be judged from subject-matter knowledge and the observed data. So the missing at random assumption is largely untestable, which is why it is assessed through reasoning about why data are missing rather than proven, and why sensitivity analyses are used to examine whether conclusions hold if the assumption fails, since the validity of missing data methods depends on an assumption that cannot be verified directly from the data.
Source: Rubin 1976
How is missing at random handled in analysis?
Missing at random is handled by methods that use the observed data to account for the missingness validly, chiefly multiple imputation, which imputes plausible values conditional on the observed variables, and likelihood-based methods such as mixed models, which use all available data. So under missing at random these principled methods give valid, approximately unbiased results, which is why they are preferred over simple approaches such as complete case analysis, and handling missing at random data well involves using the observed variables to compensate for the missingness, together with sensitivity analyses to check robustness to departures from the assumption.
Source: Rubin 1987
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
- Persistent URI
- https://healtheconomics.wiki/concept/missing-at-random
- Term code
- HE-ES-SA-118
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