Concept Architecture
Concept
Theoretically, Missing Completely at Random (MCAR) is a missing-data mechanism in which the probability that an observation is missing is entirely independent of both the observed data and the unobserved data. Introduced within the Rubin framework for missing-data theory, MCAR represents the most restrictive missingness assumption and implies that the observed data constitute a simple random sample of the complete dataset. It exists as the theoretical condition under which missing observations do not introduce systematic bias into statistical analyses.
Mathematically, Missing Completely at Random is defined by a probability model in which the missingness indicator is statistically independent of both observed and missing values. Under MCAR, the missing-data mechanism is completely unrelated to the data-generating process, allowing complete-case analyses to produce unbiased parameter estimates, although statistical efficiency is reduced because observations are discarded.
In practice, MCAR is rarely encountered in health research because missingness often depends on patient characteristics or clinical outcomes. Nevertheless, the assumption is commonly assessed using descriptive analyses and formal statistical tests such as Little's MCAR test. In health economics, MCAR provides the theoretical justification for complete-case analyses in clinical trials and observational studies involving healthcare costs, quality-adjusted life years, resource utilisation and patient-reported outcomes, although analysts should carefully evaluate whether the assumption is plausible.
Purpose
Used to characterise a missing-data mechanism in which missingness occurs entirely at random, supporting unbiased complete-case analyses and providing a benchmark assumption for missing-data methodology.
Mathematical Formulae
Primary Formula
P(R | Yobs, Ymis, ?) = P(R | ?)
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
Under MCAR:
R ? (Yobs, Ymis)
where:
- ? denotes statistical independence
Related Mathematical Methods
- Complete-Case Analysis
- Missing at Random
- Missing Not at Random
- Little's MCAR Test
- Multiple Imputation
- Maximum Likelihood Estimation
- Sensitivity Analysis
Example
A health economist analyses healthcare costs collected from 600 trial participants.
Ten questionnaires are lost during postal delivery because of an administrative error. The probability that a questionnaire is lost is unrelated to patient age, treatment allocation, healthcare costs or any other measured or unmeasured characteristic.
The missing-data mechanism is therefore considered Missing Completely at Random.
A complete-case analysis is performed using the remaining 590 participants. Because the missing observations constitute a random subset of the original sample, parameter estimates remain unbiased, although statistical precision is slightly reduced because of the smaller sample size.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| ISBLANK | =ISBLANK(B2) | Identify missing observations. |
| IF | =IF(ISBLANK(B2),1,0) | Create a missingness indicator variable. |
| COUNTIF | =COUNTIF(C2:C601,1) | Count missing observations. |
| FILTER | =FILTER(B2:F601,C2:C601=0) | Extract complete cases for analysis in supported Excel versions. |
| AVERAGE | =AVERAGE(B2:B591) | Calculate summary statistics for complete-case analyses. |
VBA (Optional)
A VBA routine can identify complete cases, summarise missing-data patterns and prepare datasets for complete-case analyses under the MCAR assumption.
Sources
- Rubin DB. Inference and Missing Data. Biometrika. 1976;63(3):581?592.
- Little RJA, Rubin DB. Statistical Analysis with Missing Data. Wiley.
- Little RJA. A Test of Missing Completely at Random for Multivariate Data with Missing Values. Journal of the American Statistical Association. 1988;83(404):1198?1202.
- Carpenter JR, Kenward MG. Multiple Imputation and Its Application. Wiley.
- 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 (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What does missing completely at random mean?
A missing data classification indicating the probability of missingness is entirely unrelated to any observed or unobserved variables, the strongest assumption.
Source: Rubin 1976
What does missing completely at random assume about the gaps in data?
Missing completely at random assumes that the chance of a value being absent has nothing to do with any variable, observed or not, so the missing observations are effectively a random sample of all the data. Under this strong assumption even simply dropping incomplete records leaves the remaining data unbiased, merely smaller. It is the most convenient case but rarely holds in practice, since missingness usually relates to something, such as sicker patients dropping out. Gaps unrelated to anything is its premise. Little and Rubin (2002) describe this.
Source: Little & Rubin 2002
How does missing completely at random differ from missing at random?
Missing completely at random means missingness depends on neither observed nor unobserved variables, while missing at random means it may depend on observed variables but not on the missing values themselves. Missing completely at random is thus the stronger, more restrictive assumption. So the two differ in whether missingness can relate to observed data, with missing completely at random requiring no dependence at all and missing at random allowing dependence on observed variables, which means missing completely at random is less often plausible, and methods valid under it are a subset of the situations covered by methods valid under the weaker missing at random assumption.
Source: Rubin 1976
What are the implications of missing completely at random?
The implications of missing completely at random are that the observed cases form a representative random subset of the data, so analyses using only complete cases are unbiased, though they may lose efficiency by discarding partial data. Under this assumption, missingness introduces no bias. So missing completely at random implies that simple approaches such as complete case analysis give unbiased results, which is its convenient consequence, but because it is a strong assumption that often does not hold, relying on it can be risky, and its plausibility must be considered, since if missingness actually depends on the data, treating it as completely at random can bias the analysis.
Source: Rubin 1976
Can missing completely at random be tested?
Missing completely at random can be partially assessed from the observed data, since it implies that the cases with and without missing values do not differ systematically on the observed variables, which can be examined and tested; however, it cannot be fully confirmed, because dependence on the unobserved values cannot be checked. So missing completely at random is more testable than missing at random, as its implication of no differences on observed variables can be examined, but it cannot be proven entirely, which is why apparent support from the observed data is treated as suggestive rather than conclusive, and the assumption is still considered in light of why the data are missing.
Source: Rubin 1976
When is the missing completely at random assumption plausible?
The missing completely at random assumption is plausible when missingness arises from causes genuinely unrelated to the data, such as accidental loss of records, equipment failure unrelated to the values, or missingness by random design. It is implausible when missingness relates to the variables, as when sicker patients drop out. So missing completely at random is plausible only in limited circumstances where the reason for missingness is truly independent of the data, which is often not the case, since dropout and non-response are frequently related to the outcomes or characteristics involved, meaning the weaker missing at random assumption and methods valid under it are usually more appropriate.
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-119
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