What MCAR Says About Complete Cases
A complete-case analysis uses only units for which every variable needed for the analysis is observed. The central implication of missing completely at random (MCAR) is direct: if complete-case status is independent of the complete-data values, then the distribution of those values among complete cases is the same as their distribution in the full study population. This is a population-level statement about selection, not a promise that every realized subset will match the population exactly.
The distinction matters in practice. Under MCAR, complete cases do not systematically over-represent units with particular values of the variables in the analysis. But excluding incomplete records still reduces the amount of data, and the resulting estimate can differ from the target simply through sampling variation. The implication also concerns the study’s complete-data distribution; it does not by itself establish that the original study sample represents some wider target population.
The statement applies to a complete-case indicator for a specified set of variables. If an analysis requires only a subset of the variables collected in a study, define \(C_i\) using that analysis’s required variables. Under MCAR for the full missingness pattern, any such indicator derived from \(R_i\) is also independent of \(Y_i\). A unit with a missing value on an irrelevant variable need not be excluded from every possible analysis.
To interpret representativeness, suppose complete-case status has positive probability, \(P(C_i=1)>0\). For any set \(A\) of possible complete-data values, independence gives \(P(Y_i\in A,C_i=1)=P(Y_i\in A)P(C_i=1)\). Dividing by \(P(C_i=1)\) shows that the complete-data distribution conditional on being a complete case is unchanged.
Thus, any feature of the complete-data distribution is preserved among complete cases at the population level: proportions, means, variances, and joint relationships. The result does not require the missingness rate to be small or constant across variables. It requires the missingness pattern to be independent of the complete-data values. MCAR also does not require the different variables’ missingness indicators to be independent of one another.
Worked Example: A Binary Outcome Among Complete Cases
Suppose, hypothetically, that 30% of a study population has a binary health outcome \(D=1\). A record is complete with probability 0.80, and the MCAR assumption says that this probability is the same whether \(D=1\) or \(D=0\). Find the outcome proportion among complete cases.
The assumed probabilities are \(P(D=1)=0.30\), \(P(D=0)=0.70\), and \(P(C=1\mid D=1)=P(C=1\mid D=0)=0.80\). The joint probability of having the outcome and being a complete case is \(P(D=1,C=1)=0.30(0.80)=0.24\). The overall probability of a complete case is \(P(C=1)=0.80\).
The complete-case outcome proportion is 0.30, the same as in the study population. This equality follows from the stipulated MCAR probabilities; it does not say that the observed proportion in any particular finite set of complete cases must be exactly 0.30.
What Is Preserved: Joint Distributions and Relationships
Representativeness applies to the joint distribution, not merely to each variable considered separately. This matters when the analysis concerns a relationship, such as an association between a predictor and an outcome. If the joint distribution is unchanged, then its population summaries—including means and covariances when they exist—are unchanged as well.
Worked Example: Preserving a Joint Distribution
Consider two hypothetical binary variables, \(X\) and \(Y\), with the following complete-data probabilities. Suppose the probability of being a complete case is 0.75 for every combination of \(X\) and \(Y\), as required here by MCAR. Find the joint distribution among complete cases and compare the means and covariance.
| \(X\) | \(Y\) | Complete-data probability |
|---|---|---|
| 0 | 0 | 0.40 |
| 0 | 1 | 0.10 |
| 1 | 0 | 0.20 |
| 1 | 1 | 0.30 |
For each cell, multiply the complete-data probability by 0.75 to get the joint probability of that cell and being complete. The resulting probabilities are 0.30, 0.075, 0.15, and 0.225, respectively. They sum to \(P(C=1)=0.75\). Dividing each by 0.75 gives the conditional cell probabilities 0.40, 0.10, 0.20, and 0.30: exactly the original joint distribution.
In the full population, \(E(X)=P(X=1)=0.20+0.30=0.50\), and \(E(Y)=P(Y=1)=0.10+0.30=0.40\). Also, \(E(XY)=P(X=1,Y=1)=0.30\). Therefore, \(\operatorname{Cov}(X,Y)=E(XY)-E(X)E(Y)=0.30-(0.50)(0.40)=0.10\). Because the joint distribution is unchanged among complete cases, these same calculations apply there: \(E(X\mid C=1)=0.50\), \(E(Y\mid C=1)=0.40\), and \(\operatorname{Cov}(X,Y\mid C=1)=0.10\). MCAR preserves the population association as well as the separate variable distributions.
Population Representativeness Is Not Exact Sample Matching
The distributional result is not a claim that the observed complete cases will have the same empirical proportions, mean, or other summary as the full data would have had. Random selection can produce an unbalanced subset, especially when the number of complete cases is small. Under standard independent sampling conditions, estimates based on complete cases target the same population quantities, but they are generally less precise because fewer units contribute.
For a mean, suppose the original units are independently sampled from a population with finite mean \(\mu\), and each unit’s complete-case status is independent of its complete-data value. Conditional on a positive number \(n_C\) of complete cases, those selected values have mean \(\mu\) under the usual independent-unit selection setup. Their sample mean therefore has expectation \(\mu\) conditional on \(n_C\). Its realized value can still differ from \(\mu\). This expectation statement depends on the sampling and selection setup; MCAR is a statement about independence of missingness and values, not a guarantee of independent observations or a particular sampling design.
Worked Example: An Unrepresentative Realization Under MCAR
Imagine a hypothetical finite set of four complete-data values for a quantitative outcome: 2, 4, 8, and 10 units. Their mean is \((2+4+8+10)/4=24/4=6\) units. Suppose exactly two units are selected as complete cases, with every pair equally likely. This is what results when each unit has the same independent MCAR selection probability, conditional on exactly two being selected.
One possible selected pair is 2 and 4, whose mean is \((2+4)/2=3\) units. That observed mean is not 6 units. The six possible pairs have means 3, 5, 6, 6, 7, and 9 units, so their average is \((3+5+6+6+7+9)/6=36/6=6\) units. The selection rule preserves the target in expectation across possible samples, even though this particular complete-case sample is not exactly representative.
The example also shows why MCAR does not prevent chance imbalance. A realized difference between a complete-case summary and the full-data summary is not, by itself, proof that MCAR is false.
Assumptions, Diagnostics, and Limits
The core representativeness implication requires that MCAR hold for the relevant complete-data vector and that there be a positive probability of obtaining complete cases. To translate it into statements about estimates, additional conditions may be needed: for example, independent sampling of units and an estimator appropriate to the target quantity. If the original sample was not drawn from the intended target population, MCAR does not repair that mismatch. Nor does MCAR make observations independent when the design has clustering or other dependence.
MCAR is an assumption about missingness relative to values that may be unobserved. The observed records cannot, in general, certify that the missingness probability was unrelated to values that were not recorded. Observed patterns can raise concerns about MCAR, but a lack of an obvious pattern is not proof. The practical question is whether the study design and subject-matter knowledge make value-independent missingness plausible.
Several common failures of interpretation are worth avoiding:
- “MCAR means the complete cases exactly match the full sample.” No. The distributions agree at the population level, while finite samples can differ by chance.
- “MCAR means missingness indicators are mutually independent.” No. MCAR requires independence of the missingness pattern from the complete-data values. Variables may have related missingness patterns.
- “MCAR makes complete-case analysis as precise as using all units.” No. Excluding incomplete records ordinarily reduces the effective sample size and can increase uncertainty, even when it does not change the target distribution.
- “If complete cases look representative, MCAR has been established.” No. Similarity in observed summaries cannot rule out selection depending on unobserved values.
The main alternative is to use methods that retain information from records that are incomplete, including variable-specific analyses or a principled missing-data model such as multiple imputation. These approaches can use more of the observed data, but they require their own assumptions and careful implementation. The comparison is not simply “complete cases are wrong, alternatives are right”: under MCAR, complete-case analysis can target the complete-data distribution, while the cost is often lost information and precision. If MCAR is not plausible, that representativeness conclusion cannot be carried over automatically; the conditions for valid analysis must be considered directly.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In one sentence, state the distributional implication of MCAR for complete cases.
- A binary characteristic has population proportion 0.40, and the probability of being a complete case is 0.70 for both characteristic groups. What is its proportion among complete cases?
- Under MCAR, can the mean in a particular complete-case sample differ from the mean in the full sample? Explain briefly.
- Does MCAR require missingness indicators for two variables to be independent of each other?
Solutions. (1) The complete-data distribution conditional on complete-case status equals the unconditional complete-data distribution, provided complete cases occur with positive probability. (2) It remains 0.40: \(P(D=1,C=1)=0.40(0.70)=0.28\), and \(P(C=1)=0.70\), so \(P(D=1\mid C=1)=0.28/0.70=0.40\). (3) Yes. MCAR gives a population-level distributional result; random variation can make a realized complete-case mean differ. (4) No. MCAR requires the missingness pattern to be independent of complete-data values, not the components of that pattern to be mutually independent.