What Does “Completely At Random” Mean?
A missing value is not, by itself, evidence for any particular missingness mechanism. To define a mechanism, describe how the pattern of observed and missing entries relates to the values that would be available if the data were complete. Missing completely at random (MCAR) is the strongest form of independence in this classification: the response pattern is independent of every complete-data value, observed or not.
Suppose there are \(n\) individuals and \(p\) variables. Let \(Y_i\) denote the complete-data vector for individual \(i\): it contains that individual’s value on all \(p\) variables, including values that may not have been recorded. Let \(R_i=(R_{i1},\ldots,R_{ip})\) be the response-indicator vector, with \(R_{ij}=1\) when variable \(j\) is observed for individual \(i\) and \(R_{ij}=0\) when it is missing. A particular vector \(r\) describes one response pattern.
The phrase “almost surely” matters. For a continuous \(Y_i\), the event \(Y_i=y\) usually has probability zero. Conditional probabilities at an exact value are represented by a conditional distribution, and that representation is only defined up to changes on sets of \(Y_i\)-probability zero. Independence therefore gives equality almost surely, not necessarily at every point in the support.
Why the Conditional-Probability Form Is Equivalent
The independence definition says that the joint distribution of \(R_i\) and \(Y_i\) factors into their marginal distributions. Since \(R_i\) has a finite set of possible response patterns, this factorization can be expressed pattern by pattern. For any pattern \(r\), let \(B\) be any measurable set of complete-data values. By the definition of conditional probability and the law of total expectation,
The second equality uses the MCAR condition. This is the definition of independence for the event \(R_i=r\) and every measurable set \(B\). Conversely, if \(R_i\) and \(Y_i\) are independent, the conditional distribution of \(R_i\) given \(Y_i\) equals its marginal distribution almost surely. Thus the conditional formulation and independence formulation state the same condition.
MCAR concerns the relationship between the whole response pattern and the complete data. It does not require the components of \(R_i\) to be independent of one another. For example, two measurements may tend to be missing together for a reason unrelated to the values of either measurement. Their joint response pattern can still be independent of the complete-data vector.
Worked Example: Checking MCAR With a Binary Complete-Data Value
Consider a hypothetical survey with a complete-data variable \(Y\) that is either A or B. Suppose \(P(Y=\mathrm{A})=0.60\) and \(P(Y=\mathrm{B})=0.40\). Let \(R=1\) mean the value is observed. The joint probabilities are:
| Complete-data value | \(P(R=1,Y)\) | \(P(R=0,Y)\) |
|---|---|---|
| A | 0.48 | 0.12 |
| B | 0.32 | 0.08 |
First calculate the response probability in each group:
The marginal response probability is \(P(R=1)=0.48+0.32=0.80\), so the response probability equals its marginal value for both possible values of \(Y\). For \(R=0\), the conditional probability is \(0.12/0.60=0.20\) in group A and \(0.08/0.40=0.20\) in group B, matching \(P(R=0)=0.20\). The condition holds for both response patterns, so this hypothetical mechanism is MCAR.
Worked Example: A Response Rate That Depends on the Complete Data
Now suppose a hypothetical binary outcome \(Y\) is low or high, with probability 0.50 for each value. Let the response probabilities be 0.90 for low values and 0.50 for high values. The corresponding joint probabilities are:
| Complete-data value | \(P(R=1,Y)\) | \(P(R=0,Y)\) |
|---|---|---|
| Low | \(0.50(0.90)=0.45\) | \(0.50(0.10)=0.05\) |
| High | \(0.50(0.50)=0.25\) | \(0.50(0.50)=0.25\) |
The marginal response probability is \(P(R=1)=0.45+0.25=0.70\). But the conditional probabilities are
Neither conditional probability equals the marginal value 0.70 in both groups. The response pattern is associated with the complete-data value, so MCAR does not hold. Notice that the overall response rate alone—70%—does not reveal this dependence; the conditional rates do.
Patterns in Multivariate Data
With several variables, there may be many response patterns. The MCAR check is still the same: for each possible pattern \(r\), its probability must be constant across complete-data values, in the almost-sure sense. The probabilities of different patterns need not be equal.
Worked Example: MCAR Even Though Variables Go Missing Together
Suppose two hypothetical measurements have complete-data vector \(Y=(Y_1,Y_2)\). For every possible value \(y\) of that vector, assume the response-pattern probabilities are as follows.
| Pattern \(R\) | Meaning | \(P(R=r\mid Y=y)\) |
|---|---|---|
| (1, 1) | Both measurements observed | 0.50 |
| (1, 0) | Only the first observed | 0.20 |
| (0, 1) | Only the second observed | 0.20 |
| (0, 0) | Both missing | 0.10 |
These probabilities sum to \(0.50+0.20+0.20+0.10=1.00\). Each is constant across \(y\), so for every pattern it equals its marginal probability. For example, \(P(R=(1,1)\mid Y=y)=0.50=P(R=(1,1))\) for all \(y\). The mechanism is MCAR.
However, the two response indicators are not independent. Their marginal probabilities of being observed are \(P(R_1=1)=0.50+0.20=0.70\) and \(P(R_2=1)=0.50+0.20=0.70\). If they were independent, the probability that both are observed would be \(0.70(0.70)=0.49\), whereas the specified probability is 0.50. This small difference is enough to show that independence among response indicators is not required for MCAR.
Continuous Data and the Almost-Sure Qualification
Worked Example: MCAR With a Continuous Complete-Data Value
Let a hypothetical complete-data measurement satisfy \(Y\sim N(50,100)\), so its mean is 50 units and its standard deviation is 10 units. Suppose a response indicator \(R\) is generated independently of \(Y\), with \(P(R=1)=0.85\). Then a version of the conditional response probability is \(P(R=1\mid Y)=0.85\) almost surely, and \(P(R=0\mid Y)=0.15\) almost surely. Both match their marginal probabilities, so the mechanism is MCAR.
As a numerical check, standardizing gives \(P(40\leq Y\leq60)=P(-1\leq Z\leq1)\approx0.6827\), where \(Z\sim N(0,1)\). Independence implies
The calculation uses a range of positive probability, not conditioning on one exact value of a continuous variable. That is why the almost-sure statement is the appropriate formal condition: changing a conditional-probability version at a single point does not change any probability for a range or any feature of the joint distribution.
For discrete complete data, each value with positive probability can be checked directly, as in the first two examples. For continuous or mixed data, the definition is a statement about conditional distributions that holds almost surely. It is not a requirement to assign a meaningful probability to every exact continuous value.
How MCAR Relates to Other Mechanisms
The factorization of data and missingness mechanisms provides a useful way to locate MCAR: it requires the missingness distribution \(P(R\mid Y)\) not to vary with \(Y\). MAR includes MCAR as a special case: MAR permits response to depend on values that are observed, while MCAR permits no dependence on any complete-data values. MNAR permits dependence on unobserved values even after conditioning on observed values. These labels describe assumptions about the mechanism, not facts established merely by looking at an overall missing-data percentage.
The practical distinction is important. Under MCAR, the observed response pattern contains no information about the complete-data values. Under the broader MAR condition, response may vary across observed groups and still satisfy MAR. Therefore, evidence that response rates differ by an observed variable rules out MCAR, but does not by itself decide whether MAR holds.
Diagnostics, Common Mistakes, and Limitations
MCAR is an assumption about the joint distribution of response patterns and complete data. Because the values that are missing are unavailable, the full relationship cannot generally be checked directly from observed records. Some diagnostics can challenge MCAR: for example, if response rates differ systematically across categories of a fully observed variable, then response is associated with at least part of the complete-data vector and MCAR is not plausible under the specified model. But finding no such difference does not prove MCAR; dependence on unobserved values may not be visible in the observed data.
- Common mistake: checking only the overall missingness percentage. An overall rate such as 20% missing gives no information about whether response probability varies with \(Y\). Compare conditional response probabilities across values or groups where possible.
- Common mistake: requiring equality at every continuous value. The correct condition is equality almost surely for a regular conditional distribution. Pointwise changes on a probability-zero set do not alter independence.
- Common mistake: assuming the response indicators must be mutually independent. MCAR requires \(R\) to be independent of \(Y\); the components of \(R\) may still be associated with one another.
- Common mistake: treating MCAR and MAR as disjoint. MCAR is contained within MAR. A mechanism can be MAR without being MCAR when response depends on observed values.
MCAR is a strong condition and may be implausible when participation, measurement failure, or item nonresponse is related to characteristics of individuals. Its value is conceptual as well as practical: it precisely defines a benchmark in which missingness carries no information about complete-data values. The definition alone does not guarantee that every analysis using incomplete data is appropriate; the analysis and its target still matter.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A discrete complete-data variable has two values, each with probability 0.50. The response probabilities are 0.75 and 0.75 for the two values. Is the mechanism MCAR? What is the marginal response probability?
- In a continuous-data setting, why is the MCAR condition stated almost surely rather than at every support point?
- For two response indicators, the four pattern probabilities do not vary with \(Y\), but \(R_1\) and \(R_2\) are associated. Can the mechanism still be MCAR?
- Response rates differ between two groups defined by a fully observed variable. What does this imply about MCAR, and does it alone rule out MAR?
Solutions. (1) Yes. Each conditional response probability is 0.75, and their probability-weighted average is \(0.50(0.75)+0.50(0.75)=0.75\), matching the marginal response probability. The conditional probability is constant over the complete-data values, so MCAR holds. (2) A conditional distribution given an exact continuous value is defined only up to changes on probability-zero sets. Independence requires equality almost surely; requiring a specific version to agree at every point would impose an unnecessary condition on values that cannot affect the joint distribution. (3) Yes. MCAR requires the joint response-pattern vector to be independent of \(Y\). It does not require the components of that vector to be mutually independent. (4) Such a difference is evidence against MCAR, because response depends on an observed component of the complete data. It does not alone rule out MAR: dependence on observed values is allowed under MAR.