MCAR In A Randomly Lost Survey Sample

When a Survey Record Is Lost at Random

A survey may collect answers successfully and then lose some records through a file-transfer error, a storage failure, or an accidental deletion. The central question is not whether the losses seem surprising, nor whether the overall loss rate is small. It is whether the probability of losing a record depends on the respondent’s complete survey values. A clearly specified random-loss process can make that independence plausible; a loss rate that varies with respondent characteristics cannot.

Here, let \(Y_i\) denote the complete vector of survey values for respondent \(i\), including values that may later be missing. Let \(R_i=1\) if that respondent’s record is available for analysis and \(R_i=0\) if it is missing. A missingness pattern is the vector of these indicators across respondents.

Definition: Missingness is Missing Completely At Random (MCAR) when the distribution of the missingness indicators does not depend on the complete data. In notation, for every possible missingness pattern \(r\) and complete-data value \(y\) in the relevant support, \(P(R=r\mid Y=y)=P(R=r)\). For one respondent, this says \(P(R_i=0\mid Y_i=y)\) is constant across possible \(y\).

The definition concerns the relationship between missingness and the complete data, not merely whether the loss was labelled “random.” A random-number procedure that selects records without using their answers can support an MCAR explanation. A process that selects records at different rates by age, income, or another survey value does not satisfy MCAR, even if a random draw is used within each group.

Why Independent Random Loss Meets the Definition

Suppose an independent technical process deletes each record with probability \(q\), using a random event that is independent of every respondent’s complete survey values. For a fixed pattern \(r\), assume the deletion events are also independent across respondents. Then the probability of that pattern is the product of the relevant deletion and retention probabilities:

P(R=r\mid Y=y) = \prod_{i=1}^{n} q^{1-r_i}(1-q)^{r_i}.

The expression contains \(q\), the pattern, and the sample size, but not \(y\). Therefore the pattern has the same probability for every possible set of complete survey values, which is precisely the MCAR condition. This justification depends on the loss mechanism actually being independent of the answers; a device that fails more often for certain respondent groups would not meet that assumption.

A fixed number of records can also be selected uniformly at random for deletion. In that design, the individual indicators \(R_i\) are generally not independent: once exactly \(k\) of \(n\) records have been lost, losing one record changes the number of losses available for the others. But any particular set of \(k\) deleted records has probability \(1/\binom{n}{k}\), regardless of the survey values attached to those records. Thus the pattern is still independent of \(Y\), and the mechanism is MCAR. Independence of indicators across respondents is not required by the MCAR definition.

Worked Example: Twenty Records Randomly Lost

Suppose a hypothetical survey has 200 completed respondent records. Before analysis, a file-handling incident causes exactly 20 records to be lost. An audit establishes that the 20 record positions were selected uniformly at random, without using respondent answers or other respondent characteristics. Is this MCAR?

There are \(\binom{200}{20}\) possible sets of 20 lost records. Under uniform selection, the probability of any specified set is

P(R=r\mid Y=y)=\frac{1}{\binom{200}{20}}.

This probability is the same for every complete survey dataset \(y\). It follows that \(P(R=r\mid Y=y)=P(R=r)\), so the loss mechanism is MCAR. Each record has marginal probability \(20/200=0.1000\) of being lost. However, the losses are not independent across records because the total number lost is fixed at 20. That dependence does not change the MCAR classification: the relevant condition is independence of the missingness pattern from the complete data, not independence among the indicators.

The classification relies on the audit’s account of how the records were selected. If the lost set had instead been formed by deleting a particular batch whose respondents differed systematically in their survey values, uniform random selection would not be a justified description of the mechanism.

Checking the Mechanism in Concrete Settings

Worked Example: A Random Deletion Chance for Every Record

Consider a hypothetical survey of 120 adults with recorded age and weekly minutes of exercise. After collection, software independently deletes each respondent’s entire record with probability \(q=0.0500\), using a random-number generator that does not access the survey fields. Compare a respondent aged 35 who exercises 240 minutes per week with one aged 72 who exercises 30 minutes per week.

For either respondent, the probability of deletion is \(0.0500\), and the probability of retention is \(1-0.0500=0.9500\). In particular,

P(R_i=0\mid \text{age}=35,\ \text{exercise}=240)=0.0500

P(R_i=0\mid \text{age}=72,\ \text{exercise}=30)=0.0500.

Because the deletion rule does not inspect age, exercise, or any other survey value, its probability is constant over the complete data. The mechanism is MCAR under the stated assumptions. For example, the probability that a specified first respondent is lost while two specified others are retained is

$$ 0.0500(0.9500)(0.9500)=0.045125. $$

The arithmetic illustrates the independent deletion design; the MCAR conclusion comes from the value-independent rule, not from this one pattern’s probability.

Worked Example: Random Selection Within Age Groups

In another hypothetical survey, there are 80 respondents under age 40 and 20 respondents aged 40 or older. A staff member uses a random draw within each age group to lose 4 records from the younger group and 5 from the older group. Although each draw is random within its group, does the overall loss mechanism meet MCAR?

The observed loss fractions are

\frac{4}{80}=0.0500 \qquad\text{and}\qquad \frac{5}{20}=0.2500.

If age is part of the complete survey data, the conditional probability of loss differs by age group: the design fixes the loss counts at those rates within the groups. For example, a younger respondent’s marginal loss probability under the groupwise random draw is \(4/80=0.0500\), while an older respondent’s is \(5/20=0.2500\). Thus loss probability depends on a respondent characteristic, and the mechanism is not MCAR.

The word “random” describes how records were chosen within each age group, but it does not erase the group-specific probabilities. This example does not, by itself, establish which alternative mechanism applies; that requires examining what information the loss depends on and what is observed.

What Can Be Diagnosed—and What Cannot

A useful diagnosis starts with the operational account of the loss: which system or staff action caused it, what information that process could access, and whether the probability of loss could vary across respondent values. Logs showing that a random deletion routine ran without access to survey fields are relevant evidence. By contrast, a hardware failure concentrated in particular collection periods may be concerning if those periods correspond to different respondent groups or survey values.

Observed data can reveal warning signs. For example, one can compare missingness rates across groups defined by values that are observed for everyone, or inspect whether loss rates differ by collection batch. A difference is evidence against a simple value-independent account when the grouping variable is part of the complete data. Yet a lack of observed differences does not prove MCAR: associations with values that are themselves missing cannot be checked directly, and finite samples may fail to reveal real differences.

The MCAR condition is about the actual mechanism, not a test result. A diagnostic test may identify evidence inconsistent with MCAR, but no comparison of observed records can establish independence from every value in the unobserved records. A small overall missingness rate is not a substitute for that independence claim: even a small rate can be concentrated among respondents with particular characteristics.

MCAR is distinct from Missing At Random (MAR) and Missing Not At Random (MNAR). Under MCAR, missingness is independent of the complete data. Under MAR, missingness may depend on data that are observed, after conditioning on those data; under MNAR, dependence remains on values that are unobserved. These names describe assumptions about the mechanism, not conclusions that follow from the size of the missingness rate. The detailed classification depends on specifying the relevant observed and unobserved information.

Common Pitfall: Calling Any Chance-Based Loss MCAR

Wrong reasoning: “The records were selected randomly, so the mechanism is MCAR.” This overlooks how the selection probabilities were defined. A random draw with different rates by age group can still make loss depend on age.

Correction: Identify the complete data values that could be related to loss, then ask whether the probability of each missingness pattern is the same across those values. Randomness in a step of the process is not sufficient; the resulting mechanism must be independent of the complete data.

Also distinguish record loss after data collection from the initial process that determines who enters a survey. A probability sample can use deliberate sampling probabilities that vary by design strata. Whether records subsequently disappear is a separate missingness question; the sampling design alone does not establish that later losses are MCAR.

Key takeaway: A randomly lost survey record supports an MCAR classification when the probability of loss is independent of the respondent’s complete survey values. State the loss rule, check whether its probabilities vary with those values, and treat observed comparisons as diagnostics rather than proof.

Check Your Understanding

For each question, state the reasoning before checking the solution.

A hypothetical survey has 50 records. Exactly 5 are chosen uniformly at random for deletion, without examining their values. Is the mechanism MCAR? Are the 50 deletion indicators independent?
A software routine independently deletes each record with probability \(0.0200\), regardless of every survey answer. What is the probability that a specified record is retained, and what mechanism does this describe?
A team deletes records at a rate of \(0.0300\) among respondents under 30 and \(0.1800\) among respondents aged 30 or older. Age is in the complete survey data. Is this MCAR?
A comparison finds no statistically detectable difference in observed characteristics between records that are and are not missing. Does this establish MCAR? Why or why not?

Solutions. (1) Yes. Each possible set of 5 deleted records has probability \(1/\binom{50}{5}\), independent of the complete survey values. The indicators are not independent because exactly 5 losses are fixed. (2) The retention probability is \(1-0.0200=0.9800\). Under the stated value-independent process, the mechanism is MCAR. (3) No. The loss probability differs by age, which is part of the complete data. (4) No. An observed comparison can detect some departures from MCAR, but a lack of detectable differences does not establish independence from complete values, including values that are missing.