Classifying MCAR, MAR, And MNAR In One Dataset

One Dataset, Three Possible Response Rules

The same people and the same variables can produce MCAR, MAR, or MNAR missingness, depending on the rule that determines whose income is reported. The classification is about that rule—not about whether the resulting table looks random, or whether the percentage missing is large or small.

Consider a hypothetical survey of eight adults. Let $Y$ denote annual income, measured in thousands of dollars, and let $X$ contain age and education. Let $R=1$ mean income is observed and $R=0$ mean income is missing. The table shows the complete values for this hypothetical dataset so the response rules can be compared. In an actual survey, incomes with $R=0$ would not be available to the analyst.

Person	Age (years)	Education	Income $Y$ ($1,000s)
A	35	No college	30
B	35	No college	90
C	50	College	55
D	50	College	48
E	28	College	72
F	60	No college	42
G	35	College	34
H	60	College	90

Definition: For this example, $X$ is the set of fully observed covariates (age and education), $Y$ is income, and $R$ indicates whether income is observed. MCAR means the response probability does not depend on $X$ or $Y$. MAR means it may depend on observed $X$, but not on $Y$ after conditioning on $X$. MNAR means it depends on $Y$ even after conditioning on $X$.

In symbols, these distinctions are:

\begin{aligned} \text{MCAR:}\quad & P(R=1\mid X,Y)=P(R=1),\\ \text{MAR:}\quad & P(R=1\mid X,Y)=P(R=1\mid X),\\ \text{MNAR:}\quad & P(R=1\mid X,Y)\text{ varies with }Y\text{ for fixed }X. \end{aligned}

These definitions describe conditional response probabilities. They do not require a particular income distribution or a particular regression model. The classification does require that we name the variables being conditioned on and correctly describe the response rule. Here age and education are observed for everyone, while income may be missing.

Worked Examples Under Alternative Rules

Worked Example: A Response Rule That Is MCAR

Suppose each person’s income is independently observed with probability $0.75$, regardless of age, education, or income. For a hypothetical realization, let the eight random numbers used by the response process be $0.12, 0.83, 0.40, 0.91, 0.22, 0.66, 0.05,$ and $0.72$, in table order. Income is observed when the random number is below $0.75$.

The resulting indicators are $R=(1,0,1,0,1,1,1,1)$: six incomes are observed and two are missing. The observed fraction is $6/8=0.75$. More importantly, the probability for any individual is the same:

P(R=1\mid X,Y)=0.75=P(R=1).

Thus this rule is MCAR. The observed fraction happens to equal the specified response probability in this small realization; that numerical agreement is not what makes the rule MCAR. Even if a different random realization produced five or seven observed incomes, the rule would still be MCAR.

Worked Example: A Response Rule That Is MAR

Now suppose income response depends on age group: people younger than 40 report with probability $0.90$, and people aged 40 or older report with probability $0.60$. Education is observed but does not affect response. The rule can be written:

P(R=1\mid X,Y)= \begin{cases} 0.90,&\text{age}<40,\\ 0.60,&\text{age}\geq 40. \end{cases}

For person A, age is 35, so the response probability is $0.90$. For person B, age is also 35, so it is also $0.90$, although their incomes are $30,000 and $90,000. For person C, age is 50, giving probability $0.60$. The probabilities change with observed age but not with income once age is fixed. Therefore,

P(R=1\mid X,Y)=P(R=1\mid X),

and the rule is MAR. There are four people under 40 and four aged 40 or older, so the expected number of observed incomes is $4(0.90)+4(0.60)=6.0$. The expected observed fraction is $6.0/8=0.75$. That expected fraction matches the MCAR example, but the mechanisms differ: response varies by an observed covariate here.

Worked Example: A Response Rule That Is MNAR

Suppose people are less likely to report income as their income increases, according to a logistic response model:

\operatorname{logit}\{P(R=1\mid X,Y)\}=2-0.04Y,

where $Y$ is measured in thousands of dollars and $\operatorname{logit}(p)=\log\{p/(1-p)\}$. This rule does not use age or education. To compare people at the same observed covariates, consider A and B: both are 35 and have no college education, but their incomes differ.

For A, $Y=30$, so the linear predictor is $2-0.04(30)=0.80$. Converting from log-odds to probability gives $e^{0.80}/(1+e^{0.80})=0.690$, to three significant figures. For B, $Y=90$, so the predictor is $2-0.04(90)=-1.60$, and the response probability is $e^{-1.60}/(1+e^{-1.60})=0.168$. The two probabilities differ even though age and education are identical. Thus response depends on income after conditioning on $X$, and this rule is MNAR.

The model implies lower response probability at higher income: the slope is $-0.04$ log-odds per additional $1,000. This statement describes the hypothetical response rule; it does not establish a causal effect of income on reporting behavior.

Why the Classification Follows From the Conditioning Set

The key comparison is between response probabilities for individuals with the same observed covariates but potentially different missing values. A short justification makes the logic explicit. Under MAR, let $q(X)$ be the response probability determined by $X$. For any two possible income values $y_1$ and $y_2$, holding $X=x$ fixed gives:

P(R=1\mid X=x,Y=y_1)=q(x) =P(R=1\mid X=x,Y=y_2).

Because changing income while holding observed covariates fixed does not change the response probability, the rule has no remaining dependence on the missing value. If the probability instead differs for $y_1$ and $y_2$, the MAR equality fails and the rule is MNAR. MCAR is the special case in which $q(x)$ is constant across all $x$ as well as all $y$.

A practical classification can therefore be made in a fixed order:

Specify the response indicator.
State exactly which variable is missing and whether $R=1$ means observed or missing.

List the observed conditioning variables.
For this example, age and education are observed for everyone; income is the potentially missing value.

Inspect the response rule.
Ask whether it uses none of the data, only observed covariates, or the income value that may be unreported.

Check dependence at fixed observed covariates.
If response probabilities still change with income, the rule is MNAR; if not, it is MAR, with MCAR when it also ignores observed covariates.

Reading Rules, Patterns, and Evidence Carefully

A response mechanism is the probability rule that generates $R$; a missingness pattern is the realized arrangement of zeros and ones. One particular pattern can arise under all three mechanisms. For instance, two missing incomes among eight people could occur under MCAR, under the age-dependent MAR rule, or under an income-dependent MNAR rule. Counting missing values alone cannot classify the mechanism.

The distinction also differs from looking for an association between response and observed covariates. If response rates differ by age, that is evidence against MCAR under the stated design and analysis, but it does not establish MNAR. The age-based rule above is MAR. Conversely, an MNAR rule can be difficult to detect from observed data because the missing income values are unavailable for direct comparison.

For a rule specified by design or by a stated model, classification is direct: examine which variables appear in the conditional response probability. For an unknown real-world mechanism, classification is more limited. The observed data reveal $X$, $R$, and $Y$ only when $R=1$; they do not directly reveal how response probability changes with unobserved $Y$ at a fixed $X$. Therefore, the observed data alone generally cannot prove MAR or exclude MNAR. Substantive knowledge about how responses are generated is important.

Response rule	Does response depend on $X$?	Does it depend on missing $Y$, given $X$?	Classification
Same probability for everyone	No	No	MCAR
Probability changes by age group	Yes	No	MAR
Probability changes with income at fixed age and education	May or may not	Yes	MNAR

This comparison is about the mechanism for the specified missing variable and conditioning set. If the set of observed covariates changes, the MAR claim must be checked again. This tutorial considers one income response indicator; other arrangements of partially observed variables require specifying the response indicators and conditioning information for those variables separately.

Common Mistakes, Diagnostics, and Limits

Calling a rule MCAR because the missingness looks scattered. A scattered pattern is compatible with any mechanism. Check the stated response probabilities, not the visual appearance of the realized indicators.
Calling any relationship with age MNAR. If age is observed and the rule depends on age but not income after conditioning on age, the rule is MAR, as in the second example.
Calling a rule MAR because observed predictors were included. Including age in a model does not establish MAR if response still depends on the unreported income after age is held fixed.
Treating equal overall missingness rates as equal mechanisms. Both the MCAR and MAR examples have an expected observed fraction of 0.75, but their conditional response probabilities differ.
Claiming observed data can verify MAR. Observed-data comparisons can reveal some departures from MCAR, but missing income values prevent a direct check of dependence on those values.

As a diagnostic, tabulate or model response rates across observed age and education groups and compare them with the proposed response rule. Such checks can flag incompatibility with MCAR or show that a claimed MAR rule has omitted an observed predictor. They cannot, by themselves, test whether response depends on the unreported income after conditioning on observed variables. The substantive limitation is therefore fundamental: distinguishing MAR from MNAR may require assumptions about the response process that are not identified by the observed data alone.

Key takeaway: Classify the rule, not the resulting pattern. No dependence on any data implies MCAR; dependence only on observed information implies MAR; dependence on a missing value after conditioning on observed information implies MNAR.

Check Your Understanding

Answer using the response rule and the stated observed variables, not the number of missing records.

Income response has probability $0.80$ for every person, independent of age, education, and income. Classify the rule.
Response probability is $0.95$ for college graduates and $0.65$ for people without college education. Education is observed for everyone, and the rule does not use income. Classify the rule.
For two people with the same age and education, income response probabilities are $0.70$ at income $40,000 and $0.30$ at income $80,000. Classify the rule.
Two datasets each have 25% of incomes missing. Must they have the same mechanism? Explain.
In observed data, response rates are lower among older people. Does this establish MNAR? Explain.

Solutions. (1) MCAR, because the response probability is constant over $X$ and $Y$. (2) MAR, because response depends on fully observed education but not on income after conditioning on education. (3) MNAR, because the probabilities differ with income for fixed observed covariates. (4) No. The same overall missing fraction can result from MCAR, MAR, or MNAR rules; the conditional response probabilities determine the classification. (5) No. The association with observed age is compatible with MAR if the rule depends on age and not on unreported income after conditioning on age. It may be evidence against MCAR, but it does not establish MNAR.