Defining Missing At Random

Why Missingness Can Depend on What Is Observed

A response rate can vary across people and still satisfy Missing At Random. For example, a survey item may be answered less often by older participants than by younger participants, even when age is recorded for everyone. In this case, the missingness mechanism depends on observed age. The key question is not whether missingness varies, but whether, for a given missingness pattern, its probability depends on values that are observed in that pattern or also on values that are missing.

This distinction is expressed with the complete-data vector $Y$ and the missingness indicators $R$. Under the course convention, $R_j=1$ means variable $Y_j$ is observed, and $R_j=0$ means it is missing. A particular value $r$ of $R$ identifies a missingness pattern. For that pattern, $Y_{\mathrm{obs}}(r)$ denotes the components of $Y$ that are observed, and $Y_{\mathrm{mis}}(r)$ denotes those that are missing.

Definition: Missing At Random (MAR) holds if, for each missingness pattern $r$, the probability of that pattern given the complete data depends only on the values observed under that pattern. Formally, for complete-data values in the relevant support, $$ P(R=r\mid Y=y)=g_r\!\left(y_{\mathrm{obs}}(r)\right) $$ for some function $g_r$. Equivalently, if $y$ and $y'$ agree on all components observed under $r$, then $$ P(R=r\mid Y=y)=P(R=r\mid Y=y'). $$

The equality is specific to a pattern. The functions $g_r$ may differ across patterns, and probabilities may differ across values of the observed data. MAR does not say that every person has the same probability of having missing data. It says that once the observed values for a particular pattern are fixed, changing the unobserved values does not change the probability of that pattern.

Reading the Pattern-Specific Condition

To apply the definition, first fix a pattern $r$. Then identify exactly which components of $Y$ are observed in that pattern. Hold those observed components fixed and compare the pattern probability across different possible values of the missing components. If the probability remains unchanged, the condition holds for that pattern. Repeat the check for each pattern in the mechanism.

For example, if $Y=(Y_1,Y_2)$ and $r=(1,0)$, then $Y_1$ is observed and $Y_2$ is missing. MAR for this pattern requires that $P(R=(1,0)\mid Y_1,Y_2)$ be a function of $Y_1$ alone. It may vary with $Y_1$, but not with $Y_2$ after $Y_1$ is fixed. If $r=(1,1)$, both variables are observed, so the pattern probability may depend on both.

Worked Example: A Response Rate That Depends on Observed Age

Suppose a hypothetical survey records age for everyone and asks for annual income. Let $A$ be age, $Y$ be income, and $R_Y=1$ if income is reported. Assume the probability of missing income is 0.10 for participants under 50 and 0.30 for participants aged 50 or older, regardless of their income. Does this mechanism satisfy MAR?

For the pattern $R_Y=0$, income is missing but age is observed. The stated mechanism gives $$ P(R_Y=0\mid A,Y)= \begin{cases} 0.10, & A<50,\\ 0.30, & A\geq 50. \end{cases} $$

For a participant aged 42, compare two possible incomes, say $Y=\$40{,}000$ and $Y=\$90{,}000$: $$ P(R_Y=0\mid A=42,Y=\$40{,}000)=0.10 \quad\text{and}\quad P(R_Y=0\mid A=42,Y=\$90{,}000)=0.10. $$ The probability is unchanged when the missing income changes while observed age stays fixed. For age 57, the corresponding probabilities are both 0.30. Thus the missing-income pattern probability is a function of observed age alone, and satisfies the MAR condition.

The mechanism is not constant across participants: the hypothetical missingness rate is 20 percentage points higher for those aged 50 or older. That variation does not contradict MAR because age is observed. This example checks the stated probability rule; it does not establish that such a rule is plausible in any actual survey.

A Short Derivation of the Equivalent Condition

The definition can be understood as a comparison among complete-data values that are indistinguishable on what is observed under a fixed pattern. Fix $r$, and suppose two complete-data values $y$ and $y'$ have the same observed components: $y_{\mathrm{obs}}(r)=y'_{\mathrm{obs}}(r)$. If the pattern probability is given by a function of those components, then

P(R=r\mid Y=y) =g_r\!\left(y_{\mathrm{obs}}(r)\right) =g_r\!\left(y'_{\mathrm{obs}}(r)\right) =P(R=r\mid Y=y').

Conversely, suppose the pattern probability is equal for every pair of complete-data values that agree on the observed components. For any possible observed value $u$, define $g_r(u)$ to be that common probability over complete-data values whose observed components equal $u$. The pairwise equality ensures that this definition does not depend on which missing-component values are used. Therefore $P(R=r\mid Y=y)=g_r(y_{\mathrm{obs}}(r))$, which is the MAR condition.

These statements concern values in the relevant support of the data. They do not require comparing impossible combinations of values. If a pattern has probability zero in a part of the support, its conditional pattern probability there is not identified by observed cases; the formal condition is understood on the support where the conditional probabilities are defined.

Worked Example: Checking a Multivariable Pattern

Consider hypothetical binary variables $Y_1$ and $Y_2$, with $R=(R_1,R_2)$. Suppose the conditional probabilities of the four patterns are specified by $$ \begin{aligned} P(R=(1,0)\mid Y_1,Y_2)&=0.20+0.10Y_1,\\ P(R=(0,1)\mid Y_1,Y_2)&=0.20+0.10Y_2,\\ P(R=(1,1)\mid Y_1,Y_2)&=0.50-0.10Y_1-0.10Y_2,\\ P(R=(0,0)\mid Y_1,Y_2)&=0.10. \end{aligned} $$ Here $(1,0)$ means $Y_1$ is observed and $Y_2$ is missing. Check the pattern-specific condition and verify the probabilities for $Y_1=1,Y_2=0$.

For pattern $(1,0)$, only $Y_1$ is observed. Its probability is $0.20+0.10Y_1$, which does not involve $Y_2$. For instance, holding $Y_1=1$ fixed gives $$ P(R=(1,0)\mid Y_1=1,Y_2=0)=0.20+0.10(1)=0.30, $$ $$ P(R=(1,0)\mid Y_1=1,Y_2=1)=0.20+0.10(1)=0.30. $$ Thus this pattern satisfies the condition.

For pattern $(0,1)$, only $Y_2$ is observed, and its probability $0.20+0.10Y_2$ depends only on that observed value. For $(1,1)$, both variables are observed, so its probability may depend on both. For $(0,0)$, neither is observed, and its probability is constant at 0.10. Each pattern therefore satisfies the pattern-specific MAR condition.

At $Y_1=1,Y_2=0$, the four probabilities are $0.30$, $0.20$, $0.40$, and $0.10$, respectively. Their sum is $$ 0.30+0.20+0.40+0.10=1.00. $$ They form a valid conditional distribution over patterns at these complete-data values. This check matters: pattern probabilities must be nonnegative and sum to one for every possible complete-data value.

What MAR Does and Does Not Say

The earlier material on Missing Completely At Random introduced the more restrictive condition that the missingness pattern is independent of the complete data. Under MAR, the pattern probability can depend on observed values, so MCAR is a special case of MAR: a constant function $g_r$ is allowed. The distinction is consequential because a changing response rate across observed groups is compatible with MAR, whereas MCAR requires no dependence on the complete data.

MAR is a statement about a probability model for $R$ given $Y$, not a description that can usually be confirmed from the observed data alone. Observed data can show, for example, that missingness rates differ by recorded age. Such a difference can be represented by dependence on observed age and is compatible with MAR. But the same observed pattern of rates does not reveal whether missingness also depends on the values that are unobserved. The condition is therefore an assumption to assess using subject-matter knowledge and sensitivity analysis, not a conclusion established merely by a nonsignificant test or by a table of observed response rates.

Diagnostics can still be useful for finding problems in a proposed account of the mechanism. Check whether missingness rates vary with recorded variables that the account says are relevant; inspect whether important observed predictors of response have been omitted; and verify that pattern probabilities are coherent. These checks can reveal that a simple proposed description is inadequate. They cannot prove that the pattern probability is invariant to the missing values, because those values are unavailable precisely where they are needed for that comparison.

Common pitfall: “The response rate differs by age, so the data are not MAR.” This reasoning is incorrect when age is observed: MAR permits missingness to depend on observed age. The right check fixes the observed age and asks whether the pattern probability also changes as the missing value changes. If it does, the stated MAR condition for that pattern fails.

MAR also does not mean that missing values are random draws from the observed values, that all variables are fully observed, or that an analysis is automatically unbiased. It is a restriction on how the pattern probability may depend on complete-data values. Whether an analysis can usefully proceed under that assumption depends on the data structure, the analysis model, and the plausibility of the assumed mechanism. The practical consequence is not that MAR can be read directly from the data, but that its exact conditioning set must be stated and defended.

Practice and Solutions

Check Your Understanding

For each question, use the pattern-specific definition. Solutions are included so that the conditioning set and conclusion can be checked.

A study records treatment group $G$ for everyone and has a missing outcome $Y$. The probability that $Y$ is missing is 0.08 in group A and 0.18 in group B, with no dependence on $Y$ within either group. Is the mechanism MAR?
For pattern $r=(1,0)$ in data $Y=(Y_1,Y_2)$, suppose $P(R=r\mid Y_1,Y_2)=0.25+0.05Y_1$, where $Y_1$ is binary. What does the condition require, and does it hold?
For a fixed pattern, two complete-data records agree on every observed component but have different missing-component values. Their conditional probabilities of that pattern are 0.22 and 0.22. What does this comparison show, and what does it not establish by itself?
A researcher says, “The missingness probability is not the same for all participants, so MAR cannot hold.” Give the correction.

Solutions. (1) Yes. For the missing-outcome pattern, $G$ is observed and the probability is a function of $G$ alone: 0.08 for group A and 0.18 for group B. (2) Under $r=(1,0)$, $Y_1$ is observed and $Y_2$ is missing. The condition requires the probability to be unchanged across values of $Y_2$ when $Y_1$ is fixed. The expression contains $Y_1$ only, so the condition holds. (3) The equality is consistent with the MAR condition for that pair of complete-data values. To establish the condition for the pattern, the probability must be invariant across all relevant missing-component values at each fixed observed value. (4) MAR allows probabilities to vary with observed values. The researcher must fix the observed components for each pattern and check whether changing the missing components changes that pattern’s probability.

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