Why the Conditioning Matters
A missing value cannot be compared directly with its own missingness indicator in the observed data: when a value is missing, its value is unavailable. The important question is therefore about the mechanism that generated the missingness. Does the chance that a value is recorded still depend on that value after accounting for the information that was observed?
This tutorial defines missing not at random (MNAR) in those conditional terms. A marginal association between missingness and a variable is not enough to establish MNAR: the association might be explained by an observed variable. Conversely, a response pattern that depends on an unobserved value even after conditioning on the observed information is MNAR. This is the distinction from missing at random (MAR) developed in the earlier material on MAR.
Definition in Pattern Notation
Let \(Y=(Y_1,\ldots,Y_p)\) denote the complete data vector for one unit, including values that may not be recorded. Let \(R=(R_1,\ldots,R_p)\) be the corresponding missingness indicators, with \(R_j=1\) if \(Y_j\) is observed and \(R_j=0\) if it is missing. A particular value \(r\) of \(R\) describes a missingness pattern. For that pattern, \(Y_{\mathrm{obs}}(r)\) denotes the components observed and \(Y_{\mathrm{mis}}(r)\) the components missing.
The definition is conditional on the observed values for the particular pattern. For example, if a score \(Y\) can be missing and \(X\) is always observed, MAR allows the response probability to vary with \(X\). It is MNAR if that probability also varies with \(Y\) after \(X\) is held fixed. In the special case where no values are missing, there is no missing part of the data on which the mechanism could depend.
MCAR is more restrictive: the probability of a missingness pattern does not depend on any component of \(Y\). Thus MCAR implies MAR, while MNAR identifies dependence on missing values that remains after conditioning on the observed values. These labels describe assumptions about the data-generating mechanism, not features that can necessarily be read off a table of observed values.
Worked Example: A Binary Value That Affects Its Own Observation
Suppose a hypothetical survey records whether each participant had a particular symptom, with \(Y=1\) for “yes” and \(Y=0\) for “no.” Let \(R=1\) mean the symptom answer was recorded. Assume the population has \(P(Y=1)=0.30\), and that the response probabilities are \(P(R=1\mid Y=1)=0.40\) and \(P(R=1\mid Y=0)=0.80\). There are no other observed variables to condition on in this example.
Classification. The response probability differs according to the value of \(Y\), which is precisely the value unavailable when \(R=0\). The probability of recording the answer therefore depends on a missing value: this is MNAR.
Check the implied proportions. The probability of an observed “yes” is \(0.30\times0.40=0.12\). The probability of an observed “no” is \(0.70\times0.80=0.56\). Thus \(P(R=1)=0.12+0.56=0.68\). Among respondents, the fraction answering “yes” is \(0.12/0.68=0.1765\), or about 17.65%. The population fraction is 30.0%, but the respondent fraction is lower because the hypothetical response probability is lower for \(Y=1\).
These calculations follow from the stipulated population prevalence and response model; they do not show that a real survey’s mechanism is MNAR. In actual data, the values for nonrespondents are not available to perform this comparison directly.
Conditioning Can Change the Classification
A variable can be associated with missingness in the marginal data even when the mechanism is MAR. If the response probability depends on an observed variable, and that observed variable is associated with the partly missing value, then the response probability can differ across values of the missing variable before conditioning. The MAR definition asks whether dependence remains after conditioning on the observed information.
Worked Example: A Marginal Association That Is Not MNAR
In a hypothetical setting, \(Y\) is a binary score that may be missing, and \(C\) is an observed clinic group, either A or B. Suppose \(P(C=A)=P(C=B)=0.50\), \(P(Y=1\mid C=A)=0.20\), and \(P(Y=1\mid C=B)=0.80\). Assume the response probabilities are \(P(R=1\mid C=A,Y)=0.90\) and \(P(R=1\mid C=B,Y)=0.50\), regardless of whether \(Y=0\) or \(Y=1\). All these probabilities are positive.
Conditional classification. Within group A, response probability is 0.90 for either value of \(Y\); within group B, it is 0.50 for either value. Therefore \(P(R=1\mid C,Y)=P(R=1\mid C)\). Because \(C\) is observed, the mechanism is MAR, not MNAR.
Marginal comparison. The joint probabilities for \(C=A,Y=1\) and \(C=B,Y=1\) are \(0.50\times0.20=0.10\) and \(0.50\times0.80=0.40\). Since \(P(Y=1)=0.50\), the conditional proportions among those with \(Y=1\) are 0.20 in group A and 0.80 in group B. Hence
For \(Y=0\), the joint probabilities for A and B are \(0.50\times0.80=0.40\) and \(0.50\times0.20=0.10\), giving conditional group proportions 0.80 and 0.20. Thus
The marginal response probabilities differ by \(0.82-0.58=0.24\), or 24 percentage points. Yet at each fixed value of the observed group \(C\), response probability does not depend on \(Y\). This marginal difference is explained by the different group compositions, so it does not establish MNAR.
A Direct Check for MNAR
For a partly observed variable \(Y\), the simplest check is to write the response probability conditional on both \(Y\) and the other observed variables \(X\). If \(R=1\) indicates that \(Y\) was recorded, the relevant comparison is between values of \(Y\) at the same \(X=x\):
If these probabilities differ for some \(y,y'\) while \(X=x\) is fixed, the stipulated mechanism is MNAR. If the response probability is the same across \(y\) for every fixed \(x\), then this part of the mechanism is MAR. For multiple variables and missingness patterns, apply the same logic pattern by pattern, holding fixed the values observed under that pattern and varying the values missing under it.
This is a definition-based comparison, not a recipe for estimating the response probabilities from observed data. When \(Y\) is missing, its value cannot be used to form the \(R=0\) conditional response probabilities without additional information or assumptions. A model that includes \(Y\) in a formula does not by itself make MNAR identifiable from the observed records.
Worked Example: Two Different Mechanisms With the Same Observed Data
Return to the hypothetical binary survey in the first example. The observed data show that 68% respond, and that the fraction of respondents with \(Y=1\) is \(0.12/0.68=0.1765\). The observed joint proportions are therefore \(P(R=1,Y=1)=0.12\) and \(P(R=1,Y=0)=0.56\); values of \(Y\) for \(R=0\) are unavailable.
One possible model has population prevalence \(P(Y=1)=0.30\), response probability 0.40 for \(Y=1\), and response probability 0.80 for \(Y=0\). It produces the observed joint proportions \(0.30\times0.40=0.12\) and \(0.70\times0.80=0.56\).
A different model has population prevalence \(P(Y=1)=0.20\), response probability 0.60 for \(Y=1\), and response probability 0.70 for \(Y=0\). It also produces \(0.20\times0.60=0.12\) and \(0.80\times0.70=0.56\). Both models therefore have the same response rate, \(0.12+0.56=0.68\), and the same distribution of \(Y\) among respondents, but they imply different population prevalences.
This calculation shows why observed data alone generally cannot determine whether the missingness mechanism is MNAR or which MNAR model is correct. Distinguishing such possibilities requires additional assumptions, external information, or sensitivity analysis; the observed response rate and respondent values do not reveal the missing values.
Comparison, Diagnostics, and Limitations
| Mechanism | What the probability of missingness may depend on | Key implication |
|---|---|---|
| MCAR | Neither observed nor missing data values | Missingness is independent of the complete data. |
| MAR | Observed values, but not missing values once observed values are held fixed | Conditional differences by observed information are allowed. |
| MNAR | At least one missing value even after conditioning on the observed values | The missingness process contains dependence not captured by the observed data. |
Observed-data summaries can help describe missingness patterns and identify which variables are associated with being observed. They cannot, on their own, establish that the mechanism is MAR rather than MNAR: the missing values needed to assess residual dependence are not observed. Nor does failure to find an association with available variables prove MCAR. Such summaries are useful for diagnosing model fit and informing assumptions, but they do not test the defining MNAR condition directly.
A practical assessment combines subject-matter reasoning with sensitivity analysis: specify plausible ways missing values could affect observation, then examine how conclusions change under those alternatives. Where possible, external measurements or follow-up information about some nonrespondents can add evidence. These approaches make the dependence assumption visible; none turns an untestable assumption into a fact merely by fitting a more elaborate model.
Common Mistakes
- Calling any association MNAR. A marginal association between \(R\) and \(Y\) may arise because response depends on an observed covariate associated with \(Y\). Check the relationship at fixed observed values.
- Assuming an observed-data test can rule out MNAR. The values needed to check residual dependence are precisely the values missing for some units. Observed-data patterns alone generally do not settle the mechanism.
- Equating MNAR with “missing values are systematically different.” That phrase is incomplete unless it specifies the conditioning information. The definition concerns dependence on missing values after conditioning on observed values.
- Treating MAR as independence. Under MAR, missingness may depend strongly on observed variables. The restriction is that it does not additionally depend on missing values once those observed variables are fixed.
Check Your Understanding
Answer each question using the conditional definition, not a marginal association alone.
- A partly observed measurement \(Y\) has \(P(R=1\mid X,Y)=0.7\) for every value of \(Y\) at fixed \(X\), but the probability varies with \(X\). Is the mechanism MAR or MNAR?
- At fixed observed \(X\), response probabilities for \(Y=0\) and \(Y=1\) are 0.85 and 0.45. Classify the mechanism for \(Y\).
- In the clinic example, why does the difference between marginal response probabilities 0.58 and 0.82 not imply MNAR?
- Can observed response rates and respondent values generally prove that a mechanism is not MNAR? Explain.
Solutions. (1) MAR: response may depend on the observed \(X\), but at fixed \(X\) it does not depend on the missing \(Y\). (2) MNAR: the response probabilities differ by \(Y\) while \(X\) is held fixed, so dependence on the potentially missing value remains. (3) The difference is due to the changing proportions of clinic groups A and B across values of \(Y\); within each observed group the response probability is constant across \(Y\). (4) Generally, no. The missing values cannot be used to check the dependence directly, and different population distributions and response models can produce the same observed response rate and respondent-value distribution.