Missing Data Patterns and Their Causes

What Is Missing, and for Whom?

An empty cell is not yet an explanation. A value may be absent because the variable does not apply to that observational unit, because a respondent did not provide it, because a measurement failed, or because a unit disappeared from follow-up. These situations differ in meaning. Before choosing an analysis, identify which values are genuinely missing, how missingness is arranged across variables and records, and what the observed data can support.

The pattern of missingness describes where values are absent. A missingness mechanism describes how the probability of absence relates to the data. Neither label, by itself, establishes the underlying real-world cause. For example, a pattern of entire records missing could result from nonparticipation, a data-collection failure, or another process.

Representing Missingness

Suppose a data set has observational units indexed by $i$ and variables indexed by $j$. Let $Y_{ij}$ denote the value that would be defined for unit $i$ on variable $j$, whether or not it was recorded. Use an indicator to record whether that value is observed.

Definition: For a value that is applicable to unit $i$, define $R_{ij}=1$ if $Y_{ij}$ is observed and $R_{ij}=0$ if it is missing. The vector $(R_{i1},\ldots,R_{ip})$ is unit $i$'s missingness pattern across $p$ variables. A structural absence is different: if a variable is not defined for that unit, it is not an applicable value that is merely unobserved.

Patterns can be described at several levels. Item nonresponse means that some applicable variables are missing while others are recorded for the same unit. Unit nonresponse means that none of the intended data for a sampled unit are available. In longitudinal data, monotone missingness occurs when, after a unit drops out, later measurements are all absent; intermittent missingness occurs when a later measurement is observed despite an earlier one being absent. These are descriptions of arrangement, not explanations of why the values are missing.

Worked Example: Separating Structural Absence From Item Missingness

A hypothetical labor survey records employment status, annual earnings, and hourly wage for three people. The records are:

Person	Employment status	Annual earnings	Hourly wage
A	Employed	$42,000	$21
B	Employed	Blank	$18
C	Not employed	$0	Not applicable

For person B, annual earnings are applicable but unrecorded, so this is item missingness. The hourly wage is observed, making B's pattern $(1,0,1)$ when the variables are listed as status, annual earnings, and hourly wage. For person C, annual earnings are observed as zero: zero is a value, not a missing entry. Hourly wage is not defined for this person under the survey's definition, so “not applicable” is a structural absence, not an unobserved wage. It should not be coded as an ordinary missing wage without retaining that distinction.

The distinction depends on the variable's definition. If the study instead defined a measure of potential hourly earnings for everyone, then a nonworker's value might be conceptually defined but unavailable. The analyst must state what was intended to be measured before classifying the cell.

Patterns and Mechanisms Are Different

The same pattern can arise under different mechanisms, and the same mechanism can produce different patterns. The standard mechanism classifications compare the probability of a missingness pattern with the complete data. Write $Y$ for the vector of all variables that would be available if there were no missingness. For a particular pattern $r$, write $Y_{\mathrm{obs}}(r)$ for the components observed under that pattern.

Definition: Missingness is missing completely at random (MCAR) if $P(R=r\mid Y)=P(R=r)$ for every pattern $r$: the pattern is independent of the complete data. It is missing at random (MAR) if $P(R=r\mid Y)=P(R=r\mid Y_{\mathrm{obs}}(r))$: conditional on the values observed under that pattern, the probability of the pattern does not depend on its unobserved values. Missingness is missing not at random (MNAR) when the probability of missingness still depends on unobserved values after conditioning on the observed values.

MAR is a conditional statement, not a claim that missingness is unrelated to all data. For example, a survey response can depend on recorded age and still be MAR if, within age groups, response does not additionally depend on the missing income. The classifications are relative to the variables included in the complete-data description. Adding a relevant observed auxiliary variable can change whether a mechanism is reasonably described as MAR.

Worked Example: Classifying Three Hypothetical Response Processes

A survey records age $X$ for everyone reached and asks for income $Y$. Let $R=1$ indicate that income is reported. Consider three hypothetical response rules.

Rule A: Every person reports income with probability $0.8$, independently of age and income. Then $P(R=1\mid X,Y)=0.8$, so the response process is MCAR.

Rule B: The response probability is $0.9$ for younger respondents and $0.6$ for older respondents, and within each age group it does not depend on income. Since age is observed, $P(R=1\mid X,Y)=P(R=1\mid X)$. This is MAR, not MCAR: the response probability varies with an observed variable.

Rule C: Holding age fixed, people with higher income are less likely to report income. Then response probability depends on the unobserved income itself, even after conditioning on age. This is MNAR.

These classifications follow from the stipulated response rules. Observing that older respondents report less often would not, on its own, establish whether the real process is Rule B or whether income also affects response. The distinction concerns dependence on values that are missing, which are not directly visible for nonrespondents.

What Observed Records Can Establish

Observed records directly reveal which cells or records are absent, the observed values, and the empirical frequencies of the recorded patterns in the available data. They do not generally reveal the values that would have appeared in the missing cells. Consequently, a summary of observed cases need not equal the corresponding full-data quantity.

A simple mean calculation shows the issue. Let $Y$ be a numerical variable with finite expectation, and let $R$ indicate whether it is observed. Write $p=P(R=1)$. When $p>0$, the mean among observed cases is $E[Y\mid R=1]$. Its difference from the full-population mean can be expressed using the covariance between the value and the observation indicator.

Express the observed-case mean.
Because $R$ is zero or one, $E[YR]=pE[Y\mid R=1]$. Thus $E[Y\mid R=1]=E[YR]/p$, provided $p>0$.

Use the covariance definition.
$\operatorname{Cov}(Y,R)=E[YR]-E[Y]E[R]=E[YR]-pE[Y]$.

Compare the means.
Dividing the covariance identity by $p$ gives $E[Y\mid R=1]-E[Y]=\operatorname{Cov}(Y,R)/p$. An observed-case mean equals the full mean exactly when this covariance is zero.

There is a useful refinement when both observed and unobserved groups exist. If $0<p<1$, define $\mu_1=E[Y\mid R=1]$ and $\mu_0=E[Y\mid R=0]$. The law of total expectation gives $E[Y]=p\mu_1+(1-p)\mu_0$. Therefore,

\operatorname{Cov}(Y,R)=p(1-p)(\mu_1-\mu_0).

Under $0<p<1$, the covariance is nonzero exactly when the observed- and unobserved-group conditional means differ. If $p=1$, every value is observed: the observed mean equals the full mean, $R$ is constant, and no unobserved-group mean is defined. If $p=0$, the observed-case mean is undefined. These statements require the relevant expectations to be finite.

Worked Example: Quantifying Selection in a Small Hypothetical Roster

A complete hypothetical roster lists annual earnings, in thousands of dollars, for four people: $10,20,30,40$. The first three earnings are observed; the fourth is missing. Treating these four values as the full roster makes it possible to compare the observed and full means directly.

The observed-case mean is \[ \bar{Y}_{\mathrm{obs}}=\frac{10+20+30}{3}=20 \] thousand dollars. The full-roster mean is \[ \bar{Y}_{\mathrm{full}}=\frac{10+20+30+40}{4}=25 \] thousand dollars. The observed-case mean is therefore $5$ thousand dollars lower for this roster.

The observation rate is $p=3/4$. Using averages over the four roster members, $E[Y]=25$, $E[R]=3/4$, and $E[YR]=(10+20+30+0)/4=15$. Thus \[ \operatorname{Cov}(Y,R)=15-(25)(0.75)=-3.75, \qquad \frac{\operatorname{Cov}(Y,R)}{p}=\frac{-3.75}{0.75}=-5. \] This matches the observed-minus-full difference $20-25=-5$ thousand dollars. In this example, the observed mean is lower because the one unobserved person has higher earnings than the observed-group mean. It illustrates a selection difference; it does not establish a general causal effect of being observed.

Why the Full-Data Quantity May Not Be Identified

Without assumptions about missing values or the missingness process, multiple complete data sets can agree with every observed record. If those complete data sets imply different values of the target quantity, the quantity is not identified by the observed records alone. This is why missingness classifications matter: they clarify the assumptions that might justify moving from observed summaries to full-data conclusions.

Worked Example: Two Full Data Sets, the Same Observed Records

Suppose a hypothetical group has 100 people. Income is observed for 50 people, all with income recorded as $0$ thousand dollars; income is missing for the other 50. The observed data—including the observation rate and all recorded incomes—are fixed.

One completion assigns income $0$ to each of the 50 people whose income is missing. The full-group mean is then \[ \frac{50(0)+50(0)}{100}=0 \] thousand dollars. A second completion assigns income $10$ thousand dollars to each person whose income is missing. Its full-group mean is \[ \frac{50(0)+50(10)}{100}=5 \] thousand dollars. Both completions produce exactly the same observed records, but their full-group means differ. Therefore, those observed records alone do not determine the full-group mean.

Assumptions such as MCAR or MAR can support particular inferences when combined with an appropriate statistical model and adequate data coverage. They are assumptions, not facts established merely by looking at the recorded cells. In particular, the observed data alone generally cannot distinguish MAR from MNAR: the difference concerns how missingness relates to values that are not observed. External information, a defensible account of data collection, or sensitivity analysis may be needed to assess conclusions that depend on such assumptions.

Common Mistakes and a Practical Checklist

Treating every blank as the same kind of missingness. First determine whether the value is defined for that unit. “Not applicable” and an unrecorded applicable value are not interchangeable.
Confusing pattern with mechanism. “Income is missing for some respondents” describes a pattern. It does not say whether missingness is MCAR, MAR, or MNAR.
Assuming MAR because missingness relates to an observed variable. MAR permits dependence on observed values; it also requires that, conditional on those values, missingness does not depend on the unobserved values.
Reading an observed-case mean as the full-data mean without justification. That equality is not automatic. In the mean identity, the difference depends on the relationship between $Y$ and $R$.
Claiming that the observed records prove a mechanism. Missingness rates and patterns are observable; dependence on missing values is generally not directly verifiable from those records alone.

Key takeaway: Describe missingness at the cell and record levels, separate structural absence from unrecorded applicable values, and state the assumptions that connect observed cases to the full-data quantity. The observed records do not, by themselves, fill in what was never observed.

Check Your Understanding

Answer each question using the definitions and distinctions in this tutorial.

A panel member has a recorded first-period outcome, a missing second-period outcome, and a recorded third-period outcome. Is this monotone or intermittent missingness?
A survey records annual earnings as zero for a person with no earnings. Why should this not automatically be coded as missing?
Income response depends on observed age, but not on income after conditioning on age. Which standard missingness mechanism describes this process?
For $0<P(R=1)<1$, what does $\operatorname{Cov}(Y,R)=0$ imply about the observed- and unobserved-group means, assuming finite expectations?
Why can observed records alone fail to identify the full-population mean when some outcomes are missing?