Missingness Rates And Missingness Mechanisms

Three Questions About Missingness

A data set can have 20% missing values for a variable, several distinct configurations of missing values across variables, and a particular probabilistic process that generated the response indicators. These are related descriptions, but they answer different questions. A missingness rate describes how often values are absent; a missingness pattern describes which variables are absent together; a missingness mechanism describes how the indicators of response are distributed in relation to the data.

Keeping these distinctions clear matters because a rate or pattern is directly calculable from the observed response indicators, while a mechanism is an assumption about a probability law. A high missingness rate does not by itself imply MNAR, and a low rate does not establish MCAR. The earlier material on missingness indicators and complete-data notation gives the basic indicator setup used here.

Definition: For record $i$ and variable $j$, let $R_{ij}=1$ when the value is observed and $R_{ij}=0$ when it is missing. Among $n$ records for which variable $j$ was intended to be measured, its empirical missingness proportion is $$ \hat{m}_j=\frac{1}{n}\sum_{i=1}^{n}(1-R_{ij}) =1-\frac{1}{n}\sum_{i=1}^{n}R_{ij}. $$ A joint missingness pattern for record $i$ is the vector $\mathbf{R}_i=(R_{i1},\ldots,R_{ip})$. Its pattern frequency is the number of records with that vector, and its pattern proportion is that frequency divided by $n$.

The rate is a marginal summary for one variable: it counts missing values in that column without distinguishing what happened to other variables in the same records. A joint pattern retains that information. For $p$ variables, there are at most $2^p$ possible response-indicator patterns, though a particular sample may contain only a subset. The all-ones vector denotes a complete record; any other vector denotes a record with at least one missing value.

Calculate Rates and Patterns Separately

Worked Example: Two Variables and Four Patterns

Suppose a hypothetical data set has $n=8$ records and two variables, $A$ and $B$. The observed response-indicator patterns are summarized below. A 1 means observed and a 0 means missing.

Pattern $(R_A,R_B)$	Number of records
(1,1)	4
(1,0)	2
(0,1)	1
(0,0)	1

Variable $A$ is missing in patterns $(0,1)$ and $(0,0)$, for $1+1=2$ records. Thus $\hat{m}_A=2/8=0.25$, or 25%. Variable $B$ is missing in patterns $(1,0)$ and $(0,0)$, for $2+1=3$ records, so $\hat{m}_B=3/8=0.375$, or 37.5%.

There are $2+1+1=4$ records with at least one missing value, so the proportion with any missing value is $4/8=0.50$, or 50%. The proportion complete on both variables is $4/8=0.50$. These are different summaries: the variable-specific rates are 25% and 37.5%, even though half the records have at least one value missing.

The pattern proportions are $4/8=0.50$, $2/8=0.25$, $1/8=0.125$, and $1/8=0.125$, respectively. They sum to 1. The marginal missingness rate for $A$, for example, is obtained by adding the proportions of patterns with $R_A=0$: $0.125+0.125=0.25$.

More generally, let $\mathcal{P}$ be the set of possible indicator vectors, and let $\hat{\pi}(\mathbf{r})$ be the observed proportion with pattern $\mathbf{r}$. Then

\hat{m}_j=\sum_{\mathbf{r}\in\mathcal{P}:r_j=0}\hat{\pi}(\mathbf{r}).

This is a bookkeeping identity: every record missing variable $j$ belongs to exactly one pattern with $r_j=0$, and no record with $r_j=1$ contributes to the rate. The marginal rates can therefore be recovered from the pattern table, but they cannot recover the full pattern table. For example, marginal rates alone do not reveal whether two variables tend to be missing on the same records.

A Mechanism Is a Probability Rule

The mechanism is not another count in the data table. It concerns the conditional distribution of the response indicators given the complete data. Write $\mathbf{Y}$ for the complete-data variables and $\mathbf{R}$ for their response indicators. In the standard factorization, the joint distribution can be expressed as $f(\mathbf{Y},\mathbf{R})=f(\mathbf{Y})f(\mathbf{R}\mid\mathbf{Y})$. The first factor describes the data; the second describes the response process.

In brief, MCAR means the response process does not depend on the complete-data values. MAR means it may depend on values that are observed, but not on the missing values once the observed values are given. MNAR means the response process depends on missing values even after conditioning on observed values. In multivariable settings these definitions apply to the relevant observed and unobserved components of the data. The crucial point here is that each label characterizes $P(\mathbf{R}\mid\mathbf{Y})$, not the observed proportion of zeros in $\mathbf{R}$.

Worked Example: The Same Overall Rate Under MCAR and MAR

Consider a hypothetical population split evenly between two groups, $G=0$ and $G=1$, where group membership is observed. Let $R=1$ indicate that a particular outcome is observed. Compare two possible response rules.

Under Rule A, the probability of a missing outcome is 0.24 for everyone, regardless of group or outcome:

P(R=0\mid G,Y)=0.24.

This is MCAR with respect to the outcome and group: the probability of response does not vary with the complete-data values. The marginal missingness probability is $P(R=0)=0.24$.

Under Rule B, suppose the missingness probabilities are 0.10 in group $G=0$ and 0.38 in group $G=1$, regardless of the outcome value within either group. Because the groups are equally common,

$$ P(R=0)=0.5(0.10)+0.5(0.38)=0.05+0.19=0.24. $$

This rule is MAR: response depends on the observed group $G$, but, by construction, does not depend on the outcome after conditioning on $G$. Both rules have the same overall missingness probability, 24%, but they describe different mechanisms. The overall rate alone cannot distinguish them. In an actual finite sample, observed proportions can also differ from these probabilities because of sampling variation.

A rate calculated from a data set is an empirical proportion. In a probability model, it can estimate a marginal probability such as $P(R_j=0)$, but it does not reveal how that probability varies with the complete data. A pattern table adds information about which indicators co-occur, yet it still records only realized response configurations. It does not, by itself, identify the conditional probability rule that generated them.

Worked Example: Equal Rates Do Not Rule Out MNAR

Suppose a hypothetical binary outcome $Y$ has $P(Y=1)=0.5$. Let the probability of a missing outcome be 0.40 when $Y=1$, and 0.10 when $Y=0$. The marginal missingness probability is

P(R=0)=P(R=0\mid Y=1)P(Y=1)+P(R=0\mid Y=0)P(Y=0)

$$ =(0.40)(0.50)+(0.10)(0.50)=0.20+0.05=0.25. $$

Thus the overall missingness rate is 25%. But the response probability differs according to $Y$, which is the value missing when $R=0$. The rule is MNAR. An MCAR rule with $P(R=0)=0.25$ would produce the same marginal rate while having a different dependence structure. The calculation is consistent with the stated assumptions: the outcome prevalence is one half, and the two conditional missingness probabilities are used with those group probabilities.

What Summaries Can and Cannot Tell You

Rates and patterns are essential descriptive checks, but neither should be treated as a mechanism test. A rate answers “how much is missing for this variable in this defined set of records?” A pattern summary answers “which combinations of variables are observed together?” Mechanism assumptions answer “how does the chance of a response depend on the data?” These summaries may inform reasoning about the response process, but the first two do not settle the third.

Summary or assumption	What it describes	What it does not establish
Variable-specific missingness rate	Fraction of records missing a particular variable	Whether missingness is MCAR, MAR, or MNAR
Joint pattern frequencies	Observed combinations of response indicators	Why those combinations occurred
Mechanism assumption	Conditional probability rule for the indicators	A directly observed label in the data table

Rates stratified by observed groups can be informative descriptions. If response rates differ across observed groups, that is relevant to whether MCAR is plausible; it does not, on its own, prove MAR. Conversely, similar rates across the groups examined do not prove MCAR: differences could involve unexamined variables or unobserved values, and finite samples can conceal real differences. This distinction is especially important when a pattern table contains many small cells. Sparse cells give unstable empirical frequencies, and a zero count does not mean the corresponding pattern has probability zero in the population.

Set the denominator.
Specify the records for which each variable was intended to be measured. Use that same clearly stated population of records when calculating its rate.

Report marginal rates.
Give the number missing and the denominator as well as the percentage, so a rate such as 25% can be interpreted with its sample size.

Summarize joint configurations.
Report pattern counts or proportions when the arrangement across variables matters. Check that the pattern counts sum to $n$ and their proportions sum to 1.

Keep mechanism claims separate.
Use substantive knowledge and appropriate analyses to assess mechanism assumptions; do not infer a mechanism merely from the size of a rate or the shape of a pattern table.

Common Mistakes and Limitations

Using the wrong denominator. A missing count is not a rate until its denominator is specified. State which records were eligible for measurement and report the numerator and denominator.
Adding variable-specific rates to get the percentage of incomplete records. A record missing two variables contributes to both column rates. The proportion with any missing value must instead count records with at least one zero in their response vector.
Calling a large missingness rate MNAR. The rate says how often values are missing, not whether missingness depends on unobserved values. Different mechanisms can have the same rate.
Calling a pattern table a mechanism model. A pattern table describes the realized indicator vectors. It does not say which data values made a pattern more or less probable.
Treating equal observed rates as proof of MCAR. Equality across a few groups cannot rule out dependence on other observed or unobserved values, and sample rates are subject to variability.

The summaries also have practical limitations. A single overall rate can hide concentration of missingness in a subgroup. A detailed pattern table can become difficult to interpret when many variables create many possible patterns, some represented by very few records. Reporting both marginal rates and the most frequent joint patterns is often more informative than relying on either alone. Neither report replaces a justified mechanism assumption.

Key takeaway: Missingness rates quantify how much is missing; pattern frequencies describe how missingness is arranged; MCAR, MAR, and MNAR describe probabilistic dependence of response on data values. The first two are observed summaries, not substitutes for the third.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

In a data set of 60 records, 15 values of variable $C$ are missing. What is the empirical missingness proportion for $C$?
For two variables, pattern counts are $(1,1): 12$, $(1,0): 3$, $(0,1): 4$, and $(0,0): 1$. Find the missingness rates for both variables and the proportion of records with any missing value.
Can a 10% overall missingness rate establish MCAR? Explain.
An observed-group-specific response rule depends on a fully observed group but not on the missing outcome after conditioning on that group. Which mechanism label fits this rule?

Solutions. (1) $\hat{m}_C=15/60=0.25$, or 25%. (2) There are $20$ records in total. Variable 1 is missing in patterns $(0,1)$ and $(0,0)$, so its rate is $(4+1)/20=25\%$. Variable 2 is missing in $(1,0)$ and $(0,0)$, so its rate is $(3+1)/20=20\%$. Any missing value occurs in $3+4+1=8$ records, giving $8/20=40\%$. (3) No. The rate is a marginal descriptive proportion and does not show whether missingness depends on complete-data values. (4) MAR, because the rule depends on observed information and, by the stated condition, not on the missing outcome once that information is given.