MCAR, MAR, and MNAR as Working Assumptions

Why the Missingness Mechanism Matters

A missing value is a feature of the recorded data, but the reason it is missing is not automatically recorded. If claim repair costs are absent more often for one submission channel, the observed costs may still be useful after accounting for that channel. If high costs themselves are more likely to be omitted, the observed costs may give a misleading picture of the full population. The distinction affects how much confidence an analyst can place in a model or summary based on the available records.

MCAR, MAR, and MNAR are assumptions about the process that produced missing values. They are not labels that can be assigned just by looking at an NA in R, nor are they three kinds of missing-value code. The earlier material on missing-value representation and missingness patterns helps identify which values are absent; the mechanism asks how that absence relates to the data.

Definition: Let \(Z\) denote the complete data that would be available if every value were recorded. Let \(R\) denote the missingness pattern, with \(R_j=1\) when variable \(Z_j\) is observed and \(R_j=0\) when it is missing. For a particular pattern \(r\), \(Z_{\text{obs}}(r)\) means the values observed under that pattern, and \(Z_{\text{mis}}(r)\) means the values missing under it. MCAR means the probability of the pattern does not depend on \(Z\). MAR means it may depend on \(Z_{\text{obs}}(r)\), but not on \(Z_{\text{mis}}(r)\) once the observed values are given. MNAR means the probability still depends on at least one unobserved value after conditioning on the observed values.

In a simple setting with one variable \(Y\) that may be missing and an always-observed predictor \(X\), write \(R=1\) if \(Y\) is recorded and \(R=0\) otherwise. The definitions become especially clear in terms of \(P(R=1\mid X,Y)\):

\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{ depends on }Y \text{ even after conditioning on }X. \end{aligned}

These expressions describe assumptions about the data-collection process, not necessarily the literal decisions made by a person. In a more complex data set, the relevant observed and missing values depend on the whole pattern \(r\). A variable that is observed in one record may be missing in another, so MAR must be understood relative to the values available for the pattern being considered.

What Each Assumption Says

Assumption	What the observation probability may depend on	What it implies for analysis
MCAR	Neither observed nor unobserved data	The observed records are not selected according to the complete-data values; missingness mainly costs information and precision.
MAR	Values that are observed for the pattern	Analysis can account for observed predictors of missingness, provided the model and required conditions are adequate.
MNAR	At least one value that is itself unobserved, even after conditioning on observed data	Observed data alone generally do not determine the missing-value process; additional assumptions or sensitivity analysis are needed.

MCAR is the strongest of these assumptions. It implies MAR, because a probability that depends on no data also does not depend on unobserved data after conditioning on observed data. MAR does not imply MCAR: missingness can vary across observed groups while remaining unrelated to the missing values within each group. MNAR describes the case where observed values do not fully account for the relationship between the data and the chance of being observed.

Why MCAR and MAR Preserve Particular Distributions

A short conditional-probability argument explains why these assumptions matter. Under MCAR, suppose the probability that \(Y\) is observed is a constant \(c\), with \(c>0\). Then for any possible value \(y\),

P(Y=y\mid R=1) =\frac{P(R=1\mid Y=y)P(Y=y)}{P(R=1)} =\frac{cP(Y=y)}{c} =P(Y=y).

Thus, under MCAR, observing \(Y\) does not change its distribution. This does not mean that a finite observed sample must exactly match the population; random variation remains. It means the selection process does not systematically favor particular values.

Under MAR, let \(X\) be observed for everyone and suppose \(P(R=1\mid X,Y)=g(X)\), where \(g(X)>0\) for the group being considered. Applying the same argument within a fixed value of \(X=x\) gives:

\begin{aligned} P(Y=y\mid X=x,R=1) &=\frac{P(R=1\mid X=x,Y=y)P(Y=y\mid X=x)} {P(R=1\mid X=x)}\\ &=\frac{g(x)P(Y=y\mid X=x)}{g(x)}\\ &=P(Y=y\mid X=x). \end{aligned}

So MAR preserves the conditional distribution of \(Y\) within observed \(X\)-groups, under the stated conditions. It does not generally preserve the unadjusted distribution of \(Y\), because some \(X\)-groups may be more likely to be observed than others. The condition \(g(x)>0\) matters: if a group has no observed \(Y\) values, its conditional distribution cannot be learned from those records alone.

Worked Example: Missing Repair Costs by Submission Channel

A hypothetical insurer records a submission channel \(X\) for each claim and tries to record repair cost \(Y\). Suppose 40% of claims arrive through a web portal and 60% by telephone. The probability that repair cost is recorded is 0.90 for web claims and 0.60 for telephone claims, regardless of the cost itself. Classify the mechanism and calculate the overall recording rate.

Step 1: Identify the observed and potentially missing values. The channel \(X\) is observed for every claim. Repair cost \(Y\) may be missing, and \(R=1\) indicates that it is recorded.

Step 2: Classify the mechanism. The stated probabilities vary with the observed channel but do not vary with \(Y\) after channel is known. Thus \(P(R=1\mid X,Y)=P(R=1\mid X)\): this is MAR. It is not MCAR because the recording probabilities differ by channel, \(0.90\ne0.60\).

Step 3: Calculate the overall recording probability. Use the law of total probability:

P(R=1) =P(X=\text{web})P(R=1\mid X=\text{web}) +P(X=\text{telephone})P(R=1\mid X=\text{telephone})

$$ P(R=1)=(0.40)(0.90)+(0.60)(0.60)=0.36+0.36=0.72. $$

The missing proportion is \(1-0.72=0.28\). The method is to condition on the observed channel before assessing the dependence. The common mistake is to call the data MCAR because repair cost itself does not affect recording; that overlooks the different rates across observed channels.

Worked Examples: Seeing the Difference in Consequences

Worked Example: When High Costs Are Less Often Recorded

In a hypothetical portfolio, 20% of claims have a high repair cost and 80% have a low repair cost. Suppose a high-cost claim has probability 0.50 of having its cost recorded, while a low-cost claim has probability 0.90. There are no other variables in this example. Classify the mechanism and calculate the proportion of high-cost claims among both the recorded and unrecorded claims.

Step 1: Classify the mechanism. Here \(Y\) indicates whether cost is high or low, and no other observed variable is available to explain the difference. The probability of recording depends directly on \(Y\), which is missing when \(R=0\). This is MNAR.

Step 2: Find the overall recorded and unrecorded proportions. The joint proportion of high-cost claims that are recorded is \((0.20)(0.50)=0.10\). For low-cost claims it is \((0.80)(0.90)=0.72\). Therefore:

P(R=1)=0.10+0.72=0.82,\qquad P(R=0)=1-0.82=0.18.

Step 3: Calculate the high-cost fraction among recorded claims.

P(\text{high}\mid R=1)=\frac{(0.20)(0.50)}{0.82} =\frac{0.10}{0.82}\approx0.1220.

Step 4: Calculate the high-cost fraction among unrecorded claims. The joint proportion of high-cost claims not recorded is \((0.20)(1-0.50)=0.10\), so:

P(\text{high}\mid R=0)=\frac{0.10}{0.18}\approx0.5556.

The checks are consistent: high and low claims among recorded claims have proportions \(0.10/0.82\) and \(0.72/0.82\), which sum to 1; among unrecorded claims the corresponding proportions are \(0.10/0.18\) and \(0.08/0.18\), which also sum to 1. The method is to calculate joint proportions first, then divide by the total in the group of interest. A common mistake is to report 20% as the high-cost proportion in the recorded data; that is the population proportion, not the selected proportion.

Worked Example: Using Observed Groups Under MAR

In a hypothetical claims study, \(X\) is a risk band observed for all claims. Three-quarters of claims are in the low band and one-quarter are in the high band. The chance that outcome \(Y=1\) occurs is 0.10 in the low band and 0.40 in the high band. Outcomes are recorded for 0.80 of low-band claims and 0.50 of high-band claims, independently of \(Y\) within each band. Calculate the overall recorded proportion, the outcome rate among recorded claims, and the population outcome rate obtained by combining the within-band rates using the full band proportions.

Step 1: Classify the mechanism. Recording depends on observed \(X\), but not on \(Y\) after \(X\) is given. The mechanism is MAR, not MCAR, because recording probabilities differ between bands.

Step 2: Calculate the overall recorded proportion.

$$ P(R=1)=(0.75)(0.80)+(0.25)(0.50)=0.60+0.125=0.725. $$

Step 3: Calculate the outcome proportion among recorded claims. The recorded-and-positive proportions from the low and high bands are \((0.75)(0.80)(0.10)=0.060\) and \((0.25)(0.50)(0.40)=0.050\). Their sum is \(0.110\). Therefore:

P(Y=1\mid R=1)=\frac{0.110}{0.725}\approx0.1517.

Step 4: Combine the within-band outcome rates using the population band proportions.

$$ P(Y=1)=(0.75)(0.10)+(0.25)(0.40)=0.075+0.100=0.1750. $$

The recorded-only rate, about 0.1517, differs from the population rate 0.1750 because the high band has both a higher outcome rate and a lower recording rate. Under the specified MAR mechanism, the within-band rates remain 0.10 and 0.40, so combining them with the full population band proportions recovers 0.1750. The check is that the population weights sum to \(0.75+0.25=1\). The common mistake is to use the band proportions among recorded claims as if they were the population proportions.

What the Assumptions Permit—and What They Do Not

Under MCAR, methods that rely on the observed cases representing the complete-data population have a clear justification from the selection mechanism, though missing records still reduce precision. Under MAR, observed variables related to missingness should be accounted for in an analysis where appropriate. A likelihood or imputation model may use those observed variables, but the result still depends on correct model specification, relevant variables being included, and adequate observations across the groups being modeled.

MAR is not a guarantee that a particular method is valid. For example, if a group has no recorded outcomes, the conditional relationship in that group cannot be estimated directly from the observed records. The positivity condition illustrated in the derivation fails there. MNAR requires more: an analyst must make assumptions about how the unobserved values affect their own chance of being missing, or assess how conclusions change under alternative plausible assumptions.

Most importantly, the mechanism is not generally identified from the observed data alone. Observed differences in missingness across recorded variables can challenge MCAR, but cannot prove MAR. A dataset with observed values may be compatible with both a MAR explanation and an MNAR explanation. Context about data collection can inform the choice of assumptions, but should not be presented as proof.

Key takeaway: Ask what the probability of being observed depends on. Dependence on neither observed nor unobserved data is MCAR; dependence only on values observed for the missingness pattern is MAR; remaining dependence on a missing value is MNAR. The more the mechanism depends on unobserved values, the more the analysis relies on assumptions that the observed data cannot verify by themselves.

Common Exam Traps

Calling a mechanism MCAR because missingness is unrelated to the missing value. If the recording rate varies with an observed predictor, it is not MCAR, even if that predictor accounts for the variation.
Calling a mechanism MAR just because it is related to an observed variable. If it also depends on the missing value after conditioning on observed variables, it is MNAR.
Treating an observed-data check as proof of MAR. Observed relationships can reveal evidence against MCAR, but generally cannot rule out dependence on unseen values.
Forgetting overlap. MAR does not make a subgroup’s missing values recoverable from observed data if that subgroup has no observed cases.
Confusing the missingness code with its cause. An R value of NA records absence; it does not reveal whether the mechanism is MCAR, MAR, or MNAR.

Check Your Understanding

For each problem, classify the mechanism from the stated assumptions and calculate the requested quantity. Probabilities are rounded to four decimal places where needed.

A hypothetical survey records a participant’s region for everyone, but an optional travel-time value is recorded with probability 0.85 in every region, independently of travel time. What mechanism is specified, and what is the overall recording probability? (Answer: MCAR; \(0.8500\).)
In a hypothetical file, 30% of records are from urban areas and 70% from rural areas. A value is recorded with probability 0.90 for urban records and 0.60 for rural records, with no dependence on the value itself within either area. Classify the mechanism and calculate the overall recording probability. (Answer: MAR, not MCAR; \(0.30(0.90)+0.70(0.60)=0.6900\).)
In a hypothetical population, 30% of cases have a high outcome. High outcomes are recorded with probability 0.40 and low outcomes with probability 0.90. Calculate the overall recording probability and the high-outcome proportion among missing cases. (Answer: MNAR; recording probability \(0.30(0.40)+0.70(0.90)=0.7500\); high among missing \(0.30(0.60)/[0.30(0.60)+0.70(0.10)]=0.7200\).)
A result \(Y\) is missing, and a fully observed group label \(X\) is available. Within each group, the recording probability is constant across all values of \(Y\), but it differs between groups. Is the mechanism MCAR, MAR, or MNAR? What conditional relationship does the assumption preserve? (Answer: MAR, not MCAR; within each group with a positive recording probability, \(P(Y\mid X,R=1)=P(Y\mid X)\).)
In a hypothetical study, recording depends only on an observed group \(X\), so the mechanism is MAR. However, no \(Y\) values are recorded for one group. Can the conditional distribution of \(Y\) in that group be estimated directly from the observed \(Y\) values? (Answer: No; that group has zero probability of observation, so the required overlap condition fails.)

What You'll Learn