Types of Missingness: MCAR

Introduction

Missing data are almost unavoidable in clinical research. Patients may withdraw from a study, miss a visit, fail to complete a questionnaire, discontinue treatment, or have an assessment that cannot be performed.

The statistical consequences of missing data depend not only on how much data are missing, but also on why the data are missing.

A central framework for understanding missing-data mechanisms divides them into three categories:

• MCAR — Missing Completely at Random
• MAR — Missing at Random
• MNAR — Missing Not at Random

MCAR is the strongest and most restrictive assumption.

The Basic Setup

Suppose a clinical trial collects an outcome \(Y\), such as change from baseline in a continuous efficacy endpoint.

For each patient, define a missingness indicator:

The variable \(R_i\) tells us whether the outcome is available.

The complete observed data can therefore be represented by the outcome values that were observed and the corresponding missingness indicators.

The Formal Definition of MCAR

The MCAR assumption states that the probability of missingness does not depend on either the observed values or the missing values.

Let \(Y_{\mathrm{obs}}\) denote the observed portion of the outcome data and \(Y_{\mathrm{mis}}\) denote the unobserved portion. The MCAR condition can be written as:

In words: the missingness mechanism is independent of the data themselves.

This is stronger than saying that missingness is unrelated to the unobserved outcome. It says that missingness is unrelated to all of the outcome information, observed or unobserved.

A Simple Example of MCAR

Imagine that a laboratory analyzer randomly fails during a study.

Suppose the analyzer has a hardware problem that causes approximately 5% of measurements to be lost, with no relationship to:

• Treatment assignment
• Patient age
• Baseline disease severity
• Previous laboratory values
• Current laboratory value

If the failures truly occur independently of all these factors, the missing laboratory measurements could plausibly be MCAR.

For example, the missing measurements might look like this:

The key question is not whether patients 003 and 005 look unusual. The key question is whether the mechanism causing their observations to be missing is independent of the data.

MCAR Is About the Missingness Mechanism

One of the most important concepts in missing-data analysis is that MCAR is a statement about the mechanism generating missingness, not merely about the percentage of missing observations.

For example, suppose 5% of observations are missing. That does not automatically mean the data are MCAR.

Likewise, 30% missingness does not automatically imply that the data are not MCAR.

MCAR Versus MAR

Under MAR, missingness can depend on information that has already been observed.

The MAR assumption can be expressed as:

Thus, after conditioning on observed information, the unobserved value itself does not provide additional information about missingness.

For example, suppose patients with poor baseline disease severity are more likely to miss their Week 24 assessment. If baseline disease severity is observed, missingness depends on an observed variable. That can be consistent with MAR.

It is not MCAR because missingness is associated with observed information.

MCAR Versus MNAR

Under MNAR, missingness depends on the value that would have been observed, even after accounting for available observed information.

For example, suppose patients with particularly poor treatment response are more likely to discontinue before the final efficacy assessment. If the probability of discontinuation remains related to the unobserved outcome after accounting for observed information, the mechanism is MNAR.

The distinction can be summarized as:

A Clinical-Trial Example

Consider a randomized clinical trial in which the primary endpoint is change from baseline in a symptom score at Week 12.

Suppose patients are randomly selected by a computer for a laboratory substudy, but the laboratory machine occasionally loses a result because of a random technical failure. If the failures are unrelated to treatment, baseline characteristics, prior outcomes, and the true Week 12 value, the missing laboratory data may satisfy MCAR.

Now change the scenario. Suppose patients with greater disease severity are more likely to miss the Week 12 visit. If disease severity is measured at baseline, missingness depends on an observed variable. That is more naturally described as MAR rather than MCAR.

Finally, suppose patients who are doing particularly poorly are more likely to drop out because they are dissatisfied with treatment. If the reason for dropout is directly related to the unobserved Week 12 outcome, MNAR may be plausible.

Why MCAR Is a Strong Assumption

MCAR requires missingness to be unrelated to essentially all relevant data. That is difficult to establish in many clinical trials.

Patients do not usually disappear from a study for purely random reasons. They may withdraw because:

• The treatment is ineffective
• Adverse events occur
• The patient feels better and no longer wants treatment
• The patient feels worse
• The visit schedule is burdensome
• The patient relocates
• The patient becomes medically unable to attend

Many of these mechanisms can be related to observed or unobserved outcomes.

What Does MCAR Imply for the Observed Sample?

If data are truly MCAR, the patients with observed outcomes are effectively a random subset of the full target population with respect to the variables affected by missingness.

Consequently, complete cases can remain representative of the underlying population for estimation of many parameters.

For example, suppose:

and the missingness indicator satisfies MCAR. Then the distribution of the observed outcomes remains centered at the same population mean:

Thus, the observed cases can provide an unbiased estimate of \(\mu\), assuming the statistical model and other required conditions are appropriate.

Complete-Case Analysis Under MCAR

A complete-case analysis excludes patients with missing values from the analysis.

Under MCAR, complete-case analysis can be statistically valid for many common estimands because the observed cases constitute a random subset.

For example, if 10% of patients have a missing endpoint completely at random, the remaining 90% can still provide an unbiased estimate of the treatment effect under suitable conditions.

The major disadvantage is loss of precision.

If the trial enrolls 500 patients but only 450 have observed outcomes, the analysis effectively uses less information than a method capable of utilizing all available data.

A Simple Numerical Illustration

Suppose a trial has 200 patients in each treatment group. The true mean treatment difference is estimated using a continuous endpoint. If 20 patients in each group have outcomes missing completely at random, the complete-case analysis uses:

rather than the original 200 per group.

The expected treatment-effect estimate may remain unbiased under MCAR, but its standard error will generally increase because less information is available.

MCAR and Randomized Clinical Trials

Randomization does not automatically make missing outcomes MCAR.

Randomization balances treatment assignment at baseline, but it does not prevent patients from subsequently withdrawing or missing assessments for systematic reasons.

For example, suppose patients receiving active treatment experience adverse events and consequently have more missing Week 24 outcomes. The treatment assignment is randomized, but the missingness mechanism is not necessarily MCAR.

This distinction is crucial:

MCAR and Treatment Groups

One useful diagnostic question is whether missingness appears to differ by treatment group.

Suppose:

These missingness rates are similar. That is supportive of MCAR, but it does not prove MCAR.

Now consider:

A strong treatment-group difference in missingness makes MCAR less plausible, because missingness appears related to treatment assignment.

Investigating MCAR in Practice

There is no universal statistical test that can definitively establish MCAR from the observed data.

This is because the mechanism can depend on values that are themselves unobserved.

Nevertheless, investigators can look for evidence against MCAR.

Useful investigations include comparing missingness across:

• Treatment groups
• Study sites
• Baseline disease severity
• Age categories
• Sex
• Baseline outcome values
• Previous outcome measurements
• Adverse-event status
• Concomitant medication use
• Visit history

Create a Missingness Indicator

A practical approach is to create an indicator variable:

The indicator can then be analyzed as an outcome. For example, logistic regression can model the probability that the outcome is observed.

missing_flag <- ifelse(is.na(y), 0, 1)

model <- glm(
  missing_flag ~ treatment + age + baseline + site,
  family = binomial,
  data = dat
)

summary(model)

If treatment or baseline variables strongly predict missingness, that provides evidence against MCAR.

Little's MCAR Test

A commonly discussed formal diagnostic is Little's MCAR test.

The test evaluates whether patterns of missingness are consistent with MCAR. The null hypothesis is:

and the alternative is:

A small p-value provides evidence against MCAR.

However, a large p-value should not be interpreted as proof that MCAR is true.

Why MCAR Cannot Usually Be Proven

Suppose patients with very poor outcomes are more likely to withdraw before their final assessment. The very outcome values that would reveal this relationship are missing. Therefore, the observed data alone cannot completely distinguish between:

• A mechanism in which missingness is unrelated to the outcome
• A mechanism in which missingness depends on the unobserved outcome

This is a fundamental limitation of missing-data inference.

Consequently, MCAR, MAR, and MNAR are partly assumptions about the data generation process rather than properties that can always be conclusively identified from the observed dataset.

MCAR and Multiple Imputation

Multiple imputation can be used under MCAR, although it is not necessary merely because data are MCAR.

Under MCAR, a correctly specified imputation model can recover information about the missing observations and improve efficiency relative to simply discarding incomplete cases.

For example, an imputation model might use:

• Treatment assignment
• Baseline outcome
• Previous outcome measurements
• Demographic characteristics
• Other prognostic variables

The important point is that under MCAR, the missingness mechanism itself does not create a systematic difference between observed and missing observations.

MCAR and Likelihood-Based Analysis

Likelihood-based methods such as maximum likelihood can also be valid under MCAR when the outcome model is correctly specified.

For example, suppose:

If the missingness process is independent of the data, the observed-data likelihood can be used without modeling a separate missingness mechanism.

This is one reason why likelihood-based methods are attractive for incomplete longitudinal clinical-trial data.

MCAR Does Not Mean "Ignore Missing Data"

A common misconception is:

"If the data are MCAR, we do not need to worry about missing data."

That is incorrect.

MCAR can protect an estimator from certain forms of selection bias, but missing observations still reduce the amount of information available.

If 30% of observations are missing completely at random, the analysis still has substantially fewer observations than the planned study.

This can affect:

• Confidence-interval width
• Statistical power
• Precision
• Variance estimates
• Study interpretation

MCAR and Statistical Power

Suppose a study was designed for 90% power with 400 evaluable patients. If 20% of outcomes are missing completely at random and no replacement or other recovery occurs, only approximately 320 patients contribute observed outcomes.

The study may therefore have substantially less power than originally planned.

The missingness mechanism may be benign with respect to bias, but the information loss remains real.

This is one reason why sample-size calculations frequently include assumptions about dropout.

MCAR in Longitudinal Clinical Trials

Longitudinal studies are particularly interesting because missingness can occur at individual visits rather than only at the end of the study.

Consider a patient with measurements at:

A single missing visit does not automatically mean the patient has dropped out. The missingness mechanism must be considered at the visit level.

If Week 8 was missed because of an unrelated random administrative error, MCAR might be plausible. If Week 8 was missed because the patient's symptoms suddenly worsened and the patient could not attend, MCAR is much less plausible.

Intermittent Missingness Versus Dropout

It is useful to distinguish:

• Intermittent missingness: a patient misses one or more assessments but later returns.
• Monotone dropout: after a particular visit, all subsequent measurements are missing.

Both patterns can theoretically be MCAR, MAR, or MNAR. The pattern itself does not determine the mechanism.

A Useful Mental Model

Imagine that the complete dataset existed in an ideal world. Now imagine that a mechanism randomly removes some observations. If the removal process could have been performed by a fair random-number generator that knows nothing about the patients or their data, you have the intuition behind MCAR.

MCAR, MAR, and MNAR: The Key Comparison

Example: Three Mechanisms, Same Trial

Consider a trial measuring a depression score at Week 12.

MCAR Scenario

A random computer failure deletes 5% of Week 12 assessments regardless of treatment, baseline score, previous symptoms, or the true Week 12 score. This is consistent with MCAR.

MAR Scenario

Patients with higher baseline depression scores are more likely to miss the Week 12 assessment. Because baseline depression is observed, this can be consistent with MAR.

MNAR Scenario

Patients whose unobserved Week 12 depression scores would have been especially poor are more likely to discontinue before the assessment, even after accounting for their observed history. This is consistent with an MNAR mechanism.

Why the Same Observed Dataset Can Fit Multiple Mechanisms

Suppose 15% of patients have missing Week 12 outcomes. From the observed data alone, you may be able to establish that:

• Missingness is somewhat higher in one treatment group.
• Missingness is associated with baseline severity.
• Missingness is associated with prior outcome measurements.

These observations can provide evidence against MCAR.

But they do not automatically tell you whether the data are MAR or MNAR. The distinction depends on whether missingness remains related to the unobserved outcome after conditioning on observed information.

R Example: Exploring Missingness

Suppose a dataset contains:

id
treatment
age
baseline
week12

Create a missingness indicator:

dat$week12_observed <-
  ifelse(is.na(dat$week12), 0, 1)

table(dat$week12_observed)

Compare missingness by treatment:

table(
  dat$treatment,
  dat$week12_observed
)

Compare baseline values:

aggregate(
  baseline ~ week12_observed,
  data = dat,
  FUN = mean
)

And model the probability that the Week 12 outcome is observed:

missing_model <- glm(
  week12_observed ~ treatment +
    age +
    baseline,
  family = binomial,
  data = dat
)

summary(missing_model)

These analyses can identify observed predictors of missingness. They cannot prove that the data are MCAR.

Using Previous Outcomes

In longitudinal trials, previous outcome measurements can be particularly informative.

Suppose patients with a poor Week 8 response are more likely to miss their Week 12 assessment. Then Week 12 missingness depends on an observed prior outcome. That provides evidence against MCAR.

A logistic missingness model might therefore include:

missing_model <- glm(
  week12_observed ~ treatment +
    baseline +
    week4 +
    week8,
  family = binomial,
  data = dat
)

If Week 8 strongly predicts whether Week 12 is observed, the MCAR assumption becomes difficult to defend.

What Happens If MCAR Is False?

The consequences depend on what mechanism actually generated the missing data.

If missingness depends on observed variables, MAR-based methods may be more appropriate.

If missingness depends on unobserved outcomes, MNAR sensitivity analyses may be needed.

The key progression is:

MCAR and Sensitivity Analysis

A primary analysis may assume MAR rather than MCAR in many modern clinical trials, with sensitivity analyses exploring departures from that assumption.

MCAR can still serve as an informative benchmark. For example, investigators may compare:

• Complete-case analysis
• Likelihood-based analysis
• Multiple imputation
• Pattern-mixture sensitivity analyses
• Reference-based imputation
• Tipping-point analyses

If conclusions remain stable across reasonable assumptions about missingness, confidence in the treatment-effect conclusion is strengthened.

MCAR Does Not Require Imputation

Another common misconception is that missing observations must always be imputed.

If the data are MCAR, complete-case analysis can often produce unbiased estimates under suitable conditions.

However, an analysis using all available information may still be more efficient.

Thus:

and:

MCAR and Standard Errors

Suppose \(Y_1,\ldots,Y_n\) are independent observations with variance \(\sigma^2\). If only \(n_{\mathrm{obs}}\) observations are available, the approximate standard error of the sample mean is:

As \(n_{\mathrm{obs}}\) decreases, the standard error increases.

Thus, even under ideal MCAR missingness:

This is the primary efficiency cost of MCAR missingness.

MCAR and Bias Versus Precision

It is useful to separate two statistical concepts.

This distinction is important. MCAR can make missingness less problematic for validity without making it irrelevant for efficiency.

Common Mistakes

1. Assuming that a low missing-data rate means MCAR. A 2% missing-data rate can still be MNAR.
2. Assuming similar missingness rates across treatment groups prove MCAR. MCAR concerns all relevant variables, not just treatment.
3. Using a nonsignificant Little's test as proof of MCAR. Failure to reject MCAR is not evidence that MCAR has been established.
4. Confusing MAR with MCAR. If missingness depends on observed baseline or previous outcomes, MCAR is already violated.
5. Assuming randomization guarantees MCAR. Random treatment assignment does not randomize post-baseline missingness.
6. Thinking MCAR means missing data have no consequences. Even completely random missingness reduces precision and power.
7. Ignoring the clinical reason for dropout. The reason a patient misses an assessment can be more informative than the overall percentage missing.
8. Assuming MCAR can always be demonstrated statistically. The unobserved outcomes cannot be directly examined to verify their role in the missingness mechanism.

A Practical MCAR Assessment Workflow

What Should Be Documented in a Clinical Trial?

A statistical analysis plan should describe how missing data will be handled before unblinded treatment comparisons are performed.

Relevant documentation includes:

• Definition of the primary estimand
• Definition of the analysis population
• Expected amount of missing data
• Expected patterns of missingness
• Reasons for discontinuation
• Rules for partial or intermittent missingness
• Primary missing-data method
• Assumptions underlying the method
• Sensitivity analyses
• Handling of patients who withdraw before endpoint assessment

MCAR and the Estimand Framework

Modern clinical-trial analysis should distinguish the missing-data mechanism from the estimand itself.

The estimand describes the treatment effect that the trial is intended to estimate. Missing-data assumptions describe how incomplete observations are handled while estimating that target.

Therefore:

A well-defined estimand should be established before deciding how missing observations will be analyzed.

MCAR in Repeated-Measures Analysis

Suppose a trial measures an outcome at baseline, Week 4, Week 8, and Week 12. A patient may contribute:

If one measurement is missing completely at random, the remaining observations may still contribute useful information to a longitudinal model.

This is one reason mixed models and likelihood-based repeated-measures approaches can be preferable to simply discarding an entire patient whenever one visit is missing.

MCAR Does Not Mean Every Patient Has Complete Data

A dataset can be MCAR even when individual patients have different numbers of observations.

For example:

The pattern of missing visits does not itself establish MCAR or any other mechanism. The cause of the missingness remains the central issue.

How to Think About MCAR in an Interview

A strong interview answer should be concise but precise.

MCAR in One Equation

If you remember only one equation from this tutorial, remember:

The left side says that missingness may depend on the observed and unobserved data. Under MCAR, that dependence disappears.

MCAR in One Sentence

The simplest verbal definition is:

MCAR vs. MAR vs. MNAR: Final Mental Model

The Most Important Concepts

There are several ideas to retain from this tutorial.

• MCAR is the most restrictive of the three common missing-data mechanisms.
• MCAR means missingness is independent of both observed and unobserved data.
• The percentage of missing observations does not determine whether data are MCAR.
• Randomization does not guarantee MCAR.
• Similar missingness rates across treatment groups do not prove MCAR.
• Little's MCAR test can provide evidence against MCAR but cannot prove it.
• Complete-case analysis can be valid under MCAR under appropriate conditions.
• Even under MCAR, information is lost and precision generally decreases.
• If missingness depends on observed information, MCAR is violated and MAR may be more appropriate.
• If missingness depends on the unobserved outcome after conditioning on observed information, MNAR may be relevant.

References

Little, R.J.A. (1988). A test of missing completely at random for multivariate data with missing values. Journal of the American Statistical Association, 83(404), 1198–1202.
Rubin, D.B. (1976). Inference and missing data. Biometrika, 63(3), 581–592.
Little, R.J.A. & Rubin, D.B. (2019). Statistical Analysis with Missing Data. 3rd ed. Wiley.
National Research Council (2010). The Prevention and Treatment of Missing Data in Clinical Trials. National Academies Press.
ICH E9(R1). Addendum on Estimands and Sensitivity Analysis to the Guideline on Statistical Principles for Clinical Trials.
Carpenter, J.R. & Kenward, M.G. (2013). Multiple Imputation and its Application. Wiley.