Missing data: MCAR, MAR and MNAR

MCAR is rare outside equipment failure and deliberate sampling. MAR is the working assumption behind most modern methods. MNAR is common in exactly the settings people most want to study: dropout from treatment, non-response on sensitive questions, and people who stop using an app because they recovered or deteriorated.

The mechanism determines which methods stay valid, and the hierarchy is strict.

This is also where GEE differs sharply from mixed models. Standard GEE requires MCAR, a stronger assumption than the MAR that mixed models tolerate. In a study with dropout related to anything observed, that difference can decide which method is defensible.

Deleting rows with any missing value is what software does unless told otherwise, which is why it rarely gets described as a decision. It is one, and it has two costs.

The first is power. With ten variables each missing five per cent at random, around forty per cent of rows can disappear, and the loss compounds silently. The second is bias. Unless data is MCAR, the complete cases are a selected subgroup, and the analysis describes them rather than your sample.

There is a narrower case where complete-case analysis is fine even under MAR: when missingness depends only on variables that are already covariates in your model and the outcome is fully observed. That is a real exemption, worth knowing, and it is not the situation most people are in.

A study measures an outcome Y in two equally sized groups defined by X. Y is missing for some people, and how often depends on X.

Missingness depends only on X, which you observed. That makes this missing at random, the condition most modern methods are built for.

Step 1. The answer you want

true mean of Y = (0.5 × 10.0) + (0.5 × 20.0) = 15.00

Step 2. What complete-case analysis gives you

Work out what fraction of the surviving records come from each group. Group X = 0 retains 80% of its half, so 0.40 of the original sample; group X = 1 retains 40% of its half, so 0.20. Together 0.60 of the sample survives.

complete-case mean = (0.40 × 10.0 + 0.20 × 20.0) / 0.60
    = 8.0 / 0.60 = 13.33

Step 3. The damage

bias = 13.33 − 15.00 = −1.67

Two things happened at once. You lost 40% of your data, which costs precision. And the surviving records over-represent the low-Y group, which costs accuracy. The second is the serious one, because it shifts the answer rather than widening the interval, and nothing in the output flags it.

Step 4. What fixes it

Because missingness depends only on X, and X is observed, both multiple imputation and inverse probability weighting recover 15.00. Imputation does so by filling in Y from its relationship with X; weighting does so by up-weighting the under-represented group. Either works; neither would work if missingness depended on Y itself.

Mean substitution and last observation carried forward still appear in submitted papers. Both are now treated as red flags by reviewers, and the second was effectively retired from regulated trial work years ago.

Multiple imputation fills the gaps several times over, with the variation between those versions carrying the uncertainty that single imputation throws away. Analyse each completed dataset, then pool using Rubin's rules, which combine the estimates and inflate the standard errors to reflect how much the imputations disagreed.

If people left because of the thing you are measuring, no complete-data method recovers the truth. What you can do is show how much the conclusion depends on the assumption.

The usual approach is a delta-adjusted sensitivity analysis: impute under MAR, then shift the imputed values by a specified amount in the plausible direction, and repeat across a range of shifts. The output is not a corrected estimate. It is a statement of how large the departure from MAR would have to be before your conclusion changes. If the answer is "a small one", that is important for your reader to know.

In app and wearable research this is routine rather than exceptional. People stop logging for reasons tied to their symptoms, so the gaps are informative by construction, and the same filtering that produces self-selection at recruitment keeps operating throughout follow-up.

R

mice is the standard package and handles the whole impute-analyse-pool cycle.

library(mice)
imp <- mice(df, m = 30, method = "pmm", seed = 123)
fit <- with(imp, lm(outcome ~ treatment + age))
summary(pool(fit))

Three details matter. m = 30 rather than the default 5, since a reasonable rule is at least as many imputations as the percentage of incomplete cases. Predictive mean matching keeps imputed values inside the observed range, which avoids impossible values. And pool() applies Rubin's rules, which is what inflates the standard errors to reflect the uncertainty in the imputation. Analysing a single completed dataset skips that entirely.

Inspect the result with densityplot(imp), comparing imputed against observed distributions.

SPSS

Multiple imputation is under Analyze, Multiple Imputation, Impute Missing Data Values.

MULTIPLE IMPUTATION outcome treatment age
  /IMPUTE METHOD = AUTO NIMPUTATIONS = 30
  /CONSTRAINTS outcome (MIN = 0)
  /OUTFILE IMPUTATIONS = imputed.

SPSS then pools automatically when you run supported procedures on the imputed file, showing the pooled row at the bottom of the output. Not every procedure supports pooling, and where it does not SPSS silently reports each imputation separately, so check for the pooled row before reporting anything.

Stata

The mi suite declares the structure, imputes, then prefixes the analysis.

mi set wide
mi register imputed outcome
mi impute chained (pmm, knn(5)) outcome = treatment age, add(30)
mi estimate: regress outcome treatment age

mi estimate applies Rubin's rules and reports the fraction of missing information, which is worth reading: a high value means the result rests heavily on the imputation model rather than on observed data.

Item 12c of the STROBE statement asks for most of this directly, and it is among the items reviewers read first. Answering it properly in the methods removes an entire line of objection before it is raised.