Missing Data — mechanisms & multiple imputation

Dr. R. Düsing · University of Osnabrück
Missing Data — Help
Example

A longitudinal study measures Social Support Z and Stress Level X (both z-standardized, t=1, fully observed; r(X,Z)=−.40) as well as Well-Being Y (0–100, t=2, partially missing). True data-generating process (DGP): Y = 60 − 6·X + 3·Z + ε. Target analysis: Y ~ X (simple regression). Z is an auxiliary variable: observed, in the imputation model, but not in the analysis model. The marginal population effect for Y~X is β_X = −7.20 (= −6 + 3·(−.40), because of r_XZ·β_Z).

The question behind it

Y isn't available for everyone. Before computing anything, you need to clarify: why are the values missing — does it depend on chance, on other observed variables, or on the missing value itself? That determines which of the three common strategies even produces unbiased estimators: listwise deletion (dropping incomplete cases), single imputation (inserting one estimated value, e.g. the mean), or multiple imputation (drawing several plausible values and pooling the results). The following three sections describe the three possible "whys" — after that, the strategies in detail.

MCAR — Missing Completely At Random

P(Y missing) is constant and independent of X, Z, and Y — e.g. a technical server outage. Listwise in Y~X produces unbiased estimators but loses efficiency. MI (imputation with X+Z, analysis Y~X) is likewise unbiased and more efficient.

MAR — Missing At Random

P(Y missing | X, Z): high stress (X↑) AND low social support (Z↓) jointly increase the dropout probability. Both are observed — the missingness pattern is fully explainable by observed variables.
Listwise in the target analysis Y ~ X is doubly biased: (1) high-X people are missing → β_X underestimated (|effect| too small); (2) low-Z people are missing → μ_Y overestimated.
MI corrects MAR because Z is included as an auxiliary predictor in the imputation model — even though Z doesn't appear in the target analysis Y~X. The key point: the imputation model must be richer than the analysis model.

MNAR — Missing Not At Random

P(Y missing | Y) depends on the missing Y itself — e.g. people with low well-being drop out precisely because of their low well-being. The missingness pattern isn't fully explainable by any observed variable. Listwise is strongly biased. MI is theoretically not valid, but in practice often reduces bias substantially — and in simulation studies is rarely worse than alternatives.

Listwise deletion — when it fails

Listwise (complete case analysis) excludes all observations with missing Y and computes only on the complete ones. Unbiased only under MCAR. Under MAR and MNAR, the remaining sample is no longer a random sample — the mean and regression coefficients can be systematically biased.

Mean imputation — why it fails

Mean imputation replaces every missing Y with the observed mean ȳ_obs. This roughly restores the mean (under MCAR), but severely distorts the variance and all correlations — because artificially many observations now sit exactly on the mean. The slope β_X is strongly underestimated under MAR and MNAR (attenuation).

Multiple imputation — the PMM algorithm

Predictive Mean Matching (PMM, the default in R's mice package) is used here. PMM always imputes real observed values — no values outside the data range, and more robust to non-normality:

For m = 1 … M: 1. Fit imputation model OLS: Ŷ = α̂ + β̂_X·X + β̂_Z·Z (on observed) 2. Perturb parameters: α̃,β̃_X,β̃_Z ~ N(estimate, SE) 3. Compute ŷ_i (missing) and ŷ_j (observed) with perturbed parameters 4. Find k=5 nearest donors j: |ŷ_i − ŷ_j| minimal 5. Draw one donor j* at random; impute y_i = y_j* 6. Analyze completed dataset with Y~X → θ̂_m (β_X, μ_Y)

Pooling per Rubin (1987):

θ̂_MI = (1/M) Σ θ̂_m Var_MI = W + (1 + 1/M)·B W = (1/M) Σ Var_m (within variance) B = Σ(θ̂_m − θ̂_MI)²/(M−1) (between variance)

B measures the additional uncertainty due to imputation. The more missing values and the weaker the imputation model, the larger B. With "◇ Show MI," the imputed Y values can be displayed in the plot as hollow diamonds (◇) — they scatter around the regression line of the imputation model.

When MI works well

1. Good imputation model: all variables that predict the missingness pattern and the outcome must be included in the imputation model — even if they're not in the analysis model. Here: X (main predictor, also in Y~X) and Z (auxiliary — generates the MAR pattern, affects Y, but not in the analysis model Y~X).
2. MAR plausible: the missingness pattern should be well explained by observed variables.
3. m ≥ 5–20: more imputations → more stable estimators, smaller between-variance share.
4. Not too much missing: above >50%, MI becomes unstable, because the imputation model is based on too little data.
5. MNAR: MI is not a perfect solution, but in practice rarely worse than listwise or mean imputation.

References

Rubin, D. B. (1987). Multiple Imputation for Nonresponse in Surveys. Wiley.
Enders, C. K. (2022). Applied Missing Data Analysis (2nd ed.). Guilford Press.
Kleinke, K., Reinecke, J., Salfrán, D. & Spiess, M. (2020). Applied Multiple Imputation: Advantages, Pitfalls, New Developments and Applications in R. Springer.
He, Y., Zhang, G. & Hsu, C.-H. (2021). Multiple Imputation of Missing Data in Practice: Basic Theory and Analysis Strategies. Chapman & Hall/CRC.
Urban, D. & Mayerl, J. (2018). Angewandte Regressionsanalyse: Theorie, Technik und Praxis (5th ed.). Springer VS.

Running example
A longitudinal study measures Stress Level X and Social Support Z completely at t1 — Well-Being Y is measured at t2, but is only available for part of the sample. Target analysis: the effect of stress on well-being, Y ~ X.
ZAuxiliary variable — fully observed, affects Y (β_Z = +3) and (under MAR) the missingness. Not part of the research question, but needed in the imputation model. XMain predictor — fully observed, r(X,Z) = −.40 YOutcome — well-being t=2 (0–100), partially missing
Y isn't available for part of the sample. Three questions: why does this even cause problems? What pattern can the missingness of Y take (MCAR, MAR, MNAR)? And what do you do about it? Answers in the fundamentals below.
Fundamentals — before you dive in
The key terms at a glance: bias (systematic distortion), the three missingness mechanisms MCAR / MAR / MNAR, the three strategies listwise deletion / single imputation / multiple imputation, and auxiliary variables in the imputation model. Already familiar? Then dive straight into the exercise below. If not — expand here:
① Bias — what is it actually about?
An estimator is biased if, averaged over many repetitions, it systematically deviates from the true population value — not just scattering randomly, but skewed in a particular direction. With missing data, bias arises when the observed subsample is no longer a random sample of the full population: if Y is preferentially missing for stressed people (high X), the remaining data systematically say something different about the X→Y relationship than the complete data would. Whether that happens depends exactly on why values are missing — hence step ②.

② The three mechanisms — why are values missing?
MCAR (Missing Completely At Random): the missingness depends on nothing — pure chance, e.g. a server outage. MAR (Missing At Random): the missingness depends on observed variables — here: high stress X and low support Z jointly. MNAR (Missing Not At Random): the missingness depends on the missing value itself — here: low well-being Y leads to dropout. Only under MCAR is the remaining data still a random sample; under MAR and MNAR bias potentially arises (step ①).

③ Three strategies for handling it
Listwise deletion: drop all incomplete cases, compute only on the rest — simple, but unbiased only under MCAR. Single imputation (e.g. mean imputation): insert a single estimated value per gap — can roughly rescue the mean, but destroys the variance. Multiple imputation (MI): draw several plausible values per gap, analyze repeatedly, and pool the results — corrects bias under MAR and yields correct standard errors.

④ Imputation & auxiliary variables
Imputation means: replacing missing values with estimated, plausible values, based on a model of the observed data. For this model to actually fix the bias, it must be richer than the target analysis — it needs auxiliary variables: observed variables that aren't part of the research question, but belong in the imputation model. Selection criterion: a good auxiliary variable correlates either with the missingness (predicts who is missing) or with the outcome itself — ideally with both. Here Z is exactly that: Z affects both who drops out (the MAR mechanism) and Y directly. Without Z in the imputation model, PMM couldn't fix the MAR bias, even though Z doesn't appear in the target analysis Y~X at all.
Try it — pick a mechanism on the left
Scatter — stress level X vs. well-being Y
MCAR: Purple circles (○) show the true Y values of missing observations — evenly distributed, no systematic pattern. Listwise and MI unbiased.
Density distribution Y — comparing the datasets
Concepts
The three mechanisms
MCAR (Missing Completely At Random): the missingness is completely random, depends on nothing. MAR (Missing At Random): the missingness depends on observed variables (here X and Z), but not on the missing value itself. MNAR (Missing Not At Random): the missingness depends on the missing value Y itself. Which mechanism holds determines which method is unbiased — and cannot be reliably tested from the data alone.
Listwise deletion (complete case)
Drops all cases with missing Y and computes only on the complete ones. Unbiased only under MCAR. Under MAR/MNAR, the remaining sample is no longer a random sample → the mean and regression coefficient can be systematically biased. Even under MCAR it always costs efficiency (smaller n, larger standard errors).
Mean imputation — why it fails
Replaces every missing Y with the observed mean. This roughly restores the mean (under MCAR), but destroys variance and correlations: artificially many points sit exactly on the mean, the spread becomes too small, and the slope βX becomes too flat (attenuation). A seeming solution that creates a new distortion.
Multiple imputation (PMM)
Instead of one replacement value, MI produces several complete datasets. Predictive Mean Matching draws, for each gap, a plausible real donor value from similar cases. You analyze each dataset and pool the results (Rubin's rules) — this way the imputation uncertainty flows into the standard errors, unlike with single imputation.
The imputation model must be richer
The key to MAR: all variables that predict the missingness pattern or Y belong in the imputation model — including an auxiliary variable Z that doesn't appear in the target analysis Y~X at all. Here Z generates the MAR pattern and affects Y; if you include Z in the imputation model, PMM finds donors with correctly low Y values and the MAR bias disappears. Imputation model ⊇ analysis model.
Which method when?
MCAR: everything unbiased, listwise only inefficient. MAR: multiple imputation (or FIML/ML) corrects — provided the relevant predictors are in the model. MNAR: no method fully corrects; you need assumptions about the mechanism and sensitivity analyses. Related is the general logic of selection effects: who ends up in the data depends on the mechanism. → Berkson's Paradox (selection)