Multiple Imputation for Missing Data in SPSS and R: MCAR, MAR, MNAR and MICE

Missing data are not a rarity that only affects poorly planned theses: they appear in longitudinal studies when participants drop out, in surveys when sensitive items are skipped, in clinical records when data were not collected. What you decide to do with them affects the validity of your conclusions, and "remove all cases that have any missing value" is not the statistically correct answer in most situations. This guide explains why and shows the alternatives step by step.

The three missing data mechanisms

How data go missing determines what you can do about it. Rubin (1976) distinguishes three mechanisms:

• MCAR (Missing Completely At Random): the probability that a value is missing does not depend on either the value of that variable or any other variable in the study. This is the most benign assumption (and the least common in practice). A computer crashing randomly during data collection produces MCAR. If data are MCAR, removing incomplete cases produces unbiased estimates, although you lose power.
• MAR (Missing At Random): the probability that a value is missing depends on other observed variables, but not on the value of the missing datum itself. For example: older men respond less often to mental health questions than younger women, and you have that information in your dataset. MAR is the assumption that modern imputation methods require and that can be defended with argumentation and the available data.
• MNAR (Missing Not At Random): the probability that a value is missing depends on the value it would have taken. The classic example: patients with greater pain do not attend the study's final assessment, so you lack their pain scores. MNAR is the most problematic and no statistical method can fully resolve the bias it produces without additional external information.

The first step, always, is to explore the pattern of missing data and argue which mechanism is most plausible. This goes in the Methods section. Ignoring it is one of the errors that Q1 journal reviewers flag most frequently.

Why deleting cases is almost always a bad idea

Listwise deletion is only valid under MCAR, which is the most restrictive and generally incorrect assumption. Under MAR, removing incomplete cases introduces bias because those who remain are systematically different from those who are removed. You also lose power by reducing the sample, and in studies where the sample size was already tight, that can make the analysis unviable.

Pairwise deletion (which SPSS uses in some correlation tables) does not solve the problem either: it uses different subsets of cases for each pair of variables, producing correlation matrices that are not positive definite, which causes CFA and other multivariate techniques to fail.

Single imputation: mean substitution and similar methods

Substituting missing values with the variable mean is simple and intuitive, but has two serious flaws: it reduces the real variance of the variable (all imputed values are identical) and it does not incorporate uncertainty into the estimate. Standard errors end up artificially small. Mean imputation is only acceptable when there are very few missing values (under 5%) under MCAR, and it is always inferior to multiple imputation.

Multiple imputation with MICE: the current standard

Multiple imputation (MI) generates several complete versions of the dataset (typically 5 to 20, depending on the percentage of missing data), analyses each version separately and combines the results using Rubin's rules. By generating several imputations, the uncertainty about the missing values is reflected in the final standard errors.

The most widely used algorithm is MICE (Multiple Imputation by Chained Equations), also called fully conditional specification. Each variable with missing data is imputed with a regression model (linear, logistic, etc.) that uses the other variables as predictors, iteratively. MICE works well with variables of different types (continuous, ordinal, dichotomous mixed together) and under the MAR assumption.

Multiple imputation in SPSS

1. Go to Analyze > Multiple Imputation > Impute Missing Data Values.
2. Move the variables with missing data (and auxiliary variables that help predict them) to the analysis panel.
3. Choose the number of imputations: minimum 5, more if the percentage of missing data exceeds 20%.
4. SPSS creates a new variable Imputation_ that identifies each imputed version (0 = original data, 1 to M = imputed versions).
5. Run your main analysis on this expanded dataset. SPSS with the imputation procedure active combines the results automatically using Rubin's rules.

Multiple imputation in R with mice

library(mice)
# Explore missing data pattern
md.pattern(data)

# Impute: m = number of imputations, method = pmm for continuous variables
imp <- mice(data, m = 10, method = "pmm", seed = 123)

# Analyse each imputed dataset
fit <- with(imp, lm(y ~ x1 + x2))

# Pool results (Rubin's rules)
summary(pool(fit))

The pmm (predictive mean matching) method is robust for non-normal continuous variables because it imputes actually observed values rather than regression-predicted values. For dichotomous variables use logreg; for polytomous variables, polyreg.

How many imputations do you need?

The classic rule of 5 imputations is conservative. Von Hippel (2020) proposes that the number of imputations should be at least equal to the percentage of missing data: if you have 20% missing data, use at least 20 imputations. With fewer, the estimation efficiency drops and the combined standard errors are not stable across runs.

How to report in the Methods section

Describe: (1) the percentage of missing data per variable, (2) the assumed mechanism (MCAR, MAR) and how you evaluated it (for example, Little's test for MCAR), (3) the imputation method (MICE with M imputations), (4) the auxiliary variables included in the imputation model, and (5) the software. Example: "8% of values were missing on the anxiety scale. Little's test did not reject the MCAR assumption (chi2(34) = 38.1, p = .29), although given the response pattern MAR was considered more plausible. Multiple imputation was performed using MICE (M = 10) with the mice 3.16 package in R (R Core Team, 2024), including age, sex and depression score as auxiliary variables."

The Spanish article on how to handle missing data expands on the technical details and application conditions of each method.

Once your data are cleaned with multiple imputation, the next step is typically to verify the internal consistency of your instruments. The McDonald's omega calculator and the Cronbach's alpha calculator let you check it directly from your scale statistics.