MCAR / MAR / MNAR, listwise deletion, indicators, and three imputation strategies
Most introductory examples treat a dataset as if every row is complete and every value is right. Real datasets never are. Sensors fail, forms get skipped, wireless signals drop, and high-income survey respondents leave the “income” field blank. The question is not whether you’ll have missing data — it’s what kind of missingness, and what to do about it.
Rule of thumb: before touching any imputation method, ask why the data is missing. The fix depends entirely on the cause.
Formal acronyms are not always used, but the distinction is crucial:
| Acronym | What it means | Example |
|---|---|---|
| MCAR (Missing Completely At Random) | Missingness unrelated to any variable | Sensor glitched for an hour; transmission dropped a packet |
| MAR (Missing At Random) | Missingness depends on observed variables | Men skip the “weight” field more often than women (depends on gender, which we have) |
| MNAR (Missing Not At Random) | Missingness depends on the unobserved value itself | High-income respondents skip the income field; date-of-death missing for living patients (censoring) |
Why it matters: discarding rows is safe under MCAR, risky under MAR, and dangerous under MNAR. Let’s see the bias in action.
set.seed(2026)
n <- 1000
age <- runif(n, 25, 65)
job_yrs <- runif(n, 0, 40)
# True income: depends on age and years of experience
income <- 20000 + 1500 * age + 800 * job_yrs + rnorm(n, 0, 10000)
dat <- data.frame(age, job_yrs, income)
head(dat) age job_yrs income
1 52.94694 35.19696 145283.62
2 47.26122 22.77992 118119.53
3 30.60560 25.17761 99354.62
4 36.42893 17.05002 84790.44
5 47.21476 15.46030 113812.23
6 26.00525 39.73042 84592.66We have the truth. Now let’s hide some of it three different ways.
# MCAR: drop 20% of incomes completely at random
mcar <- dat
mcar$income[sample(n, 200)] <- NA
# MAR: older people skip more often (missingness depends on age, which we observe)
p_miss_mar <- plogis(-5 + 0.1 * dat$age)
mar <- dat
mar$income[as.logical(rbinom(n, 1, p_miss_mar))] <- NA
# MNAR: high-income people skip more often (missingness depends on income itself)
p_miss_mnar <- plogis(-6 + 0.00008 * dat$income)
mnar <- dat
mnar$income[as.logical(rbinom(n, 1, p_miss_mnar))] <- NA
mean_true <- mean(dat$income)
mean_mcar <- mean(mcar$income, na.rm = TRUE)
mean_mar <- mean(mar$income, na.rm = TRUE)
mean_mnar <- mean(mnar$income, na.rm = TRUE)
round(c(True = mean_true, MCAR = mean_mcar, MAR = mean_mar, MNAR = mean_mnar)) True MCAR MAR MNAR
103128 103267 96162 81560 MCAR barely shifts the mean (noise). MAR is only a modest shift because the observed variable (age) has a weak link to income. MNAR noticeably biases the mean downward because the highest-income values are the ones most likely to be missing.
Common pitfall: students love to assume missing data is random and “just drop the rows.” Discarding rows is only safe under MCAR. Under MNAR (the most common real-world case), your model becomes biased and you won’t know unless you specifically look.
The simplest option is also the most dangerous. Do use listwise deletion when:
Don’t use it when missingness is driven by the value you care about.
# Listwise deletion is just na.omit in R
complete_cases_mnar <- na.omit(mnar)
cat("Original rows: ", nrow(mnar), "\n")Original rows: 1000 cat("After listwise drop: ", nrow(complete_cases_mnar), "\n")After listwise drop: 184 cat("Biased mean income: ", round(mean(complete_cases_mnar$income)), "\n")Biased mean income: 81560 Instead of discarding, encode “this value is missing” as part of the model. For a categorical variable, just add a new level. For a numeric variable, the move is:
is_missing.Step 3 is what makes this work. Without the interactions, the coefficients on the other variables get pulled in weird directions to compensate. With them, you’re effectively fitting two parallel models — one for rows with complete data, one for rows where this variable is missing — a “tree with one branch.”
# Start with the MAR dataset
mar_ind <- mar
mar_ind$income_missing <- as.integer(is.na(mar_ind$income))
mar_ind$income[is.na(mar_ind$income)] <- 0
# Without interactions — income coefficient gets contaminated by the zeros
fit_bad <- lm(job_yrs ~ age + income + income_missing, data = mar_ind)
# With interactions — effectively two models
fit_good <- lm(job_yrs ~ age + income + income_missing +
income:income_missing + age:income_missing,
data = mar_ind)
summary(fit_good)$coefficients |> round(4) Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.6077 1.9731 -0.3080 0.7581
age -0.8883 0.0581 -15.2880 0.0000
income 0.0006 0.0000 19.3068 0.0000
income_missing 20.4973 3.2655 6.2770 0.0000
age:income_missing 0.8828 0.0763 11.5738 0.0000Why interact with every other variable? Because the missing-indicator alone only shifts the intercept. To let the model respond differently to the other predictors when income is missing — which is what you actually want when missingness correlates with other features — you need the interactions.
Imputation means estimating the missing values and filling them in. The three methods covered here trade off simplicity vs realism.
Simplest possible: replace each missing value with the mean (numeric), median (numeric, robust to skew), or mode (categorical).
mean_impute <- function(x) {
x[is.na(x)] <- mean(x, na.rm = TRUE)
x
}
imputed_mean <- mnar
imputed_mean$income <- mean_impute(imputed_mean$income)
cat("True mean income: ", round(mean_true), "\n")True mean income: 103128 cat("Mean-imputed mean income:", round(mean(imputed_mean$income)), "\n")Mean-imputed mean income: 81560 Mean imputation preserves the mean of the imputed column but collapses all variance in the missing subset to zero, which distorts correlations and downstream models.
Build a regression model on the rows without missing data that predicts the missing factor from the other factors. Use it to impute.
# Fit a model on the complete rows
complete_rows <- mnar[!is.na(mnar$income), ]
missing_rows <- mnar[is.na(mnar$income), ]
impute_model <- lm(income ~ age + job_yrs, data = complete_rows)
# Predict for the missing rows
predicted_income <- predict(impute_model, newdata = missing_rows)
imputed_reg <- mnar
imputed_reg$income[is.na(imputed_reg$income)] <- predicted_income
cat("Mean of imputed incomes (regression):", round(mean(predicted_income)), "\n")Mean of imputed incomes (regression): 101417 cat("Range:",
round(min(predicted_income)), "to",
round(max(predicted_income)), "\n")Range: 59703 to 137734 This usually gives a better individual estimate than the global mean, but it still collapses legitimate variation — every 45-year-old with 15 years on the job gets the same predicted income, even though in the real population their incomes span a range.
Fix the variance-collapse problem by adding random noise calibrated to the regression residual standard error:
# Residual standard error from the imputation model
res_sd <- summary(impute_model)$sigma
set.seed(2026)
perturbed <- predicted_income + rnorm(length(predicted_income), 0, res_sd)
imputed_pert <- mnar
imputed_pert$income[is.na(imputed_pert$income)] <- perturbed
cat("Regression-only SD in imputed values: ", round(sd(predicted_income)), "\n")Regression-only SD in imputed values: 16727 cat("Regression+perturbation SD in imputed vals:", round(sd(perturbed)), "\n")Regression+perturbation SD in imputed vals: 18744 cat("Full dataset SD (true): ", round(sd(dat$income)), "\n")Full dataset SD (true): 21424 The perturbation looks like it makes each individual estimate worse (adding noise), but it preserves the spread of the imputed values. When variance matters downstream, this matters.
Rule of thumb: don’t impute more than about 5% of any single factor. Past that, switch to the indicator-plus-interaction approach from Strategy 2.
Why? Because imputation introduces error, and adding that error to more than a small slice of the data compounds. At that point you’re better off modeling the missingness explicitly rather than hiding it behind a guess.
You’re worried: “I’ve imputed values with error, now I’m feeding those into a model that also has error. How do I quantify the total error?”
The honest answer: there isn’t a clean way. Use a held-out test set to evaluate overall performance and accept that the test set itself probably has some missingness too.
Here’s the practical reframe: every dataset has errors — sensors drift, humans type wrong numbers, digital thermometers give different readings 10 seconds apart. The question is not “is my imputation perfect” but “is my imputation worse than the typical error already baked into the data?” Usually it isn’t.
| Situation | Best approach |
|---|---|
| Missingness tiny, clearly random | Listwise deletion (na.omit) |
| Missingness correlated with other variables | Indicator + interactions (Strategy 2) |
| Missingness correlated with the missing variable itself | Indicator + interactions (Strategy 2). Do NOT drop rows. |
| Small amount (<5%), value-dependent | Regression imputation, with perturbation if variance matters |
| >5% missing in one factor | Indicator approach, not imputation |
Common pitfalls recap: