Missing Data Mechanisms: MCAR, MAR, MNAR (with a concrete simulation)

Introduction

Missing values are not all “the same problem”.
The reason why data is missing matters, because it affects:

• whether your estimates are biased,
• whether an imputer can “recover” information,
• how trustworthy your conclusions are.

There are three classic mechanisms:

• MCAR Missing Completely At Random
• MAR Missing At Random (but conditional on observed data)
• MNAR Missing Not At Random (depends on unobserved value / missing value itself)

Below we simulate each mechanism on the California Housing dataset.

import numpy as np
import pandas as pd
from sklearn.datasets import fetch_california_housing

data = fetch_california_housing(as_frame=True)
X = data.data.copy()
y = data.target.copy()

# Add a categorical feature (to reuse your "categorical variables" lecture)
# Example: region bucket based on longitude
X["Region"] = pd.qcut(X["Longitude"], q=5, labels=["W", "SW", "C", "SE", "E"])

X.head()

Why add Region?

Because real datasets often have mixed types (numeric + categorical), and missingness can affect both.

The simulation: MCAR, MAR, MNAR-like

We create a copy X_miss and inject missingness with specific rules.

rng = np.random.default_rng(42)
X_miss = X.copy()

n = len(X_miss)

1) MCAR — Missing Completely At Random

A value is missing for reasons unrelated to anything in the dataset (neither observed nor unobserved).

• Example: a sensor randomly drops readings.
• Example: a survey page randomly fails to save.

mcar_mask = rng.uniform(size=n) < 0.15
X_miss.loc[mcar_mask, "MedInc"] = np.nan

• Each row has an independent 15% chance to be missing in MedInc.
• The probability does not depend on HouseAge, Longitude, the value of MedInc, etc.

Practical consequence

• If data is truly MCAR, then dropping rows does not systematically bias estimates.
• But you still lose data with the implication of bigger variance (and therefore worse performance).

Typical strategy under MCAR

• SimpleImputer (mean/median/mode) often works fine.
• Dropping rows/columns can be acceptable if missingness is small.

2) MAR — Missing At Random (conditional on observed variables)

Missingness depends on something you observed, but not on the missing value itself once you condition on observed data.

• Example: older buildings have more incomplete surveys.
• Example: people in rural areas skip certain questions more often.

prob_mar = (X_miss["HouseAge"] - X_miss["HouseAge"].min()) / (X_miss["HouseAge"].max() - X_miss["HouseAge"].min())
mar_mask = rng.uniform(size=n) < (0.05 + 0.25 * prob_mar)
X_miss.loc[mar_mask, "AveRooms"] = np.nan

What’s happening:

1. prob_mar rescales HouseAge to roughly [0, 1].

2. Missingness probability becomes 0.05 + 0.25 * prob_mar:

   • new houses → around 5% missing
   • old houses → up to around 30% missing

So AveRooms is missing more often for old houses.

Practical consequence

• If you ignore the dependency (e.g. mean impute without using HouseAge), you can introduce bias.
• But MAR is fixable with methods that use other observed features.

Typical strategy under MAR

• Prefer imputers that use other columns:
  • KNNImputer (needs scaling, can be strong)
  • IterativeImputer (models each feature from the others; often best but slower)
• Add missingness indicators sometimes helps (especially for linear models):
  • SimpleImputer(add_indicator=True)

3) MNAR — Missing Not At Random

Missingness depends on the value that is missing (or on something unobserved strongly linked to it).

• Example: people with very high income refuse to answer income questions.
• Example: extreme values are censored for privacy.

pop_scaled = (X_miss["Population"] - X_miss["Population"].min()) / (X_miss["Population"].max() - X_miss["Population"].min())
mnar_mask = rng.uniform(size=n) < (0.02 + 0.35 * pop_scaled)
X_miss.loc[mnar_mask, "Population"] = np.nan

• Higher Population -> higher chance of being missing.
• This is “MNAR-like” because in a real MNAR scenario you wouldn’t have the true values for the missing ones—here we simulate the mechanism using available values.

Practical consequence (the important one)

MNAR is the hard case:

• No imputer can fully “solve” MNAR from the observed data alone, because the missingness mechanism itself hides information.
• You often need:
  • domain knowledge about the missingness process,
  • explicit modeling of missingness,
  • sensitivity analysis (best practice in applied work).

Typical strategy under MNAR

• Be honest: “MNAR cannot be guaranteed-correctly imputed without assumptions.”

• Practical mitigations:

  • keep missingness indicators,
  • compare multiple imputers + report sensitivity,
  • use domain-informed rules (e.g. censored models, bounds, or custom imputations),
  • if possible, collect extra variables that explain missingness (turn MNAR -> MAR).

Missingness in categorical variables

cat_mask = rng.uniform(size=n) < 0.10
X_miss.loc[cat_mask, "Region"] = np.nan

X_miss.isna().mean().sort_values(ascending=False)

This is a simple random 10% missingness on a categorical column.

Common handling

• SimpleImputer(strategy="most_frequent") for categories
• or fill with a dedicated label like "Missing" (often a good teaching trick)

Key mental model

• MCAR: missingness is “random noise” -> simple methods OK; dropping not biased (but wastes data).
• MAR: missingness is explainable by observed features -> use imputers that leverage other columns.
• MNAR: missingness is tied to the hidden value -> requires assumptions/modeling; results depend on what you assume.

Conclusion

Missing data is not just a preprocessing detail: it’s an assumption about how your dataset was generated.

• If missingness is MCAR, you mostly lose efficiency (more variance), and simple baselines often work.
• If missingness is MAR, you can often do much better by using imputers that exploit relationships among observed features.
• If missingness is MNAR, there is no free lunch: any imputation requires extra assumptions, so the right approach is usually transparency + sensitivity analysis.

In practice, you rarely know the true mechanism. A good workflow is:

1. Diagnose patterns (missingness rates + correlations with observed features).
2. Start with simple baselines.
3. Compare stronger imputers under cross-validation.
4. If MNAR is plausible, report uncertainty.