Missing Data — MCAR, MAR, MNAR, Imputation

Missing data is ubiquitous in real-world research. Understanding the mechanism that generates missing values is critical for choosing appropriate handling methods.

Types of Missingness

MCAR — Missing Completely at Random

Missingness is completely unrelated to any data (observed or missing). Like data?? being lost in the mail.

MAR — Missing at Random

Missingness depends on observed data but not on the missing values themselves.

MNAR — Missing Not at Random

Missingness depends on the unobserved values themselves. The hardest mechanism to handle.

Comparison

| Mechanism | Missingness depends on | Example |

|-----------|----------------------|---------|

| MCAR | Nothing | Data entry errors; random equipment failure |

| MAR | Observed variables only | Young people skip income questions |

| MNAR | Missing values themselves | Depressed people don't report depression |

Handling Missing Data

Listwise Deletion

Delete any row with missing values.

| Pros | Cons |

|------|------|

| Simple; unbiased under MCAR | Loses data; reduces power |

| | Biased under MAR and MNAR |

Mean Imputation

Replace missing values with the observed mean.

Multiple Imputation

Multiple Imputation: Rubin's Rules

Predictive Mean Matching (PMM)

The most popular imputation method. For each missing value:

1. Fit a regression predicting the variable from other variables

2. Find observed values with similar predicted values

3. Use the observed value as the imputation

Python Implementation

import numpy as np

import pandas as pd

from sklearn.experimental import enable_iterative_imputer

from sklearn.impute import IterativeImputer, SimpleImputer

import matplotlib.pyplot as plt



np.random.seed(42)



# Simulate data with missing values

n = 500

X1 = np.random.randn(n)

X2 = 0.7 * X1 + np.random.randn(n) * 0.5

X3 = 0.3 * X1 + 0.4 * X2 + np.random.randn(n) * 0.8



# MAR: X1 missing depends on X2

missing_prob = 1 / (1 + np.exp(-(-1 + 0.5*X2)))

R = np.random.binomial(1, missing_prob)

X1_obs = X1.copy()

X1_obs[R == 1] = np.nan



df = pd.DataFrame({'X1': X1_obs, 'X2': X2, 'X3': X3})

print(f"Missing in X1: {df['X1'].isna().sum()} ({df['X1'].isna().mean():.1%})")



# Listwise deletion

complete = df.dropna()

print(f"\nListwise deletion: n={len(complete)}")

print(f"X1 mean (complete): {complete['X1'].mean():.3f} (true: {X1.mean():.3f})")



# Multiple Imputation

mice = IterativeImputer(random_state=42, max_iter=10)

imputed = pd.DataFrame(mice.fit_transform(df), columns=df.columns)

print(f"\nMICE imputation:")

print(f"X1 mean (imputed): {imputed['X1'].mean():.3f} (true: {X1.mean():.3f})")



# Visualize

fig, axes = plt.subplots(1, 2, figsize=(12, 5))

axes[0].hist(X1, bins=30, alpha=0.5, label='True')

axes[0].hist(complete['X1'], bins=30, alpha=0.5, label='Listwise')

axes[0].legend()

axes[0].set_title('Listwise Deletion')



axes[1].hist(X1, bins=30, alpha=0.5, label='True')

axes[1].hist(imputed['X1'], bins=30, alpha=0.5, label='MICE')

axes[1].legend()

axes[1].set_title('Multiple Imputation')

plt.tight_layout()

plt.show()

Worked Example

Key Takeaways

Related Topics

• See Multiple Imputation for detailed MI methods

• See Propensity Score Matching for handling selection bias

• See Causal Inference for related topics