Multiple Imputation Theory

Why Multiple Imputation?

Problem with complete case analysis:

  • Wastes data

  • Can introduce bias under MAR/MNAR

Problem with single imputation:

  • Treats imputed values as if they were observed

  • Underestimates uncertainty

  • Standard errors too small, p-values too optimistic

Multiple imputation solution:

  • Creates multiple plausible versions of complete data

  • Properly accounts for imputation uncertainty

  • Produces valid inference under MAR

The Three-Step Process

  1. Imputation: Create m complete datasets with different imputed values

  2. Analysis: Analyze each dataset separately

  3. Pooling: Combine results using Rubin’s rules

What is MICE?

MICE (Multiple Imputation by Chained Equations) imputes one variable at a time:

  1. Initialize: Fill missing values with simple method (mean/sample)

  2. Iterate: For each incomplete variable:

    • Set its imputed values to missing

    • Fit model using other variables as predictors

    • Predict and fill missing values

    • Repeat for all variables

  3. Converge: Continue iterations until chains stabilize

  4. Repeat: Create m complete datasets

Key Points

Assumptions:

  • Data is MAR

  • Imputation models are correctly specified

  • Sufficient iterations for convergence

Advantages:

  • Flexible (different methods for different variables)

  • Handles mixed data types

  • No joint distribution required

Limitations:

  • Assumes MAR (problems if MNAR)

  • Requires convergence checking

  • Conditional models may not have joint distribution

When MICE Works

✓ Data is MAR ✓ Clear relationships between variables ✓ Moderate missingness (<30-40%) ✓ Adequate sample size

See MICE Overview for usage details.