Exploring Missing Data & M.I.C.E.

Oregon State University

Brian Cervantes Alvarez

Tuesday, December 3, 2024

Missing Data

How Missing Data Occurs & Why It Matters

Common Causes of Missing Data

• Human Error: Mistakes during entry or recording.
  
• System Failures: Equipment malfunctions.
  
• Non-response: Participants can’t or won’t respond.
  
• Deliberate Omission: Excluding unreliable data.

Why It Matters

• Bias: Missing data skews parameter estimates and inferences.
  
• Reduced Accuracy: Less reliable predictions and results.
  
• Power Loss: Harder to detect meaningful effects.

Types of Missing Data & MICE Assumptions

Types of Missing Data

• MCAR: Missingness is random, no bias.
  
• MAR: Related to observed data; handled with MICE.
  
• MNAR: Linked to unobserved data; requires advanced models.

MICE assumes data is MAR, making it crucial to identify the missingness type to apply the right method.

M.I.C.E

What is M.I.C.E.?

MICE = Multiple Imputation by Chained Equations

• A statistical method to handle missing data by creating multiple plausible datasets.
• Generates several estimates to reflect uncertainty, rather than a single imputation.

For each variable with missing data X_j:

X_j^{(\text{missing})} = f_j(X_{-j}) + \varepsilon_j

• X_{-j}: All other variables
• f_j: Predictive model (e.g., regression function)
• \varepsilon_j: Random error term capturing residuals

(van Buuren, 2018)

Another way to put it

Handling missing data with MICE is like trying to complete a puzzle with missing pieces. Instead of guessing one fixed piece to fill a gap, you evaluate several options that could reasonably fit, based on the surrounding picture. Each plausible “piece” represents a potential imputation, reflecting the uncertainty of the missing data.

How Does MICE Work?

1. Initial Imputation
   • Fill missing values with initial guesses (mean, median, or mode).
2. Iterative Process
   • For each variable with missing data X_j:
     • Step 1: Treat X_j as the target variable.
     • Step 2: Use other variables X_{-j} to build a predictive model f_j (e.g., linear regression).
     • Step 3: Predict and replace the missing values in X_j using f_j(X_{-j}).
3. Repeat Cycles
   • Iterate over all variables multiple times until the imputations converge (usually around 10 cycles).

MICE models each variable conditional on others, preserving multivariate relationships and producing statistically valid imputations.

Study Overview

Goal: Assess the impact of missing data patterns on regression analysis and demonstrate how MICE recovers accurate results.

Dataset: Product Sales and Returns
Problem: Missing values simulated in Refunds column.

Steps:
1. Data Prep: Selected Refunds, Purchased Item Count, Total Revenue, Category.
2. Simulating Missingness: Applied patterns (MCAR, MAR, MNAR) at 10–70% levels.
3. MICE Imputation: Used mice package with predictors to fill missing Refunds.
4. Regression Analysis: Evaluated the effect of missing patterns and imputation on results.

Plot 1

Plot 2

Plot 3

Conclusion

• MICE effectively handles missing data by leveraging multivariate relationships.
• Maintains data integrity and provides unbiased parameter estimates.

Hence, understanding missing data types is crucial for selecting appropriate imputation methods.

References

• van Buuren, S. (2018). Flexible Imputation of Missing Data (2nd ed.). Chapman and Hall/CRC. https://doi.org/10.1201/9780429492259

Questions?

Appendix

Evaluation Criteria for Multiple Imputation

• Raw Bias (RB):
  • Difference between the expected estimate and the true value.
  • \text{RB} = E(\bar{Q}) - Q
• Percent Bias (PB):
  • Relative bias expressed as a percentage.
  • \text{PB} = 100 \times \left| \frac{E(\bar{Q}) - Q}{Q} \right|
• Coverage Rate (CR):
  • Proportion of confidence intervals that contain the true value.
  • Goal: Should be close to the nominal level (e.g., 95%).
• Average Width (AW):
  • Average width of confidence intervals.
  • Indicates statistical efficiency; narrower intervals are more efficient but must maintain adequate coverage.
• Root Mean Squared Error (RMSE):
  • Combines bias and variance.
  • \text{RMSE} = \sqrt{E\left( \bar{Q} - Q \right)^2}

(van Buuren, 2018)

Low Bias and High Coverage: Indicates randomization-valid methods.

Efficiency: Shorter confidence intervals (AW) are better if coverage (CR) is adequate.

Univariate vs. Multivariate Imputation

• Univariate Imputation:
  • Replaces missing values using only the variable itself (mean, median).
  • Limitation: Ignores relationships between variables; may distort distributions and underestimate variability.
• Multivariate Imputation (like MICE):
  • Uses other variables to estimate missing values.
  • Advantage: Preserves covariance structure and maintains data integrity.

Advantages of MICE

• Flexibility:
  • Compatible with various data types (numerical, categorical).
  • Allows for different imputation models per variable.
• Statistical Validity:
  • Accounts for uncertainty by creating multiple imputed datasets.
  • Provides unbiased estimates under the MAR assumption.
• Preserves Data Structure:
  • Maintains natural variability and multivariate relationships.
  • Reflects the true covariance among variables.
• Reduces Bias:
  • More accurate than univariate methods, especially when data is not MCAR.

Limitations of MICE

• Computationally Intensive:
  • Requires more processing time and resources, especially with large datasets.
• Complexity:
  • Implementation and tuning can be challenging; requires statistical expertise.
• Assumptions:
  • Relies on the data being Missing at Random (MAR).
  • Violations of MAR can lead to biased imputations.
• Model Dependency:
  • Quality depends on the correctness of specified models for each variable.
  • Mis-specified models can introduce errors.

When Not to Use Multiple Imputation

• Multiple Imputation is powerful but not always needed.

• Complete-Case Analysis (also known as Listwise Deletion):

  • What is it?
    • Using only the data entries (rows) that have no missing values in the variables you’re analyzing.
    • You exclude any rows that contain missing data and perform your analysis on the remaining complete cases.

When is Complete-Case Analysis Appropriate?

• If missingness occurs only in Y and the data are Missing Completely at Random (MCAR).
• Analyzing only the complete cases can be as effective as multiple imputation but is simpler.

Be Careful:

• Using only complete cases can lead to biased results when data is MAR or MNAR and these conditions aren’t met.
• Choosing this method should be a thoughtful decision, and you should clearly explain why you’re using it, considering how the missing data might affect your analysis.

(van Buuren, 2018)