Convergence Diagnostics

After running MICE, it’s crucial to check whether the algorithm has converged. This guide explains what convergence means and how to assess it.

What is Convergence?

Convergence means the MICE algorithm has stabilized—the imputed values are no longer systematically changing from one iteration to the next.

Why It Matters

If MICE hasn’t converged:

  • Imputed values may be unreliable

  • Statistical inferences may be biased

  • More iterations are needed

How MICE Converges

During each iteration, MICE updates the imputed values for each variable. Initially, these updates cause substantial changes, but as the algorithm proceeds, changes should become smaller and stabilize.

Chain Statistics

MICE tracks two key statistics across iterations for each variable:

Chain Mean

The mean of the imputed values at each iteration

Chain Variance

The variance of the imputed values at each iteration

These “chains” should:

  1. Start from initial values

  2. Potentially drift in early iterations

  3. Stabilize after some iterations (convergence!)

Visualizing Convergence

The primary tool for checking convergence is plotting the chain statistics:

from plotting.diagnostics import plot_chain_stats

# After running MICE
mice = MICE(df)
mice.impute(n_imputations=5, maxit=20)

# Plot convergence
plot_chain_stats(
    chain_mean=mice.chain_mean,
    chain_var=mice.chain_var,
    save_path='convergence.png'
)

Interpreting the Plots

What to look for:

✓ Stable horizontal lines: Means and variances have stabilized ✓ No systematic trends: Values aren’t consistently increasing or decreasing ✓ Mixing of chains: If multiple imputations are shown, they should overlap

✗ Trending lines: Values still changing systematically ✗ Unstable oscillations: Large swings even in later iterations ✗ Separated chains: Different imputations have very different patterns

Numerical Assessment

While visual inspection is primary, you can also check numerically:

import numpy as np

# Get chain means for a specific variable
var_name = 'income'
chain = mice.chain_mean[var_name]

# Check if last few iterations are stable
last_5 = chain[-5:]
variation = np.std(last_5) / np.mean(last_5)  # Coefficient of variation

if variation < 0.01:  # Less than 1% variation
    print(f"{var_name}: Converged")
else:
    print(f"{var_name}: May need more iterations")

What to Do If Not Converged

Increase Iterations

The simplest solution:

# Try more iterations
mice.impute(n_imputations=5, maxit=50)  # Increased from 10 to 50

# Check again
plot_chain_stats(mice.chain_mean, mice.chain_var)

Most convergence issues are resolved by running more iterations.

Adjust Initial Values

Try different initial imputation:

# Use mean instead of sample for initialization
mice.impute(n_imputations=5, maxit=20, initial='mean')

Simplify Predictor Matrix

Too many predictors or multicollinearity can slow convergence:

from imputation.utils import quickpred

# Use automatic selection with higher threshold
predictor_matrix = quickpred(df, mincor=0.3)
mice.impute(predictor_matrix=predictor_matrix, maxit=20)

Change Method

Some methods converge faster than others:

# Try a different method
mice.impute(method='cart', maxit=20)  # Instead of PMM

How Many Iterations?

Default: 10 iterations

Sufficient for many datasets

Recommendation: 15-20 iterations

Safer choice, check convergence diagnostics

Complex data: 30-50+ iterations

  • High missingness (>30%)

  • Many variables

  • Complex relationships

Rule of thumb: Run until chains are flat for at least 5 iterations

Convergence by Variable

Different variables may converge at different rates:

# Check each variable separately
for var in mice.chain_mean.keys():
    chain = mice.chain_mean[var]
    plt.figure()
    plt.plot(chain)
    plt.title(f'Convergence: {var}')
    plt.xlabel('Iteration')
    plt.ylabel('Mean')
    plt.savefig(f'convergence_{var}.png')
    plt.close()

Variables with more missingness or weaker predictive relationships typically need more iterations.

Other Diagnostic Checks

Compare Observed vs Imputed

Even if chains converge, check that imputed values are reasonable:

from plotting.diagnostics import stripplot, densityplot

missing_pattern = df.notna().astype(int)

# Stripplot: visual check
stripplot(mice.imputed_datasets, missing_pattern)

# Density plot: distributional check
densityplot(mice.imputed_datasets, missing_pattern)
Look for:

  • Imputed values (red) within range of observed values (blue)

  • Similar distributions between observed and imputed

  • No impossible values

Check Variability Between Imputations

Multiple imputations should differ from each other:

# For a specific variable
var = 'income'
imputed_values = [dataset[var] for dataset in mice.imputed_datasets]

# Check standard deviation across imputations
sd_across = np.std(imputed_values, axis=0)

print(f"Mean SD across imputations: {sd_across.mean()}")

If imputations are nearly identical, you may need more iterations or a less deterministic method.

Common Convergence Issues

Slow Convergence

Symptoms: Chains still changing after many iterations

Causes:

  • High dimensionality

  • Weak predictor relationships

  • High missingness

  • Multicollinearity

Solutions:

  • Use quickpred to select predictors

  • Increase ridge parameter in PMM

  • Try different method (CART/RF)

  • More iterations

Non-Convergence

Symptoms: Chains never stabilize, even after 50+ iterations

Causes:

  • Perfect multicollinearity

  • Circular dependencies

  • Insufficient data

  • Model misspecification

Solutions:

  • Check for perfectly correlated variables

  • Remove redundant predictors

  • Simplify predictor matrix

  • Consider different imputation strategy

Oscillating Chains

Symptoms: Chains oscillate rather than stabilize

Causes:

  • Conflicting information from different predictors

  • Overfitting with complex methods

Solutions:

  • Use simpler method (PMM instead of RF)

  • Regularize more strongly

  • Reduce predictor complexity

Separated Chains

Symptoms: Different imputation chains don’t mix

Causes:

  • Insufficient iterations

  • Bimodal or complex distributions

  • Categorical variables with many levels

Solutions:

  • More iterations

  • Check if true multimodality exists

  • Use method appropriate for data type

Best Practices

  1. Always check convergence: Never skip this step

  2. Visual inspection first: Plots are more informative than statistics

  3. Be conservative: If unsure, run more iterations

  4. Check all variables: Don’t just look at your outcome variable

  5. Look at early iterations: They can reveal problems with initialization

  6. Compare multiple runs: Rerun with different seeds to check stability

Quick Convergence Checklist

Before finalizing your imputation:

☐ Chain plots show stable horizontal lines for all variables ☐ No systematic trends in the last 5-10 iterations ☐ Imputed values are in reasonable range ☐ Distributions of observed and imputed values are similar ☐ Multiple imputations show appropriate variability ☐ Convergence achieved with acceptable number of iterations (<50)

If all checked, your imputation is ready for analysis!

Next Steps