MICE: Multiple Imputation by Chained Equations

• MICE is a statistical method that imputes missing data by iteratively modeling each variable using the observed values of others, and is widely applied in social sciences, epidemiology, and machine learning.
• It employs chained equations of univariate conditional models (typically main-effect GLMs), but may suffer from model misspecification when handling high-cardinality or complex interaction effects.
• Empirical studies show that alternative approaches like MI-CART and MI-DPM reduce bias and improve confidence interval coverage, offering robust options over default MI‐GLM in challenging settings.

Multiple Imputation by Chained Equations (MICE) is a widely adopted framework for handling missing data in multivariate settings, particularly when the variables are of mixed types (continuous, categorical, ordinal). MICE iteratively constructs conditional univariate models for each variable with missing values, drawing imputations sequentially in a process akin to a Gibbs sampler until convergence. The method’s flexibility, simplicity, and software support have contributed to its broad use in domains such as social sciences, epidemiology, and machine learning. However, simulation studies and theoretical research have highlighted substantial limitations of default MICE implementations, especially when applied to categorical or high-dimensional data, motivating the development and empirical evaluation of alternative engines such as classification and regression trees (CART) and fully Bayesian mixture models.

1. Methodological Framework of MICE

The core principle of MICE is the construction of a system of univariate conditional models, cycling through the set of variables with missing data. For each variable $Y_{(j)}$ in a dataset with $p$ variables, the conditional model is specified as

$$P(Y_{(j)} = y \mid Y_{(1)}, \dots, Y_{(j-1)}, Y_{(j+1)}, \dots, Y_{(p)})$$

with available predictors typically restricted to main effects. For categorical variables, the default choices in the MI‐GLM implementation (standard MICE) are:

• Logistic regression for binary outcomes,
• Multinomial logistic regression for unordered categorical outcomes,
• Cumulative logistic regression for ordered categorical responses.

The algorithm cycles through variables, imputing missing values using conditional draws, commonly starting from random initial fills. Multiple (typically $L$) completed datasets are produced. For each parameter of interest,

$$\bar{q} = \frac{1}{L} \sum_{l=1}^L q_l,\quad T = \left(1 + \frac{1}{L}\right) b + \bar{u},$$

with $b$ and $\bar{u}$ representing between-imputation and within-imputation variances, respectively: $$b = \frac{1}{L-1}\sum_{l=1}^L \left(q_l - \bar{q}\right)^2, \quad \bar{u} = \frac{1}{L} \sum_{l=1}^L u_l.$$ Final inference is performed using Rubin’s Rules based on pooled point estimates and $t$-based confidence intervals.

2. Theoretical Guarantees and Limitations

The fully conditional specification (FCS) underlying MICE offers modeling flexibility but generates a collection of univariate conditionals that are not always compatible—that is, not necessarily the set of conditionals from any joint probability distribution (Murray, 2018). Incompatibility can lead to ambiguous convergence behavior or order-dependence of stationary distributions. Sufficient conditions for proper convergence involve compatibility and regularity in Bayesian univariate regression models, but practical implementations may often violate these assumptions, especially in complex or high-dimensional applications. Diagnosing convergence (e.g., via trace plots) and performing multiple runs with variable ordering are recommended for practitioners (Murray, 2018).

Moreover, MICE's default MI‐GLM implementation specifies only main effects in the conditional models, which can result in model misspecification and inability to capture complex dependencies, particularly for categorical variables with multiple levels or in the presence of interactions (Akande et al., 2015). Such misspecification translates into increased bias, sub-nominal coverage, and potential instability (e.g., singularity or non-convergence in multinomial logistic regression).

3. Empirical Performance: MI-GLM vs. MI-CART and MI-DPM

Simulation studies based on high-dimensional categorical data (e.g., sourced from the American Community Survey) systematically compared the default MICE (MI‐GLM) with tree-based chained equations (MI‐CART) and a fully Bayesian Dirichlet process mixture (MI‐DPM) (Akande et al., 2015):

Method	Model Type	Key Strengths	Noted Weaknesses
MI‐GLM	Chained main effect GLMs (MICE)	Simplicity; widespread availability	High bias, poor coverage, unstable for high-cardinality variables
MI‐CART	Chained classification trees	Captures nonlinearities/interactions	Slightly low coverage in some cases
MI‐DPM	Bayesian Dirichlet mixture model	Robust coverage, especially in small/sparse samples	Longer lower tail for coverage distribution

• MI‐GLM: Consistently showed higher bias, higher relative mean squared error (Rel.MSE), and frequent undercoverage. Models often failed or produced unstable imputations for nominal predictors with many categories.
• MI‐CART: Yielded lower Rel.MSE and produced nearly nominal coverage, leveraging its ability to capture nonlinearity and complex interactions automatically.
• MI‐DPM: Produced robust coverage, frequently exceeding nominal rates (overcoverage), particularly advantageous in small sample or high-missingness contexts due to its shrinkage from the latent class structure.
• Both MI‐CART and MI‐DPM dominated default MI‐GLM, with performance differences primarily reflecting differences in bias.

This pattern has been corroborated in multiple empirical applications and simulations, where MI‐CART also outperformed deep learning and generative models in terms of bias and preservation of joint distributions, particularly in large survey applications (Wang et al., 2021).

4. Practical Implications for Categorical Data

The empirical findings articulate clear limitations of default MICE (MI‐GLM) for high-dimensional categorical data (Akande et al., 2015):

• Instability in multinomial logistic regression models when categories are numerous,
• Inability to automatically detect or model higher order interactions,
• Sub-nominal or inconsistent coverage and increased bias.

Practitioners are advised to either enrich the MICE model (e.g., by explicitly coding interaction terms, a process that is nontrivial and labor-intensive in large data sets) or to use alternative engines:

• MI‐CART: Particularly suitable when accurate point estimation and low bias are priorities; requires minimal model tuning.
• MI‐DPM: Recommended for situations prioritizing robust interval coverage, small or sparse samples, or when global distributional coverage is paramount.

The choice of method should be guided by the researcher's analysis priorities and the data structure: MI‐CART for accurate point estimates, MI‐DPM for robust global coverage.

5. Trade-offs, Limitations, and Model Tuning

Default MICE, implemented via chained equations of main-effect GLMs, is attractive for its accessibility, interpretability, and software support. However, its practical operation is highly sensitive to model specification (Akande et al., 2015), and sophisticated tuning (e.g., inclusion of interactions) is rarely feasible at scale. While tree-based and Bayesian mixture alternatives increase computational demands, their automation and robustness often justify their selection in contemporary applications (large-scale surveys, complex dependency structures, high-dimensional categorical data).

Performance differences are mainly driven by bias, not variance. In challenging scenarios (high missingness, sparse tables, nominal data with many categories), tree-based chained equations such as MI‐CART and joint mixture models (MI‐DPM) represent more robust choices.

6. Application and Interpretation of MICE Results

Analysts employing MICE for categorical data should:

• Avoid reliance on default MI‐GLM for data sets with complex dependency structure or high-cardinality categorical predictors,
• Consider MI‐CART or MI‐DPM as default engines, especially for moderate or high missingness and smaller samples,
• Ensure alignment between the analysis and imputation model (congeniality) to avoid bias in parameter estimation and inference (Murray, 2018),
• Closely inspect diagnostics including convergence and the realized coverage of confidence intervals.

Improved performance in terms of bias, efficiency, and confidence interval accuracy has practical import in applied settings, especially in preserving the associations and correlation structure of substantive interest (e.g., survey data relationships among predictors).

7. Summary and Forward Directions

Default MICE, via chained equations of main-effect GLMs (MI-GLM), is generally outperformed by tree‐based or Bayesian mixture alternatives for the imputation of categorical data, with MI‐CART dominating in terms of point estimate accuracy and MI‐DPM in terms of robust coverage (Akande et al., 2015). The critical determinant of performance is bias introduced by misspecification or failure to model interactions and high-cardinality structures. Both alternative approaches—MI‐CART and MI‐DPM—remain competitive and suitable as default engines depending on the application context and analytical priority.

For future development, integration of flexible nonparametric models (tree- or mixture-based) within the chained equations framework, strategies for automatic model selection (including interaction discovery), and improved diagnostics for model fit and convergence represent key areas for methodological refinement. Robust imputation in complex categorical data must balance computational scalability and validity of inferences, with empirical evidence guiding the default choices toward more flexible, modern engines within the MICE paradigm.