Imputation Uncertainty in Interpretable Machine Learning Methods

AI summary

Overview

Research area: Interpretable machine learning (IML) applied to incomplete data — specifically how missing-value imputation affects the uncertainty of model explanations.

Technical level: Intermediate. The paper assumes familiarity with IML concepts (PFI, PD plots, Shapley values), missing-data mechanisms (MCAR, MAR, MNAR), and standard imputation terminology (single vs. multiple imputation, Rubin's rules).

Scope: The paper compares single and multiple imputation methods according to how well the confidence intervals of permutation feature importance, partial dependence plots, and global SHAP feature importance achieve nominal coverage, using simulation studies plus a real wine-quality dataset.

What This Paper Is About

When datasets contain missing values, analysts must fill them in before training and interpreting most machine learning models. Prior work has shown that the choice of imputation method changes what IML explanations look like, but has focused on bias while ignoring the extra uncertainty that imputation itself introduces. This paper asks whether the confidence intervals built around IML explanations are trustworthy — that is, whether they actually cover the true explanation at the stated rate — and shows that single imputation systematically produces intervals that are too narrow, whereas multiple imputation brings coverage close to the nominal level.

Key Contributions

1. Extends learner uncertainty to include imputation uncertainty. The authors build on the learner-Ψ framework of Molnar et al., which decomposes IML estimation error into bias and learner variance, by adding imputation uncertainty as an additional error source in the model and explanation estimates (illustrated in their Figure 1).

2. First systematic comparison of CI coverage probabilities for IML methods under missing data. The authors state they are not aware of any prior study comparing confidence interval coverage probabilities for IML methods with missing data; earlier work examined bias and explanation differences only.

3. Simulation study across two data-generating processes, three missingness mechanisms, three missingness proportions, four imputation methods, two learners, and two resampling strategies. The design covers linear and non-linear DGPs, MCAR/MAR/MNAR, missingness of 0.1, 0.2 and 0.4, mean imputation, MissForest, MICE PMM and MICE RF, linear model and XGBoost, and bootstrap and subsampling.

4. A real-data demonstration on the UCI wine dataset. Simulating 40% MCAR missingness under mean imputation versus MICE PMM shows how PFI, PD and SHAP interpretations — and their confidence intervals — change.

Main Findings

• Single imputation underestimates variance. Mean imputation and MissForest ignore imputation uncertainty; mean imputation in particular leads to extremely poor coverage and even reduces confidence interval width compared to the complete dataset, because missing values are replaced by the variable mean.

• Multiple imputation is close to nominal coverage in most cases. MICE-based approaches provide drastically improved variance estimation; with a nominal rate of 0.95, only multiple imputation generally approaches the nominal level. MICE PMM performs best for the linear DGP and MICE RF for the non-linear DGP.

• MICE often exceeds the coverage of the complete-data model. The authors explain this by MICE overestimating variance while bootstrap underestimates it, so the two effects cancel out. They caution that in practice these effects might not cancel and could lead to over- or underestimation.

• Unadjusted variance estimation gives poor coverage. Consistent with Molnar et al. for complete data, the naive variance estimator without the Nadeau and Bengio correction performed poorly.

• Coverage generally decreases with the missingness rate. Coverage of MICE methods remains relatively stable across missingness proportions — particularly MICE PMM for the linear DGP and MICE RF for the non-linear DGP — while MissForest is close to complete-data coverage at 0.1 missingness and mean imputation is considerably lower (except for PD).

• Missingness mechanism matters least, MNAR slightly worse. Plots across MCAR, MAR and MNAR were very similar, though coverage could get poorer under MNAR.

• Mean imputation shows the largest positive bias for PFI and SHAP in the linear DGP, growing strongly with the missingness rate; MissForest shows an increasing negative bias; MICE RF stays very close to zero, with MICE PMM also close to zero but with a negative tendency. PD bias was very small across all imputation methods.

• Non-linear DGP biases are generally smaller when accounting for the different scale. For PFI there, MissForest and MICE RF had the smallest bias; for SHAP, MICE RF had the smallest bias with MICE PMM and MissForest second.

• CI width grows with missingness for all methods except mean imputation for PFI, where it decreases. MICE methods generally produced the widest intervals because they account for imputation uncertainty; MissForest was closest to ground truth at 0.1 missingness and had the smallest increase except for PFI.

• Bootstrapping and subsampling behave similarly, with subsampling generally giving equal or lower coverage and slightly smaller width.

• With the linear model learner, results for SHAP and PFI were similar to XGBoost, though coverage decreased more for MICE RF and mean imputation with missingness rate in the linear DGP, and non-linear DGP coverage was slightly closer to nominal than XGBoost. For PD, MissForest and mean imputation gave poorer coverage, particularly for the linear DGP.

• Real wine data: mean squared error averaged over refits was 0.41 with complete data, 0.48 with mean imputation and 0.42 with MICE — performance was barely affected, but interpretation was. Mean imputation drastically underestimated the importance of alcohol and, less extremely, sulphates; confidence intervals were wider with multiple imputation and narrower with mean imputation. The MICE PD curve stayed close to the complete-data PD, while mean imputation deviated most at very low or very high alcohol values.

• Proposed explanation for MissForest's overestimation: unlike the other methods, MissForest includes the target y in the imputation, which the authors suggest might cause it to overfit to important features.

Methodology in Plain English

The authors define a ground-truth explanation Ψ_f (what an IML method would report if applied to the true data-generating function) and treat the explanation calculated on a trained model, Ψ_f̂, as an estimator of it. Estimation error splits into bias and learner variance. Learner variance is estimated by averaging explanations over k different model fits and applying a variance formula with a correction term c; the paper uses the Nadeau and Bengio correction c = n_test/n_train because resampled training and test sets are not independent.

Simulations proceed in stages: sample n = 1000 points from either a linear DGP (Y = X₁ − X₂ + ε) or a non-linear DGP (Y = X₁ − √(1 − X₂) + X₃X₄ + (X₄/10)² + ε) with ε ~ N(0,1) and features drawn from a multivariate Gaussian with a Toeplitz covariance Σ_ij = 0.5^|i−j|; introduce missingness under MCAR, MAR or MNAR at proportions 0.1, 0.2 and 0.4; impute with mean imputation, MissForest, MICE PMM or MICE RF; apply bootstrap or subsampling with 20 resampling iterations on each imputed dataset; fit a linear model or XGBoost (maximum 20 boosting iterations, maximum tree depth 2); compute PD, PFI and SHAP on test data — TreeSHAP for XGBoost and exact calculation for linear regression — and pool results with Rubin's rules. For multiple imputation, the number of imputations increases with missingness: 10, 20 and 40 for proportions 0.1, 0.2 and 0.4, following White et al. Each setting is repeated 1000 times, coverage is assessed against a nominal rate of 0.95, and ground truth is computed by averaging over 10,000 replications without missing values. CI width and bias are examined at 15 refits, as recommended in the literature. The real-data analysis uses the UCI wine dataset (1599 observations, 12 numerical variables, predicting red vinho verde quality on a 0–10 scale) with XGBoost, 40% simulated MCAR missingness, mean versus MICE PMM imputation with 40 imputations, and adjusted bootstrapping over 15 refits.

Why This Matters

Impact on research: The paper shows that reporting IML explanations from single-imputed data without accounting for imputation uncertainty yields overconfident conclusions — confidence intervals that look precise but fail to cover the truth. It provides a concrete methodological extension (learner-Ψ plus imputation uncertainty) and releases code at https://github.com/bips-hb/iml_imputation_paper for reproduction.

Real-world applications:

• Healthcare and epidemiology, where missing patient data is common and where the authors' own institution (Leibniz Institute for Prevention Research & Epidemiology – BIPS) works.
• Scientific studies using survey or registry data with item non-response, where feature importance rankings inform policy or further research.
• Any tabular ML deployment where stakeholders are shown feature importance, PD curves or SHAP summaries derived from incomplete records.
• Model auditing and regulatory review, where overstated confidence in explanations could mislead decision-makers.

Industry relevance: Practitioners routinely pick a convenient single imputation method such as mean imputation or MissForest because it produces one clean dataset. This work quantifies the interpretive cost of that convenience and supports the recommendation to use multiple imputation when explanations, not just predictions, matter.

Future Directions

• Investigating settings where the variance-overestimation of MICE and the variance-underestimation of bootstrap do not cancel, which the authors note could lead to over- or underestimation of variance in practice.
• Addressing the target-inclusion problem in MissForest, which the authors suggest may explain its overestimated importance, and reconciling this with the need to include the target for valid multiple imputation inference.
• Extending the evaluation of imputation uncertainty beyond the global methods studied here (PD, PFI, global SHAP) — for example to local explanations or other IML techniques.
• Performance under MNAR, where the authors observed considerably worse performance of all methods and which remains the hardest mechanism to handle.

Target Audience

Researchers and practitioners working at the intersection of missing data and interpretable machine learning: statisticians and methodologists studying uncertainty quantification for XAI, applied data scientists in epidemiology, healthcare and the social sciences who interpret models trained on incomplete tabular data, and IML developers deciding how to report confidence intervals for explanations. Readers should already be comfortable with concepts like permutation importance, partial dependence, Shapley values, MCAR/MAR/MNAR, and Rubin's rules.

Authors’ abstract

In real data, missing values occur frequently, which affects the interpretation with interpretable machine learning (IML) methods. Recent work considers bias and shows that model explanations may differ between imputation methods, while ignoring additional imputation uncertainty and its influence on variance and confidence intervals. We therefore compare the effects of different imputation methods on the confidence interval coverage probabilities of the IML methods permutation feature importance, partial dependence plots and Shapley values. We show that single imputation leads to underestimation of variance and that, in most cases, only multiple imputation is close to nominal coverage.