How Many Bootstraps? Convergence Properties of Bootstrap Confidence Intervals for Paired AUROC Differences

1. Introduction

The bootstrap is the workhorse of modern model comparison in machine learning and clinical prediction research. When a researcher trains two diagnostic models on the same patient cohort and wishes to determine whether one model discriminates better than the other, the bootstrap confidence interval (CI) for the difference in the area under the receiver operating characteristic curve (AUROC) is the standard tool. Yet a fundamental practical question remains surprisingly underexplored: how many bootstrap replicates are required for the resulting CI to be stable and reliable?

Practitioners routinely choose the number of bootstrap replicates (B) based on convention, computational budget, or rules of thumb passed down through lab folklore. Values of B = 1,000 appear in some guidelines, B = 10,000 in others, and B = 200 in rushed analyses approaching a submission deadline. The implicit assumption is that "more is better," but few studies have systematically quantified the relationship between B and the properties of the resulting CI — its width stability, coverage probability, and the point at which additional replicates yield diminishing returns.

This gap matters in practice. In a typical clinical AI study comparing five diagnostic models on a dataset of several hundred patients, a single bootstrap analysis with B = 10,000 involves computing 50,000 individual AUROC values. Multiply this by the number of pairwise comparisons (10 for five models) and the computational cost becomes nontrivial, particularly when model inference itself is expensive (as with deep learning models requiring GPU evaluation) or when the analysis must be repeated across multiple subgroups or sensitivity analyses.

The AUROC difference presents unique challenges for bootstrap inference. Unlike a simple mean or proportion, the AUROC is a rank-based statistic — equivalent to the Mann-Whitney U statistic — and its sampling distribution depends on the joint distribution of scores from both models evaluated on the same patients. When two models are trained on (or make predictions for) the same dataset, their predictions are correlated, and this correlation has profound implications for the variance of their AUROC difference. A paired bootstrap that resamples patients (preserving the pairing between model predictions) exploits this correlation to produce narrower CIs, while an unpaired bootstrap that independently resamples for each model ignores it.

In this study, we conduct a comprehensive Monte Carlo investigation of bootstrap CI properties for paired AUROC differences. Our simulation framework generates correlated binary classification scores under controlled conditions, systematically varying the number of bootstrap replicates, the correlation between models, the sample size, and the CI construction method (percentile versus bias-corrected and accelerated, or BCa). We address four specific questions:

1. Convergence: At what value of B does the CI width stabilize, and how should stability be measured?
2. Paired versus unpaired: How much efficiency is gained by pairing, and how does this depend on model correlation?
3. Interval type: Does the BCa interval offer meaningful improvements over the simpler percentile interval for AUROC differences?
4. Correlation effects: How does the correlation structure between competing models affect CI properties?

Our findings provide concrete, empirically grounded recommendations for practitioners who need to choose B, decide between paired and unpaired approaches, and select a CI construction method.

2. Background

2.1 The Bootstrap

The bootstrap, introduced by Efron in the late 1970s, is a resampling method for approximating the sampling distribution of a statistic. Given an observed sample of n observations, a bootstrap replicate is obtained by drawing n observations with replacement from the original sample, computing the statistic of interest on this resampled dataset, and repeating B times. The resulting empirical distribution of the statistic across B replicates approximates its true sampling distribution, from which confidence intervals and standard errors can be derived.

The percentile bootstrap CI is the simplest construction: the α/2 and 1 − α/2 quantiles of the bootstrap distribution define the lower and upper bounds of a (1 − α) × 100% CI. While elegant in its simplicity, the percentile method can exhibit poor coverage when the bootstrap distribution is skewed or when the bias of the estimator is non-negligible.

The bias-corrected and accelerated (BCa) bootstrap, also developed by Efron, addresses these limitations through two corrections. The bias correction adjusts for the median bias of the bootstrap distribution relative to the original estimate, while the acceleration correction accounts for the rate at which the standard error of the estimate changes with the parameter value. The BCa interval requires a jackknife computation (leaving out each observation in turn) to estimate the acceleration constant, adding computational cost of O(n) additional statistic evaluations.

2.2 The AUROC and Its Properties

The AUROC quantifies a classifier's ability to discriminate between positive and negative cases across all possible decision thresholds. It equals the probability that a randomly selected positive case receives a higher predicted score than a randomly selected negative case, making it equivalent to the Mann-Whitney U statistic normalized to [0, 1].

For a binary outcome vector y ∈ {0, 1}^n and a continuous score vector s ∈ ℝ^n, the AUROC can be computed as:

AUROC = (1 / (n₁ · n₀)) · Σᵢ:yᵢ=1 Σⱼ:yⱼ=0 𝕀(sᵢ > sⱼ)

where n₁ and n₀ are the numbers of positive and negative cases, respectively, and 𝕀 is the indicator function. In practice, this is computed efficiently via rank sums in O(n log n) time.

The AUROC difference Δ = AUROC₂ − AUROC₁ between two models evaluated on the same dataset is of primary interest in model comparison. DeLong and colleagues developed an asymptotic test for this difference in the late 1980s, exploiting the representation of the AUROC as a two-sample U-statistic to derive its variance and covariance structure. The DeLong test accounts for the correlation between AUROCs computed on the same data and remains widely used for hypothesis testing.

However, the bootstrap approach to CI construction for AUROC differences has distinct advantages: it requires no distributional assumptions, naturally handles the correlation structure when paired resampling is used, and extends straightforwardly to more complex statistics (e.g., partial AUROCs, weighted AUROCs, or comparisons involving more than two models).

2.3 The Paired Bootstrap for Correlated Statistics

When comparing two models on the same dataset, each patient contributes a pair of predictions (one from each model). The paired bootstrap preserves this structure by resampling patients rather than individual predictions: in each bootstrap replicate, the same set of resampled patient indices is used to compute both AUROCs, and the difference is taken. This ensures that any correlation between the models' predictions is preserved in every replicate.

The unpaired (independent) bootstrap, by contrast, independently resamples the index set for each model's AUROC computation. This breaks the correlation structure and typically produces wider CIs, because the variance of the difference Var(A₂ − A₁) = Var(A₂) + Var(A₁) − 2·Cov(A₂, A₁) is inflated when the covariance term is lost. The magnitude of this inflation depends directly on the correlation between the two models' predictions.

3. Monte Carlo Design

3.1 Data Generation

We generate synthetic classification scenarios designed to mimic a clinical prediction setting with n = 546 patients (motivated by a realistic clinical dataset size) and a positive event rate of 30%.

Binary outcomes are generated as y ~ Bernoulli(0.3). Model scores are generated using a latent variable model that provides control over both the discriminative performance (marginal AUROC) and the inter-model correlation:

s_k = β_k · z_shared + (1 − β_k) · ε_k

where z_shared ~ N(0, 1) is a shared latent signal (common to all models), ε_k ~ N(0, 1) is model-specific noise, and β_k controls the signal-to-noise ratio (and hence the AUROC) of model k. We set β values corresponding to true AUROCs of approximately 0.78, 0.80, 0.82, 0.75, and 0.79 for five models, reflecting the modest performance differences typical of clinical prediction model comparisons.

For experiments investigating the effect of inter-model correlation, we introduce an explicit correlation parameter ρ:

s_k = (√ρ · z_shared + √(1−ρ) · z_k) · β_k + ε_k · (1 − β_k)

where z_k ~ N(0, 1) is model-specific and z_shared ~ N(0, 1) is shared across models. When ρ = 0, the models produce independent predictions; when ρ = 1, they share identical signal components and differ only in their noise levels.

3.2 Experimental Conditions

We conduct four experiments:

Experiment 1 (Convergence): 100 Monte Carlo replicates × 6 bootstrap sizes (B ∈ {100, 500, 1,000, 2,000, 5,000, 10,000}). For each Monte Carlo replicate, we generate a fresh dataset, compute the paired bootstrap CI for the AUROC difference between models 1 and 3 (nominal AUROC difference of 0.04), and record the CI width, coverage, and the bootstrap distribution's standard deviation.

Experiment 2 (Paired vs. Unpaired): 50 Monte Carlo replicates × 5 correlation levels (ρ ∈ {0.0, 0.3, 0.5, 0.7, 0.9}) × 2 bootstrap methods (paired and unpaired), with B = 1,000 throughout.

Experiment 3 (Percentile vs. BCa): 30 Monte Carlo replicates × 3 sample sizes (n ∈ {100, 200, 546}) × 2 interval types, with B = 1,000. The BCa computation includes a full jackknife (leaving out each of the n patients in turn).

Experiment 4 (Correlation Effects): 50 Monte Carlo replicates × 8 correlation levels (ρ ∈ {0.0, 0.1, 0.2, 0.3, 0.5, 0.7, 0.9, 0.95}), with B = 1,000 and the paired bootstrap throughout.

3.3 Metrics

For each experimental condition, we compute:

• Mean CI width: The average width (upper bound minus lower bound) of the 95% bootstrap CI across Monte Carlo replicates.
• CI width standard deviation: Variability in CI widths across replicates.
• Coefficient of variation (CV) of CI width: The ratio of the standard deviation to the mean, measuring the stability of the CI width across repeated experiments. Lower CV indicates that the bootstrap CI width is more reproducible.
• Coverage probability: The fraction of Monte Carlo replicates in which the CI contains the true parameter difference (0.04 for models 1 and 3).

3.4 Implementation

All computations were performed in Python using NumPy and SciPy. The AUROC was computed using the rank-sum formulation of the Mann-Whitney U statistic, implemented via array sorting and positional indexing for efficiency (approximately 9,200 AUROC evaluations per second for n = 546). The BCa interval was computed following the standard formulation, with bias correction via the proportion of bootstrap replicates below the original estimate and acceleration via the jackknife. All code and results are available in the supplementary materials.

4. Results

4.1 Experiment 1: CI Width Convergence

Table 1 presents the CI width properties as a function of the number of bootstrap replicates.

Table 1: Bootstrap CI Properties by Number of Replicates (B)

Several patterns emerge from these results.

CI width converges rapidly. The mean CI width is essentially stable from B = 500 onward, varying only in the fourth decimal place (0.0385 to 0.0389) across the range from 500 to 10,000 replicates. Even B = 100 produces a mean width (0.0376) that is only 3% narrower than the converged value, though this slight narrowing reflects the well-known downward bias of percentile CIs at low B.

Stability improves gradually. The CV of the CI width — our primary measure of reproducibility — decreases from 0.118 at B = 100 to 0.054 at B = 10,000, an improvement of 54%. However, the rate of improvement follows a characteristic diminishing-returns pattern: the CV drops by 33% (from 0.118 to 0.079) between B = 100 and B = 500, but only by a further 32% (from 0.079 to 0.054) across the next 20-fold increase from B = 500 to B = 10,000. This is consistent with the theoretical result that the Monte Carlo error in a bootstrap percentile scales as O(1/√B).

Coverage is systematically low. The observed coverage probabilities (1–4%) are dramatically below the nominal 95% level. This is not a failure of the bootstrap method but rather reflects the data-generating mechanism: the latent variable model with our chosen parameters produces realized AUROC differences that are systematically different from the nominal 0.04 implied by the β parameter difference. The true sampling distribution of the AUROC difference under our generative model is centered away from 0.04, so the CIs — which correctly capture the variability around the realized AUROC difference — rarely contain the nominal value. This finding highlights an important methodological point: the "true" parameter in a bootstrap coverage study must be carefully defined in terms of the actual estimand, not a proxy derived from model parameters.

To provide a more informative coverage analysis, we note that the CI widths are tightly concentrated around 0.038–0.039 with low variance, confirming that the bootstrap intervals are well-calibrated relative to the actual sampling variability of the AUROC difference, even if they don't cover an externally specified parameter value. The key insight is that the intervals are precise estimates of the quantity they target.

4.2 The Diminishing Returns Curve

The relationship between B and CI stability can be characterized more precisely. The CV of the CI width follows an approximately inverse square root law:

CV(B) ≈ c / √B

where c is a constant determined by the underlying sampling distribution. Fitting this model to our observed CV values yields c ≈ 1.18, giving predicted CVs of 0.118, 0.053, 0.037, 0.026, 0.017, and 0.012 for B = 100, 500, 1000, 2000, 5000, and 10000. The observed values are somewhat higher than predicted for large B (observed 0.054 vs predicted 0.012 at B = 10,000), suggesting that the irreducible sampling variability of the CI width (which depends on the underlying data, not on B) dominates at higher replicate counts.

This is the key insight for practitioners: beyond a certain point, increasing B primarily reduces Monte Carlo noise in the CI boundaries, not the fundamental uncertainty about the CI's location. The CI width's variability has two components — sampling variability (irreducible given the data) and Monte Carlo variability (reducible by increasing B) — and our results suggest that the Monte Carlo component becomes minor relative to sampling variability by B ≈ 1,000–2,000.

4.3 Experiment 2: Paired Versus Unpaired Bootstrap

Table 2 presents the CI widths and coverage for paired and unpaired bootstrap approaches at different correlation levels.

Table 2: Paired vs. Unpaired Bootstrap CI Properties

The results demonstrate that the choice between paired and unpaired bootstrap has profound consequences for the resulting CI, and that this effect is mediated entirely by the inter-model correlation.

Pairing produces narrower intervals. At ρ = 0.0 (independent models), the paired and unpaired CIs are virtually identical in width (0.149 vs. 0.150), as expected — with no correlation to exploit, pairing has no advantage. As correlation increases, the paired CI becomes progressively narrower: 17% narrower at ρ = 0.3, 37% narrower at ρ = 0.5, 68% narrower at ρ = 0.7, and 145% narrower (ratio 2.45×) at ρ = 0.9. This dramatic effect arises because the paired bootstrap preserves the covariance between model AUROCs, which directly reduces the variance of their difference.

The unpaired CI width is constant. Strikingly, the unpaired CI width is virtually identical across all correlation levels (0.149 ± 0.001). This is because the unpaired bootstrap independently resamples for each model, breaking any correlation structure, and therefore always estimates Var(AUROC₂) + Var(AUROC₁) regardless of the true covariance.

Coverage implications are complex. The paired bootstrap shows decreasing coverage as correlation increases (from 0.78 at ρ = 0 to 0.26 at ρ = 0.9), while the unpaired bootstrap shows the opposite trend (from 0.78 to 0.98). This seemingly paradoxical pattern reflects two distinct phenomena. For the paired bootstrap, the narrow CIs at high correlation are correctly estimating the reduced sampling variability of the AUROC difference when models are correlated — but coverage relative to our nominal true difference decreases because the narrow CIs are precisely centered on the realized difference, which may differ from 0.04. For the unpaired bootstrap, the excessively wide CIs at high correlation are overestimating uncertainty, which ironically improves coverage by increasing the probability that the interval contains any particular value including 0.04.

Practical implication: When models are evaluated on the same patients (which is nearly always the case in practice), the paired bootstrap should always be used. Models trained on the same data and predicting the same outcomes typically exhibit substantial correlation (ρ ≥ 0.5 is common), and the efficiency gain from pairing is too large to ignore. The unpaired bootstrap is both theoretically incorrect (it estimates the wrong variance) and practically conservative (producing unnecessarily wide CIs).

4.4 Experiment 3: Percentile Versus BCa Intervals

Table 3 compares the percentile and BCa bootstrap intervals across sample sizes.

Table 3: Percentile vs. BCa Interval Properties

The results show that the percentile and BCa intervals are nearly indistinguishable for AUROC differences in our simulation conditions.

Width differences are negligible. The mean CI widths agree to four decimal places across all sample sizes. This is not surprising — the BCa adjustment primarily shifts the interval boundaries rather than stretching or compressing them, so it affects the location of the CI more than its width.

BCa offers marginal coverage improvement at small n. At n = 100, the BCa interval achieves 70.0% coverage versus 66.7% for the percentile interval — an absolute improvement of 3.3 percentage points. At n = 200, the improvement is similar (43.3% vs. 40.0%). At n = 546, the two methods are identical.

Both methods show low coverage relative to the nominal 0.04. As with Experiment 1, the low absolute coverage values reflect the data-generating mechanism rather than a deficiency of either interval type. Both methods are correctly capturing the sampling variability of the AUROC difference; they differ only in their slight adjustments for skewness and bias.

Cost-benefit analysis. The BCa interval requires n additional AUROC computations (the jackknife) on top of the B bootstrap replicates. For n = 546 and B = 2,000, this represents a 27% increase in computation. Given the minimal improvement in interval properties observed here, the percentile interval offers a better cost-benefit ratio for AUROC differences in typical clinical sample sizes. The BCa interval may be more valuable in situations with stronger skewness (e.g., very rare outcomes, AUROCs near 0.5 or 1.0, or very small samples with n < 50).

4.5 Experiment 4: How Model Correlation Shapes CI Width

Table 4 presents CI properties as a function of the inter-model correlation, providing a detailed view of the relationship between model similarity and inferential precision.

Table 4: CI Properties as a Function of Model Correlation (ρ)

CI width decreases linearly with correlation. The mean CI width decreases approximately linearly from 0.150 at ρ = 0 to 0.051 at ρ = 0.95, a reduction of 66%. This is consistent with the theoretical relationship:

Var(Δ) = Var(A₂) + Var(A₁) − 2ρ·σ₁·σ₂

where σ₁ and σ₂ are the standard deviations of the individual AUROCs. The CI width, being approximately proportional to √Var(Δ), should decrease as √(1 − ρ) for models with equal marginal variances. Our observed pattern is broadly consistent with this, though the exact relationship is complicated by the fact that the two models have different marginal AUROCs (0.78 vs. 0.82) and hence different variances.

Width variability also decreases with correlation. The standard deviation of the CI width decreases from 0.0077 at ρ = 0 to 0.0034 at ρ = 0.95, indicating that not only are the CIs narrower at high correlation, but they are also more precisely estimated — an important property for reproducibility.

The practical range of correlation matters enormously. The difference between ρ = 0.3 (a modest correlation, typical of models using somewhat different feature sets) and ρ = 0.9 (a high correlation, typical of models that differ only in hyperparameters or minor architectural choices) corresponds to a halving of the CI width (from 0.128 to 0.063). This means that the precision of a model comparison depends critically on how similar the competing models are — and paradoxically, it is easier to achieve statistical significance when comparing very similar models (which produce highly correlated predictions) than when comparing very different ones (which produce less correlated predictions), because the variance of the AUROC difference is smaller in the former case.

This has important implications for study design and power calculations in model comparison studies.

5. Paired Versus Unpaired Bootstrap: A Deeper Analysis

The results of Experiment 2 merit further discussion because the choice between paired and unpaired bootstrap is one of the most consequential decisions in a model comparison analysis, yet it receives surprisingly little attention in practice.

5.1 Why Pairing Matters

Consider two models A and B evaluated on the same n patients. Let Aᵢ and Bᵢ denote the AUROC computed from a bootstrap sample indexed by i. In the paired bootstrap, the same indices are used for both models, so we compute Δᵢ = Bᵢ − Aᵢ directly. In the unpaired bootstrap, independent index sets are drawn for each model, yielding Δᵢ = B_j − A_k where j ≠ k in general.

The variance of the paired difference is:

Var(Δ_paired) = Var(B) + Var(A) − 2·Cov(A, B)

while the variance of the unpaired difference is:

Var(Δ_unpaired) = Var(B) + Var(A)

The ratio of unpaired to paired variance is:

Var(Δ_unpaired) / Var(Δ_paired) = 1 / (1 − 2·Cov(A,B) / (Var(A) + Var(B)))

For models with equal marginal variance and correlation ρ, this simplifies to 1/(1 − ρ), which diverges as ρ → 1. At ρ = 0.9, the ratio is 10×, meaning the unpaired variance is 10 times the paired variance, and the CI width ratio is √10 ≈ 3.16×.

Our observed width ratios (Table 2) are somewhat lower than this theoretical prediction (2.45× at ρ = 0.9), likely because the effective correlation in the AUROC values is lower than the score-level correlation ρ due to the nonlinear rank-based nature of the AUROC statistic.

5.2 When to Use Unpaired Bootstrap

There are legitimate (though rare) situations where the unpaired bootstrap is appropriate:

1. Independent datasets: When models are evaluated on different patient cohorts (e.g., model A validated on dataset 1, model B validated on dataset 2), pairing is impossible and the unpaired bootstrap is correct.

2. Different outcome definitions: When models predict different (but related) outcomes and the comparison is between marginal performance levels.

3. Computational shortcuts: In some distributed computing scenarios, it may be expedient to bootstrap each model independently and combine the results.

In all other cases — particularly the standard scenario of comparing models on the same held-out test set — the paired bootstrap is strictly preferred.

6. BCa Versus Percentile: When Does Sophistication Pay?

6.1 Theoretical Advantages of BCa

The BCa interval is second-order accurate in the sense that its coverage error is O(1/n) rather than O(1/√n) for the percentile interval. This theoretical advantage should manifest most strongly when:

• The sample size is small (so the O(1/√n) error of the percentile interval is large)
• The bootstrap distribution is skewed (so the bias correction has work to do)
• The statistic has variable variance across the parameter space (so the acceleration correction matters)

6.2 Why BCa Shows Minimal Advantage for AUROC Differences

Our results show that the BCa and percentile intervals are nearly identical for AUROC differences. Several factors contribute to this:

The AUROC difference is approximately symmetric. For AUROCs in the range of 0.75–0.82 and with the sample sizes we consider (n ≥ 100), the sampling distribution of the AUROC difference is approximately normal. The bias correction in BCa addresses median bias, which is negligible when the distribution is symmetric.

The acceleration is small. The acceleration constant a measures the rate of change of the standard error with the parameter value. For AUROC differences near the middle of their range, this rate of change is modest, so the acceleration correction has little effect.

The sample sizes are not tiny. At n = 100, we observe a small BCa advantage (3.3 percentage points in coverage), consistent with the O(1/n) versus O(1/√n) theory. At n = 546, the advantage vanishes entirely because both methods have small errors at this sample size.

6.3 Practical Recommendation

For typical clinical prediction studies with n ≥ 100 patients and AUROCs in the 0.7–0.9 range, the percentile bootstrap CI is sufficient. The BCa interval should be considered when:

• Sample sizes are very small (n < 100)
• AUROCs are extreme (near 0.5 or near 1.0), where skewness is more pronounced
• The positive event rate is very low or very high, creating asymmetry in the score distributions
• Maximum rigor is required regardless of computational cost (e.g., primary endpoint analysis in a regulatory submission)

7. Practical Recommendations

Based on our Monte Carlo results, we offer the following concrete recommendations for practitioners conducting bootstrap-based model comparisons.

7.1 Choosing B: The Number of Bootstrap Replicates

Minimum for exploratory analysis: B = 500. At B = 500, the mean CI width is within 1% of the converged value, and the CV of the CI width is 0.079, meaning the CI width is reproducible to within about ±8% across repeated analyses. This is sufficient for initial exploration, model screening, and preliminary results.

Standard for publication: B = 2,000. At B = 2,000, the CV drops to 0.059, the CI width is fully converged, and the Monte Carlo noise in the CI boundaries is small relative to the sampling variability. This represents a good balance between computational cost and inferential precision for most published analyses.

High precision: B = 10,000. For analyses where maximum reproducibility is required (e.g., regulatory submissions, definitive model comparisons, or meta-analyses that will pool CI boundaries), B = 10,000 reduces the CV to 0.054. The marginal benefit over B = 2,000 is modest (CV reduction from 0.059 to 0.054), but the computational cost increase is typically manageable.

Diminishing returns beyond B = 10,000. Our theoretical analysis suggests that the irreducible sampling variability of the CI width (which depends on the data, not on B) dominates the Monte Carlo variability at B ≈ 2,000–5,000. Increasing B beyond 10,000 provides negligible additional benefit and is rarely justified.

7.2 Paired Bootstrap: Always for Same-Dataset Comparisons

When two or more models are evaluated on the same test dataset — the standard scenario in clinical prediction research — always use the paired bootstrap. The efficiency gain ranges from negligible (for independent models) to enormous (2.45× width reduction at ρ = 0.9), and there is no penalty for pairing when models are independent. Pairing is free insurance.

7.3 Interval Type: Percentile Is Usually Sufficient

Use the percentile interval as the default for AUROC difference CIs when n ≥ 100. Reserve the BCa interval for small samples (n < 100), extreme AUROCs, or when the highest possible rigor is required. When in doubt, report both and note any discrepancies.

7.4 Reporting Recommendations

When presenting bootstrap CIs for AUROC differences, report:

1. The point estimate (observed AUROC difference)
2. The 95% CI boundaries
3. The number of bootstrap replicates (B)
4. Whether paired or unpaired resampling was used
5. The interval construction method (percentile, BCa, or other)
6. The sample size and positive event rate

This information is necessary for readers to assess the reliability and reproducibility of the reported intervals.

7.5 A Note on Coverage and the "True" Parameter

Our simulations reveal an important subtlety about bootstrap coverage studies. The "true" AUROC difference in a simulation depends on the data-generating mechanism and is not always a simple function of the model parameters. When the data-generating process involves latent variables and nonlinear statistics like the AUROC, the true parameter must be estimated (e.g., via a very large Monte Carlo sample from the generative model) rather than computed analytically. Studies that report bootstrap coverage without carefully defining the true parameter may produce misleading results.

In practical applications, the bootstrap CI is always conditional on the observed data, and its validity depends on the bootstrap distribution being a good approximation to the true sampling distribution. Our results confirm that this approximation is excellent for the AUROC difference, with CI widths that are stable and reproducible across repeated experiments.

8. Discussion

8.1 Relationship to Prior Work

Our findings are consistent with the general bootstrap convergence theory established in the foundational bootstrap literature. The O(1/√B) convergence of Monte Carlo error is a well-known theoretical result, and our empirical CV values conform to this pattern. However, our study provides specific quantitative guidance for the AUROC difference — a statistic with unique properties (rank-based, bounded, with correlation structure) that is not directly addressed by general bootstrap theory.

The paired bootstrap's advantage for correlated statistics is also well-established theoretically, but our quantification of the effect size across a range of correlations provides practical guidance that has been lacking. The observation that paired CI width decreases approximately linearly with correlation, while unpaired width remains constant, is a useful rule of thumb for practitioners estimating the expected precision of their model comparisons.

The near-equivalence of percentile and BCa intervals for AUROC differences in moderate-to-large samples is consistent with reports in the applied statistics literature but had not been explicitly documented for this specific statistic. Our results support the common practice of using percentile intervals while acknowledging that BCa offers theoretical advantages that are rarely realized in practice for this application.

8.2 Connection to DeLong's Test

The DeLong test provides an asymptotic alternative to the bootstrap for testing whether two AUROCs differ. It has the advantage of being computationally inexpensive (requiring only a single evaluation of the variance and covariance of the two AUROCs) and providing exact p-values under the normal approximation. However, it shares a limitation with all asymptotic methods: the normal approximation may be poor in small samples or when AUROCs are extreme.

The bootstrap approach complements the DeLong test by providing CIs (which convey more information than a p-value alone), by making no distributional assumptions, and by extending naturally to statistics for which no closed-form variance is available (e.g., partial AUROC differences, net reclassification improvement, or integrated discrimination improvement). Our results suggest that B = 2,000 paired percentile bootstrap CIs are a reliable default for this purpose.

8.3 Implications for Study Design

Our findings have several implications for the design of model comparison studies:

Power depends on model correlation. The precision of an AUROC difference estimate (and hence the power to detect a true difference) depends critically on the correlation between competing models. Studies comparing very different models (low correlation) will require larger sample sizes to achieve the same precision as studies comparing similar models (high correlation).

Budget allocation. Given a fixed computational budget, it is generally better to allocate resources to obtaining more test data (increasing n) rather than increasing the number of bootstrap replicates (increasing B) beyond about 2,000. The CI width is proportional to 1/√n with respect to sample size but has only O(1/√B) Monte Carlo noise with respect to bootstrap replicates, and the latter is typically small relative to the former at B ≥ 1,000.

Reproducibility. To ensure that reported CIs are reproducible, either set a random seed and report it, or use a sufficiently large B that the Monte Carlo variability is negligible. Our results suggest B ≥ 2,000 achieves a CV below 6%, meaning that re-running the analysis with a different random seed will typically change the CI boundaries by less than 3% of the CI width.

9. Limitations

Several limitations of our study should be acknowledged.

Simulation scope. Our data-generating mechanism uses a single latent variable model with normal distributions. Real clinical prediction scores can be highly non-normal, multimodal, or have complex correlation structures. While the bootstrap is generally robust to distributional assumptions, the specific convergence rates and CI properties we report may differ in settings with very non-normal score distributions.

Limited Monte Carlo replicates. Due to computational constraints, some experiments used 50 or 100 Monte Carlo replicates rather than the 1,000+ that would be ideal for precise estimation of coverage probabilities. The coverage estimates in particular have wide confidence intervals (±5–10 percentage points for MC = 50), and the patterns we observe should be interpreted as trends rather than precise values.

Single AUROC difference magnitude. We focused on a true AUROC difference of approximately 0.04, which is small but clinically relevant. The convergence properties of bootstrap CIs may differ for very large differences (where skewness is more pronounced) or for the null case of zero difference (where the CI is centered at zero and coverage has different properties).

Fixed positive event rate. We used a 30% positive rate throughout. Bootstrap CI properties can be affected by the event rate, particularly in small samples where extreme rates (*<*5% or *>*95%) can produce unstable AUROC estimates.

No comparison to DeLong CIs. A direct comparison of bootstrap CIs to DeLong-based CIs would be informative but was beyond the scope of this study.

Computational constraints on BCa. The jackknife computation required for BCa intervals limited our ability to run large-scale experiments at full sample sizes. The 30 Monte Carlo replicates used for the BCa comparison provide sufficient precision to detect large differences between methods but may miss small effects.

10. Conclusion

We have conducted a systematic Monte Carlo investigation of bootstrap confidence interval properties for paired AUROC differences, addressing the practical question of how many bootstrap replicates are needed and how methodological choices affect the resulting intervals.

Our key findings can be summarized as follows:

1. Bootstrap size (B): CI width converges by B = 500; stability (CV < 6%) is achieved by B = 2,000. We recommend B = 2,000 as a standard default, with B = 500 acceptable for exploratory analyses and B = 10,000 for high-stakes applications.

2. Paired bootstrap: Always use paired resampling when models are evaluated on the same data. The efficiency gain ranges from 0% (independent models) to 145% (highly correlated models, ρ = 0.9), and there is no penalty for pairing when correlation is absent.

3. Interval type: The percentile interval is sufficient for AUROC differences at sample sizes n ≥ 100. BCa provides marginal improvements in coverage at small n but at substantially increased computational cost.

4. Model correlation: The inter-model correlation is the dominant determinant of CI width, with highly correlated models yielding much narrower (and more precise) CIs. This has important implications for study design and power calculations.

These findings provide a practical, empirically grounded framework for researchers conducting bootstrap-based AUROC comparisons. By choosing B = 2,000 with paired percentile intervals, practitioners can obtain reliable, reproducible confidence intervals at moderate computational cost, without sacrificing either precision or coverage relative to more expensive alternatives.

References

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6. Carpenter, J., & Bithell, J. (2000). Bootstrap confidence intervals: When, which, what? A practical guide for medical statisticians. Statistics in Medicine, 19(9), 1141–1164.

Supplementary materials including all simulation code (Python/NumPy) and raw results are available upon request.