Variance-calibrated cross-individual bootstrapping for small-sample neuroscience

Abstract

Small sample sizes are a pervasive limitation in experimental neuroscience and related biological fields, frequently compromising statistical power and the reliability of inference. Conventional resampling techniques, such as stratified or hierarchical bootstrapping, often fail to balance coverage accuracy with interval precision, particularly when data are skewed or violate strict exchangeability assumptions. Here, we introduce Variance-Calibrated Cross-Individual Bootstrapping (CIB-VC), a robust resampling framework designed for datasets characterized by few subjects but distinct within-subject trial structures. The method constructs synthetic individuals by recombining trials across subjects to maximize informational use, coupled with a variance calibration step that restores empirical between-subject variability. Using extensive Monte Carlo simulations across Gaussian, heavy-tailed, and lognormal distributions, we demonstrate that CIB-VC consistently achieves near-nominal 95% coverage. In contrast, stratified bootstrapping exhibits severe under-coverage under distributional skew (coverage ), while hierarchical bootstrapping yields overly conservative confidence intervals. Crucially, empirical validation on behavioral tracking data from weakly electric fish () confirmed the practical utility of the method: CIB-VC produced confidence intervals approximately 23% narrower than standard hierarchical bootstrapping without introducing bias. By providing a statistically principled solution that preserves Type-I error control without sacrificing power, CIB-VC offers a practical tool for improving reproducibility in studies where increasing the number of subjects is ethically or logistically constrained.

Introduction

Reliable statistical inference in the biological and behavioral sciences is frequently constrained by the difficulty of obtaining large sample sizes. In fields ranging from experimental neuroscience and psychophysics to biomechanics, ethical considerations, logistical costs, and the complexity of data acquisition often limit studies to a small number of independent subjects (N). Yet, these same studies often generate rich datasets within individuals, characterized by large numbers of repeated trials or observations (M). The resulting data structure is hierarchical, creating a fundamental tension in analysis: while trial-level data is abundant, population-level inference remains statistically underpowered^1,2. Standard analytical approaches typically resolve this by averaging responses across trials to produce a single metric per subject, effectively discarding intra-individual variability and reducing the dataset to N data points. When N is small (e.g., ), this aggregation leads to unstable parameter estimates, inflated Type-II error rates, and a pervasive crisis of reproducibility^3,4.

Bootstrap resampling offers a powerful, nonparametric alternative to classical parametric inference, allowing researchers to estimate sampling distributions without rigid assumptions about normality or homoscedasticity^5,6. To address the hierarchical nature of experimental data, several bootstrap variants have been proposed. The stratified bootstrap resamples trials within each subject, preserving individual-specific variability, but often yields anti-conservative confidence intervals when data are skewed or when N is small⁷. Conversely, the hierarchical bootstrap (or cluster bootstrap) resamples at both the subject and trial levels^8,9. While this approach respects the nested design, it tends to be overly conservative in small-N settings, producing confidence intervals that are often prohibitively wide for detecting subtle physiological or behavioral effects¹⁰.

A critical gap therefore remains for a resampling framework that can leverage the richness of trial-level data to stabilize population estimates without succumbing to the biases of stratified resampling or the excessive conservatism of hierarchical methods. This need is particularly acute when data violate standard assumptions of exchangeability, such as in the presence of session-level drifts¹¹, heavy-tailed noise distributions¹², or heteroscedasticity across subjects.

To address this challenge, we introduce Variance-Calibrated Cross-Individual Bootstrapping (CIB-VC), a robust resampling framework designed to maximize the utility of small-N datasets. The method generates “synthetic individuals” by recombining trials drawn across different subjects, a process that intentionally disrupts within-subject nesting to maximize the exploration of the between-subject state space. To correct for the variance compression inherent in this synthetic construction, we integrate a variance calibration step that aligns the bootstrap distribution with the empirical between-subject variability.

We validate the CIB-VC framework through a dual approach combining extensive Monte Carlo simulations and empirical application. First, using simulations across Gaussian, Student’s t, and log-normal distributions, we demonstrate that CIB-VC achieves near-nominal 95% confidence interval coverage while exhibiting superior robustness to experimental stressors where traditional methods fail (e.g., session-level bias and noise imbalance). Second, we demonstrate the practical utility of the method by applying it to behavioral tracking data from weakly electric fish (). In this empirical setting, CIB-VC yields confidence intervals that are significantly narrower than those produced by hierarchical bootstrapping without introducing bias. By bridging the gap between individual-level richness and population-level rigor, CIB-VC provides a practical pathway toward more reliable and statistically efficient inference in data-limited experimental disciplines.

Results

To rigorously validate the Variance-Calibrated Cross-Individual Bootstrap (CIB-VC), we executed a comprehensive validation pipeline spanning both extensive Monte Carlo simulations and empirical biological application. Our analysis was designed to benchmark CIB-VC against the current standard practices in the field: Stratified Bootstrapping, which represents the high-precision but often anti-conservative lower bound of interval estimation^6,13, and Hierarchical Bootstrapping, which represents the theoretically valid but often overly conservative upper bound^8,9. We also evaluated an Uncalibrated Cross-Individual Bootstrap to serve as an ablation control, isolating the specific contribution of the variance calibration step.

The simulation framework scrutinized performance across three outcome distributions chosen to mimic common neurobiological data structures: standard Gaussian inputs (baseline for aggregate potentials²), Student’s (mimicking heavy-tailed measurement noise or artifacts^12,14), and Lognormal distributions (mimicking skewed physiological metrics such as firing rates or reaction times^15,16). Furthermore, to assess reliability under non-ideal conditions, we subjected the methods to two specific stress tests: Session Bias (violating temporal stability) and Noise Imbalance (violating homoscedasticity). The results are structured to demonstrate the method’s performance hierarchy: first establishing statistical validity, then demonstrating robustness to stress, quantifying efficiency gains, and finally validating utility on real behavioral tracking data.

Validity under distributional skew and heavy tails

Under ideal conditions with Gaussian data, all resampling methods approximated the nominal coverage probability () reasonably well. However, as sample sizes decreased to the critical thresholds common in neuroscience (^17–21), distinct failure modes emerged that highlight the limitations of conventional approaches.

Hierarchical bootstrapping proved consistently over-conservative, often yielding coverage probabilities approaching . While technically “valid” (in that it does not exceed the Type-I error rate), this inflation indicates confidence intervals that are prohibitively wide. This conservatism arises from the discreteness of resampling small sets: with only subjects, the bootstrap distribution becomes “lumpy,” dominated by the specific combination of subjects selected, which artificially expands the interval and dilutes the statistical power to detect real effects.

In contrast, standard Stratified bootstrapping displayed excessive fragility to distributional shape. While performing acceptably for Gaussian inputs, its performance degraded significantly for heavy-tailed () and skewed (Lognormal) distributions. In these non-Gaussian regimes, the method consistently underestimated the true population variance, leading to under-coverage. This failure confirms that preserving within-subject trial structure alone is insufficient for population inference when between-subject variability is non-normal.

CIB-VC, however, maintained near-nominal coverage () across all three distribution types. By generating synthetic individuals, CIB-VC effectively smoothes the discreteness that hampers hierarchical methods; by calibrating the variance, it corrects the underestimation that plagues stratified methods. This result demonstrates that the framework effectively decouples the estimation of distribution shape (via synthetic mixing) from distribution scale (via calibration), ensuring validity regardless of the underlying data topology.

Robustness to experimental stressors

Real-world experimental data rarely satisfy perfect exchangeability assumptions^11,22,23. To evaluate method performance under realistic constraints, we introduced two stressors: Session Bias, representing a systematic baseline drift () for specific subjects, and Noise Imbalance, representing subject-specific heterogeneity in recording quality.

As shown in Fig. 1, CIB-VC exhibited “Robust Invariance,” maintaining valid coverage () regardless of the stressor or underlying distribution. In stark contrast, Stratified bootstrapping exhibited severe under-coverage under stress, revealing a critical vulnerability in standard resampling practices. Notably, we observed a “paradox of variance” where Stratified methods performed worse on clean Gaussian data than on noisy heavy-tailed data. For example, under Session Bias (), Stratified coverage dropped to for Gaussian data but remained significantly higher () for Student’s data. This counter-intuitive result arises because the heavy tails of the distribution naturally inflate the within-subject residual variance, inadvertently widening the confidence intervals enough to encompass the biased mean. In contrast, the “clean” Gaussian data yields high precision around the biased subject means, resulting in intervals that exclude the true population parameter entirely.

Fig. 1.

Methodological Robustness under Experimental Stress. Dot plot showing the coverage probability of 95% confidence intervals for Stratified (blue), Hierarchical (green), and CIB-VC (purple) methods across Gaussian, , and Lognormal distributions (). While Hierarchical and CIB-VC maintain coverage within the safety zone (shaded grey, ), Stratified bootstrapping fails catastrophically under Session Bias, particularly for Lognormal distributions (labeled “False Precision”). CIB-VC demonstrates robust invariance, maintaining validity across all conditions.

However, the most severe methodological failure occurred under the combination of skew and bias. For Lognormal data with Session Bias, Stratified coverage plummeted to (Fig. 1, right panel). This phenomenon, which we term False Precision, occurs because the skewed distribution pulls the bootstrap distribution tightly around the sample median, while the session bias shifts the true population mean into the tail. Consequently, Stratified bootstrapping produces extremely narrow confidence intervals that are precisely located at the wrong value, effectively guaranteeing Type-I errors. CIB-VC, by pooling information across the population to estimate variance, correctly interprets these subject-specific drifts as sources of between-subject uncertainty (), maintaining coverage even in this worst-case scenario.

Statistical efficiency: the precision gain

While Hierarchical bootstrapping provides a safe (conservative) alternative to Stratified methods^8,9, it achieves validity at the cost of precision. Because Hierarchical methods resample whole subjects with replacement, they are limited by the discrete combinations of subjects available (combinatorial granularity)^10,24. When N is small, this results in “lumpy” bootstrap distributions and prohibitively wide confidence intervals that may obscure true physiological effects.

CIB-VC is designed to mitigate this conservatism by smoothing the sampling space. Figure 2 illustrates the “Efficiency Map,” plotting coverage probability against confidence interval width relative to the Hierarchical baseline (ratio indicates narrower intervals). Across the simulated scenarios, CIB-VC consistently occupied the optimal region: maintaining coverage near the nominal target while reducing interval width by approximately compared to Hierarchical bootstrapping.

Fig. 2.

Relative Efficiency Map. Scatter plot of Coverage Probability versus Relative CI Width (normalized to the Hierarchical mean width of 1.0). The green shaded region represents the target accuracy (Coverage and Width Ratio ). The red shaded region indicates invalid inference (Coverage ). The inset zooms in on the efficiency gain, showing that CIB-VC (purple) systematically reduces interval width compared to Hierarchical bootstrapping (green) while avoiding the under-coverage of Stratified methods (blue).

This reduction represents a pure gain in statistical efficiency. By generating synthetic individuals, CIB-VC explores the variance structure between the discrete points accessible to hierarchical resampling. The inset (Fig. 2, bottom middle) highlights this systematic shift, showing CIB-VC estimates migrating to the left (increased precision) and down (corrected conservatism) relative to the Hierarchical centroid. In contrast, Stratified bootstrapping (far left) produced the narrowest intervals but fell deep into the “Invalid Zone” (shaded red). This confirms that the extreme precision of stratified methods is illusory–a form of False Precision that excludes the true population mean in favor of a tightly clustered but biased estimate.

Power and type-I error control

Finally, we evaluated the critical trade-off between statistical power (sensitivity) and Type-I error rate (false positives) for detecting a mean difference between two conditions (). In small-sample studies, methods often achieve high power only by sacrificing validity; the goal of CIB-VC is to break this dependency.

Figure 3 reveals the hidden cost of the apparent high power observed in Stratified methods. As shown in the inset (“The Graveyard”), Stratified estimates cluster far to the right, with Type-I error rates frequently exceeding . This disqualifies the method entirely: its ability to detect effects is not due to sensitivity, but to a systematic bias toward rejecting the null hypothesis, rendering its results statistically inadmissible.

Fig. 3.

Power versus Validity. Scatter plot comparing Statistical Power (y-axis) against Type-I Error rate (x-axis). The main plot focuses on the valid zone (), while the inset shows the full scale. Connector lines link the Hierarchical result (green) to the CIB-VC result (purple) for identical datasets; vertical lines indicate a pure power gain with no loss of validity. Stratified methods (blue, inset) exhibit prohibitively high false-positive rates, disqualifying them despite their high apparent power.

Focusing on the valid methods (Fig. 3, main plot), both Hierarchical and CIB-VC effectively clustered within the “Valid Zone,” controlling Type-I error rates near the nominal . However, a clear “Efficiency Gap” emerged between the two. CIB-VC demonstrated a consistent “vertical lift” in power: for the exact same dataset configurations, it achieved higher sensitivity than Hierarchical bootstrapping. This result confirms that the gain in interval precision (observed in Fig. 2) directly translates into improved hypothesis testing capability. By producing narrower valid intervals, CIB-VC is able to exclude the null hypothesis in cases where the wider Hierarchical intervals would fail to reject, effectively recovering statistical power that is otherwise lost to conservative approximation.

Empirical validation: tracking performance in weakly electric fish

To demonstrate the practical utility of CIB-VC in a realistic experimental setting, we applied the method to behavioral tracking data from weakly electric fish (Eigenmannia virescens). This dataset represents a quintessential small-N neuroscience study: data were collected from only subjects, but each subject performed multiple trials (M) of a refuge-tracking task requiring precise sensorimotor integration. We assessed the impact of sensory conflict (High vs. Low Illumination) on motor performance, quantified by the Frequency Tracking Error (FTE).

Figure 4A illustrates the comparison between standard Hierarchical bootstrapping and the proposed CIB-VC framework (parametric implementation). While both methods centered on the same mean difference estimate–indicating that CIB-VC introduces no bias relative to the standard approach–the precision of the inference differed markedly. Hierarchical bootstrapping yielded a wide confidence interval, reflecting the conservatism inherent in resampling only 5 subjects. In contrast, CIB-VC leveraged the rich trial-level structure to produce a significantly tighter interval, achieving a 22.8% reduction in interval width compared to the hierarchical baseline. This efficiency gain allows for more precise localization of the biological effect size, effectively increasing statistical power without the ethical cost of adding more animals.

Fig. 4.

Empirical Validation on Weakly Electric Fish Data(). (A) Comparison of 95% confidence intervals for the mean difference in tracking error (FTE) between high and low illumination conditions. CIB-VC (purple) yields a 22.8% narrower interval compared to standard Hierarchical bootstrapping (green), demonstrating improved precision in a small-sample biological context. (B) Convergence analysis showing the stability of the 95% CI width as a function of the number of synthetic individuals generated (K). Data were restricted to a subset ( trials) to stress-test the method. (C) The Monte Carlo error (coefficient of variation of the CI width) decreases as K increases, dropping below the 1% threshold (red line) for , indicating robust numerical stability.

We further scrutinized the numerical stability of the estimator by conducting a sensitivity analysis on a restricted subset of the data ( trials per condition). As shown in Fig. 4B, the precision of the confidence interval boundaries stabilized rapidly as the number of synthetic individuals (K) increased. While small synthetic populations () exhibited minor fluctuations due to Monte Carlo noise, the estimator achieved robust convergence at higher replicates. The coefficient of variation for the interval width dropped below the threshold at (Fig. 4C), confirming that the default setting of ensures numerical reliability even in data-limited regimes.

Discussion

The reproducibility crisis in the biological sciences is often attributed to low statistical power and the misapplication of analytical methods to small datasets^1,3. In this study, we introduced and validated Variance-Calibrated Cross-Individual Bootstrapping (CIB-VC), a resampling framework specifically engineered for the hierarchical structure of neuroscience data–where subjects are few (N), but observations per subject are many (M). Our results demonstrate that CIB-VC effectively decouples the trade-off between statistical validity and power. By generating synthetic individuals and calibrating their variance, the method achieves the nominal coverage of conservative hierarchical models while delivering the statistical sensitivity required to detect subtle effects in small samples.

The perils of stratified resampling in small-N studies

A central finding of our simulations is the severe under-coverage of standard stratified bootstrapping under conditions of skew and experimental bias. While stratified resampling is theoretically sound for estimating within-subject stability, it fundamentally fails to account for between-subject sampling error when used for population inference, a methodological pitfall often termed pseudoreplication¹³.

Our data revealed a “paradox of variance”: stratified methods appeared more robust under heavy-tailed noise () than under Gaussian noise in the presence of session bias. This occurs because heavy tails naturally inflate the within-subject residual variance¹⁴, inadvertently widening the confidence intervals enough to encompass the biased mean. In contrast, the “clean” Gaussian data yields high precision around the biased subject means, resulting in intervals that miss the true population parameter entirely. However, this protection in the heavy-tailed case is illusory. When the outcome distribution was skewed (Lognormal) and biased–a common scenario in reaction time or firing rate data–stratified coverage collapsed to near zero (). This phenomenon, which we term False Precision, arises because the method resamples the bias of the few available subjects with high precision. These findings serve as a cautionary tale: narrow confidence intervals in small-N studies should be viewed with skepticism unless the resampling method explicitly accounts for the hierarchical structure of the variance²⁵.

Efficiency gains via synthetic construction

While hierarchical (cluster) bootstrapping addresses the validity problem by resampling subjects, it does so inefficiently. By selecting whole subjects with replacement, hierarchical bootstrapping creates “lumpy” resampled datasets; if an outlier subject is selected twice, the bootstrap mean shifts dramatically¹⁰. This results in overly wide confidence intervals (often coverage in our simulations) and reduced power²⁶.

CIB-VC mitigates this by constructing synthetic individuals–composites of trials drawn from different original subjects. This process “smooths” the bootstrap distribution, allowing for a more granular exploration of the sampling space⁵. The theoretical advantage of this smoothing was confirmed by our empirical validation on weakly electric fish data (). In a direct comparison using the full pooled dataset, CIB-VC produced a confidence interval for behavioral tracking error that was narrower than that produced by hierarchical bootstrapping (Fig. 4A), without altering the point estimate. Crucially, our sensitivity analysis (Fig. 4B–C) demonstrated that this precision is numerically stable; the estimator converges rapidly, with Monte Carlo error dropping to negligible levels () when the synthetic population size .

Implications for experimental design and ethics

The ability to extract valid, high-power inference from small sample sizes has profound ethical and logistical implications. In animal research, the principle of Reduction (one of the 3Rs) mandates minimizing the number of animals used^27,28. However, reducing N often compromises statistical rigor¹. While Linear Mixed Models (LMMs) represent the parametric standard for hierarchical data²⁹, they frequently suffer from singular fits (variance collapse) or convergence failures when the number of clusters is small ()^30,31. CIB-VC offers a methodological pathway to satisfy both ethical constraints and statistical requirements without the instability of complex parametric models.

As demonstrated by our real-data application, CIB-VC extracted significantly higher precision from the same 5 animals than standard methods could. To achieve equivalent precision using hierarchical bootstrapping, a researcher would likely need to increase the sample size by , directly conflicting with ethical reduction goals²⁸. Similarly, in human neuroimaging or clinical studies involving rare patient populations, where increasing N is often impossible due to disease rarity or cost^4,32, CIB-VC provides a robust mechanism to validate findings that might otherwise be dismissed as anecdotal.

Limitations and assumptions

Our framework relies on the assumption that trials are exchangeable across subjects to the extent that they belong to the same experimental condition. While we demonstrated robustness to mild non-exchangeability (Session Bias, Noise Imbalance), a critical boundary condition is population homogeneity. If the subject pool contains distinct subpopulations (e.g., distinct clusters of “responders” and “non-responders”), the synthetic mixing process will average these effects, potentially obscuring biological heterogeneity. In such cases, the variance calibration step will widen the interval to reflect the high variance, but it will not reveal the multimodal structure. Researchers should therefore inspect individual subject distributions before applying this framework. Additionally, while the method is computationally more demanding than standard bootstrapping, our convergence results indicate that standard desktop computing resources ( replicates) are sufficient for reliable inference.

Conclusion

Small sample sizes remain an unavoidable reality in many frontiers of neuroscience. CIB-VC provides a statistically principled, computationally efficient, and robust alternative to traditional resampling methods. By acknowledging the hierarchical nature of experimental data while leveraging the richness of within-subject observations, this framework offers a practical and rigorous framework for reliable inference in data-limited environments.

Methods

Variance-calibrated cross-individual bootstrap (CIB-VC)

The CIB-VC framework addresses the limitations of small sample sizes (N) by leveraging the richness of within-subject observations (M) to construct a more representative bootstrap distribution. The algorithm proceeds in two stages: (1) generation of synthetic individuals via cross-individual resampling to maximize combinatorial diversity, and (2) variance calibration to restore the appropriate population-level variability structure.

Synthetic individual construction

Let denote the experimental dataset, where indexes subjects and indexes trials. The standard population mean is estimated as , where is the subject-level mean.

To estimate the sampling distribution of , CIB-VC generates a distribution of B synthetic individuals. Each synthetic individual mean () is constructed by randomly selecting exactly one trial from each of the N original subjects without replacement within the draw. Specifically:

1

where is a random index selected uniformly from for subject i.

This mixing process serves two theoretical functions: it preserves the structure of between-subject variability (as every synthetic point contains a contribution from every subject), and it breaks the serial correlation of within-subject trials to smooth the granularity of the bootstrap distribution. For the main simulations, we generated bootstrap replicates.

Variance calibration and parametric construction

While cross-individual resampling increases combinatorial diversity, it treats trials from different subjects as exchangeable components of a single synthetic observation. This assumes that the variance of the synthetic mean is driven solely by trial-level noise, which systematically underestimates the standard error in the presence of true biological heterogeneity. We correct for this via variance calibration.

Let be the empirical standard deviation of the distribution of synthetic means . We calculate a target standard error, , based on the empirical between-subject variance of the original data:

2

where is the unbiased sample variance of the original subject means . A calibration factor is computed as . The final adjusted standard error is .

To ensure robust coverage at small N, we implement a Parametric CIB-VC approach. The confidence interval is constructed as:

3

where is the critical value from the Student’s t-distribution with degrees of freedom (). This usage of t-statistics explicitly accounts for the uncertainty in the variance estimate when , preventing the under-coverage typical of standard normal approximations in small samples. For , standard normal z-scores may be used.

Bootstrapping for paired differences

For experimental designs comparing two conditions (e.g., Condition A vs. Condition B) within subjects, CIB-VC is applied to the difference scores. Let and represent the data matrices for the two conditions. We first compute the trial-level difference matrix D. If trials are strictly temporally paired, we compute . If no strict trial-to-trial correspondence exists (as is common in behavioral tasks where trial counts may vary or temporal alignment is irrelevant), we generate difference scores by randomly pairing trials between conditions within each subject. The synthetic resampling and variance calibration steps described above are then applied directly to the difference matrix D to estimate the confidence interval for the mean difference .

Simulation framework

We validated the method using extensive Monte Carlo simulations ( iterations per scenario) across a grid of sample sizes () and trial counts ().

Outcome distributions

Data were generated from three probability distributions chosen to represent distinct neurobiological data types:

• Gaussian: , representing ideal experimental conditions or averaged potentials.

• Student’s : , representing heavy-tailed biological noise or datasets containing measurement artifacts.

• Lognormal: , representing positively skewed metrics such as reaction times, firing rates, or synaptic weights.

In all cases, subject means were drawn from a standard normal distribution to introduce between-subject variability.

Experimental stress tests (non-exchangeability)

To evaluate robustness against violations of exchangeability, we modeled two specific stressors:

1. Session Bias: A systematic baseline drift () added to the first of trials for all subjects, mimicking temporal effects such as fatigue, sensor drift, or learning.

2. Noise Imbalance: Heteroscedasticity introduced by doubling the trial-level noise variance () for a random of subjects, mimicking variable recording quality.

Real data analysis

To assess performance on empirical data, we analyzed behavioral tracking error (FTE) from adult weakly electric fish (Eigenmannia virescens) performing a refuge-tracking task under Low versus High Illumination conditions.

Subjects and Ethics: Fish (7–15 cm; 15–30 g; 8–18 months old; either sex) were obtained from certified commercial vendors and housed in individual tanks on a 12:12 light:dark cycle. Water conductivity was maintained between and temperature at . All experimental procedures were approved by the Hacettepe University Animal Experiments Ethics Committee (Approval No: 2023/33) and conducted by certified personnel (I.U.). Methods were carried out in accordance with the ethical principles of the European Parliament Directive 2010/63/EU and associated national regulations. Furthermore, we confirm that this study is reported in accordance with the ARRIVE guidelines.

Experimental Protocol: Experiments were strictly non-invasive; no anesthesia, sedation, analgesia, or surgical procedures were used. Fish were transferred to the experimental setup and allowed to acclimate for hours prior to data collection. To minimize stress, 2-minute rest intervals were provided between trials. Animals were returned to their home tanks immediately following the experiments with no observed harm.

Pooled Analysis (Fig. 4A): To benchmark efficiency on a realistic dataset, we pooled trials by concatenating data across secondary conditions (Conductivity and Refuge), resulting in trials per subject per illumination level. We compared the width of 95% confidence intervals generated by Hierarchical Bootstrapping versus Parametric CIB-VC (using t-distribution critical values as defined above, with ).

Sensitivity Analysis (Fig. 4B–C): To stress-test the stability of the method, we restricted the dataset to a strict subset of trials (, holding Conductivity, Window, and Refuge fixed). We systematically varied the number of synthetic individuals K (equivalent to bootstrap replicates B) from 50 to 5000. For this specific convergence analysis, we utilized a Percentile CIB-VC approach (using the quantiles of the calibrated bootstrap distribution rather than the parametric t-interval) to visualize the raw stability of the bootstrap distribution shape without the smoothing assumption of the t-distribution.

Comparative analysis

CIB-VC was benchmarked against three methods: Stratified Bootstrap (resampling trials within subjects), Hierarchical Bootstrap (resampling subjects then trials), and Uncalibrated CIB (ablation control). Performance was assessed via three metrics:

• Coverage Probability: The proportion of 95% confidence intervals containing the true population mean (Target: 0.95).

• Relative CI Width: The ratio of the mean confidence interval width of a given method to that of the Hierarchical bootstrap (Values indicate improved precision).

• Statistical Power: The probability of correctly rejecting the null hypothesis () when the true difference was . Rejection was defined as the 95% confidence interval excluding zero. Type-I error rates were similarly assessed at .

Author contributions

I.U. conceived the study, developed the methodology, performed the simulations, analyzed the results, and wrote the manuscript.

Funding

This research was supported by The Scientific and Technological Research Council of Türkiye (TUBITAK) under Grants No. 123E155 awarded to Ismail Uyanik.

Data availability

All statistical analyses and simulations were implemented in MATLAB (R2025b, The MathWorks Inc., Natick, MA). All data and custom analysis code supporting the findings of this study are publicly available via Aperta – Türkiye Open Archive (https://aperta.ulakbim.gov.tr) and can be accessed via the following DOI: 10.48623/aperta.286921.

Declarations

Competing interests

The authors declare no competing interests.