New confidence interval methods for Shannon index

Abstract: Several factors affect the structure of communities, including biological, physical and chemical phenomena, impacting the quantification of biodiversity, measured by diversity indexes such as Shannon's entropy. Then, once a point estimate is obtained, confidence intervals methods such as the bootstrap ones are often used. These methods, however, can have different performances, which many authors have revealed in the last decade. Furthermore, problems such as the asymmetry of the distribution of estimates and the possibility of Shannon's diversity index estimator bias can lead to incorrect recommendations to the research community. Thus, we propose two methods and compare them with seven others using their performances to face these problems. The first idea uses the credible interval (CI) method to build a bootstrap confidence interval. The second one starts by correcting the bias and then uses an asymptotic approach. We considered 27 community structures representing scenarios with high dominance, high codominance or moderate dominance, the number of species equal to 4, 20 or 80 and 10, 50 or 500 individuals to compare their performances. Then, we generated 1000 samples, built 95% confidence intervals, and calculated the percentage of times they included the community diversity index (coverage percentage) for each community structure. Our results showed the feasibility of both proposed methods to estimate Shannon's diversity. The simulation study revealed the bootstrap-t technique had the best performance, i.e., best coverage percentage, compared with the other methods. Finally, we illustrate the methodology by applying it to an original aphid and parasitoid species dataset. We recommend the bootstrap-t when the community structure analysed is similar to the simulated ones. Also, the methods provided high performance for the high dominance scenarios.

• The paper introduces two novel confidence interval methods – a credible interval approach and an asymptotically corrected method – that enhance estimation accuracy over standard techniques.
• By employing extensive simulations across 27 ecological community structures, the study shows that the bootstrap-t method consistently achieves near-nominal coverage even under challenging sampling regimes.
• Application to empirical data underscores how improved CI estimation for the Shannon index supports more reliable biodiversity inference, benefiting conservation and pest management efforts.

Confidence Interval Estimation for the Shannon Index: Methodological Advances and Simulation-Based Evaluation

Introduction

The paper "New confidence interval methods for Shannon index" (2204.10073) addresses the persistent challenges in statistical inference for biodiversity measurement, notably the construction of accurate confidence intervals (CIs) for the Shannon diversity index, $H$. The Shannon index remains pivotal for quantifying α-diversity, yet estimation uncertainty is often inadequately treated due to bias in the standard plug-in estimator and the suboptimal performance of classical bootstrap and asymptotic CI procedures under realistic ecological sampling regimes. This work proposes two alternative CI methods—one rooted in credible intervals (CrI) for skewed bootstrap distributions, and another based on analytical bias correction and asymptotic normality—and rigorously compares them and several established bootstrap methods using simulations on 27 representative community structures.

Methodological Innovations

Two novel methods are articulated. The first, a credible interval-based approach, leverages the empirical bootstrap distribution of $H$ to construct intervals with minimal width covering at least a specified proportion of the distribution mass. Its advantage over traditional bootstrap percentile methods lies in robustness to asymmetry and multimodality, both commonly encountered in ecological abundance data.

The second, an analytically corrected asymptotic (AC) interval, implements explicit bias correction based on higher-order expansions for the expectation and variance of the plug-in entropy estimator. The adjusted estimator, $\widehat{H}^{\prime}$, incorporates terms reflecting species richness ($k$) and sample size ($n$), yielding improved frequentist properties, particularly in moderate-to-large samples or where the asymptotic normality of $\widehat{H}^{\prime}$ is a credible approximation.

These innovations are juxtaposed with the traditional battery of bootstrap CIs: the percentile, bias-corrected (BC), bias-corrected and accelerated (BCA), standard percentile (SPerc), empirical corrected (EC), and the bootstrap-$t$ method, the latter using nested resampling to estimate a variance-stabilized pivotal statistic. For all methods, inferential target performance is anchored on CI coverage probability (proportion intervals contain the true $H$) and average width.

Simulation Design and Results

A comprehensive simulation framework underpins the method comparison. Community structures are parameterized by three regimes of dominance (high, high codominance, moderate dominance) and varying species richness ($K = 4, 20, 80$). For each setting, samples of size $n = 10, 50, 500$ (1,000 replicates each) are evaluated. The underlying species abundance distributions reflect real ecological scenarios, with codominance and evenness controlled to produce biologically meaningful comparisons. Bootstrap methods utilize 1,000 iterations per sample.

The bootstrap-$t$ method is shown to yield the most reliable coverage: for 89% of simulated scenarios, its empirical coverage percentage was at or above the nominal 95% level, outperforming all alternatives. The AC method and the bootstrap BCA also exhibit strong but consistently inferior performance. Notably, in high dominance communities—a frequent real-world pattern—the bootstrap-$t$ and BCA are especially robust. Notably, methods relying purely on the percentile or uncorrected bootstrap approaches are frequently anti-conservative or yield intervals with substantial undercoverage.

Numerical evaluation also revealed that the average CI width, while often narrower for the simple percentile methods, does not correlate with empirical coverage; narrow intervals commonly fail to include the true $H$. Thus, coverage is retained as the primary criterion for method selection, confronting prior literature which privileged interval width or computational efficiency.

Application to Empirical Ecological Data

The methodology is validated on a real dataset tracking aphid and parasitoid communities in wheat crop ecosystems. The sampling structure in the empirical data strongly mirrors simulation parameters, and the recommended bootstrap-$t$ method provides interpretable and robust CI estimates, reflecting the underlying community dominance structure across temporal replicates.

Implications and Prospective Directions

This work reinforces the necessity of tailoring inferential methodology to the specifics of ecological sampling, species abundance distribution, and sample size. Recommendations for CI construction for the Shannon index should be context-dependent: the bootstrap-$t$ is recommended for typical local-scale communities, especially those characterized by dominance or codominance, as often encountered in agroecosystems and fragmented landscapes. The newly proposed AC and CrI approaches merit use where computational scalability or skewness of the estimator is of particular concern, although neither surpasses bootstrap-$t$ in overall frequentist performance for most community structures.

Theoretically, the findings emphasize the limitations of percentile and simple bootstrap methods in non-idealized sampling regimes, and underline the need for rigorous performance benchmarking using realistic simulation settings. Practically, improved CI estimation for diversity indices enables more reliable inference in conservation biology, pest management, and community ecology, influencing biological interpretation and subsequent decision-making.

Future work should extend the simulation framework to encompass more complex community architectures (e.g., spatially structured metacommunities, under-sampled rare taxa, zero-inflated regimes), assess Bayesian and nonparametric alternatives, and directly address interval estimation for related diversity measures (e.g., Simpson or Chao indices).

Conclusion

The systematic comparison of classical and novel CI methods for Shannon's diversity index evidences that the bootstrap-$t$ approach achieves the best balance of nominal coverage and interpretability across a broad range of ecological scenarios. The credible interval and asymptotic correction strategies provide further options, particularly where distributional irregularities or computational efficiency are paramount. This work advances the statistical toolkit available for biodiversity inference and provides empirically justified guidelines for applied researchers (2204.10073).