Mathematical biases in the calculation of the Living Planet Index lead to overestimation of vertebrate population decline

The study addresses why the widely publicized Living Planet Index (LPI) reports a ~69% average decline in vertebrate populations since 1970 while several analyses of the same underlying population time series report roughly balanced increases and decreases. The authors scrutinize the LPI’s methodological pipeline to identify biases arising from weighting by species richness across taxa and realms, inclusion of very short and sparse time series, smoothing choices, treatment of zeros in population series, and hierarchical averaging combined with time propagation of early-year changes. Given the LPI’s prominence for policy (CBD, IPBES) and public communication, understanding and correcting these biases is critical to accurately represent biodiversity change.

Prior critiques have highlighted several issues: (1) Weighting by estimated species richness of taxa and regions can amplify declines because species-rich (often tropical) but data-poor regions dominate the weighted index (McRae et al. 2017; also noted here with larger declines in weighted vs unweighted LPI). (2) Buschke et al. (2021) showed that random fluctuations and GAM smoothing can induce a spurious ~9.6% decline, and that LPI can decrease under certain simulation setups where populations diverge from an initial point. The present paper argues those simulations are non-stationary on the arithmetic scale and have an absorbing boundary at zero, so a decline is expected; under stationary distributions or symmetry on the logarithmic scale, LPI remains stable. (3) Puurtinen et al. (2022) argued geometric averaging is inappropriate for cross-population aggregation; the authors counter that multiplicative population changes justify geometric means and that stationarity can still be reflected. (4) Other works have raised concerns about short time series, sampling error, site-selection bias, and data heterogeneity (e.g., Wauchope et al. 2019; Daskalova et al. 2020; Hébert & Gravel 2023). This study builds on and extends these critiques by auditing the code, decomposing sources of bias, and quantifying their effects on LPI.

Data: Publicly available time series from the Living Planet Database (LPD) as of 01/2022, including 22,175 populations from 4,777 vertebrate species across terrestrial, freshwater, and marine ecosystems with varying lengths, irregular sampling, and different abundance proxies. Code and computation: The authors inspected the R package rlpi (v0.1.0) provided by ZSL for LPI calculation, identified errors that can bias results for subsets (taxa/realms), and provided corrected code (Supplementary Software). They replicated and adjusted the LPI using alternative settings, focusing on sensitivity to key choices. Standard LPI pipeline (as implemented in rlpi and detailed here): - If any zero occurs in a population time series, add a constant equal to 1% of the series’ mean (from non-zero values) to all values (special handling for all-zero series, which were removed in this study). - Estimate missing annual values by: - GAM smoothing (log-scale, then back-transform) if the series has ≥6 records and the fit is adequate (smoothing parameter set to half the series length), else - Chain method (log-linear interpolation) for series with <6 records or poor GAM fit. - Log10-transform population values. - Compute annual population growth lambda as the log10 ratio of consecutive years: λ = log10(Nt+1/Nt). - Hierarchical arithmetic averaging (equivalent to geometric means on the original scale) per year: 1) average populations within species and realm; 2) average species within taxa and realm; 3) average taxa within a realm using species-richness weights (diversity-weighted LPI) or unweighted alternative; 4) average realms within an ecosystem using realm weights; 5) for the global LPI, average across ecosystems (weights adjusted by 1/3 when forming a global index; this step was not implemented in code and is noted by the authors). - Index recursion: It = It-1 × g (g is the final geometric mean of growth rates); I1970 = 1. Bootstrap CIs via 100 species resamples per taxon. Sensitivity analyses and adjustments performed: - Series fullness and duration: recomputed indices requiring at least 3, 5, or 10 recorded points; also applied minimum series duration thresholds (≥3, ≥5, ≥10 years). - Treatment of zeros: Recalculated LPI after removing zeros. For internal zeros, split series at zeros; for terminal zeros, truncate accordingly. Compared to original approach (adding 1% of mean to avoid division/log issues). - Weighting: Compared diversity-weighted vs unweighted indices globally and by ecosystem. - Early-period sensitivity: Examined impacts of sparse 1970s data, hierarchical averaging, and single-population representatives (e.g., early Palearctic herptiles represented solely by a declining viper population). - Grouping choices: Tested alternative groupings (e.g., merging realms/taxa such as herptiles, Indo-Pacific realm) and assessed how grouping granularity affects the resulting means. Simulations and conceptual demonstrations: - Demonstrated conditions under which LPI remains stable (stationary abundance distributions; symmetric multiplicative/log-scale fluctuations) versus declines (non-stationary, arithmetic-scale symmetry with absorbing boundary at zero). - Illustrated downward bias from short series via sampling error for low, discrete counts and paired two-record examples showing multiplicative asymmetry. All analyses used corrected code, with outputs and scripts provided in Supplementary Software.

- Diversity weighting amplifies declines relative to unweighted indices: - The weighted global LPI shows a substantially greater decline than the unweighted version (reported earlier as ~38% greater decline; quantified here as a 44.5% difference for the analyzed dataset). Ecosystem-specific effects: terrestrial +14.8% decline, freshwater +47.3%, marine +83.5% in weighted vs unweighted (Supplementary Table 1). - Very short/sparse time series bias the LPI downward: - Excluding two-record series (i.e., requiring ≥3 records) reduces the global decline by 14.3%; requiring ≥5 and ≥10 records reduces the decline by 14.7% and 26.4%, respectively (Supplementary Table 1). - Short series processed without smoothing retain sampling error, producing asymmetric multiplicative changes that bias the LPI downward, especially for low, discrete counts. - Treatment of zeros is a major source of bias: - Replacing zeros by adding 1% of the mean imposes arbitrary order-of-magnitude changes; since zeros are more common at the end of series, this biases the index downward. - Removing zeros reduces the apparent global decline by 19.2% (Global index from 0.327 to 0.519). By ecosystem: terrestrial reduction 33.8% (0.368 to 0.706), freshwater 19.3% (0.181 to 0.373), marine <1% (0.526 to 0.532). Confidence intervals often widen and can overlap 1 after zero removal, indicating uncertainty about net declines. - Early-year sensitivity and hierarchical averaging cause long-lasting effects: - Because the index multiplies year-on-year, early declines persist and later growth cannot easily raise the index when it is already low. - Single-population representatives can dominate early periods and propagate their effect through the entire trajectory. Example: a single declining viper population (Vipera berus, 1974–1977) caused an 89.5% greater decrease in the Palearctic LPI (index shifts from 0.826 to 1.721 when those four records are removed) and a 3.3% greater decrease in the terrestrial LPI, demonstrating extreme sensitivity. - Stationarity and fluctuation scale matter conceptually: - LPI is stable under stationary distributions or symmetric multiplicative (log-scale) fluctuations; it declines under non-stationary arithmetic-scale fluctuations with absorption at zero. - Additional noted issues: - GAM edge effects and smoothing can misestimate series ends; prior work estimated a ~9.6% spurious decline from such effects. - Grouping choices (number/definition of taxa and realms) affect results; merging/altering groups can reduce apparent declines (e.g., a 6.3% reduced decline in terrestrial LPI under coarser grouping).

The findings show that the LPI’s headline global decline is sensitive to methodological choices and data idiosyncrasies: weighting by species richness, inclusion of very short series, treatment of zeros, and early-period sparsity combined with hierarchical averaging all contribute to overestimating declines. While the LPI can validly represent mean multiplicative changes when assumptions are met (stationary distributions or symmetric log-scale fluctuations), in practice the combination of data heterogeneity and algorithmic choices introduces systematic bias. Removing zeros, enforcing minimum record counts, testing alternative groupings, and sensitivity analyses to early single-population representatives substantially alter the index and often reduce the apparent decline, with increased uncertainty. The authors argue that colonization and extinction dynamics should be analyzed separately from population fluctuation-based indices, as zeros fundamentally break the consecutive-year multiplicative link underlying the LPI. They recommend routine sensitivity analyses for grouping and weighting, exclusion or downweighting of sparse series, potential shifts in reference year to mitigate early-period sparsity, and exploration of resampling or reshuffling time series to gauge robustness. Given potential sampling and site-selection biases in the LPD (non-standardized, unevenly distributed in space/time and often focused on protected or particular sites), even a bias-corrected LPI may not reflect true global trends. The study underscores the need for standardized surveys and careful methodological design, especially for regional/national LPIs where biases can be amplified.

This study identifies and quantifies multiple mathematical and data-driven biases in the LPI that tend to overestimate global vertebrate population declines. By auditing the code and systematically varying key settings, the authors show that: (i) removing zeros, (ii) excluding very short series or enforcing a minimum number of records, (iii) reconsidering diversity weighting, and (iv) testing sensitivity to grouping choices and early single-population representatives collectively yield markedly less negative indices and larger uncertainty bounds. They provide corrected code and advocate for companion analyses that separately treat colonization/extinction dynamics. Despite methodological improvements, the reliability of any single global index remains constrained by heterogeneous, non-standardized data and uneven geographic/taxonomic coverage. Future work should prioritize standardized monitoring, robust sensitivity frameworks, and potentially alternative indicators less sensitive to these biases.

- Data representativeness: LPD time series are not from standardized systematic surveys; spatial, taxonomic, and site-selection biases may persist and differ in direction and magnitude, so corrected LPI may still under- or overestimate real changes. - Early-period sparsity: Few series in the 1970s create strong sensitivity to early declines; this is difficult to rectify without sacrificing temporal coverage or changing the reference year. - Zeros handling: The LPI framework based on ratios/logs cannot accommodate zeros without either arbitrary replacement or exclusion; both choices influence results, and colonization/extinction must be treated separately. - Short time series: Excluding them improves bias but reduces geographic/taxonomic representativeness; including them introduces downward bias via sampling error and lack of smoothing. - Weighting and grouping: Diversity weighting and grouping definitions (taxa/realms) materially affect outcomes; while sensitivity analysis helps, there is no uniquely correct choice. - Code/version dependence: Errors found in rlpi v0.1.0 have negligible global impact but can bias subset indices; results may depend on code versions and settings.