Linking Indices for Biodiversity Monitoring to Extinction Risk Theory

Biodiversity indices often combine data from different species when used in monitoring programs. Heuristic properties can suggest preferred indices, but we lack objective ways to discriminate between indices with similar heuristics. Biodiversity indices can be evaluated by determining how well they reflect management objectives that a monitoring program aims to support. For example, the Convention on Biological Diversity requires reporting about extinction rates, so simple indices that reflect extinction risk would be valuable. We developed 3 biodiversity indices that are based on simple models of population viability that relate extinction risk to abundance. We based the first index on the geometric mean abundance of species and the second on a more general power mean. In a third index, we integrated the geometric mean abundance and trend. These indices require the same data as previous indices, but they also relate directly to extinction risk. Field data for butterflies and woodland plants and experimental studies of protozoan communities show that the indices correlate with local extinction rates. Applying the index based on the geometric mean to global data on changes in avian abundance suggested that the average extinction probability of birds has increased approximately 1% from 1970 to 2009. Conectando Índices para el Monitoreo de la Biodiversidad con la Teoría de Riesgo de Extinción Resumen Los índices de biodiversidad combinan frecuentemente los datos de diferentes especies cuando se usan en los programas de monitoreo. Las propiedades heurísticas pueden sugerir índices preferidos, pero carecemos de medios objetivos para discriminar a los índices con propiedades heurísticas similares. Los índices de biodiversidad pueden evaluarse al determinar qué tan bien reflejan los objetivos de manejo que un programa de monitoreo busca apoyar. Por ejemplo, la Convención sobre la Diversidad Biológica requiere reportar las tasas de extinción, así que los índices que reflejan el riesgo de extinción serían valiosos. Desarrollamos 3 índices de biodiversidad que se basan en modelos sencillos de viabilidad de población y que relacionan el riesgo de extinción con la abundancia. Basamos el primer índice en la media geométrica de la abundancia de especies, y el segundo en una media de poder más general. En el tercer índice integramos la media geométrica y la tendencia. Estos índices requieren los mismos datos que índices previos, pero también se relacionan directamente con el riesgo de extinción. La información de campo sobre mariposas y plantas de bosque, y los estudios experimentales de comunidades protozoarias, muestran que los índices se correlacionan con las tasas locales de extinción. Al aplicar el índice basado en la media geométrica sobre los datos globales de los cambios en la abundancia de aves, sugirió que la probabilidad de extinción promedio de aves ha incrementado aproximadamente 1% desde 1970 hasta 2009. Palabras Clave Índice de biodiversidad, media geométrica, medida de la biodiversidad, riesgo de extinción

(assuming two generations per year). For the second index, this results in b ≈ 6.6 assuming  = 0.1. Greater variability in the population dynamics (i.e.,  > 0.1) would lead to smaller values for b. The data were also analysed when the data were restricted to species that were genuine grassland specialists, but as the results were qualitatively the same, only results with the larger number of species are reported.
The composition and abundance of all native plant species of ten grassy woodland patches in western Victoria, Australia were surveyed in 2006 (Sutton & Morgan 2009). These data were compared to the species composition and population abundance of the same patches observed in 1975 to determine patch extinction rates. To reduce false absences, sites were surveyed on three occasions, with the time spent searching being proportional to each site's size and heterogeneity. To further minimize chances of missed detections, we restricted our analyses to perennial species. Finally, we limit variation in extinction risk due to idiosyncratic differences among species by considering only species that were present in at least eight of the ten patches in the analysis.
Abundances of the plants was estimated in 1975 using a four-point scale: "Very Rare" (less than two dozen individuals seen across the site), "Rare" (appearing in dozens), "Common" (appearing in hundreds), "Very Common" (appearing in thousands). The discretised and censored data meant we were unable to calculate the indices directly. Instead, we fitted a Pareto distribution to the abundance data, using maximum likelihood methods, and calculated the indices from the parameters of the estimated distribution. To fit the Pareto distribution, we assumed that the four abundance class were distinguished by threshold values of 24, 100 and 1000 (i.e., "Very Rare" was assumed to be <24 individuals, "Rare" was 24-100, etc).
We chose to fit a Pareto distribution since, for the deterministic model with negative growth rate, if the mean time to extinction (T) is exponentially distributed with parameter , abundance (x = exp[T/k]) is a Pareto random variable with scale parameter x m = 1, and shape parameter α = k (Krishnamorrthy 2006). We also fitted a log-normal distribution instead of a Pareto distribution. The results obtained were very similar in both cases, so we only present results for the Pareto. Assuming =0.1 and T=30 leads to b = 3.4 in the power mean index I b .
Experimental protozoan communities were assembled with 4 ciliate species, at 2 temperatures (15 °C and 20 °C), and sampled for abundance data 3 times a week for 163 days (Clements et al. 2013). Four 3-species communities and one 4-species community were replicated at each temperature giving a total of 10 communities (a "community" in this case being a particular combination of species and temperatures). Extinction events were driven by either competitive exclusion or starvation. Population trends between days 5 and 9, abundance at day 9, and the proportion of species extinct by day 163, were calculated and averaged across replicates. Day 9 was chosen as the initial date because this day occurred prior to all but one extinction event, and effects of initial conditions on the community dynamics had attenuated. Each community was replicated 5 times each, except for one community in which a replicate was excluded when a species went extinct prior to day 9. The

Appendix S2. Method for simulating expected correlation
Even if the probability of extinction of species in a community were perfectly correlated with a particular index of extinction, the observed proportion of species going extinct would vary randomly around the actual probability of extinction because there are a finite number of species in each community. This level of variation will tend to increase as the number of species declines. The consequence of this variation is that the observed correlations between the indices and the proportion of species going extinct would be less than one even if the indices were perfectly correlated with the probability of extinction. We evaluated the distribution of measured correlations that would be expected under the assumption that the indices were perfectly correlated with extinction probability to examine whether the measured correlation coefficients for the datasets were smaller than would be expected.
The analysis was conducted by simulating extinction of species in each of the communities.
For the indices based on the geometric mean (I g ) and the power mean (I b ), we assumed that the probability of extinction of species in each community (p) was proportional to the index for that community. That is, p = bI j , where I j is the particular index being examined. The constant of proportionality was set for each dataset such that the means of p and I j , were the same as those in the dataset (i.e., b = j I p / ). For the index based on the trend, the index sometimes took negative values, so we could not use this approach. Instead, we set p = a + bI t , with a and b estimated from linear regression of the observed relationship.
For each species in each community, we simulated extinction by randomly determining whether extinction occurred given the specified probability of extinction p. That is, the number of extinctions, y, in each community was assumed to be drawn from a binomial distribution with parameters p and n, where n was the number of species originally in that community. The simulated proportion of species in each community that went extinction was then p = y/n. For each dataset, we then calculated the correlation between p and I j , which was recorded. This process was iterated 1 million times to generate a distribution of expected correlations under the assumption of perfect correlation between p and I j given random variation in the observed values of p. The observed correlation coefficient can then be compared to this distribution. If observed correlation coefficient lies outside the 95% prediction interval based on the simulated distribution of correlation coefficients, then it suggests the observed correlation is unexpected given the assumptions. Comparing the observed correlation to the simulated 95% interval is equivalent to testing a null hypothesis that the indices were perfectly correlated with extinction risk.
These comparisons indicate that the observed correlations are often within the anticipated range (Fig. S1). In the case of the index based on the power mean (I b ), the observed correlations are substantially smaller than expected for the butterfly and protozoan dataset. In the other cases, the observed correlations are generally consistent with the range of values that might be expected. The one other exception was the index based on the geometric for the butterfly dataset, where the observed correlation coefficient was actually larger than expected.