A new method for choosing sample size for confidence interval-based inferences

Scientists often need to test hypotheses and construct corresponding confidence intervals. In designing a study to test a particular null hypothesis, traditional methods lead to a sample size large enough to provide sufficient statistical power. In contrast, traditional methods based on constructing a confidence interval lead to a sample size likely to control the width of the interval. With either approach, a sample size so large as to waste resources or introduce ethical concerns is undesirable. This work was motivated by the concern that existing sample size methods often make it difficult for scientists to achieve their actual goals. We focus on situations which involve a fixed, unknown scalar parameter representing the true state of nature. The width of the confidence interval is defined as the difference between the (random) upper and lower bounds. An event width is said to occur if the observed confidence interval width is less than a fixed constant chosen a priori. An event validity is said to occur if the parameter of interest is contained between the observed upper and lower confidence interval bounds. An event rejection is said to occur if the confidence interval excludes the null value of the parameter. In our opinion, scientists often implicitly seek to have all three occur: width, validity, and rejection. New results illustrate that neglecting rejection or width (and less so validity) often provides a sample size with a low probability of the simultaneous occurrence of all three events. We recommend considering all three events simultaneously when choosing a criterion for determining a sample size. We provide new theoretical results for any scalar (mean) parameter in a general linear model with Gaussian errors and fixed predictors. Convenient computational forms are included, as well as numerical examples to illustrate our methods.     

Comparing species richness among assemblages using sample units: why not use extrapolation methods to standardize different sample sizes?

Comparisons of species richness among assemblages using different sample sizes may produce erroneous conclusions due to the strong positive relationship between richness and sample size. A current way of handling the problem is to standardize sample sizes to the size of the smallest sample in the study. A major criticism about this approach is the loss of information contained in the larger samples. A potential way of solving the problem is to apply extrapolation techniques to smaller samples, and produce an estimated species richness expected to occur if sample size were increased to the same size of the largest sample. We evaluated the reliability of 11 potential extrapolation methods over a range of different data sets and magnitudes of extrapolation. The basic approach adopted in the evaluation process was a comparison between the observed richness in a sample and the estimated richness produced by estimators using a sub-sample of the same sample. The Log-Series estimator was the most robust for the range of data sets and sub-sample sizes used, followed closely by Negative Binomial, SO-J1, Logarithmic, Stout and Vandermeer, and Weibull estimators. When applied to a set of independently replicated samples from a species-rich assemblage, 95% confidence intervals of estimates produced by the six best evaluated methods were comparable to those of observed richness in the samples. Performance of estimators tended to be better for species-rich data sets rather than for those which contained few species. Good estimates were found when extrapolating up to 1.8-2.0 times the size of the sample. We suggest that the use of the best evaluated methods within the range of indicated conditions provides a safe solution to the problem of losing information when standardizing different sample sizes to the size of the smallest sample.     

Nonparametric Lower Bounds for Species Richness and Shared Species Richness under Sampling without Replacement

Summary A number of species richness estimators have been developed under the model that individuals (or sampling units) are sampled with replacement. However, if sampling is done without replacement so that no sampled unit can be repeatedly observed, then the traditional estimators for sampling with replacement tend to overestimate richness for relatively high-sampling fractions (ratio of sample size to the total number of sampling units) and do not converge to the true species richness when the sampling fraction approaches one. Based on abundance data or replicated incidence data, we propose a nonparametric lower bound for species richness in a single community and also a lower bound for the number of species shared by multiple communities. Our proposed lower bounds are derived under very general sampling models. They are universally valid for all types of species abundance distributions and species detection probabilities. For abundance data, individuals' detectabilities are allowed to be heterogeneous among species. For replicated incidence data, the selected sampling units (e.g., quadrats) need not be fully censused and species can be spatially aggregated. All bounds converge correctly to the true parameters when the sampling fraction approaches one. Real data sets are used for illustration. We also test the proposed bounds by using subsamples generated from large real surveys or censuses, and their performance is compared with that of some previous estimators.     

Estimating the immunity coverage required to prevent epidemics in a community of households

An estimation of the immunity coverage needed to prevent future outbreaks of an infectious disease is considered for a community of households. Data on outbreak size in a sample of households from one epidemic are used to derive maximum likelihood estimates and confidence bounds for parameters of a stochastic model for disease transmission in a community of households. These parameter estimates induce estimates and confidence bounds for the basic reproduction number and the critical immunity coverage, which are the parameters of main interest when aiming at preventing major outbreaks in the future. The case when individuals are homogeneous, apart from the size of their household, is considered in detail. The generalization to the case with variable infectivity, susceptibility and/or mixing behaviour is discussed more briefly. The methods are illustrated with an application to data on influenza in Tecumseh, Michigan.     

Rarefaction and extrapolation of species richness using an area‐based Fisher's logseries

Fisher's logseries is widely used to characterize species abundance pattern, and some previous studies used it to predict species richness. However, this model, derived from the negative binomial model, degenerates at the zero‐abundance point (i.e., its probability mass fully concentrates at zero abundance, leading to an odd situation that no species can occur in the studied sample). Moreover, it is not directly related to the sampling area size. In this sense, the original Fisher's alpha (correspondingly, species richness) is incomparable among ecological communities with varying area sizes. To overcome these limitations, we developed a novel area‐based logseries model that can account for the compounding effect of the sampling area. The new model can be used to conduct area‐based rarefaction and extrapolation of species richness, with the advantage of accurately predicting species richness in a large region that has an area size being hundreds or thousands of times larger than that of a locally observed sample, provided that data follow the proposed model. The power of our proposed model has been validated by extensive numerical simulations and empirically tested through tree species richness extrapolation and interpolation in Brazilian Atlantic forests. Our parametric model is data parsimonious as it is still applicable when only the information on species number, community size, or the numbers of singleton and doubleton species in the local sample is available. Notably, in comparison with the original Fisher's method, our area‐based model can provide asymptotically unbiased variance estimation (therefore correct 95% confidence interval) for species richness. In conclusion, the proposed area‐based Fisher's logseries model can be of broad applications with clear and proper statistical background. Particularly, it is very suitable for being applied to hyperdiverse ecological assemblages in which nonparametric richness estimators were found to greatly underestimate species richness.     

Plant species diversity and environmental heterogeneity: spatial scale and competing hypotheses

Questions: What is the observed relationship between plant species diversity and spatial environmental heterogeneity? Does the relationship scale predictably with sample plot size? What are the relative contributions to diversity patterns of variables linked to productivity or available energy compared to those corresponding to spatial heterogeneity? Methods: Observational and experimental studies that quantified relationships between plant species richness and within‐sample spatial environmental heterogeneity were reviewed. Effect size in experimental studies was quantified as the standardized mean difference between control (homogeneous) and heterogeneous treatments. For observational studies, effect sizes in individual studies were examined graphically across a gradient of plot size (focal scale). Relative contributions of variables representing spatial heterogeneity were compared to those representing available energy using a response ratio. Results: Forty‐one observational and 11 experimental studies quantified plant species diversity and spatial environmental heterogeneity. Observational studies reported positive species diversity‐spatial heterogeneity correlations at all points across a plot size gradient from ～1.0 × 10^?1 to ～1.0 × 10¹¹ m², although many studies reported spatial heterogeneity variables with no significant relationships to species diversity. The cross‐study effect size in experimental studies was not significantly different from zero. Available energy variables explained consistently more of the variance in species richness than spatial heterogeneity variables, especially at the smallest and largest plot sizes. Main conclusions: Species diversity was not related to spatial heterogeneity in a way predictable by plot size. Positive heterogeneity‐diversity relationships were common, confirming the importance of niche differentiation in species diversity patterns, but future studies examining a range of spatial scales in the same system are required to determine the role of dispersal and available energy in these patterns.     

When species accumulation curves intersect: implications for ranking diversity using small samples

Diversity in biological communities frequently is compared using species accumulation curves, plotting observed species richness versus sample size. When species accumulation curves intersect, the ranking of communities by observed species richness depends on sample size, creating inconsistency in comparisons of diversity. We show that species accumulation curves for two communities are expected to intersect when the community with lower actual species richness has higher Simpson diversity (probability that two random individuals belong to different species). This may often occur when comparing communities that differ in habitat heterogeneity or disturbance, as we illustrate using data from neotropical butterflies. In contrast to observed species richness, estimated Simpson diversity always produces a consistent expected ranking among communities across sample sizes, with the statistical accuracy to confidently rank communities using small samples. Simpson diversity should therefore be particularly useful in rapid assessments to prioritize areas for conservation.     

Models and estimators linking individual-based and sample-based rarefaction,extrapolation and comparison of assemblages

Aims In ecology and conservation biology, the number of species counted in a biodiversity study is a key metric but is usually a biased underestimate of total species richness because many rare species are not detected. Moreover, comparing species richness among sites or samples is a statistical challenge because the observed number of species is sensitive to the number of individuals counted or the area sampled. For individual-based data, we treat a single, empirical sample of species abundances from an investigator-defined species assemblage or community as a reference point for two estimation objectives under two sampling models: estimating the expected number of species (and its unconditional variance) in a random sample of (i) a smaller number of individuals (multinomial model) or a smaller area sampled (Poisson model) and (ii) a larger number of individuals or a larger area sampled. For sample-based incidence (presence–absence) data, under a Bernoulli product model, we treat a single set of species incidence frequencies as the reference point to estimate richness for smaller and larger numbers of sampling units.Methods The first objective is a problem in interpolation that we address with classical rarefaction (multinomial model) and Coleman rarefaction (Poisson model) for individual-based data and with sample-based rarefaction (Bernoulli product model) for incidence frequencies. The second is a problem in extrapolation that we address with sampling-theoretic predictors for the number of species in a larger sample (multinomial model), a larger area (Poisson model) or a larger number of sampling units (Bernoulli product model), based on an estimate of asymptotic species richness. Although published methods exist for many of these objectives, we bring them together here with some new estimators under a unified statistical and notational framework. This novel integration of mathematically distinct approaches allowed us to link interpolated (rarefaction) curves and extrapolated curves to plot a unified species accumulation curve for empirical examples. We provide new, unconditional variance estimators for classical, individual-based rarefaction and for Coleman rarefaction, long missing from the toolkit of biodiversity measurement. We illustrate these methods with datasets for tropical beetles, tropical trees and tropical ants.Important findings Surprisingly, for all datasets we examined, the interpolation (rarefaction) curve and the extrapolation curve meet smoothly at the reference sample, yielding a single curve. Moreover, curves representing 95% confidence intervals for interpolated and extrapolated richness estimates also meet smoothly, allowing rigorous statistical comparison of samples not only for rarefaction but also for extrapolated richness values. The confidence intervals widen as the extrapolation moves further beyond the reference sample, but the method gives reasonable results for extrapolations up to about double or triple the original abundance or area of the reference sample. We found that the multinomial and Poisson models produced indistinguishable results, in units of estimated species, for all estimators and datasets. For sample-based abundance data, which allows the comparison of all three models, the Bernoulli product model generally yields lower richness estimates for rarefied data than either the multinomial or the Poisson models because of the ubiquity of non-random spatial distributions in nature.     

The challenge of accurately documenting bee species richness in agroecosystems: bee diversity in eastern apple orchards

Bees are important pollinators of agricultural crops, and bee diversity has been shown to be closely associated with pollination, a valuable ecosystem service. Higher functional diversity and species richness of bees have been shown to lead to higher crop yield. Bees simultaneously represent a mega‐diverse taxon that is extremely challenging to sample thoroughly and an important group to understand because of pollination services. We sampled bees visiting apple blossoms in 28 orchards over 6 years. We used species rarefaction analyses to test for the completeness of sampling and the relationship between species richness and sampling effort, orchard size, and percent agriculture in the surrounding landscape. We performed more than 190 h of sampling, collecting 11,219 specimens representing 104 species. Despite the sampling intensity, we captured <75% of expected species richness at more than half of the sites. For most of these, the variation in bee community composition between years was greater than among sites. Species richness was influenced by percent agriculture, orchard size, and sampling effort, but we found no factors explaining the difference between observed and expected species richness. Competition between honeybees and wild bees did not appear to be a factor, as we found no correlation between honeybee and wild bee abundance. Our study shows that the pollinator fauna of agroecosystems can be diverse and challenging to thoroughly sample. We demonstrate that there is high temporal variation in community composition and that sites vary widely in the sampling effort required to fully describe their diversity. In order to maximize pollination services provided by wild bee species, we must first accurately estimate species richness. For researchers interested in providing this estimate, we recommend multiyear studies and rarefaction analyses to quantify the gap between observed and expected species richness.     

Species richness in soil bacterial communities: a proposed approach to overcome sample size bias

Estimates of species richness based on 16S rRNA gene clone libraries are increasingly utilized to gauge the level of bacterial diversity within various ecosystems. However, previous studies have indicated that regardless of the utilized approach, species richness estimates obtained are dependent on the size of the analyzed clone libraries. We here propose an approach to overcome sample size bias in species richness estimates in complex microbial communities. Parametric (Maximum likelihood-based and rarefaction curve-based) and non-parametric approaches were used to estimate species richness in a library of 13,001 near full-length 16S rRNA clones derived from soil, as well as in multiple subsets of the original library. Species richness estimates obtained increased with the increase in library size. To obtain a sample size-unbiased estimate of species richness, we calculated the theoretical clone library sizes required to encounter the estimated species richness at various clone library sizes, used curve fitting to determine the theoretical clone library size required to encounter the "true" species richness, and subsequently determined the corresponding sample size-unbiased species richness value. Using this approach, sample size-unbiased estimates of 17,230, 15,571, and 33,912 were obtained for the ML-based, rarefaction curve-based, and ACE-1 estimators, respectively, compared to bias-uncorrected values of 15,009, 11,913, and 20,909.     

Rethinking patch size and isolation effects: the habitat amount hypothesis

I challenge (1) the assumption that habitat patches are natural units of measurement for species richness, and (2) the assumption of distinct effects of habitat patch size and isolation on species richness. I propose a simpler view of the relationship between habitat distribution and species richness, the ‘habitat amount hypothesis’, and I suggest ways of testing it. The habitat amount hypothesis posits that, for habitat patches in a matrix of non‐habitat, the patch size effect and the patch isolation effect are driven mainly by a single underlying process, the sample area effect. The hypothesis predicts that species richness in equal‐sized sample sites should increase with the total amount of habitat in the ‘local landscape’ of the sample site, where the local landscape is the area within an appropriate distance of the sample site. It also predicts that species richness in a sample site is independent of the area of the particular patch in which the sample site is located (its ‘local patch’), except insofar as the area of that patch contributes to the amount of habitat in the local landscape of the sample site. The habitat amount hypothesis replaces two predictor variables, patch size and isolation, with a single predictor variable, habitat amount, when species richness is analysed for equal‐sized sample sites rather than for unequal‐sized habitat patches. Studies to test the hypothesis should ensure that ‘habitat’ is correctly defined, and the spatial extent of the local landscape is appropriate, for the species group under consideration. If supported, the habitat amount hypothesis would mean that to predict the relationship between habitat distribution and species richness: (1) distinguishing between patch‐scale and landscape‐scale habitat effects is unnecessary; (2) distinguishing between patch size effects and patch isolation effects is unnecessary; (3) considering habitat configuration independent of habitat amount is unnecessary; and (4) delineating discrete habitat patches is unnecessary.     

A new index of interactivity in parasite communities.

A new index of interactivity which allows objective evaluation and comparison of interactivity in communities between different host species is presented. The index is derived from the equations for species-accumulation curves generated using non-linear regression (with the Levenberg-Marquardt algorithm) of sample infracommunity richness data. It is advantageous in that it requires only presence/absence data to calculate, is applicable to all parasite taxa (including asexual species), is largely independent of sample size and allows objective comparison of parasite communities while correcting for differences in total richness. Iterative randomisation of infracommunity richness values to generate a mean value for the index avoids spurious results which may be generated by heterogeneity in infracommunity richness and the variation this produces in the non-linear regression results.     

Effects of the number of presences on reliability and stability of MARS species distribution models: the importance of regional niche variation and ecological heterogeneity

Question: What are the effects of the number of presences on models generated with multivariate adaptive regression splines (MARS)? Do these effects vary with data quality and quantity and species ecology? Location: Spain and Ecuador. Methods: We used two data sets: (1) two trees from Spain, representing high‐occurrence number data sets with real absences and unbalanced prevalence; (2) two herbs from Ecuador, representing low‐occurrence number data sets without real absences and balanced prevalence. For model quality, we used two different measures: reliability and stability. For each sample size, different replicates were generated at random and then used to generate a consensus model. Results: Model reliability and stability decrease with sample size. Optimal minimum sample size varies depending on many factors, many of which are unknown. Regional niche variation and ecological heterogeneity are critical. Conclusions: (1) Model predictive power improves greatly with more than 18‐20 presences. (2) Model reliability depends on data quantity and quality as well as species ecological characteristics. (3) Depending on the number of presences in the data set, investigators must carefully distinguish between models that should be treated with skepticism and those whose predictions can be applied with reasonable confidence. (4) For species combining few initial presences and wide environmental range variation, it is advisable to generate several replicate models that partition the initial data and generate a consensus model. (5) Models of species with a narrow environmental range variation can be highly stable and reliable, even when generated with few presences.     

Ordered multiple comparisons with the best and their applications to dose-response studies

This article considers the problem of comparing several treatments (dose levels, interventions, etc.) with the best, where the best treatment is unknown and the treatments are ordered in some sense. Order relations among treatments often occur quite naturally in practice. They may be ordered according to increasing risks, such as tolerability or safety problems with increasing dose levels in a dose-response study, for example. We tackle the problem of constructing a lower confidence bound for the smallest index of all treatments being at most marginally less effective than the (best) treatment having the largest effect. Such a bound ensures at confidence level 1 -alpha that all treatments with lower indices are relevantly less effective than the best competitor. We derive a multiple testing strategy that results in sharp confidence bounds. The proposed lower confidence bound is compared with those derived from other testing strategies. We further derive closed-form expressions for power and sample size calculations. Finally, we investigate several real data sets to illustrate various applications of our methods.     

Helminth parasite richness among vertebrates

The numbers of intestinal helminth species (parasite richnesS) recorded from each of 488 vertebrate host species are compared using data compiled from the published literature. Associations between parasite richness, sampling effort, host size and host habitat (aquatic versus terrestrial) are assessed using a method designed to control for phylogenetic association. Parasite richness increases with the number of surveys on which each estimate of parasite richness is based (sampling effort). When the effects of sampling effort are controlled for, there remains a strong positive relationship between parasite richness and host body size. There is no tendency for aquatic hosts to harbour more parasite species than terrestrial hosts independently of differences in sampling effort and body size. The results are interpreted in the context of hosts representing habitats for parasite colonization, resource allocation between parasite species, and the age of the major mammalian radiations.     

Life history, ecology and parasite community structure in Soviet birds

The paper describes an investigation of parasite richness in relation to host life history and ecology using data from an extensive survey of helminth parasites (cestodes, trematodes and nematodes) in Soviet birds. Correlates of parasite richness (number of parasite species per host species) were sought among 13 life-history variables, 13 ecological variables and one non-biological variable (number of host individuals examined) across a sample of 158 species of host. A statistical method to control for the effects of phylogenetic association was adopted throughout. Parasite richness correlates positively with the number of hosts examined (sample size) in all three parasite groups. Positive correlations (after controlling for the effects of sample size) were also found between host body weight and parasite richness for trematodes and nematodes, but not for cestodes.
A number of ecological variables were associated with parasite richness. However, when the effects of sample size and body weight were controlled for, only a single significant correlation (an association between trematode richness and aquatic habitat) remained. Similarly, a number of significant correlates of parasite richness were found among the life-history variables examined. Though several of these were robust to the confounding effects of sample size, all could be explained by the co-variation between life-history traits and body weight among the host species under investigation.     

Uncertainty in detecting the disturbance history of forest ecosystems using dendrochronology

QuestionsUncertainty in detecting disturbance histories has long been ignored in dendrochronological studies in forest ecosystems. Our goal was to characterize this uncertainty in relation to the key parameters of forest ecosystems and sample size. In addition, we aimed to provide a method to define uncertainty bounds in specific forest ecosystems with known parameters, and to provide a required (conservative) minimal sample size to achieve a pre-defined level of uncertainty if no actual key forest parameters are known.LocationTraining data were collected from Žofínský Prales (48°40′N, 14°42′E, 735–830 m a.s.l., granite, Czech Republic).MethodsWe used probability theory and expressed uncertainty as the length (the difference between the upper and lower bounds) of the 95% confidence interval. We studied the uncertainty of (i) the initial growth of trees – if they originated under canopy or in a gap; and (ii) the responses to disturbance events during subsequent growth – on the basis of release detection in the radial growth of trees. These two variables provide different information, which together give a picture of the disturbance history. While initial growth date the existence of a gap in a given decade (recent as well as older gaps are included), release demonstrates the moment of a disturbance event.ResultsWith the help of general mathematical deduction, we have obtained results valid across vegetation types. The length of a confidence interval depends on the sample size, proportion of released trees in a population, as well as on the variability of tree layer features (e.g., crown area of suppressed and released trees).ConclusionsMost studies to date have evaluated the initial growth of trees with higher uncertainty than for canopy disturbed area. The length of the 95% confidence interval for detecting initial growth has been rarely shorter than 0.1 (error ± 5%) and has mostly been much longer. To reach 95% confidence interval length of 0.1 (error ± 5%) when detecting the canopy disturbed area, at least 485 tree cores should be evaluated in studied time period, while to reach a 0.05 interval length (error ± 2.5%) at least 1925 tree cores are required. Our approach can be used to find the required sample size in each specific forest ecosystem to achieve pre-defined levels of uncertainty while detecting disturbance history.     

No evidence that sexual selection is an 'engine of speciation' in birds

Abstract Sexual selection has been implicated as having a role in promoting speciation, as it should increase the rate of evolution of reproductive isolation, and there is some comparative evidence that sexual selection may be related to imbalances in clade size seen in resolved phylogenies. By employing a new comparative method we are able to investigate the role of sexual selection in explaining the patterns of species richness across birds. We used data for testes size as an index of post‐mating sexual selection, and sexual size dimorphism and sexual dichromatism as indices of pre‐mating sexual selection. These measures were obtained for 1031 species representing 467 genera. None of the variables investigated explained the patterns of species richness. As sexual selection may also increase extinction rates, the net effect on species richness in any given clade will depend on the balancing effects of sexual selection upon speciation and extinction rates. We suggest that variance across clades in this balance may have resulted in the lack of a relationship between species richness and sexual selection seen in birds.     

Local versus landscape-scale effects of anthropogenic land-use on forest species richness

The study investigated the effects of human-induced landscape patterns on species richness in forests. For 80 plots of fixed size, we measured human disturbance (categorized as urban/industrial and agricultural land areas), at ‘local’ and ‘landscape’ scale (500 m and 2500 m radius from each plot, respectively), the distance from the forest edge, and the size and shape of the woody patch. By using GLM, we analyzed the effects of disturbance and patch-based measures on both total species richness and the richness of a group of specialist species (i.e. the ‘ancient forest species’), representing more specific forest features. Patterns of local species richness were sensitive to the structure and composition of the surrounding landscape. Among the landscape components taken into account, urban/industrial land areas turned out as the most threatening factor for both total species richness and the richness of the ancient forest species. However, the best models evidenced a different intensity of the response to the same disturbance category as well as a different pool of significant variables for the two groups of species. The use of groups of species, such as the ancient forest species pool, that are functionally related and have similar ecological requirements, may represent an effective solution for monitoring forest dynamics under the effects of external factors. The approach of relating local assessment of species richness, and in particular of the ancient forest species pool, to land-use patterns may play an important role for the science-policy interface by supporting and strengthening conservation and regional planning decision making.     

Drivers of symbiont diversity in freshwater snails: a comparative analysis of resource availability,community heterogeneity,and colonization opportunities

Decades of community ecology research have highlighted the importance of resource availability, habitat heterogeneity, and colonization opportunities in driving biodiversity. Less clear, however, is whether a similar suite of factors explains the diversity of symbionts. Here, we used a hierarchical dataset involving 12,712 freshwater snail hosts representing five species to test the relative importance of potential factors in driving symbiont richness. Specifically, we used model selection to assess the explanatory power of variables related to host species identity, resource availability (average body size, host density), ecological heterogeneity (richness of hosts and other taxa), and colonization opportunities (wetland size and amount of neighboring wetland area) on symbiont richness in 146 snail host populations in California, USA. We encountered a total of 23 taxa of symbionts, including both obligatory parasites such as digenetic trematodes as well as more commensal, mutualistic, or opportunistic groups such as aquatic insect larvae, annelids, and leeches. After validating richness estimates per host population using species accumulative curves, we detected positive effects on symbiont richness from host body size, total richness of the aquatic community, and colonization opportunities. Neither snail density nor the richness of snail species accounted for significant variation in symbiont diversity. Host species identity also affected symbiont richness, with higher gamma and average alpha diversity among more common host species with higher local abundances. These findings highlight the importance of multiple, concurrent factors in driving symbiont richness that extend beyond epidemiological measures of host abundance or host diversity alone.