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1m78OSu} BEe C Appl. This option has the advantages of leveraging all observed reads, comparing estimates of the actual parameter of interest (taxonomic richness), and accounting for experimental noise. The library sizes can dominate the biology in determining the result of the diversity analysis (Lande, 1996). The unique property of microbiome experiments and alpha diversity analysis is that samples do not faithfully represent the entire microbial community under study. However, this is not generally true, because environments can be identical with respect to one alpha diversity metric, but the different abundance structures will induce different biases when rarefied. Copyright 2019 Willis. If the measurement error on the machine was random (e.g., with 0 mean and variance of 1 unit for all amendments), this would not affect any particular amendment. Alpha diversity metrics summarize the structure of an ecological community with respect to its richness (number of taxonomic groups), evenness (distribution of abundances of the groups), or both. The same is not true for other alpha diversity metrics. In meta-analyses, larger studies need to be given more weight in determining the overall effect size, and this is incorporated into a meta-analysis via the smaller standard errors on the effect size estimates. The author also thanks Thea Whitman and two referees for many thoughtful suggestions on the manuscript. To illustrate, consider the following example where the alpha diversity metric of interest is strain-level richness of a microbial community (the total number of strain variants present in the environment). The author is grateful to Berry Brosi, the MBL, the STAMPS course directors, and the STAMPS participants for countless discussions on this topic. I then take a sample from Environment B, count the number of different taxa in that sample, and compare it to the number of taxa in Environment A. I am likely to observe higher numbers of different taxa in the sample with more microbial reads. If the variance in the measurement error was 1 unit for amendment A but 5 units for amendment B, we would similarly adjust with a measurement error model. Estimating the number of species in microbial diversity studies. I describe the state of the statistical literature for addressing these problems, focusing on the analysis of microbial diversity. This article is based on course notes presented by the author at the Marine Biological Laboratory at the STAMPS course in 2013, 2014, 2015, 2016, 2017, and 2018. The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. doi: 10.1080/10485252.2016.1190357, Keywords: bioinformatics, computational biology, ecological data analysis, latent variable model, reproducibility, measurement error, Citation: Willis AD (2019) Rarefaction, Alpha Diversity, and Statistics. Nat. 11, 16. The first practice is using biased estimates of alpha diversity indices. 0000002526 00000 n
12:42. doi: 10.2307/1411, Hurlbert, S. H. (1971). 11, 19641974. Hoboken, NJ: Wiley-Interscience. The set-up where an estimate of a quantity converges to the correct value as more samples are obtained is also well understood in statistics. Unfortunately, we do not have knowledge of every microbe. 0000000016 00000 n
doi: 10.1111/rssc.12206, Willis, A. D., and Martin, B. D. (2018). 0000001988 00000 n
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Bell Syst. Am. doi: 10.1214/16-AOAS944, Arora, T., Seyfried, F., Docherty, N. G., Tremaroli, V., le Roux, C. W., Perkins, R., et al. {tX1cw'BjDEA&?f50~|Q doi: 10.1111/biom.12332, Willis, A. D., Bunge, J., and Whitman, T. (2016). (2016). There are currently two commonly used methods for comparing alpha diversity. 151 0 obj<>stream
Modeling parameters observed with estimation error is not a new suggestion: this approach is from the field of statistical meta-analysis, where the results of multiple studies estimating the same effect size is compared (Demidenko, 2004; Willis et al., 2016; Washburne et al., 2018). Alpha diversity could be compared exactly, because we would know entire microbial populations with perfect precision. 0000007248 00000 n
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However, there are two incorrect practices surrounding alpha diversity that are preventing the uptake of statistically-motivated methodologies. 0000011550 00000 n
Bias in Estimating and Comparing Alpha Diversity, Creative Commons Attribution License (CC BY), Department of Biostatistics, University of Washington, Seattle, WA, United States. Appl. Adjusting for sample size when comparing different groups of observations without discarding data is widely prevalent in the sciences, and discarding data to adjust for unequal sample sizes is the exception. Measurement Error and Variance in Microbiome Studies, 3. Biometrics 58, 531539.
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However, richness estimation has a well-studied statistical literature, and richness estimators that are adapted to microbiome data exist (see Bunge et al., 2014 for a review). The second method is to generate a normalized, or rarefied sample by randomly discarding reads from all samples until each sample has nA1 reads (the number of reads in the smallest sample), Figure 1C. 1, 427445. endstream
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In the setting of Figure 1A, this leads to the erroneous conclusion that Environment A has lower richness than Environment B. endstream
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To decide if measurement error must be accounted for when observations are made in an experiment, it is necessary to consider the effect of repeating the observational process on the same experimental unit. The resulting rarefied richness levels are then cA1, cA2, cB1, and cB2. doi: 10.1101/305045, Zhang, Z., and Grabchak, M. (2016). I argue that latent variable models can address issues with variance, but bias corrections need to be utilized as well. Ecol. Received: 19 August 2019; Accepted: 07 October 2019; Published: 23 October 2019. Z., Peddada, S., Amir, A., Bittinger, K., Gonzalez, A., et al. Furthermore, this discussion applies equally to diversity analyses performed at the strain, species, or other taxonomic level. 0000006078 00000 n
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Unfortunately, determining how to meaningfully estimate and compare alpha diversity is not trivial. 0000014469 00000 n
This manuscript has been released as a preprint via bioRxiv (Willis, 2017). This leads to the conclusion that Environment A and Environment B do not have significantly different richnesses, and the estimates of richness are far below the actual richnesses of each ecosystem (there is substantial negative bias in the estimates), prohibiting comparison of richness across different experiments. Unfortunately, rarefaction is neither justifiable nor necessary, a view framed statistically by McMurdie and Holmes (2014) in the context of comparison of relative abundances. This is sometimes justified by claiming that rarefied estimates are equally biased. Ann. J. We currently do not account for measurement error in microbial diversity studies. Interactions between soil- and dead wood-inhabiting fungal communities during the decay of Norway spruce logs. We take samples from environments, and investigate the microbial community present in the sample. Estimating diversity via frequency ratios. 10:2407. doi: 10.3389/fmicb.2019.02407. I encourage microbial ecologists to use estimates of alpha diversity that account for unobserved species, and to use the variance of the estimates in measurement error models to compare diversity across ecosystems. Front. 0000048969 00000 n
ISME J. The samples are not of particular interest, except that they reflect the environment from which they were sampled. In the case where the environments have equal richness (Figures 1EH), this approach correctly detects equal richness, even when the abundance structures differ. Based on these subsamples of equal size, diversity metrics can be calculated that can contrast ecosystems fairly, independent of differences in sample sizes (Weiss et al., 2017). xref
doi: 10.2307/3545743, Lande, R., DeVries, P. J., and Walla, T. R. (2000). Suppose I conduct an experiment in which I take a sample from Environment A and count the number of different microbial taxa present in my sample. 0000028841 00000 n
Environ. While the example discussed here is richness, this approach to estimating and comparing alpha diversity using a bias correction (incorporating unobserved taxa) and a variance adjustment (measurement error model) could apply to any alpha diversity metric. H|TKs0W%q-t:2(-ulJ8aj_rc6vCr&o[mOr9.-r* Stat. To assess if the amendments affect the flux, we would fit a regression-type model (such as ANOVA) to flux with amendment as an explanatory variable. However, detecting a difference between the effects of amendment on flux would be more challenging statistically: we would require more samples to detect a true difference compared to the case without measurement error. 0000001884 00000 n
To illustrate this distinction, I contrast microbial diversity experiments with a non-microbial experiment. 0000004938 00000 n
11, 20352046. These estimates are then used for modeling and hypothesis testing (see, for example, Arora et al., 2017). Similarly, when comparing the response of different treatment groups in clinical trials, the number of subjects in each treatment group is accounted for in a comparison of the overall treatment effect. Because many perturbations to a community affect the alpha diversity of a community, summarizing and comparing community structure via alpha diversity is a ubiquitous approach to analyzing community surveys. The first method, Figure 1B, is to use the estimates cA1, cA2, cB1, and cB2, and perform modeling and hypothesis testing (such as ANOVA) as if both the bias and variance of these estimates were zero (see, for example, Makipaa et al., 2017). 0000004882 00000 n
For example, the Chao-Bunge (Chao and Bunge, 2002) and breakaway (Willis and Bunge, 2015) estimators of taxonomic richness provide variance estimates, account for unobserved taxa, and are not overly sensitive to the singleton count (the number of species observed once). I describe statistical methodology for alpha diversity analysis that adjusts for missing taxa, which should be used in place of existing common approaches to diversity analysis in ecology. There is unadjusted error in using our samples as proxies for the entire community. We would adjust for the measurement error by adding 5 units to each measurement before comparing them. 0000003702 00000 n
0
In order to draw meaningful conclusions about the entire microbial community, it is necessary to adjust for inexhaustive sampling using statistically-motivated parameter estimates for alpha diversity. PLoS Comput. Methods for phylogenetic analysis of microbiome data. doi: 10.1002/0471728438, Fisher, R. A., Corbet, A. S., and Williams, C. B. Imagine that we had complete knowledge of every microbe in existence, including identity, abundance and location. (2017). Furthermore, not all information collected from the samples was used in making the comparison. doi: 10.1023/A:1026096204727, Demidenko, E. (2004). 0000005481 00000 n
I introduce a statistical perspective on the estimation of alpha diversity, and argue that a common view of diversity indices is causing fundamental issues in comparing samples. Plug-in estimates of many alpha diversity indices (including richness and Shannon diversity) are negatively biased for the environment's alpha diversity parameter, that is, they underestimate the true alpha diversity (Lande, 1996). Rarefying samples to the same number of reads can also lead to incorrect conclusions (C,G). Oikos 89, 601605. %%EOF
Statistics and partitioning of species diversity, and similarity among multiple communities. J. R. Stat. QKjhjZF`N_$ xOV 10, 14961516. Annu. *7]9rQ(_Eh%;K) [8)JR=W-&z%/q b<5mD:;3[\.z6H-Aa&9WD\h+(*0,8OuNOd*B&jr'J
V ^o |o7\;lW N6p*n:K;tK{ DG%9gHs6 Implicitly, this model acknowledges that we can assess the flux with high precision; that is, the margin for error for determining flux is negligible. Because technical replicates in microbiome experiments yield different numbers of reads, different community compositions, and different levels of alpha diversity, we have measurement error in microbial experiments. Ecol. While measurement error in microbiome studies affects all analyses of microbiome data, alpha diversity is particularly affected because commonly used estimates of alpha diversity are heavily biased compared to other estimation problems in microbial ecology (such as estimating relative abundances). Arbel, J., Mengersen, K., and Rousseau, J. 102, 243282. bioRxiv 123. 28, 563575. Marine benthic diversity: a comparative study. ^vB+
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To account for the additional experimental noise, we would use a model that would account for measurement error in assessing differences between amendments. HlT0W(RpL*T\RqB@gF1USKbnw^~0Zlr%,t`S&Jf{K@M`| ! Normalization and microbial differential abundance strategies depend upon data characteristics. In this way, both sample richness and rarefied richness are driven by artifacts of the experiment (library size), and not purely the microbial community structure. Bayesian nonparametric dependent model for partially replicated data: the influence of fuel spills on species diversity. doi: 10.1371/journal.pcbi.1003531, Sanders, H. L. (1968). The strategy outlined here for modeling richness after adjusting for missing species adjusts for both bias and variance, thus accounting for library size differences and incomplete microbial surveys. Stat. ISME J. 166KK@D$ISuH@IIY+2f#P+c1pY m@].iiNsAl-mtD Observing small samples from a large population is not an experimental set-up unique to microbial ecology: it is almost universal in statistics. 119 33
Comparing sample taxonomic richness can therefore often lead to incorrect conclusions about true richness (B,F). 10, 429443. ISME J. The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. (2002). %PDF-1.6
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This means that as we increase sampling, our calculation of any diversity metric [e.g., richness (Fisher et al., 1943), Shannon index (Shannon, 1948), and Simpson index (Simpson, 1949)] approaches the value of that diversity metric as calculated using the entire population. Extensive literature discusses different methods for describing diversity and documenting its effects on ecosystem health and function. Let cij be the observed richness of environment i on replicate j. The second practice is treating alpha diversity estimates as precisely observed quantities that do not have measurement error. doi: 10.1038/ismej.2017.70, PubMed Abstract | CrossRef Full Text | Google Scholar, Bunge, J., Willis, A., and Walsh, F. (2014). .b lVehxWr=y3(o!!Mwzom9Wg6R.c-x.-s@Pd3'77h(Cpz \u But what happens when we have random measurement error? To compare microbial diversity, we would define specific environments (e.g., the distal gut of women aged 35 living in the contiguous U.S.) and compare diversity metrics across different ecological gradients (e.g., with or without irritable bowel syndrome diagnoses). (2018). Soc. H|TMo0W4CHa-KBev#Hv'Oo8b+Zrvb-Q%
Z97C8z:.wW>Co\8 doi: 10.1111/j.0006-341X.2002.00531.x, Chao, A., and Shen, T.-J. Consider the setting in Figure 1A, where we are investigating 2 different environments, and Environment A's richness (call it CA) is higher than Environment B's richness (CB). <<9BA3DB7AEA9C6C4BA71E5272DAA5A3D1>]>>
Microbiol. To this criticism, I add misapplying statistical tools is undermining many analyses of alpha diversity. jt$gZ
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The editor and reviewer's affiliations are the latest provided on their Loop research profiles and may not reflect their situation at the time of review. (2003). Nat. endstream
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No use, distribution or reproduction is permitted which does not comply with these terms. Mixed Models: Theory and Applications. Rarefaction is a method that adjusts for differences in library sizes across samples to aid comparisons of alpha diversity. While the example employed here concerns microbial richness, the same argument applies to macroecological richness, as well as other alpha diversity indices. doi: 10.1002/j.1538-7305.1948.tb01338.x, Simpson, E. H. (1949). Microbiome 5:27. doi: 10.1186/s40168-017-0237-y, Willis, A. While alpha diversity estimation for microbiomes is an active area of research in statistics (Arbel et al., 2016; Zhang and Grabchak, 2016; Willis and Martin, 2018), there remain many features of microbial ecosystems (such as crosstalk between samples and spatial organization of microbes) that are not yet incorporated into statistical methodology for alpha diversity estimation. I encourage ecologists to use estimates of diversity that account for unobserved species, and to use measurement error models to compare diversity across ecosystems. 3:652. doi: 10.1038/s41564-018-0156-0, Weiss, S., Xu, Z. Waste not, want not: why rarefying microbiome data is inadmissible. Without measurement error in the observations, we would consistently observe the same flux measurement, while if we had random measurement error, we would most likely observe slightly different flux measurements. In microbial ecology, analyzing the alpha diversity of amplicon sequencing data is a common first approach to assessing differences between environments. Suppose we have two biological replicates of samples from each environment: nA1 and nA2 reads from Environment A, nB1 and nB2 reads from Environment B, and nA1 < nB1 < nA2 < nB2. gh78?PFj#HfHi:?hsk8f`i9Xjgry2I0o4)~CKCa*s~]Ir$&z4
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Understanding the drivers of diversity is a fundamental question in ecology. When species accumulation curves intersect: implications for ranking diversity using small samples. It has recently been argued that studying microbial diversity without context is distracting us from gaining insight into ecological mechanisms (Shade, 2016). J. Nonparametr. 0000013146 00000 n
Nature 163:688. doi: 10.1038/163688a0, Washburne, A. D., Morton, J. T., Sanders, J., McDonald, D., Zhu, Q., Oliverio, A. M., et al. Expected sample taxonomic richness increases with number of reads (A,E). While the focus of the examples is microbiome data analysis, the issues and discussion are equally applicable to macroecological data analysis. doi: 10.1038/ismej.2016.118, Shannon, C. E. (1948). xb```b``Qa`e`` l,|{5,A/tXxf=~** 6" .}|oyzYETY_?#2eCStfi~4A}`i6N6*tlljQ4GT6.G{Dd\jb3_K%MU(^%P-|%)Hp(Zz.@5@JxY@at!k[d4\N,IX)ar"SKk1. (1943). (2016). J. Anim. Without advocating for any particular model of microbial sampling, I suggest a general approach to comparing microbial diversity, one which accounts for uncertainty in estimating diversity metrics. However, it is widely believed that diversity depends on the intensity of sampling. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). 0000012940 00000 n
doi: 10.1101/231878, Willis, A., and Bunge, J. (2017). !Fh{T$zCwJR?Oh,zy,UQ[vb]2A 0000007622 00000 n
Stat. The nonconcept of species diversity: a critique and alternative parameters. 4k^p
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Tech. Despite this, alpha diversity estimates that account for unobserved taxa and provide variance estimates are vastly preferable to both plug-in and rarefied estimates, which do not account for unobserved taxa nor provide variance estimates. Diversity is the question, not the answer. As we sample more and more of the environment using larger samples, we get closer to understanding the true and total microbial community of interest. AW wrote the manuscript and performed the data analysis. 0000005354 00000 n
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Stat. 27, 379423. (2015). The relation between the number of species and the number of individuals in a random sample of an animal population. Biol. (2017). Biometrics 71, 10421049. Entropic representation and estimation of diversity indices. Montana State University System, United States. startxref
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Here I advocate for a third strategy: adjust the sample richness of each ecosystem by adding to it an estimate of the number of unobserved species, estimate the variance in the total richness estimate, and compare the diversities relative to these errors (Figure 1D). 66, 963977. 0000005143 00000 n
Microbiol. In the flux experiment, this would involve measuring the flux of the same soil sites again using the same experimental conditions. Now suppose we knew that our flux-measuring machine consistently underestimated flux by exactly 5 units. We use our findings about the sample to draw inferences about the environment that we are truly interested in. 0000015025 00000 n
bioRxiv 18. 10:e1003531. To clarify this discussion, I will focus on taxonomic richness (the simplest case), and later generalize the argument to other alpha diversity metrics. I discuss a statistical perspective on diversity, framing the diversity of an environment as an unknown parameter, and discussing the bias and variance of plug-in and rarefied estimates. Nonparametric estimation of Shannon's index of diversity when there are unseen species in sample. Rev. Estimating the number of species in a stochastic abundance model. However, since estimates for alpha diversity metrics are heavily biased when taxa are unobserved, comparing alpha diversity using either raw or rarefied data should not be undertaken. 0000009023 00000 n
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AF`F=g`, Attempting to address this problem using rarefaction actually induces more bias. For example, Figure 1E shows two environments with different abundance structures but equal richness; rarefying gives the false impression of unequal richness (see also Lande et al., 2000).