Abstract
Classical gradient analysis continues to be used to explore and test theories and models in community ecology. Yet the foundations of classical gradient analysis were developed at a time when computational power was limited, relative to current computational power. I argue that this history has left a lasting legacy on the field. Consequently, many gradient analyses do not to take advantage of current computer technology. Here I show how to use computationally intensive Markov-chain Monte Carlo methods to improve gradient analyses of presence–absence community data. The methods that I use were developed by quantitative social scientists in the early 1990s, and therefore tested and efficient software already exists for practical data analysis. As an example, I analyze the classic dune meadow vegetation data. A main advantage of the Bayesian approach to indirect gradient analysis is that, unlike essentially all classical indirect methods, it is able to make empirically testable probabilistic predictions of observed species occurrence patterns. The Bayesian approach also poses challenges for statistical ecology. In particular, the development of Markov-chain Monte Carlo methods for a wider class of Bayesian indirect gradient analysis models would permit more flexible approaches to generating probabilistic predictions.
Similar content being viewed by others
Notes
Proof: \({\mathbf {XQQ}}^\top {\mathbf {B}}^\top = {\mathbf {XIB}}^\top = {\mathbf {XB}}^\top\). The first equality is a property of orthogonal matrices and the second is a property of the identity matrix, \({\mathbf {I}}\).
References
Albert A, Anderson J (1984) On the existence of maximum likelihood estimates in logistic regression models. Biometrika 71(1):1–10
Albert JH (1992) Bayesian estimation of normal ogive item response curves using Gibbs sampling. J Educ Stat 17(3):251–269
Austin MP (1987) Models for the analysis of species’ response to environmental gradients. Vegetatio 69:35–45
Bafumi J, Gelman A, Park DK, Kaplan N (2005) Practical issues in implementing and understanding Bayesian ideal point estimation. Polit Anal 13:171–187
Baker FB (1992) Item response theory: parameter estimation techniques. Marcel Dekker, New York
Batterink M, Wijffels G (1983) Ein vergelijkend vegetatiekundig onderzoek naar de typologie en invloeden van het beheer van 1973 tot 1982 in de duinweilanden op Terschelling. Report Agricultural University,Department of Vegetation Science, Plant Ecology and Weed Science, Wageningen
Bolker BM (2008) Ecological models and data in R. Princeton University Press, Princeton
Bray JR, Curtis J (1957) An ordination of the upland forest communities of southern Wisconsin. Ecol Monogr 27(4):326–349
Brown JH, Gillooly JF, Allen AP, Savage VM, West GB (2004) Toward a metabolic theory of ecology. Ecology 85(7):1771–1789
Clark JS (2005) Why environmental scientists are becoming Bayesians. Ecol Lett 8:2–14
Clark JS (2007) Models for ecological data. Princeton University Press, Princeton
Clark JS, Gelfand AE, Woodall CW, Zhu K (2014) More than the sum of the parts: forest climate response from joint species distribution models. Ecol Appl 24(5):990–999
Cottenie K (2005) Integrating environmental and spatial processes in ecological community dynamics. Ecol Lett 8:1175–1182
Dorazio RM, Royle JA (2005) Estimating the size and composition of biological communities by modeling the occurrence of species. J Am Stat Assoc 100(470):389–398
Duong T (2014) ks: kernel smoothing R package version 1.9.3 http://CRAN.R-project.org/package=ks
Fournier DA, Skaug HJ, Ancheta J, Ianelli J, Magnusson A, Maunder MN, Nielsen A, Sibert J (2012) AD model builder: using automatic differentiation for statistical inference of highly parameterized complex nonlinear models. Optim Methods Softw 27:233–249
Gauch H, Whittaker R (1972) Coenocline simulation. Ecology 53(3):446–451
Gauch HG (1982) Multivariate analysis in community ecology. Cambridge University Press, Cambridg
Gauch HG, Chase GB, Whittaker RH (1974) Ordination of vegetation samples by Gaussian species distributions. Ecology 55(6):1382–1390
Gelman A, Meng XL, Stern H (1996) Posterior predictive assessment of model fitness via realized discrepancies. Stat Sin 6:733–807
Guisan A, Edwards TC, Hastie T (2002) Generalized linear and generalized additive models in studies of species distributions: setting the scene. Ecol Model 157:89–100
Harris DJ (2015) Building realistic assemblages with a joint species distribution model. Method Ecol Evol, online early
Hubbell SP (2001) The unified neutral theory of biodiversity and biogeography, monographs in population biology, vol 32. Princeton University Press, Princeton
Hui FKC, Taskinen S, Pledger S, Foster SD, Warton DI (2014) Model-based approaches to unconstrained ordination. Method Ecol Evol. doi:10.1111/2041-210X.12236
Ives AR, Helmus MR (2011) Generalized linear mixed models for phylogenetic analyses of community structure. Ecol Monogr 81:511–525
Jackson DA, Harvey HH (1997) Qualitative and quantitative sampling of lake fish communities. Can J Fish Aquat Sci 54:2807–2813
Jackson DA, Somers KM (1991) Putting things in order: the ups and downs of detrended correspondence analysis. Am Nat 137(5):704–712
Kooperberg, C (2013) logspline: Logspline density estimation routines. R package version 2.1.5 http://CRAN.R-project.org/package=logspline
Legendre P, Legendre L (2012) Numerical ecology, 3 English edn. Elsevier, Amsterdam
Leibold M, Holyoak M, Mouquet N, Amarasekare P, Chase J, Hoopes M, Holt R, Shurin J, Law R, Tilman D, Loreau M, Gonzalez A (2004) The metacommunity concept: a framework for multi-scale community ecology. Ecol Lett 7:601–613
Lord F (1952) A theory of test scores. Psychometric Society, New York
Lord F, Novick M (1968) Statistical theories of mental test scores. Addison-Wesley, Reading
Loreau M, Naeem S, Inchausti P, Bengtsson J, Grime J, Hector A, Hooper D, Huston M, Raffaelli D, Schmid B, Tilman D, Wardle D (2001) Biodiversity and ecosystem functioning: current knowledge and future challenges. Science 294:804–808
Lunn DJ, Thomas A, Best N, Spiegelhalter D (2000) WinBUGS—a Bayesian modelling framework: concepts, structure, and extensibility. Stat Comput 10:325–337
Martin AD, Quinn KM, Park JH (2009) MCMCpack: Markov chain Monte Carlo in R. J Stat Softw 42(9):1–22
McGill BJ, Enquist BJ, Weiher E, Westoby M (2006) Rebuilding community ecology from functional traits. Trends Ecol Evol 21(4):178–185
Minchin PR (1987) An evaluation of the relative robustness of techniques for ecological ordination. Vegetatio 69:89–107
Nelder JA, Wedderburn RWM (1972) Generalized linear models. J R Stat Soc Ser A 135:370–384
Ovaskainen O, Hottola J, Siitonen (2010) Modeling species co-occurrence by multivariate logistic regression generates new hypotheses on fungal interactions. Ecology 91(9):2514–2521
Ovaskainen O, Soininen J (2011) Making more out of sparse data: hierarchical modeling of species communities. Ecology 92(2):289–295
Oksanen J, Guillaume BF, Kindt R, Legendre P, Minchin PR, O’Hara RB, Simpson GL, Solymos P, Stevens MHH, Wagner H (2013) vegan: Community Ecology Package. R package version 2.0-10 http://CRAN.R-project.org/package=vegan
Plate T, Heiberger R (2011) abind: Combine multi-dimensional arrays R package version 1.4-0 http://CRAN.R-project.org/package=abind
Pledger S, Arnold R (2014) Multivariate methods using mixtures: correspondence analysis, scaling and pattern-detection. Comput Stat Data Anal 71:241–261
Plummer M (2003) JAGS: a program for analysis of Bayesian graphical models using Gibbs sampling. Proceedings of the 3rd International Workshop on Distributed Statistical Computing
Plummer M, Best N, Cowles K, Vines K (2009) coda: output analysis and diagnostics for MCMC. R package version 0.13-4
Pollock LJ, Morris WK, Vesk PA (2012) The role of functional traits in species distributions revealed through a hierarchical model. Ecography 35:716–725
Pollock LJ, Tingley R, Morris WK, Golding N, O’Hara RB, Parris KM, Vesk PA, McCarthy MA (2014) Understanding co-occurrence by modelling species simultaneously with a joint species distribution model (JSDM). Method Ecol Evol 5:397–406
R Development Core Team (2014) R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, http://www.R-project.org/
Schnakenberg KE, Fariss CJ (2014) Dynamic patterns of human rights practices. Polit Sci Res Methods 2:1–31
Schönemann PH (1966) A generalized solution of the orthogonal Procrustes problem. Psychometrika 31:1–10
Stan Development Team (2014) Stan: a C++ library for probability and sampling version 2.5.0 http://mc-stan.org/
Swan J (1970) An examination of some ordination problems by use of simulated vegetational data. Ecology 51(1):89–102
ter Braak CJF (1985) Correspondence analysis of incidence and abundance data: properties in terms of a unimodal response model. Biometrics 41(4):859–873
ter Braak CJF, Prentice IC (1988) A theory of gradient analysis. Adv Ecol Res 18:271–317
Venables WN, Ripley BD (2002) Modern applied statistics with S. Springer, New York
Walker SC, Jackson DA (2011) Random-effects ordination: describing and predicting multivariate correlations and co-occurrences. Ecol Monogr 81(4):635–663
Warton DI (2011) Regularized sandwich estimators for analysis of high-dimensional data using generalized estimating equations. Biometrics 67:116–123
Warton DI, Foster SD, De’ath G, Stoklosa J, Dunstan PK (2014) Model-based thinking for community ecology. Plant Ecol. doi:10.1007/s11258-014-0366-3
Webb CO, Ackerly DD, McPeek MA, Donoghue MJ (2002) Phylogenies and community ecology. Annu Rev Ecol Syst 33:475–505
Whittaker R (1967) Gradient analysis of vegetation. Biol Rev 42:207–264
Wickham H (2007) Reshaping data with the reshape Package. J Stat Softw 21(12):1–20
Wickham H (2009) ggplot2: elegant graphics for data analysis. Springer, New York
Wickham H (2011) The split-apply-combine strategy for data analysis. J Stat Softw 40(11):1–29
Yee TW (2004) A new technique for maximum-likelihood canonical gaussian ordination. Ecol Monogr 74(4):685–701
Yee TW (2010) The VGAM package for categorical data analysis. J Stat Softw 32(10):1–34
Yee TM, Hadi AF (2014) Rowcolumn interaction models, with an R implementation. Comput Stat 29(6):1427–1445
Acknowledgments
I thank Donald Jackson, Marie-Josee Fortin, Ben Bolker, Laura Timms, Keith Somers and Nick Collins for comments on various earlier drafts. I gratefully acknowledge funding from the Natural Sciences and Engineering Research Council of Canada (NSERC) and an Ontario Graduate Scholarship (OGS).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Peter R. Minchin and Jari Oksanen.
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Walker, S.C. Indirect gradient analysis by Markov-chain Monte Carlo. Plant Ecol 216, 697–708 (2015). https://doi.org/10.1007/s11258-015-0467-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11258-015-0467-7