Molecular Ecology (2014) 23, 2887–2889
NEWS AND VIEWS
PERSPECTIVE
All models are wrong MICHAEL J. HICKERSON*† *Department of Biology, City College of New York, 160 Convent Ave., MR 526, New York, NY 10031, USA, †The Graduate Center of the City University of New York, Subprogram in Ecology, Evolution and Behavior, 365 5th Ave, New York, NY 10016, USA
Abstract As the field of phylogeography has continued to move in the model-based direction, researchers continue struggling to construct useful models for inference. These models must be both simple enough to be tractable yet contain enough of the complexity of the natural world to make meaningful inference. Beyond constructing such models for inference, researchers explore model space and test competing models with the data on hand, with the goal of improving the understanding of the natural world and the processes underlying natural biological communities. Approximate Bayesian computation (ABC) has increased in recent popularity as a tool for evaluating alternative historical demographic models given population genetic samples. As a thorough demonstration, Pelletier & Carstens (2014) use ABC to test 143 phylogeographic submodels given geographically widespread genetic samples from the salamander species Plethodon idahoensis (Carstens et al. 2004) and, in so doing, demonstrate how the results of the ABC model choice procedure are dependent on the model set one chooses to evaluate. Keywords: amphibians, model choice, phylogeography, population genetics – empirical, population genetics – theoretical Received 26 April 2014; revised 9 May 2014; accepted 12 May 2014 Historians have shied away from the quantitative methods that are commonplace in the historically focused scientific fields such as geology and evolutionary biology. Although there have been occasional attempts in using historical data to build and test alternative quantitative historical models (Turchin 2003, 2012), the resistance by most historians is usually based on arguing that the models used in such endeavours lack the necessary complexity and generality (Thomas 2014). As clearly outlined in this issue of MolecuCorrespondence: Michael James Hickerson, Fax: 212 650 8585; E-mail: [email protected]
© 2014 John Wiley & Sons Ltd
lar Ecology by Pelletier & Carstens (2014), there has been a similar tension within the field of phylogeography and historical population genetic inference (Pelletier & Carstens 2014). The field of phylogeography has continued to move in the model-based direction in recent years, and although there are various methods for statistical model comparison in phylogeography (Carstens & Knowles 2010; Carstens et al. 2013), ABC has gained recent popularity as a tool for evaluating alternative historical demographic models given population genetic data (Sunn aker et al. 2013). Using ABC to evaluate competing models has various hazards and comes with recommended precautions (Robert et al. 2011), and unsurprisingly, many if not most researchers have a healthy scepticism as these tools continue to mature. Rather than a ‘garbage in, garbage out’ problem of questionable model-based inferences coming from questionable data, researchers are often unsatisfied with the models themselves, due to the necessity of collapsing the natural world into a useful yet simplified representation. Indeed, as scientists, we are never ‘sure’ of anything (Feynman 1999), and ‘essentially all models are wrong, but some are useful’ (Box & Draper 1987) has become the norm in science as all models only approximate reality at best. Estimating parameters requires use of probabilistic models, and these estimates are only as good as the models that are used, yet researchers in phylogeography often find it challenging and/or unsatisfying to build and use models that are approximations of reality on the one hand while able to make useful inference on the other. Regardless of this resistance, phylogeography is undergoing an exciting growth phase, both with opportunities to collect genomicscale data (McCormack et al. 2012) and the opportunity to test a bewildering array of complex demographic models (Excoffier et al. 2013). Instead of using ABC to evaluate a diverse set of models, Pelletier & Carstens (2014) undertake a more focused investigation into the ‘innards’ of ABC using it to evaluate the likelihood of three of the simplest and most widely used coalescent-based models in phylogeography: the panmictic model, the two-population island model and the two-population divergence/isolation model (Fig. 1). Rather than using ABC with weighted model averaging to obtain the three corresponding posterior model probabilities while allowing for the handful of model parameters (h, s, c, Μ) to be estimated under each model conditioned on each model’s posterior probability, these three models are sliced up into 143 ‘submodels’ according to various parameter ranges. For example, models designated as 1020 and 1010 are both divergence/isolation models that only differ by the direction of migration. Following the strategy outlined in the paper, one would estimate the posterior
2888 N E W S A N D V I E W S : P E R S P E C T I V E
Fig. 1 Patterns of nucleotide variation in the DNA of Plethodon idahoensis were used to evaluate multiple models of historical demography. After data exploration and consideration of a range of plausible population genetic parameters, a model that parameterized population divergence between southern and northern refugia with population expansion in each and gene flow into the northern populations was identified as representing the best relative fit to the data. Image provided by Bryan C. Carstens.
probability of each of the two divergence/isolation models instead of using a single unconstrained divergence/isolation model and estimating both migration parameters. Although in practice one would take the former strategy of estimating parameters, the systematic exploration in the latter approach undertaken by Pelletier & Carstens (2014) does result in valuable insight. After this comprehensive exploration, Pelletier & Carstens (2014) found a model that parameterized population divergence between southern and northern refugia with population expansion in each and gene flow into the northern populations to be the model representing the best relative fit to the data. This operation involves estimating a single model index parameter discreetly ranging from 1 to 143, corresponding to the various sub-models. As a point of reference, the authors first start with five models that are based on prior understanding of the salamander species Plethodon idahoensis (Carstens et al. 2009) and subsequently demonstrate that inference is limited by not exploring a wider range of models. Second with the full set of 143 sub-models, they demonstrate that outcomes in model posterior probabilities are dependent on how many of the 143 are compared under the ABC model choice procedure. Specifically, they demonstrate that as greater numbers of the 143 sub-models are evaluated, the inference from their ABC model choice procedure becomes increasingly. In these cases, with greater numbers of models being compared, the most probable
models are assigned diminishing levels of posterior probability. This is an expected result, and implementation of the Bayes factor comparing the most probable model with all others can help account for this by weighting posterior probabilities by the numbers of prior models. Instead, the Bayes factor implemented in Pelletier & Carstens (2014) made pairwise comparisons between pairs of models having the highest and second highest posterior probabilities regardless of the number of models. This pairwise Bayes factor will always be more conservative except in cases when the posterior probabilities are equal for all models that are less probable than the most probable model. To illustrate given a case of five models with posterior probabilities of 0.7, 0.15, 0.15, 0.0 and 0.0, the pairwise Bayes factor would be ~4.7, whereas the “all-model” Bayes factor would be ~9.33, with the former being a more conservative value (i.e. moderately favouring the model with highest posterior probability instead of strongly favouring this model). Likewise, if the posterior probabilities are equal for all models that are less probable than the most probable model (i.e. posterior probabilities of 0.7, 0.075, 0.075, 0.075, 0.075), both Bayes factors would be ~9.33. In general, using the Bayes factor, incorporating all models is perhaps more consistent with the Bayesian approach of incorporating all uncertainty associated with the ABC model choice procedure. After all, one uses ABC model choice to evaluate all specified models rather than two of the total number of specified models, and this useful illustration by Pelletier & Carstens (2014) is a powerful reminder that the outcome of model comparison with ABC or other similar likelihoodbased methods will always be dependent on the composition of the candidate model set. The three general models explored by Pelletier & Carstens (2014) could be simple enough to allow the use of full-likelihood methods rather than ABC, and their exploration is by no means a recommendation for how such data should be analysed. Although ABC remains useful under complex models (Prado-Martinez et al. 2013; Robinson et al. 2013), using it for model choice comes with challenges and scaling up to genomic-scale data sets (i.e. next-generation sequencing; NGS) is computationally expensive (Sousa et al. 2011, but see Roux et al. 2013). There are non-ABC options that can accommodate and evaluate complex models given data sets that are typical using NGS including new methods exploiting the site frequency spectrum that can accommodate models of arbitrary complexity such as large numbers of populations, ghost populations, size change and asymmetrical migration (Excoffier et al. 2013). Another part of phylogeographic model space that is not easily explored is the spatially explicit component of genetic samples, yet there has been some exciting developments that may allow the nonspatial approximation to be relaxed (Barton et al. 2010; Brown & Knowles 2012; Kelleher et al. 2013). As suggested by Pelletier & Carstens (2014), there might be reluctance to infer phylogeographic history using models due to the lack of prior knowledge about how one
© 2014 John Wiley & Sons Ltd
N E W S A N D V I E W S : P E R S P E C T I V E 2889 might parameterize putative demographic histories, yet there are fortunately a number of model-based clustering algorithms that can help estimate how many populations the genetic samples of a species come from (Pritchard et al. 2000; Dawson & Belkhir 2001; Frichot et al. 2014), as well as ‘species delimitation’ methods if one is interested in assigning putative species names to populations having degrees of isolation that satisfy particular species models (Fujita et al. 2012). Provided that there are sufficient data, these delineation procedures can provide the model scaffolding for subsequent parameter estimation under a model or mixture of models that are in turn weighted by their relative posterior probabilities. Although the outcome of model comparison with ABC or other similar likelihood-based methods will always be dependent on the composition of the model set, and parameter estimates will only be as good as the models that are used, model-based inference provides a number of benefits. Not only can one re-evaluate models against newly constructed models in subsequent studies, these evaluations are explicit and provide a standard measure of repeatability. As researchers in phylogeography realize that using models is a fundamental component of historical inference rather than an bothersome requirement for publication, this maturation will allow phylogeography to achieve some of its original goals in bringing the analysis of aggregate population genetic data to better understand the history of biological communities (Avise 2009).
References Avise JC (2009) Phylogeography: retrospect and prospect. Journal of Biogeography, 36, 3–15. Barton NH, Kelleher J, Etheridge AM (2010) A new model for extinction and recolonization in two dimensions: quantifying phylogeography. Evolution, 64, 2701–2715. Box GEP, Draper NR (1987) Empirical Model-building and Response Surfaces. John Wiley & Sons, Oxford, UK. Brown JL, Knowles LL (2012) Spatially explicit models of dynamic histories: examination of the genetic consequences of Pleistocene glaciation and recent climate change on the American Pika. Molecular ecology, 21, 3757–3775. Carstens B, Knowles LL (2010) Navigating the unknown: model selection in phylogeography. Molecular Ecology, 19, 4581–4582. Carstens BC, Stevenson AL, Degenhardt JD, Sullivan J (2004) Testing nested phylogenetic and phylogeographic hypotheses in the Plethodon vandykei species group. Systematic Biology, 53, 781–792. Carstens B, Stoute H, Reid N (2009) An information-theoretical approach to phylogeography. Molecular Ecology, 18, 4270–4282. Carstens BC, Brennan RS, Chua V et al. (2013) Model selection as a tool for phylogeographic inference: an example from the willow Salix melanopsis. Molecular Ecology, 22, 4014–4028. Dawson KJ, Belkhir K (2001) A Bayesian approach to the identification of panmictic populations and the assignment of individuals. Genetics Research, 78, 59–77.
© 2014 John Wiley & Sons Ltd
Excoffier L, Dupanloup I, Huerta-S anchez E, Sousa VC, Foll M (2013) Robust demographic inference from genomic and SNP data. (ed. Akey JM). PLoS genetics, 9, e1003905. Feynman RP (1999) “The Value of Science,” address to the National Academy of Sciences (Autumn 1955). In: In The Pleasure of Finding Things Out: The Best Short Works of Richard, P. Feynman (ed. Robbins J), pp. 141–150. Perseus Publishing, Cambridge, Massachusetts. Frichot E, Mathieu F, Trouillon T, Bouchard G, Francßois O (2014) Fast and efficient estimation of individual ancestry coefficients. Genetics, 196, 973–983. Fujita MK, Leache AD, Burbrink FT, McGuire JA, Moritz C (2012) Coalescent-based species delimitation in an integrative taxonomy. Trends in ecology & evolution, 27, 480–488. Kelleher J, Barton NH, Etheridge AM (2013) Coalescent simulation in continuous space. Bioinformatics, 29, 955–956. McCormack JE, Maley JM, Hird SM et al. (2012) Next-generation sequencing reveals phylogeographic structure and a species tree for recent bird divergences. Molecular Phylogenetics and Evolution, 62, 397–406. Pelletier TA, Carstens BC (2014) Model choice for phylogeographic inference using a large set of models. Molecular ecology, 23, 3028– 3043. Prado-Martinez J, Sudmant PH, Kidd JM et al. (2013) Great ape genetic diversity and population history. Nature, 499, 471–475. Pritchard JK, Stephens M, Donnelly P (2000) Inference of population structure using multilocus genotype data. Genetics, 155, 945. Robert CP, Cornuet J-M, Marin J-M, Pillai NS (2011) Lack of confidence in approximate Bayesian computation model choice. Proceedings of the National Academy of Sciences, 108, 15112– 15117. Robinson JD, Hall DW, Wares JP (2013) Approximate Bayesian estimation of extinction rate in the Finnish Daphnia magna metapopulation. Molecular ecology, 22, 2627–2639. Roux C, Tsagkogeorga G, Bierne N, Galtier N (2013) Crossing the species barrier: genomic hotspots of introgression between two highly divergent Ciona intestinalis species. Molecular biology and evolution, 30, 1574–1587. Sousa VC, Grelaud A, Hey J (2011) On the nonidentifiability of migration time estimates in isolation with migration models. Molecular Ecology, 20, 3956–3962. Sunn aker M, Busetto AG, Numminen E et al. (2013) Approximate Bayesian computation. PLoS Comput Biology, 9, e1002803. Thomas RC (2014) Does diffusion of horse-related military technologies explain spatiotemporal patterns of social complexity 1500 BCE–AD 1500? Proceedings of the National Academy of Sciences, 111, E414. Turchin P (2003) Historical Dynamics. Princeton University Press, Princeton. Turchin P (2012) Dynamics of political instability in the United States, 1780–2010. Journal of Peace Research, 51, 577–591.
M.H. wrote this article. doi: 10.1111/mec.12794
NEWS AND VIEWS
PERSPECTIVE
All models are wrong MICHAEL J. HICKERSON*† *Department of Biology, City College of New York, 160 Convent Ave., MR 526, New York, NY 10031, USA, †The Graduate Center of the City University of New York, Subprogram in Ecology, Evolution and Behavior, 365 5th Ave, New York, NY 10016, USA
Abstract As the field of phylogeography has continued to move in the model-based direction, researchers continue struggling to construct useful models for inference. These models must be both simple enough to be tractable yet contain enough of the complexity of the natural world to make meaningful inference. Beyond constructing such models for inference, researchers explore model space and test competing models with the data on hand, with the goal of improving the understanding of the natural world and the processes underlying natural biological communities. Approximate Bayesian computation (ABC) has increased in recent popularity as a tool for evaluating alternative historical demographic models given population genetic samples. As a thorough demonstration, Pelletier & Carstens (2014) use ABC to test 143 phylogeographic submodels given geographically widespread genetic samples from the salamander species Plethodon idahoensis (Carstens et al. 2004) and, in so doing, demonstrate how the results of the ABC model choice procedure are dependent on the model set one chooses to evaluate. Keywords: amphibians, model choice, phylogeography, population genetics – empirical, population genetics – theoretical Received 26 April 2014; revised 9 May 2014; accepted 12 May 2014 Historians have shied away from the quantitative methods that are commonplace in the historically focused scientific fields such as geology and evolutionary biology. Although there have been occasional attempts in using historical data to build and test alternative quantitative historical models (Turchin 2003, 2012), the resistance by most historians is usually based on arguing that the models used in such endeavours lack the necessary complexity and generality (Thomas 2014). As clearly outlined in this issue of MolecuCorrespondence: Michael James Hickerson, Fax: 212 650 8585; E-mail: [email protected]
© 2014 John Wiley & Sons Ltd
lar Ecology by Pelletier & Carstens (2014), there has been a similar tension within the field of phylogeography and historical population genetic inference (Pelletier & Carstens 2014). The field of phylogeography has continued to move in the model-based direction in recent years, and although there are various methods for statistical model comparison in phylogeography (Carstens & Knowles 2010; Carstens et al. 2013), ABC has gained recent popularity as a tool for evaluating alternative historical demographic models given population genetic data (Sunn aker et al. 2013). Using ABC to evaluate competing models has various hazards and comes with recommended precautions (Robert et al. 2011), and unsurprisingly, many if not most researchers have a healthy scepticism as these tools continue to mature. Rather than a ‘garbage in, garbage out’ problem of questionable model-based inferences coming from questionable data, researchers are often unsatisfied with the models themselves, due to the necessity of collapsing the natural world into a useful yet simplified representation. Indeed, as scientists, we are never ‘sure’ of anything (Feynman 1999), and ‘essentially all models are wrong, but some are useful’ (Box & Draper 1987) has become the norm in science as all models only approximate reality at best. Estimating parameters requires use of probabilistic models, and these estimates are only as good as the models that are used, yet researchers in phylogeography often find it challenging and/or unsatisfying to build and use models that are approximations of reality on the one hand while able to make useful inference on the other. Regardless of this resistance, phylogeography is undergoing an exciting growth phase, both with opportunities to collect genomicscale data (McCormack et al. 2012) and the opportunity to test a bewildering array of complex demographic models (Excoffier et al. 2013). Instead of using ABC to evaluate a diverse set of models, Pelletier & Carstens (2014) undertake a more focused investigation into the ‘innards’ of ABC using it to evaluate the likelihood of three of the simplest and most widely used coalescent-based models in phylogeography: the panmictic model, the two-population island model and the two-population divergence/isolation model (Fig. 1). Rather than using ABC with weighted model averaging to obtain the three corresponding posterior model probabilities while allowing for the handful of model parameters (h, s, c, Μ) to be estimated under each model conditioned on each model’s posterior probability, these three models are sliced up into 143 ‘submodels’ according to various parameter ranges. For example, models designated as 1020 and 1010 are both divergence/isolation models that only differ by the direction of migration. Following the strategy outlined in the paper, one would estimate the posterior
2888 N E W S A N D V I E W S : P E R S P E C T I V E
Fig. 1 Patterns of nucleotide variation in the DNA of Plethodon idahoensis were used to evaluate multiple models of historical demography. After data exploration and consideration of a range of plausible population genetic parameters, a model that parameterized population divergence between southern and northern refugia with population expansion in each and gene flow into the northern populations was identified as representing the best relative fit to the data. Image provided by Bryan C. Carstens.
probability of each of the two divergence/isolation models instead of using a single unconstrained divergence/isolation model and estimating both migration parameters. Although in practice one would take the former strategy of estimating parameters, the systematic exploration in the latter approach undertaken by Pelletier & Carstens (2014) does result in valuable insight. After this comprehensive exploration, Pelletier & Carstens (2014) found a model that parameterized population divergence between southern and northern refugia with population expansion in each and gene flow into the northern populations to be the model representing the best relative fit to the data. This operation involves estimating a single model index parameter discreetly ranging from 1 to 143, corresponding to the various sub-models. As a point of reference, the authors first start with five models that are based on prior understanding of the salamander species Plethodon idahoensis (Carstens et al. 2009) and subsequently demonstrate that inference is limited by not exploring a wider range of models. Second with the full set of 143 sub-models, they demonstrate that outcomes in model posterior probabilities are dependent on how many of the 143 are compared under the ABC model choice procedure. Specifically, they demonstrate that as greater numbers of the 143 sub-models are evaluated, the inference from their ABC model choice procedure becomes increasingly. In these cases, with greater numbers of models being compared, the most probable
models are assigned diminishing levels of posterior probability. This is an expected result, and implementation of the Bayes factor comparing the most probable model with all others can help account for this by weighting posterior probabilities by the numbers of prior models. Instead, the Bayes factor implemented in Pelletier & Carstens (2014) made pairwise comparisons between pairs of models having the highest and second highest posterior probabilities regardless of the number of models. This pairwise Bayes factor will always be more conservative except in cases when the posterior probabilities are equal for all models that are less probable than the most probable model. To illustrate given a case of five models with posterior probabilities of 0.7, 0.15, 0.15, 0.0 and 0.0, the pairwise Bayes factor would be ~4.7, whereas the “all-model” Bayes factor would be ~9.33, with the former being a more conservative value (i.e. moderately favouring the model with highest posterior probability instead of strongly favouring this model). Likewise, if the posterior probabilities are equal for all models that are less probable than the most probable model (i.e. posterior probabilities of 0.7, 0.075, 0.075, 0.075, 0.075), both Bayes factors would be ~9.33. In general, using the Bayes factor, incorporating all models is perhaps more consistent with the Bayesian approach of incorporating all uncertainty associated with the ABC model choice procedure. After all, one uses ABC model choice to evaluate all specified models rather than two of the total number of specified models, and this useful illustration by Pelletier & Carstens (2014) is a powerful reminder that the outcome of model comparison with ABC or other similar likelihoodbased methods will always be dependent on the composition of the candidate model set. The three general models explored by Pelletier & Carstens (2014) could be simple enough to allow the use of full-likelihood methods rather than ABC, and their exploration is by no means a recommendation for how such data should be analysed. Although ABC remains useful under complex models (Prado-Martinez et al. 2013; Robinson et al. 2013), using it for model choice comes with challenges and scaling up to genomic-scale data sets (i.e. next-generation sequencing; NGS) is computationally expensive (Sousa et al. 2011, but see Roux et al. 2013). There are non-ABC options that can accommodate and evaluate complex models given data sets that are typical using NGS including new methods exploiting the site frequency spectrum that can accommodate models of arbitrary complexity such as large numbers of populations, ghost populations, size change and asymmetrical migration (Excoffier et al. 2013). Another part of phylogeographic model space that is not easily explored is the spatially explicit component of genetic samples, yet there has been some exciting developments that may allow the nonspatial approximation to be relaxed (Barton et al. 2010; Brown & Knowles 2012; Kelleher et al. 2013). As suggested by Pelletier & Carstens (2014), there might be reluctance to infer phylogeographic history using models due to the lack of prior knowledge about how one
© 2014 John Wiley & Sons Ltd
N E W S A N D V I E W S : P E R S P E C T I V E 2889 might parameterize putative demographic histories, yet there are fortunately a number of model-based clustering algorithms that can help estimate how many populations the genetic samples of a species come from (Pritchard et al. 2000; Dawson & Belkhir 2001; Frichot et al. 2014), as well as ‘species delimitation’ methods if one is interested in assigning putative species names to populations having degrees of isolation that satisfy particular species models (Fujita et al. 2012). Provided that there are sufficient data, these delineation procedures can provide the model scaffolding for subsequent parameter estimation under a model or mixture of models that are in turn weighted by their relative posterior probabilities. Although the outcome of model comparison with ABC or other similar likelihood-based methods will always be dependent on the composition of the model set, and parameter estimates will only be as good as the models that are used, model-based inference provides a number of benefits. Not only can one re-evaluate models against newly constructed models in subsequent studies, these evaluations are explicit and provide a standard measure of repeatability. As researchers in phylogeography realize that using models is a fundamental component of historical inference rather than an bothersome requirement for publication, this maturation will allow phylogeography to achieve some of its original goals in bringing the analysis of aggregate population genetic data to better understand the history of biological communities (Avise 2009).
References Avise JC (2009) Phylogeography: retrospect and prospect. Journal of Biogeography, 36, 3–15. Barton NH, Kelleher J, Etheridge AM (2010) A new model for extinction and recolonization in two dimensions: quantifying phylogeography. Evolution, 64, 2701–2715. Box GEP, Draper NR (1987) Empirical Model-building and Response Surfaces. John Wiley & Sons, Oxford, UK. Brown JL, Knowles LL (2012) Spatially explicit models of dynamic histories: examination of the genetic consequences of Pleistocene glaciation and recent climate change on the American Pika. Molecular ecology, 21, 3757–3775. Carstens B, Knowles LL (2010) Navigating the unknown: model selection in phylogeography. Molecular Ecology, 19, 4581–4582. Carstens BC, Stevenson AL, Degenhardt JD, Sullivan J (2004) Testing nested phylogenetic and phylogeographic hypotheses in the Plethodon vandykei species group. Systematic Biology, 53, 781–792. Carstens B, Stoute H, Reid N (2009) An information-theoretical approach to phylogeography. Molecular Ecology, 18, 4270–4282. Carstens BC, Brennan RS, Chua V et al. (2013) Model selection as a tool for phylogeographic inference: an example from the willow Salix melanopsis. Molecular Ecology, 22, 4014–4028. Dawson KJ, Belkhir K (2001) A Bayesian approach to the identification of panmictic populations and the assignment of individuals. Genetics Research, 78, 59–77.
© 2014 John Wiley & Sons Ltd
Excoffier L, Dupanloup I, Huerta-S anchez E, Sousa VC, Foll M (2013) Robust demographic inference from genomic and SNP data. (ed. Akey JM). PLoS genetics, 9, e1003905. Feynman RP (1999) “The Value of Science,” address to the National Academy of Sciences (Autumn 1955). In: In The Pleasure of Finding Things Out: The Best Short Works of Richard, P. Feynman (ed. Robbins J), pp. 141–150. Perseus Publishing, Cambridge, Massachusetts. Frichot E, Mathieu F, Trouillon T, Bouchard G, Francßois O (2014) Fast and efficient estimation of individual ancestry coefficients. Genetics, 196, 973–983. Fujita MK, Leache AD, Burbrink FT, McGuire JA, Moritz C (2012) Coalescent-based species delimitation in an integrative taxonomy. Trends in ecology & evolution, 27, 480–488. Kelleher J, Barton NH, Etheridge AM (2013) Coalescent simulation in continuous space. Bioinformatics, 29, 955–956. McCormack JE, Maley JM, Hird SM et al. (2012) Next-generation sequencing reveals phylogeographic structure and a species tree for recent bird divergences. Molecular Phylogenetics and Evolution, 62, 397–406. Pelletier TA, Carstens BC (2014) Model choice for phylogeographic inference using a large set of models. Molecular ecology, 23, 3028– 3043. Prado-Martinez J, Sudmant PH, Kidd JM et al. (2013) Great ape genetic diversity and population history. Nature, 499, 471–475. Pritchard JK, Stephens M, Donnelly P (2000) Inference of population structure using multilocus genotype data. Genetics, 155, 945. Robert CP, Cornuet J-M, Marin J-M, Pillai NS (2011) Lack of confidence in approximate Bayesian computation model choice. Proceedings of the National Academy of Sciences, 108, 15112– 15117. Robinson JD, Hall DW, Wares JP (2013) Approximate Bayesian estimation of extinction rate in the Finnish Daphnia magna metapopulation. Molecular ecology, 22, 2627–2639. Roux C, Tsagkogeorga G, Bierne N, Galtier N (2013) Crossing the species barrier: genomic hotspots of introgression between two highly divergent Ciona intestinalis species. Molecular biology and evolution, 30, 1574–1587. Sousa VC, Grelaud A, Hey J (2011) On the nonidentifiability of migration time estimates in isolation with migration models. Molecular Ecology, 20, 3956–3962. Sunn aker M, Busetto AG, Numminen E et al. (2013) Approximate Bayesian computation. PLoS Comput Biology, 9, e1002803. Thomas RC (2014) Does diffusion of horse-related military technologies explain spatiotemporal patterns of social complexity 1500 BCE–AD 1500? Proceedings of the National Academy of Sciences, 111, E414. Turchin P (2003) Historical Dynamics. Princeton University Press, Princeton. Turchin P (2012) Dynamics of political instability in the United States, 1780–2010. Journal of Peace Research, 51, 577–591.
M.H. wrote this article. doi: 10.1111/mec.12794
