Journal of Biogeography (J. Biogeogr.) (2014) 41, 1414–1427 ORIGINAL Assessing model sensitivity in ancestral ARTICLE area reconstruction using LAGRANGE:a case study using the Colchicaceae family Juliana Chacon* and Susanne S. Renner Department of Biology, University of Munich, ABSTRACT 80638 Munich, Germany Aim Likelihood analyses of ancestral ranges require a parameterized model that consists of a time-calibrated phylogeny, an ‘adjacency matrix’ of allowed or forbidden area connections, and an ‘area–dispersal’ matrix with probabilities for discrete periods of time. The approach is implemented in the software Lag- range. Because it can incorporate information about past continental posi- tions, the approach has been used in historical biogeographical studies of relatively old clades. Surprisingly, no study has evaluated the interactions among these input matrices. Here we use the lily family Colchicaceae and arti- ficial data to study the relative effect of the input matrices on final estimates. Location Africa, Australia, Eurasia, North America and South America. Methods Using eight plastid, mitochondrial and nuclear DNA regions from 85 of the c. 280 species of Colchicaceae (representing all genera and the entire geographical range) and relevant outgroups, we obtained a well-resolved phy- logeny dated with a molecular clock. We then assigned species to six geograph- ical distributions and carried out 22 Lagrange runs in which we modified the adjacency and dispersal matrices, the latter with zero, two or four time periods and one, three or five dispersal probabilities. For a second data set, the areas at deep nodes in the empirical tree were modified by shuffling species distribu- tions. Models were compared based on global log-likelihoods. Results The adjacency matrix strongly determined the outcome, while time slices and dispersal probability categories had minor effects. Ancestral areas reconstructed at most nodes were unaffected by the different input matrices. Colchicaceae are likely to have originated in Cretaceous East Gondwana, ini- tially diversified in Australia (c. 67 Ma), reached southern Africa during the Palaeocene–Eocene, and from there extended their range to Southeast Asia (probably through Arabia) and then North America (through Beringia). Main conclusions At least in small data sets, Lagrange models should be tested with sensitivity analyses as carried out here, concentrating on con- strained versus unconstrained adjacency matrices, and it should be good prac- tice to report the set-up of both input matrices, not just the dispersal matrix, which is the less decisive of the two. Keywords *Correspondence: Juliana Chacon, Systematic Adjacency matrix, ancestral area reconstruction, area–dispersal matrix, chron- Botany and Mycology, University of Munich, Menzinger Str. 67, 80638 Munich, Germany. ogram, Colchicaceae, geographical range evolution, Gondwana, historical bio- E-mail: [email protected] geographical methods, likelihood models, palaeogeography. 1414 http://wileyonlinelibrary.com/journal/jbi ª 2014 John Wiley & Sons Ltd doi:10.1111/jbi.12301 When to keep ancestral area reconstruction models simple will only be preferred over a dispersal scenario if it is congru- INTRODUCTION ent with the a priori accepted geological context (Ree et al., The rise of molecular clock dating as a tool in historical bio- 2005). An absence of expansion into another area could be geographical analysis has been accompanied by the develop- just that or could be a result of extinction in that area; both ment of new methods of ancestral area reconstruction (AAR). are captured by extremely low dispersal probability values. A The most sophisticated of these methods, Likelihood Analysis user can build as many dispersal matrices for different of Geographic Range Evolution or Lagrange (Ree et al., periods of time (‘time slices’) as deemed appropriate. 2005; Ree & Smith, 2008), is model-based and has been the The components described above imply that Lagrange method of choice for deep-time biogeographical studies requires more ad hoc parameter values than other biogeo- because it allows the incorporation of palaeogeographical graphical methods. Studies using the program have differed data. This is achieved through the dispersal–extinction–clado- considerably both in model details and in the reporting of genesis (DEC) model (Ree & Smith, 2008), which uses four model parameterization (Table 1 in Nauheimer et al., 2012, components: (1) a fully resolved chronogram; (2) a species provides an overview). For example, studies have left adja- distribution matrix denoting a species’ presence in a set of cency matrices unconstrained (Carlson et al., 2012) or con- geographical areas; (3) an adjacency matrix specifying allowed strained (Clayton et al., 2009), but without testing how this and forbidden ranges; and (4) an area–dispersal matrix speci- interacted with the other input matrices or how an alterna- fying dispersal probabilities between areas. The DEC model tive treatment would have impacted the global model likeli- assumes that two stochastic processes underlie the range evo- hood. Similarly, the probability of dispersal between lution of a species in the absence of lineage divergence: range Australia and South America during the Cretaceous (145– expansion through dispersal between areas and range contrac- 66 Ma) in different studies was assigned a probability of tion through extinction within an area. Therefore all models P = 1 (Buerki et al., 2011: time slices before 60 and before have two free parameters: a mean rate of dispersal between 80 Ma), P = 0.5 (Mao et al., 2012: time slice between 105 the areas i and j, Dij, and a mean rate of extinction in the and 70 Ma) or P = 0.01 (Nauheimer et al., 2012: time slices area i, Ei (Ree & Smith, 2008). The likelihood function then of 150–90 Ma and 90–30 Ma). The number of probability integrates over the conditional likelihoods of all ancestral categories also has differed from author to author, so far states at every internal node weighted by their prior probabil- ranging from five (Mao et al., 2012; P = 0.1, 0.25, 0.5, 0.75 ity (set by the user-defined input matrices), proceeding and 1) to three (Buerki et al., 2011; P = 0.01, 0.5 and 1). backwards from the tips of the tree to its root. We know of five studies that have attempted to use As regards component (1), the input chronogram provides model comparison to assess model fit. A problem here is the time-calibrated nodes and branches for which the proba- that there is no test statistic suitable for assessing the per- bility of ancestor–descendant area change is calculated. formance of DEC models because they have the same num- Component (2), the species distribution matrix, allows users ber of free parameters and are not hierarchically nested. to define areas appropriate for their clade and research ques- Therefore, likelihood ratio tests or the Akaike information tion, with the limitation that the number of biogeographical criterion (AIC) are not applicable (see Posada & Buckley, parameters to estimate from the data increases exponentially 2004). As a workaround, workers have used the global like- with the number of areas, decreasing the inferential power of lihood calculated by Lagrange to compare models and the the model (Ree & Sanmartın, 2009). Studies have used from rule that a two log-likelihood unit difference between mod- three to 15 geographical areas (see Table 1 in Nauheimer els indicates significance (Edwards, 1992). Using this et al., 2012), seeking a balance between the dispersion of tips approach, Couvreur et al. (2011) and Baker & Couvreur [species] across areas (hence the potential inferred ‘area (2013) found that their models with zero time slices were switches’ at nodes deep in the tree) and the risk of having significantly more likely than constrained models with five many singletons (areas occupied by a single tip taxon). Com- time slices. In a large data set of Cupressaceae, Mao et al. ponent (3), the adjacency matrix, is a presence–absence (2012) similarly compared the likelihoods of models with matrix in which a user defines composite ranges allowed or four, five, six, seven or eight time slices. The migration forbidden in the model (for example, the combined continent probabilities ranged from 0.1 for well-separated areas to 1.0 Laurasia but not a combined Asia and Australia). It is similar for contiguous landmasses. They found that the most com- to the cost matrix used in the program diva (Ronquist, plex (eight-time-slice) model had the best global likelihood. 1997), except that diva does not have a configuration option By contrast, in a large Araceae matrix, the likelihood of a for excluding discontinuous ranges. For component (4), the simpler (three-time-slice) model was higher than that of a area–dispersal matrix, the user specifies values (such as 1, 0.5, more complex (four-time-slice) model (Nauheimer et al., 0.01 or 0) for dispersal probabilities between areas based on 2012). The only studies to test the effects of constrained prior notions about range expansion. These values become and unconstrained adjacency matrices are an analysis of area-specific scaling factors for the program’s calculation of Psychotria in Hawaii (Ree & Smith, 2008) and one of the average rate of dispersal. Vicariance scenarios are not Cyrtandra in the Pacific Islands (Clark et al., 2008); both favoured a priori. If descendants are restricted to separate found that a constrained matrix fitted the data better than areas of the ancestral range, a vicariant speciation scenario an unconstrained one. Journal of Biogeography 41, 1414–1427 1415 ª 2014 John Wiley & Sons Ltd J. Chacon and S. S. Renner To advance the field of likelihood-based historical bioge- (Vinnersten & Reeves, 2003; Vinnersten & Manning, 2007; ography, we decided to investigate the interactions among del Hoyo & Pedrola-Monfort, 2008; Persson et al., 2011), the input matrices, number of time slices, dispersal probabil- while the best circumscription of Wurmbea is still unclear ity categories, and node/area/time slice ratio in an empirical (Thi et al., 2013).