Dev Genes Evol DOI 10.1007/s00427-016-0539-2 REVIEW Size, shape, and form: concepts of allometry in geometric morphometrics Christian Peter Klingenberg1 Received: 28 October 2015 /Accepted: 29 February 2016 # The Author(s) 2016. This article is published with open access at Springerlink.com Abstract Allometry refers to the size-related changes of mor- isotropic variation of landmark positions, they are equivalent phological traits and remains an essential concept for the study up to scaling. The methods differ in their emphasis and thus of evolution and development. This review is the first system- provide investigators with flexible tools to address specific atic comparison of allometric methods in the context of geo- questions concerning evolution and development, but all metric morphometrics that considers the structure of morpho- frameworks are logically compatible with each other and logical spaces and their implications for characterizing allom- therefore unlikely to yield contradictory results. etry and performing size correction. The distinction of two main schools of thought is useful for understanding the differ- Keywords Allometry . Centroidsize . Conformation . Form . ences and relationships between alternative methods for Geometric morphometrics . Multivariate regression . Principal studying allometry. The Gould–Mosimann school defines al- component analysis . Procrustes superimposition . Shape . lometry as the covariation of shape with size. This concept of Size correction allometry is implemented in geometric morphometrics through the multivariate regression of shape variables on a measure of size. In the Huxley–Jolicoeur school, allometry Introduction is the covariation among morphological features that all con- tain size information. In this framework, allometric trajecto- Variation in size is an important determinant for variation in ries are characterized by the first principal component, which many other organismal traits. Developmental processes are is a line of best fit to the data points. In geometric morpho- accompanied by a dramatic growth in size in developing or- metrics, this concept is implemented in analyses using either ganisms, and evolutionary diversification often involves dif- Procrustes form space or conformation space (the latter also ferentiation of body size among related taxa. Accordingly, known as size-and-shape space). Whereas these spaces differ allometry has been an important concept for evolutionary bi- substantially in their global structure, there are also close con- ology and related disciplines for much of the last century nections in their localized geometry. For the model of small (Huxley 1924, 1932;Cock1966;Gould1966; Calder 1984; Schmidt-Nielsen 1984). During this time, the methods for quantifying morphological variation underwent momentous Communicated by Nico Posnien and Nikola-Michael Prpic change, from the development of multivariate approaches to This article is part of the Special Issue BSize and Shape: Integration of the emergence of the discipline of morphometrics (Jolicoeur morphometrics, mathematical modelling, developmental and evolutionary and Mosimann 1960;Jolicoeur1963; Sneath and Sokal 1973; biology^, Guest Editors: Nico Posnien—Nikola-Michael Prpic. Oxnard 1974;Pimentel1979; Reyment et al. 1984). Finally, the rise of geometric morphometrics in the 1980s and 1990s * Christian Peter Klingenberg has established the current methods for analyzing variation in [email protected] organismal shape (Bookstein 1986, 1991; Rohlf 1990;Rohlf and Bookstein 1990; Marcus et al. 1993, 1996; Rohlf and 1 Faculty of Life Sciences, University of Manchester, Michael Smith Marcus 1993; Monteiro and dos Reis 1999; Klingenberg Building, Oxford Road, Manchester M13 9PT, UK 2010; Zelditch et al. 2012; Adams et al. 2013;Mitteroecker Dev Genes Evol et al. 2013). Throughout this history of different frameworks allometry: the Huxley–Jolicoeur school, which emphasizes for quantifying morphological variation, allometry has played the covariation among traits as a consequence of variation in a more or less prominent role. size, and the Gould–Mosimann school, which defines allom- Along with the ways of characterizing morphological var- etry as covariation of size and shape (Klingenberg 1998). A iation in general, the concept of allometry and the methods for fundamental difference between the two concepts of allometry analyzing it have changed drastically as well. In some ap- is that the framework of the Huxley–Jolicoeur school does not proaches, the main emphasis is on covariation among different involve a distinction of size and shape, which is the central traits (Huxley 1924, 1932; Jolicoeur 1963), whereas others element in the framework of the Gould–Mosimann school. focus on the covariation between size and shape (Mosimann As it turns out, the distinction between these two schools is 1970; Monteiro 1999). Another difference is whether the also useful for understanding the differences between different methods separate size and shape (Mosimann 1970; allometric approaches currently used in geometric morpho- Bookstein 1986; Goodall 1991) or whether they reject this metrics. Therefore, this section provides a brief overview of distinction and consider morphological form as a single uni- the two schools of thought and contrasts them directly with fied feature (Lele and Richtsmeier 1991;Mitteroeckeretal. each other in some key aspects. The purpose of this section is 2004). As a consequence of the different concepts of allome- solely to provide a background for the comparison of the try, the various methods also differ in the way in which they methods currently used in geometric morphometrics. It is carry out corrections for the effects of size on morphological therefore not a complete historical survey of allometry, but variation, which is one of the most used applications of allom- inevitably leaves out many concepts and methods. etry (Burnaby 1966;Sidlauskasetal.2011). A common feature of both schools of thought is the treat- A number of review papers have provided overviews of the ment of size. In all the different frameworks, allometry is biological concepts related to allometry (Cock 1966; Gould variation in various traits that is explained by or associated 1966; Klingenberg 1998) and the statistical methods for allo- with variation in size (Gould 1966;Mosimann1970; metric analyses mainly in the context of traditional morpho- Bookstein et al. 1985;Klingenberg1998; Mitteroecker et al. metrics (Bookstein 1989; Klingenberg 1996b). There is no 2013). The origin of the size variation depends on the context comparable survey, however, for allometric analyses in the of the study, and according to this context, different levels of context of geometric morphometrics (but see Mitteroecker variation can be defined (Cock 1966; Gould 1966; Cheverud et al. 2013). This paper surveys the methods for analyzing 1982; Klingenberg and Zimmermann 1992a; Klingenberg allometry in geometric morphometrics. To appreciate the 2014). Many studies have analyzed the changes associated range of current concepts and their interrelations, it is helpful with the dramatic size increases over individual growth, or to take a historical perspective that considers the origin of ontogenetic allometry (Huxley 1924, 1932; Loy et al. 1996; ideas before explaining their role in currently used methods Bulygina et al. 2006; Rodríguez-Mendoza et al. 2011; in geometric morphometrics. This article adopts this approach Mitteroecker et al. 2013; Murta-Fonseca and Fernandes and therefore starts by revisiting the concepts of allometry, 2016). Others have focused on the consequences of size var- size, and shape as they have been used traditionally, before iation within a single ontogenetic stage, or static allometry, employing these concepts to compare the different frame- most often based on samples of adults from a population works that are currently used in geometric morphometrics. (Rosas and Bastir 2002; Drake and Klingenberg 2008; Considering the structure of shape spaces has proven helpful Weisensee and Jantz 2011; Freidline et al. 2015). Evolution for comparing morphometric methods (Rohlf 1996, 2000)and can also alter the size of organisms and produce associated serves in this paper as a framework for comparing different morphological changes due to evolutionary allometry methods for analyzing allometry. A particular focus of atten- (Cardini and Polly 2013; Klingenberg and Marugán-Lobón tion is how the different allometric concepts are applied for 2013;Martín-Serraetal.2014; Sherratt et al. 2014). These size correction in morphometric data. Finally, the relationships three levels of allometry, and sometimes others as well, have among alternative methods are discussed. been compared in a range of studies (Cheverud 1982; Leamy and Bradley 1982; Klingenberg and Zimmermann 1992a; Klingenberg et al. 2012; Pélabon et al. 2013; Freidline et al. Allometry, size, and shape in different morphometric 2015; Strelin et al. 2016). These three classical levels of al- frameworks lometry are not the only levels of variation where allometry can apply, but others exist as well and may be worth investi- There are several concepts of allometry, which all concern the gating (Klingenberg 2014). For example, fluctuating asymme- effect of size on morphological variation, but differ in the try of shape may have a component of allometry, where asym- specific definitions of terms and in the aspects of morphology metry of shape is an allometric consequence