A simple procedure for the comparison of covariance matrices

BMC Evolutionary Biology

Carlos Garcia¹ 

[1] Department Xenética, CIBUS Campus Sur, Universidade de Santiago de Compostela, Santiago de Compostela, Galicia 15782, Spain

关键词: Hybrid zone;    Matrix shape;    Matrix orientation;    Littorina Saxatilis;    Principal component analysis;    Eigenvectors;   

Others  :  1140031
DOI  :  10.1186/1471-2148-12-222

 received in 2012-06-18, accepted in 2012-11-02,  发布年份 2012

【 摘 要 】

Background

Comparing the covariation patterns of populations or species is a basic step in the evolutionary analysis of quantitative traits. Here I propose a new, simple method to make this comparison in two population samples that is based on comparing the variance explained in each sample by the eigenvectors of its own covariance matrix with that explained by the covariance matrix eigenvectors of the other sample. The rationale of this procedure is that the matrix eigenvectors of two similar samples would explain similar amounts of variance in the two samples. I use computer simulation and morphological covariance matrices from the two morphs in a marine snail hybrid zone to show how the proposed procedure can be used to measure the contribution of the matrices orientation and shape to the overall differentiation.

Results

I show how this procedure can detect even modest differences between matrices calculated with moderately sized samples, and how it can be used as the basis for more detailed analyses of the nature of these differences.

Conclusions

The new procedure constitutes a useful resource for the comparison of covariance matrices. It could fill the gap between procedures resulting in a single, overall measure of differentiation, and analytical methods based on multiple model comparison not providing such a measure.

【 授权许可】

   
2012 Garcia; licensee BioMed Central Ltd.

【 图 表 】

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【 参考文献 】

• [1]Cheverud JM: A comparison of genetic and phenotypic correlations. Evolution 1988, 42:958-968.
• [2]Ackermann RR, Cheverud JM: Morphological integration in primate evolution. In Phenotypic Integration: Studying the Ecology and Evolution of Complex Phenotypes. Edited by Pigliucci M, Preston K. New York: Oxford University Press; 2004:302-319.
• [3]Game ET, Caley J: The stability of P in coral reef fishes. Evolution 2006, 60:814-823.
• [4]Kolbe JJ, Revell LJ, Szekely B, Brodie ED III, Losos JB: Convergent evolution of phenotypic integration and its alignment with morphological diversification in Caribbean Anolis ecomorphs. Evolution 2011, 65:3608-3624.
• [5]Jones AG, Arnold JS, Burger R: Evolution and stability of the G-matrix on a landscape with a moving optimum. Evolution 2004, 58:1639-1654.
• [6]Arnold SJ, Bürger R, Hohenlohe PA, Ajie BC, Jones AG: Understanding the evolution and stability of the G-matrix. Evolution 2008, 62:2451-2461.
• [7]Eroukhmanoff F, Svensson EI: Evolution and stability of the G-matrix during the colonization of a novel environment. J evol Biol 2011, 24:1363-1373.
• [8]Colautti RI, Barrett SCH: Population divergence along lines of genetic variance and covariance in the invasive plant Lythrum salicaria in eastern North America. Evolution 2011, 65:2514-2529.
• [9]Robinson MR, Wilson AJ, Pilkington JG, Clutton-Brock TH, Pemberton JM, Kruuk LEB: The impact of environmental heterogeneity on genetic architecture in a wild Population of Soay sheep. Genetics 2009, 181:1639-1648.
• [10]Monteiro LR, Nogueira MR: Adaptive radiations, ecological specialization, and the evolutionary integration of complex morphological structures. Evolution 2010, 64:724-744.
• [11]Mallitt KL, Bonser SP, Hunt J: The plasticity of phenotypic integration in response to light and water availability in the pepper grass, Lepidium bonariense. Evol Ecol 2010, 24:1321-1337.
• [12]Barker BS, Phillips PC, Arnold SJ: A test of the conjecture that B-matrices are more stable than G-matrices. Evolution 2010, 64:2601-2613.
• [13]Campbell DR, Weller SG, Sakai AK, Culley TM, Dang PN, Dunbar-Wallis AK: Genetic variation and covariation in floral allocation of two species of Schiedea with contrasting levels of sexual dimorphism. Evolution 2011, 65:757-770.
• [14]Steppan SJ, Phillips PC, Houle D: Comparative quantitative genetics: evolution of the G matrix. Trends Ecol Evol 2002, 17:320-327.
• [15]Hohenlohe PA, Arnold S: MIPoD: A hypothesis-testing framework for microevolutionary inference from patterns of divergence. Am Nat 2008, 171:366-385.
• [16]Ovaskainen O, Cano JM, Merilä J: A Bayesian framework for comparative quantitative genetics. Proc R Soc B 2008, 275:669-678.
• [17]Kohn LAP, Atchley WR: How similar are genetic correlation structures? Data from mice and rats. Evolution 1988, 42:467-481.
• [18]Atchley WR, Cowley DE, Vogl C, McLellan T: Evolutionary divergence, shape change, and genetic correlation structure in the rodent mandible. Syst Biol 1995, 41:196-221.
• [19]Roff DA, Mousseau TA, Howard DJ: Variation in genetic architecture of calling song among populations of Allonemobius socius, A. fasciatus, and a hybrid population: drift or selection? Evolution 1999, 53:216-224.
• [20]Begin M, Roff DA: From micro- to macroevolution through quantitative genetic variation: positive evidence from field crickets. Evolution 2004, 58:2287-2304.
• [21]Roff D: Comparing G matrices: a Manova approach. Evolution 2002, 56:1286-1291.
• [22]Cheverud JM: Quantitative genetic analysis of cranial morphology in the cotton-top (Sagainus oedipas) and saddle-back (S. fuscicollis) tamarins. J evol Biol 1996, 9:5-42.
• [23]Calsbeek B, Goodnight CJ: Empirical comparison of G matrix test statistics: finding biologically relevant change. Evolution 2009, 63:2627-2635.
• [24]Phillips PC, Arnold SJ: Hierarchical comparison of genetic variance-covariance matrices. I. Using the Flury hierarchy. Evolution 1999, 53:1506-1515.
• [25]Flury B: Common Principal Components and Related Multivariate Models. New York: John Wiley; 1988.
• [26]Cheverud JM, Marroig G: Comparing covariance matrices: random skewers method compared to the common principal components model. Genet Mol Biol 2007, 30:461-469.
• [27]Houle D, Mezey J, Galpern P: Interpretation of the results of common principal components analyses. Evolution 2002, 56:433-440.
• [28]Björklund M: Processes generating macroevolutionary patterns of morphological variation in birds: a simulation study. J evol Biol 1994, 7:727-742.
• [29]Cruz R, Vilas C, Mosquera J, García C: The close relationship between estimated divergent selection and observed differentiation supports the selective origin of a marine snail hybrid zone. J evol Biol 2004, 17:1221-1229.
• [30]Carballo M, Caballero A, Rolán-Alvarez E: Habitat-dependent ecotype micro-distribution at the mid-shore in natural populations of Littorina saxatilis. Hydrobiologia 2005, 548:307-311.
• [31]Hansen TF, Houle D: Measuring and comparing evolvability and constraint in multivariate characters. J evol Biol 2008, 21:1201.1219.
• [32]Hine E, Chenoweth S, Rundle HD, Blows MW: Characterizing the evolution of genetic variance using genetic covariance tensors. Phil Trans R Soc B 2009, 364:1567-1578.
• [33]Mitteroecker M, Bookstein F: The ontogenetic trajectory of the phenotypic covariance matrix, with examples from craniofacial shape in rats and humans. Evolution 2009, 63:727-737.
• [34]Phillips P: CPC – Common principal Components Analysis Program. http://pages.uoregon.edu/pphil/programs/cpc/cpc.htm webcite
• [35]Steppan SJ: Phylogenetic analysis of phenotypic covariance structure. II. Reconstructing matrix evolution. Evolution 1997, 51:587-594.
• [36]Johannesson K, Johannesson B, Rolán-Alvarez E: Morphological differentiation and genetic cohesiveness over a micro environmental gradient in the marine snail Littorina saxatilis. Evolution 1993, 47:1770-1787.
• [37]Cruz R, Rolán-Álvarez E, García C: Sexual selection on phenotypic traits in a hybrid zone of Littorina saxatilis (Olivi). J evol Biol 2001, 14:773-785.