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Once upon Multivariate Analyses: When They Tell Several Stories about Biological Evolution.

Renaud S, Dufour AB, Hardouin EA, Ledevin R, Auffray JC - PLoS ONE (2015)

Bottom Line: Standardizing within-group variance, as performed in the CVA, distorts the relationships among groups, an effect that is particularly strong if the variance is similarly oriented in a comparable way in all groups.Without arguing about a method performing 'better' than another, it rather emerges that working on the total or between-group variance (PCA and bgPCA) will tend to put the focus on the role of direction of main variance as line of least resistance to evolution.Standardizing by the within-group variance (CVA), by dampening the expression of this line of least resistance, has the potential to reveal other relevant patterns of differentiation that may otherwise be blurred.

View Article: PubMed Central - PubMed

Affiliation: Laboratoire de Biom├ętrie et Biologie Evolutive, UMR5558, CNRS, University Lyon 1, 69622 Villeurbanne, France.

ABSTRACT
Geometric morphometrics aims to characterize of the geometry of complex traits. It is therefore by essence multivariate. The most popular methods to investigate patterns of differentiation in this context are (1) the Principal Component Analysis (PCA), which is an eigenvalue decomposition of the total variance-covariance matrix among all specimens; (2) the Canonical Variate Analysis (CVA, a.k.a. linear discriminant analysis (LDA) for more than two groups), which aims at separating the groups by maximizing the between-group to within-group variance ratio; (3) the between-group PCA (bgPCA) which investigates patterns of between-group variation, without standardizing by the within-group variance. Standardizing within-group variance, as performed in the CVA, distorts the relationships among groups, an effect that is particularly strong if the variance is similarly oriented in a comparable way in all groups. Such shared direction of main morphological variance may occur and have a biological meaning, for instance corresponding to the most frequent standing genetic variation in a population. Here we undertake a case study of the evolution of house mouse molar shape across various islands, based on the real dataset and simulations. We investigated how patterns of main variance influence the depiction of among-group differentiation according to the interpretation of the PCA, bgPCA and CVA. Without arguing about a method performing 'better' than another, it rather emerges that working on the total or between-group variance (PCA and bgPCA) will tend to put the focus on the role of direction of main variance as line of least resistance to evolution. Standardizing by the within-group variance (CVA), by dampening the expression of this line of least resistance, has the potential to reveal other relevant patterns of differentiation that may otherwise be blurred.

No MeSH data available.


Effect of homogeneizing variances on the pattern of differentiation provided by PCA, bgPCA and CVA.The initial groups were bootstrapped, with the condition that the variance of all shape variables should follow a similar uniform distribution. The 999 resulting data sets were analysed using PCA, bgPCA and CVA. The resulting configurations on the first four axes of each analysis were compared using a Protest, providing a Procrustes distance estimating how much the configurations differ. For each pairwise comparison between analyses (PCA vs. bgPCA, PCA vs. CVA, bgPCA vs. CVA), this provided a simulated distribution (histograms in grey) for data sets with homogeneous variances. The observed distances (red lines) between analyses based on the real data set, characterized by strongly anisotropic variances for each group, were compared to these simulated distributions.
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pone.0132801.g005: Effect of homogeneizing variances on the pattern of differentiation provided by PCA, bgPCA and CVA.The initial groups were bootstrapped, with the condition that the variance of all shape variables should follow a similar uniform distribution. The 999 resulting data sets were analysed using PCA, bgPCA and CVA. The resulting configurations on the first four axes of each analysis were compared using a Protest, providing a Procrustes distance estimating how much the configurations differ. For each pairwise comparison between analyses (PCA vs. bgPCA, PCA vs. CVA, bgPCA vs. CVA), this provided a simulated distribution (histograms in grey) for data sets with homogeneous variances. The observed distances (red lines) between analyses based on the real data set, characterized by strongly anisotropic variances for each group, were compared to these simulated distributions.

Mentions: Since the CVA standardizes the within-group variance, it should be more impacted than the PCA and bgPCA by the occurrence of Pmax shared among groups. To validate this hypothesis, simulations were performed. The initial groups were bootstrapped, with the condition that the variance of all shape variables should follow a similar uniform distribution. The different groups should thus no longer share parallel Pmax. The resulting simulated configurations were compared using Protest (Fig 5). The simulated and the observed configurations provided by the PCA and the bgPCA were all very close. This shows that the PCA and bgPCA provided similar pictures of the morphometric differentiation, and that this picture was not impacted by Pmax. For the CVA cases, simulated datasets displayed configurations closer by 4 to 5 times compared to those provided by the PCA and the bgPCA than the original configuration. This is in agreement with the fact that the occurrence of a marked direction of main variance Pmax, parallel among groups, strongly impacts the pattern of differentiation as revealed by a CVA. The CVA assumes isotropic variation. In the case of anisotropic variance oriented along similar Pmax, this assumption is not fulfilled.


Once upon Multivariate Analyses: When They Tell Several Stories about Biological Evolution.

Renaud S, Dufour AB, Hardouin EA, Ledevin R, Auffray JC - PLoS ONE (2015)

Effect of homogeneizing variances on the pattern of differentiation provided by PCA, bgPCA and CVA.The initial groups were bootstrapped, with the condition that the variance of all shape variables should follow a similar uniform distribution. The 999 resulting data sets were analysed using PCA, bgPCA and CVA. The resulting configurations on the first four axes of each analysis were compared using a Protest, providing a Procrustes distance estimating how much the configurations differ. For each pairwise comparison between analyses (PCA vs. bgPCA, PCA vs. CVA, bgPCA vs. CVA), this provided a simulated distribution (histograms in grey) for data sets with homogeneous variances. The observed distances (red lines) between analyses based on the real data set, characterized by strongly anisotropic variances for each group, were compared to these simulated distributions.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4507858&req=5

pone.0132801.g005: Effect of homogeneizing variances on the pattern of differentiation provided by PCA, bgPCA and CVA.The initial groups were bootstrapped, with the condition that the variance of all shape variables should follow a similar uniform distribution. The 999 resulting data sets were analysed using PCA, bgPCA and CVA. The resulting configurations on the first four axes of each analysis were compared using a Protest, providing a Procrustes distance estimating how much the configurations differ. For each pairwise comparison between analyses (PCA vs. bgPCA, PCA vs. CVA, bgPCA vs. CVA), this provided a simulated distribution (histograms in grey) for data sets with homogeneous variances. The observed distances (red lines) between analyses based on the real data set, characterized by strongly anisotropic variances for each group, were compared to these simulated distributions.
Mentions: Since the CVA standardizes the within-group variance, it should be more impacted than the PCA and bgPCA by the occurrence of Pmax shared among groups. To validate this hypothesis, simulations were performed. The initial groups were bootstrapped, with the condition that the variance of all shape variables should follow a similar uniform distribution. The different groups should thus no longer share parallel Pmax. The resulting simulated configurations were compared using Protest (Fig 5). The simulated and the observed configurations provided by the PCA and the bgPCA were all very close. This shows that the PCA and bgPCA provided similar pictures of the morphometric differentiation, and that this picture was not impacted by Pmax. For the CVA cases, simulated datasets displayed configurations closer by 4 to 5 times compared to those provided by the PCA and the bgPCA than the original configuration. This is in agreement with the fact that the occurrence of a marked direction of main variance Pmax, parallel among groups, strongly impacts the pattern of differentiation as revealed by a CVA. The CVA assumes isotropic variation. In the case of anisotropic variance oriented along similar Pmax, this assumption is not fulfilled.

Bottom Line: Standardizing within-group variance, as performed in the CVA, distorts the relationships among groups, an effect that is particularly strong if the variance is similarly oriented in a comparable way in all groups.Without arguing about a method performing 'better' than another, it rather emerges that working on the total or between-group variance (PCA and bgPCA) will tend to put the focus on the role of direction of main variance as line of least resistance to evolution.Standardizing by the within-group variance (CVA), by dampening the expression of this line of least resistance, has the potential to reveal other relevant patterns of differentiation that may otherwise be blurred.

View Article: PubMed Central - PubMed

Affiliation: Laboratoire de Biom├ętrie et Biologie Evolutive, UMR5558, CNRS, University Lyon 1, 69622 Villeurbanne, France.

ABSTRACT
Geometric morphometrics aims to characterize of the geometry of complex traits. It is therefore by essence multivariate. The most popular methods to investigate patterns of differentiation in this context are (1) the Principal Component Analysis (PCA), which is an eigenvalue decomposition of the total variance-covariance matrix among all specimens; (2) the Canonical Variate Analysis (CVA, a.k.a. linear discriminant analysis (LDA) for more than two groups), which aims at separating the groups by maximizing the between-group to within-group variance ratio; (3) the between-group PCA (bgPCA) which investigates patterns of between-group variation, without standardizing by the within-group variance. Standardizing within-group variance, as performed in the CVA, distorts the relationships among groups, an effect that is particularly strong if the variance is similarly oriented in a comparable way in all groups. Such shared direction of main morphological variance may occur and have a biological meaning, for instance corresponding to the most frequent standing genetic variation in a population. Here we undertake a case study of the evolution of house mouse molar shape across various islands, based on the real dataset and simulations. We investigated how patterns of main variance influence the depiction of among-group differentiation according to the interpretation of the PCA, bgPCA and CVA. Without arguing about a method performing 'better' than another, it rather emerges that working on the total or between-group variance (PCA and bgPCA) will tend to put the focus on the role of direction of main variance as line of least resistance to evolution. Standardizing by the within-group variance (CVA), by dampening the expression of this line of least resistance, has the potential to reveal other relevant patterns of differentiation that may otherwise be blurred.

No MeSH data available.