The numerical results first present the eigenvalue decomposition of the matrix of
co-inertia on two axes (rows): eigenvalues (eig), covariance (covar), standard deviation (sdX and sdY) of the two sets of site scores on the co-inertia axes and
correlations between the two sets of site scores, computed using the Pearson
correlation coefficient.
The second and third blocks of results compare the inertia of the (cumulated)
projections of the data tables, called X and Y in the function, as they are projected in
the co-inertia analysis (“inertia”), compared to the maximum cumulated inertia of the
axes of the separate ordinations (“max”). It also gives the ratio between these values
as a measure of concordance between the two projections.
The RV coefficient is the ratio of the total co-inertia to the square root of the
product of the total inertias of the separate analyses (Robert and Escoufier, 1976).
Ranged between 0 (independent) and 1 (homothetic), it measures the closeness
between the two sets of points derived from the separate ordinations of X and
Y. For two simple variables x 1 and x 2 , RV is the square of their Pearson correlation
coefficient.
These results show that the first eigenvalue, which represents 98.9% of the total
variation, is overwhelmingly larger than the second one. Most of the common
structure of the two data matrices is therefore to be sought along the first axis. The
circular plots in Fig. 6.17 show that axes 1 of the two PCAs are almost perfectly
aligned on the first CoIA axis. The upper right-hand plot (normed site scores) shows
the positions of the sites on the co-inertia axes using the chemistry (origins of the
arrows) and physiography (arrowheads) co-inertia weights. The shorter an arrow is,
the better the concordance between the two projections of the point. The lower righthand pair of plots shows the contributions of the two groups of variables to the
canonical space. Vectors pointing in the same direction are correlated and longer
vectors contribute more to the structure. Oxygen (oxy) correlates positively with
slope (slo), phosphates (pho) negatively with slope (slo); nitrates (nit), hardness (har, label masked by nitrates) and biological oxygen demand (bod) are all
negatively correlated with elevation (ele) since these variables have higher values
downstream, and positively with discharge (dis), which increases downstream.
An extension of CoIA called RLQ analysis (Dolédec et al. 1996; Dray et al. 2002)
relates species traits to environmental variables by means of three tables: site-byspecies (table L), site-by-environment (table R), and species-by-traits (table Q).
Another related method, developed by Legendre et al. (1997) and Dray and Legendre (2008), is also an answer to what these authors have called the fourth-corner
problem. The RLQ and fourth-corner analyses are presented in Sect. 6.11.
6.9 Co-inertia Analysis (CoIA)
281
co-inertia on two axes (rows): eigenvalues (eig), covariance (covar), standard deviation (sdX and sdY) of the two sets of site scores on the co-inertia axes and
correlations between the two sets of site scores, computed using the Pearson
correlation coefficient.
The second and third blocks of results compare the inertia of the (cumulated)
projections of the data tables, called X and Y in the function, as they are projected in
the co-inertia analysis (“inertia”), compared to the maximum cumulated inertia of the
axes of the separate ordinations (“max”). It also gives the ratio between these values
as a measure of concordance between the two projections.
The RV coefficient is the ratio of the total co-inertia to the square root of the
product of the total inertias of the separate analyses (Robert and Escoufier, 1976).
Ranged between 0 (independent) and 1 (homothetic), it measures the closeness
between the two sets of points derived from the separate ordinations of X and
Y. For two simple variables x 1 and x 2 , RV is the square of their Pearson correlation
coefficient.
These results show that the first eigenvalue, which represents 98.9% of the total
variation, is overwhelmingly larger than the second one. Most of the common
structure of the two data matrices is therefore to be sought along the first axis. The
circular plots in Fig. 6.17 show that axes 1 of the two PCAs are almost perfectly
aligned on the first CoIA axis. The upper right-hand plot (normed site scores) shows
the positions of the sites on the co-inertia axes using the chemistry (origins of the
arrows) and physiography (arrowheads) co-inertia weights. The shorter an arrow is,
the better the concordance between the two projections of the point. The lower righthand pair of plots shows the contributions of the two groups of variables to the
canonical space. Vectors pointing in the same direction are correlated and longer
vectors contribute more to the structure. Oxygen (oxy) correlates positively with
slope (slo), phosphates (pho) negatively with slope (slo); nitrates (nit), hardness (har, label masked by nitrates) and biological oxygen demand (bod) are all
negatively correlated with elevation (ele) since these variables have higher values
downstream, and positively with discharge (dis), which increases downstream.
An extension of CoIA called RLQ analysis (Dolédec et al. 1996; Dray et al. 2002)
relates species traits to environmental variables by means of three tables: site-byspecies (table L), site-by-environment (table R), and species-by-traits (table Q).
Another related method, developed by Legendre et al. (1997) and Dray and Legendre (2008), is also an answer to what these authors have called the fourth-corner
problem. The RLQ and fourth-corner analyses are presented in Sect. 6.11.
6.9 Co-inertia Analysis (CoIA)
281