Features of C O V (m a n C O V a)
 COV is still an alpha version. Please send me your comments, critics, or suggestions.
 ONE SINGLE (not subdivided) MATRIX
 – Some simple matrix operations (transposition, inversion, determinant,etc.)
 – Tranposed matrix X multiplied by matrix X
 – PCA on total variancecovariance matrix
 – PCA on the pooled variancecovariance matrix
 – PARTIAL and RELATIVE WARPS scores on a single matrix (menu ‘single matrix’)
 – Two Block Partial Least Squares (on a concatenated matrix)
 PLS is a method for exploring and visualizing the patterns of covariance between two sets of variables. Here the analysis is under the menu “One single matrix” as long as the two sets of variables have been concatenated by columns into a single matrix. The same analysis can be performed on the separate two sets (see next menu “TWO MATRICES”).
 – METRIC DISPARITY
 – After successive entries of a single matrix, COV compares MD among samples by using non parametric tests (bootstrapping, see Zelditch et al., 2004. Geometric Morphometrics for Biologists. A primer).
 – Euclidian and Procrustes distances between rows of any data matrix
 – Matrix multiplication
 – Matrix substraction
 – Tranposed matrix X multiplied by matrix Y
 – “b” od “Y = b X + epsilon”
 – Correlation coefficients between each column of X and each column of Y
 – Correlation coefficients between corresponding columns of X and Y
 – T2 Hotelling test comparing two matrices
 ANGLE between HYPERPLANES (menu ‘angle between matrices’)
 – COV computes this angle, its confidence interval, pseudomean and standard error by bootrapping;
 – COV compares this angle among samples on a subdivided matrix (menu ‘subdivided matrix’), by using non parametric tests also (bootstrapping);
 – All these calculations are slow. They are following the formulas from Zelditch et al., 2004. Geometric Morphometrics for Biologists. A primer.
 PLS, Partial Least Squares
 It is a method for exploring and visualizing the patterns of covariance between two sets of variables; it finds pairs of new axes which account for the maximum amount of covariance between two sets of variables. The pattern of covariance can be analyzed one pair of PLS axes at a time.
 – Submatrix: CREATE
 – Submatrix: DESCRIPTIVE STATISTICS
 This option allows to build the distance matrix between subdivisions (groups), and from there to build a Neighbor Joining TREE if distances are Procrustes distances, an UPGMA tree if distances are Euclidean ones. See the video :
 Procrustes distances : the input file must contain raw coordinates (the file “…_format.txt”)
 Euclidean distances : the input file contains partial or relative warps (the file “…_PW.txt”, or “…_RW.txt”)
 METRIC DISPARITY (menu ‘single matrix’, and menu ‘subdivided matrix’)
 – COV computes MD (metric disparity) of shape (PW scores), its confidence interval, pseudomean and standard error by bootrapping
 – COV compares MD among sample, after successive entries (menu ‘single matrix’) or on a subdivided matrix (menu ‘subdivided matrix’), by using non parametric tests (bootstrapping).
 – COV computes Partial metric disparities (menu ‘subdivided matrix’)
 – All these calculations according to Zelditch et al., 2004. Geometric Morphometrics for Biologists. A primer.
 – Qst
 Qst – Q for ‘quantitative’ – Qst partitions quantitative genetic variation in a manner analogous to Fst for single gene markers (see Spitze, K. 1993. Population structure in Daphnia obtusa: quantitative genetic and allozymic variation. Genetics 135: 367374).
 COV computes Qst for Centroid size and each of selected Relative Warps, as well as for their average.
 Permutations are performed to test for significance among groups.
 Bootstrap is performed to compute standard deviations.
 DIFFERENT VERSUS COMMON SLOPE MODELS (menu Different versus common slope allometric model’ )
 After computing multivariate regression with (centroid) size as independent variable, and shape variables (partial warps scores, including uniform component) as dependent variables, COV will test for :
 – residual allometry within shape (PW) by multivariate regression (and permutation test for statistical significance)
 – a “common allometric slope” versus a “different slopes” model among groups
 – in case of a valid common model, COV would make it possible to compute shape differences among groups after fixing size to a unique value (i.e., shape changes are completely free of size variation), thus fixing covariate as in a classic MANCOVA.
 To allow for comparisons with existing software, COV also computes the xdesign matrix in the .nts format as needed by TPSregr to compare both common and different slopes models (see the HELP of TPSregr)
 ANGLE between, and MAGNITUD of, VECTORS separating subgroups (menu ‘differences in magnitud and direction of change’)
 – COV computes the AMOUNT of difference ‘Da’ between two subgroups A1 and A2 of group A, it computes the AMOUNT of difference ‘Db’ between two subgroups B1 and B2 of group B, and estimates the statistical significance of the difference DaDb by non parametric tests (permutation of residuals);
 – COV computes the ANGLE between the two vectors having dimensions Da and Db, and estimates its statistical significance by non parametric tests (permutation of residuals);
 – All these calculations according to Collyer, M. L. and Adams, D. C. 2007. Analysis of twostate multivariate phenotypic change in ecological studies.
 EXTERNAL SOFTWARE
 The PHYLIP ‘neighbor’ and ‘fitch’ modules (J. Felsenstein) may be called directly from COV (as well as from PAD)
 After answering the PHYLIP interface questions (by the letters ‘N’, ‘L’ and ‘Y’)
 … the PHYLIP outfile is directly printed within the COV report.
 Direct access is also given to editing tree software like njplot (see: Perriere, G. and Gouy, M. (1996) WWWQuery: An online retrieval system for biological sequence banks. Biochimie, 78, 364369).
 A module is used from TREEDYN (F. Chevenet) to modify the automatic numbering of groups, giving them full names instead of numbers.
