covDist | R Documentation |

calculates PC-coordinates of covariance matrices by using the Riemannian metric in their respective space.

```
covDist(s1, s2)
covPCA(
data,
groups,
rounds = 1000,
bootrounds = 0,
lower.bound = 0.05,
upper.bound = 0.95
)
```

`s1` |
m x m covariance matrix |

`s2` |
m x m covariance matrix |

`data` |
matrix containing data with one row per observation |

`groups` |
factor: group assignment for each specimen |

`rounds` |
integer: rounds to run permutation of distances by randomly assigning group membership |

`bootrounds` |
integer: perform bootstrapping to generate confidence intervals (lower boundary, median and upper boundary) for PC-scores. |

`lower.bound` |
numeric: set probability (quantile) for lower boundary estimate from bootstrapping. |

`upper.bound` |
numeric: set probability (quantile) for upper boundary estimate from bootstrapping. |

`covDist`

calculates the Distance between covariance matrices while `covPCA`

uses a MDS (multidimensional scaling) approach to obtain PC-coordinates
from a distance matrix derived from multiple groups. P-values for pairwise
distances can be computed by permuting group membership and comparing actual
distances to those obtained from random resampling. To calculate confidence intervals for PC-scores, within-group bootstrapping can be performed.

`covDist`

returns the distance between s1 and s2

`covPCA`

returns a list containing:

if `scores = TRUE`

`PCscores` |
PCscores |

`eigen` |
eigen decomposition of the centered inner product |

if `rounds > 0`

`dist` |
distance matrix |

`p.matrix` |
p-values for pairwise distances from permutation testing |

if `bootrounds > 0`

`bootstrap` |
list containing the lower and upper bound of the confidence intervals of PC-scores as well as the median of bootstrapped values. |

`boot.data` |
array containing all results generated from bootstrapping. |

Stefan Schlager

Mitteroecker P, Bookstein F. 2009. The ontogenetic trajectory of the phenotypic covariance matrix, with examples from craniofacial shape in rats and humans. Evolution 63:727-737.

Hastie T, Tibshirani R, Friedman JJH. 2013. The elements of statistical learning. Springer New York.

`prcomp`

```
cpca <- covPCA(iris[,1:4],iris[,5])
cpca$p.matrix #show pairwise p-values for equal covariance matrices
## Not run:
require(car)
sp(cpca$PCscores[,1],cpca$PCscores[,2],groups=levels(iris[,5]),
smooth=FALSE,xlim=range(cpca$PCscores),ylim=range(cpca$PCscores))
data(boneData)
proc <- procSym(boneLM)
pop <- name2factor(boneLM, which=3)
## compare covariance matrices for PCscores of Procrustes fitted data
cpca1 <- covPCA(proc$PCscores, groups=pop, rounds = 1000)
## view p-values:
cpca1$p.matrix # differences between covariance matrices
# are significant
## visualize covariance ellipses of first 5 PCs of shape
spm(proc$PCscores[,1:5], groups=pop, smooth=FALSE,ellipse=TRUE, by.groups=TRUE)
## covariance seems to differ between 1st and 5th PC
## for demonstration purposes, try only first 4 PCs
cpca2 <- covPCA(proc$PCscores[,1:4], groups=pop, rounds = 1000)
## view p-values:
cpca2$p.matrix # significance is gone
## End(Not run)
#do some bootstrapping 1000 rounds
cpca <- covPCA(iris[,1:4],iris[,5],rounds=0, bootrounds=1000)
#plot bootstrapped data of PC1 and PC2 for first group
plot(t(cpca$boot.data[1,1:2,]),xlim=range(cpca$boot.data[,1,]),
ylim=range(cpca$boot.data[,2,]))
points(t(cpca$PCscores[1,]),col="white",pch=8,cex=1.5)##plot actual values
for (i in 2:3) {
points(t(cpca$boot.data[i,1:2,]),col=i)##plot other groups
points(t(cpca$PCscores[i,]),col=1,pch=8,cex=1.5)##plot actual values
}
```

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