testmeanshapes function

Tests for mean shape difference, including permutation and bootstrap tests

Carries out tests to examine differences in mean shape between two independent populations, for $m=2$ or $m=3$ dimensional data. Tests are carried out using tangent co-ordinates.

H : Hotelling $T^2$ statistic (see Dryden and Mardia, 2016, equ.(9.4))

G : Goodall's F statistic (see Dryden and Mardia, 2016, equ.(9.9))

J : James $T^2$ statistic (see Amaral et al., 2007)

p-values are given based on resampling (either a bootstrap test or a permutation test) as well as the usual table based p-values. Bootstrap tests involve sampling with replacement under H0 (as in Amaral et al., 2007).

Note when the sample sizes are low (compared to the number of landmarks) some minor regularization is carried out. In particular if Sw is a singular within group covariance matrix, it is replaced by Sw + 0.000001 (Identity matrix) and a `*' is printed in the output.

testmeanshapes(A, B, resamples = 1000, replace = FALSE, scale= TRUE)

Arguments

• A: The random sample for group 1: k x m x n1 array of data, where k is the number of landmarks and n1 is the sample size. (Alternatively a k x n1 complex matrix for 2D)
• B: The random sample for group 2: k x m x n2 array of data, where k is the number of landmarks and n2 is the sample size. (Alternatively a k x n2 complex matrix for 2D)
• resamples: Integer. The number of resampling iterations. If resamples = 0 then no resampling procedures are carried out, and the tabular p-values are given only.
• replace: Logical. If replace = TRUE then bootstrap resampling is carried out with replacement within each group. If replace = FALSE then permutation resampling is carried out (sampling without replacement in pooled samples).
• scale: Logical. Whether or not to carry out Procrustes with scaling in the procedure.

Returns

A list with components - H: The Hotelling statistic (F statistic)

• H.pvalue: p-value for the Hotelling test based on resampling

• H.table.pvalue: p-value for the Hotelling test based on the null F distribution, assuming normality and equal covariance matrices

• J: The James $T^2$ statistic

• J.pvalue: p-value for the James $T^2$ test based on resampling

• J.table.pvalue: p-value for the James $T^2$ test based on the null F distribution, assuming normality but unequal covariance matrices

• G: The Goodall $F$ statistic

• G.pvalue: p-value for the Goodall test based on resampling

• G.table.pvalue: p-value for the Goodall test based on the null F distribution, assuming normality and equal isotropic covariance matrices)

References

Amaral, G.J.A., Dryden, I.L. and Wood, A.T.A. (2007) Pivotal bootstrap methods for $k$-sample problems in directional statistics and shape analysis. Journal of the American Statistical Association. 102, 695-707.

Dryden, I.L. and Mardia, K.V. (2016). Statistical Shape Analysis, with applications in R (Second Edition). Wiley, Chichester. Chapter 9.

Goodall, C. R. (1991). Procrustes methods in the statistical analysis of shape (with discussion). Journal of the Royal Statistical Society, Series B, 53: 285-339.

Author(s)

Ian Dryden

See Also

resampletest

Examples

#2D example : female and male Gorillas

data(gorf.dat)
data(gorm.dat)

A<-gorf.dat
B<-gorm.dat
testmeanshapes(A,B,resamples=100)