p.perm | R Documentation |
This function runs a permutation test to assign the statistical significance of canonical correlation coefficients. Wilks' Lambda, the Hotelling-Lawley Trace, the Pillai-Bartlett Trace, or of Roy's Largest Root can be used as a test statistic.
p.perm(X, Y, nboot = 999, rhostart = 1, type = "Wilks")
X |
array containing the independent variables, with N rows (number of observations) and p columns (number of independent variables). |
Y |
array containing the dependent variables, with N rows (number of observations) and q columns (number of dependent variables). |
nboot |
number of permutation resamples calculated. |
rhostart |
index of the largest canonical correlation coefficient included in the calculation of the test statistic (see Details). |
type |
test statistic to be used. One of "Wilks" (default), "Hotelling", "Pillai", or "Roy". |
Permutation tests are based on resampling of the original data without replacement. To test the hypothesis of no correlation
between two sets (X, Y) of variables, the values of one variable (Y) are randomly reassigned. Permutation tests do not require
specific population distributions of the variables such as normal distribution.
Canonical correlation analysis (CCA) calculates m = min(p,q) correlation coefficients, see p.asym
.
The coefficients are arranged in descending order of magnitude: rho[1] > rho[2] > rho[3] > ... > rho[m].
In p.perm
, the parameter rhostart determines how many correlation coefficients are included in the calculation
of the test statistic:
choosing rhostart=1, all canonical correlations are included,
choosing rhostart=2, the biggest canonical correlation (rho[1]) is excluded,
choosing rhostart=3, both rho[1] and rho[2] are excluded, etc.
On principle, Roy's Largest Root takes only rho[1] into account, hence only rhostart=1 can be chosen.
stat0 |
original value of the statistic (without resampling). |
stat |
nboot values of the statistic, one for each permutation resampling. |
nexcess |
number of permutation resamplings that resulted in a more extreme value of the statistic than stat0. |
p.value |
p-value, derived from nexcess. |
Permutation tests do not require a specific distribution of the variables. Tests based on random resampling do generally not yield the same results when repeated. The canonical correlation coefficients are statistically significant if Wilks' Lambda is smaller than a critical value.
Uwe Menzel <uwemenzel@gmail.com>
Efron, B. and Tibshirani, R. J. (1994) An Introduction to the Bootstrap, Chapman & Hall, New York.
Moore, D. S. amd McCabe, G. P. (2006) Introduction to the Practice of Statistics, Chapter 14, W. H. Freeman, New York.
See the function cancor
or the CCA
package for the calculation of canonical correlation coefficients.
## Load the CCP package: library(CCP) ## Simulate example data: X <- matrix(rnorm(150), 50, 3) Y <- matrix(rnorm(250), 50, 5) ## Run permutation test using Wilks Lambda (default) as test statistic; ## include different numbers of canonical correlations: p.perm(X, Y, nboot = 999, rhostart = 1) p.perm(X, Y, nboot = 999, rhostart = 2) p.perm(X, Y, nboot = 999, rhostart = 3) ## Run permutation tests based on different test statistics: p.perm(X, Y, nboot = 999, rhostart = 1, type = "Wilks") p.perm(X, Y, nboot = 999, rhostart = 1, type = "Hotelling") p.perm(X, Y, nboot = 999, rhostart = 1, type = "Pillai") p.perm(X, Y, nboot = 999, rhostart = 1, type = "Roy") ## Plot the permutation distribution ## with the value of the original statistic marked: out <- p.perm(X, Y, nboot = 999, rhostart = 3, type = "Hotelling") plt.perm(out)
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