p.asym {CCP}

Asymptotic tests for the statistical significance of canonical correlation coefficients

Description

This function runs asymptotic tests to assign the statistical significance of canonical correlation coefficients. F-approximations of Wilks' Lambda, the Hotelling-Lawley Trace, the Pillai-Bartlett Trace, or of Roy's Largest Root can be used as a test statistic.

Usage

p.asym(rho, N, p, q, tstat = "Wilks")

Arguments

rho

vector containing the canonical correlation coefficients.

N

number of observations for each variable.

p

number of independent variables.

q

number of dependent variables.

tstat

test statistic to be used. One of "Wilks" (default), "Hotelling", "Pillai", or "Roy".

Details

Canonical correlation analysis (CCA) measures the degree of linear relationship between two sets of variables. The number of correlation coefficients calculated in CCA is equal to the number of variables in the smaller set: m = min(p,q). The coefficients are arranged in descending order of magnitude: rho[1] > rho[2] > rho[3] > ... > rho[m]. Except for tstat = "Roy", the function p.asym calculates m p-values for each test statistic: the first p-value is calculated including all canonical correlation coefficients, the second p-value is calculated by excluding rho[1], the third p-value is calculated by excluding rho[1] and rho[2] etc., therewith allowing assessment of the statistical significance of each individual correlation coefficient. On principle, Roy's Largest Root takes only rho[1] into account, hence one p-value is calculated only.

Value

stat

value of the statistic, i.e. the value of either Wilks' Lambda, the Hotelling-Lawley Trace, the Pillai-Bartlett Trace, or Roy's Largest Root.

approx

value of the corresponding F-approximation for the statistic.

df1

numerator degrees of freedom for the F-approximation.

df2

denominator degrees of freedom for the F-approximation.

p.value

p-value

Note

Usage of asymptotic approximations requires multivariate normality of the variables, or a large number of observations. Canonical correlation is sensitive to outliers. The F-approximation for Roy's Largest Root is an upper bound, and the significance level is therefore optimistically small. The canonical correlation coefficients are statistically significant if Wilks' Lambda is smaller than a critical value.

Author(s)

Uwe Menzel <[e-mail]>

References

Wilks, S. S. (1935) On the independence of k sets of normally distributed statistical variables. Econometrica, 3 309–326.

Rao, C. R. (1973) Linear Statistical Inference and It's Applications (2nd ed.). John Wiley and Sons, New York, 533–543, 555–556.

Pillai, K. C. W. (1956) On the distribution of the largest or the smallest root of a matrix in multivariate analysis. Biometrika, 43 122–127.

Muller, K. E. and Peterson B. L. (1984) Practical Methods for computing power in testing the multivariate general linear hypothesis. Computational Statistics & Data Analysis, 2 143–158.

Anderson, T. W. (1984) An introduction to Multivariate Statistical Analysis. John Wiley and Sons, New York.

See Also

See the function cancor or the CCA package for the calculation of canonical correlation coefficients.

Examples

## Load the CCP package:
library(CCP)



## Simulate example data:
X <- matrix(rnorm(150), 50, 3)
Y <- matrix(rnorm(250), 50, 5)


## Calculate canonical correlations:
rho <- cancor(X,Y)$cor

## Define number of observations, 
## and number of dependent and independent variables:
N = dim(X)[1]       
p = dim(X)[2]   
q = dim(Y)[2]

## Calculate p-values using F-approximations of some test statistics:
p.asym(rho, N, p, q, tstat = "Wilks")
p.asym(rho, N, p, q, tstat = "Hotelling")
p.asym(rho, N, p, q, tstat = "Pillai")
p.asym(rho, N, p, q, tstat = "Roy")

## Plot the F-approximation for Wilks' Lambda, 
## considering 3, 2, or 1 canonical correlation(s):
res1 <- p.asym(rho, N, p, q)
plt.asym(res1,rhostart=1)
plt.asym(res1,rhostart=2)
plt.asym(res1,rhostart=3)