ade4 (version 1.7-15)

dagnelie.test: Dagnelie multinormality test

Description

Compute Dagnelie test of multivariate normality on a data table of n objects (rows) and p variables (columns), with n > (p+1).

Usage

dagnelie.test(x)

Arguments

x

Multivariate data table (matrix or data.frame).

Value

A list containing the following results:

Shapiro.Wilk

W statistic and p-value

dim

dimensions of the data matrix, n and p

rank

the rank of the covariance matrix

D

Vector containing the Mahalanobis distances of the objects to the multivariate centroid

Details

Dagnelie's goodness-of-fit test of multivariate normality is applicable to multivariate data. Mahalanobis generalized distances are computed between each object and the multivariate centroid of all objects. Dagnelie<U+2019>s approach is that, for multinormal data, the generalized distances should be normally distributed. The function computes a Shapiro-Wilk test of normality of the Mahalanobis distances; this is our improvement of Dagnelie<U+2019>s method. The null hypothesis (H0) is that the data are multinormal, a situation where the Mahalanobis distances should be normally distributed. In that case, the test should not reject H0, subject to type I error at the selected significance level.

Numerical simulations by D. Borcard have shown that the test had correct levels of type I error for values of n between 3p and 8p, where n is the number of objects and p is the number of variables in the data matrix (simulations with 1 <= p <= 100). Outside that range of n values, the results were too liberal, meaning that the test rejected too often the null hypothesis of normality. For p = 2, the simulations showed the test to be valid for 6 <= n <= 13 and too liberal outside that range. If H0 is not rejected in a situation where the test is too liberal, the result is trustworthy.

Calculation of the Mahalanobis distances requires that n > p+1 (actually, n > rank+1). With fewer objects (n), all points are at equal Mahalanobis distances from the centroid in the resulting space, which has min(rank,(n-1)) dimensions. For data matrices that happen to be collinear, the function uses ginv for inversion.

This test is not meant to be used with univariate data; in simulations, the type I error rate was higher than the 5% significance level for all values of n. Function shapiro.test should be used in that situation.

References

Dagnelie, P. 1975. L'analyse statistique a plusieurs variables. Les Presses agronomiques de Gembloux, Gembloux, Belgium.

Legendre, P. and L. Legendre. 2012. Numerical ecology, 3rd English edition. Elsevier Science BV, Amsterdam, The Netherlands.

Examples

# NOT RUN {
 # Example 1: 2 variables, n = 100
 n <- 100; p <- 2
 mat <- matrix(rnorm(n*p), n, p)
 (out <- dagnelie.test(mat))

 # Example 2: 10 variables, n = 50
 n <- 50; p <- 10
 mat <- matrix(rnorm(n*p), n, p)
 (out <- dagnelie.test(mat))

 # Example 3: 10 variables, n = 100
 n <- 100; p <- 10
 mat <- matrix(rnorm(n*p), n, p)
 (out <- dagnelie.test(mat))
 # Plot a histogram of the Mahalanobis distances
hist(out$D)

 # Example 4: 10 lognormal random variables, n = 50
 n <- 50; p <- 10
 mat <- matrix(round(exp(rnorm((n*p), mean = 0, sd = 2.5))), n, p)
 (out <- dagnelie.test(mat))
 # Plot a histogram of the Mahalanobis distances
 hist(out$D)

# }