barttest

Bartlett’s test

Syntax

ndim = barttest(x,alpha)

[ndim,prob,chisquare] = barttest(x,alpha)

Description

ndim = barttest(x,alpha) returns the number of dimensions necessary to explain the nonrandom variation in the data matrix x at the alpha significance level.

[ndim,prob,chisquare] = barttest(x,alpha) also returns the significance values for the hypothesis tests prob, and the χ² values associated with the tests chisquare.

Examples

Determine Dimensions Needed to Explain Nonrandom Data Variation

Generate a 20-by-6 matrix of random numbers from a multivariate normal distribution with mean mu = [0 0] and covariance sigma = [1 0.99; 0.99 1].

rng default  % for reproducibility
mu = [0 0];
sigma = [1 0.99; 0.99 1];
X = mvnrnd(mu,sigma,20);  % columns 1 and 2
X(:,3:4) = mvnrnd(mu,sigma,20);  % columns 3 and 4
X(:,5:6) = mvnrnd(mu,sigma,20);  % columns 5 and 6

Determine the number of dimensions necessary to explain the nonrandom variation in data matrix X. Report the significance values for the hypothesis tests.

[ndim, prob] = barttest(X,0.05)

ndim = 3

prob = 5×1

    0.0000
    0.0000
    0.0000
    0.5148
    0.3370

The returned value of ndim indicates that three dimensions are necessary to explain the nonrandom variation in X.

Input Arguments

x — Input data

matrix of scalar values

Input data, specified as a matrix of scalar values.

Data Types: single | double

alpha — Significance level

0.05 (default) | scalar value in the range (0,1)

Significance level of the hypothesis test, specified as a scalar value in the range (0,1).

Example: 0.1

Data Types: single | double

Output Arguments

ndim — Number of dimensions

positive integer value

Number of dimensions, returned as a positive integer value. The dimension is determined by a series of hypothesis tests. The test for ndim = 1 tests the hypothesis that the variances of the data values along each principal component are equal, the test for ndim = 2 tests the hypothesis that the variances along the second through last components are equal, and so on. The null hypothesis is that the number of dimensions is equal to the number of the largest unequal eigenvalues of the covariance matrix of x.

prob — Significance value

vector of scalar values in the range (0,1)

Significance value for the hypothesis tests, returned as a vector of scalar values in the range (0,1). Each element in prob corresponds to an element of chisquare.

chisquare — Test statistics

vector of scalar values

Test statistics for each dimension’s hypothesis test, returned as a vector of scalar values.

Introduced before R2006a