# barttest

Bartlett’s test

## 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.

example

[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.

example

## Examples

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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

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Input data, specified as a matrix of scalar values.

Data Types: single | double

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

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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.

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.

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

## Version History

Introduced before R2006a