Test whether two vectors are independent:

The -values are typically large when the vectors are independent:

The -values are typically small when there are dependencies:

Test whether two matrices are independent:

The -values are typically small for dependent matrices:

The -values are typically large when matrices are independent:

Using Automatic applies the most powerful appropriate test for linear independence:

The property "AutomaticTest" can be used to determine which test was chosen:

Perform a particular test for independence:

Any number of tests can be performed simultaneously:

Perform all tests appropriate to the data simultaneously:

Use the property "AllTests" to identify which tests were used:

Create a HypothesisTestData object for repeated property extraction:

The properties available for extraction:

Extract some properties from the HypothesisTestData object:

The -value and test statistic from the "PearsonCorrelation" test:

Extract any number of properties simultaneously:

The -value and test statistic from a Spearman rank test:

Tabulate the results from a selection of tests:

A full table of all appropriate test results:

A table of selected test results:

Retrieve the entries from a test table for customized reporting:

The -values are above 0.05, so there is not enough evidence to reject at that level:

Tabulate -values for a test or group of tests:

The -value from the table:

A table of -values from all appropriate tests:

A table of -values from a subset of tests:

Report the test statistic from a test or group of tests:

The test statistic from the table:

A table of test statistics from all appropriate tests:

A two-sided test is performed by default:

Perform a two-sided test or a one-sided alternative:

A two-sided test:

The two one-sided alternatives:

Some tests are inherently two-sided:

This is due to the shape of the null distribution:

Set the maximum number of iterations to use for multivariate nonparametric tests:

Setting the value too low may result in failed convergence but produce faster timings:

By default, -values are computed using asymptotic test statistic distributions:

The -value can be obtained using permutation methods:

Set the number of permutations to use:

By default, random permutations are used:

Set the seed used for generating random permutations:

Set the significance level for diagnostic tests:

By default, 0.05 is used. The message shows 0.025 because two tests were performed:

Setting the significance level may alter which test is automatically chosen:

A nonparametric test would have been chosen by default:

The significance level is also used for "TestConclusion" and "ShortTestConclusion":

By default, normality is tested when appropriate:

Diagnostics can be controlled as a group using All or None:

Verify all assumptions:

Check no assumptions:

Diagnostics can be controlled independently:

Check for normality:

Explicitly set the diagnostic result:

It is often useful to bypass diagnostic tests for simulation purposes:

The assumptions of the test hold by design, so a great deal of time can be saved:

The results are identical: