Editing Canonical correlation (section)

==Hypothesis testing==
Each row can be tested for significance with the following method. Since the correlations are sorted, saying that row <math>i</math> is zero implies all further correlations are also zero.  If we have <math>p</math> independent observations in a sample and <math>\widehat{\rho}_i</math> is the estimated correlation for <math>i = 1,\dots, \min\{m,n\}</math>. For the <math>i</math>th row, the test statistic is:

:<math>\chi^2 = - \left( p - 1 - \frac{1}{2}(m + n + 1)\right) \ln \prod_{j = i}^{\min\{m,n\}} (1 - \widehat{\rho}_j^2),</math>

which is asymptotically distributed as a [[chi-squared distribution|chi-squared]] with <math>(m - i + 1)(n - i + 1)</math> [[degrees of freedom (statistics)|degrees of freedom]] for large <math>p</math>.<ref>{{Cite book
 | author = [[Kanti V. Mardia]], J. T. Kent and J. M. Bibby
 | title = Multivariate Analysis
 | year = 1979
 | publisher = [[Academic Press]]
}}</ref>  Since all the correlations from <math> \min\{m,n\}</math> to <math>p</math> are logically zero (and estimated that way also) the product for the terms after this point is irrelevant.

Note that in the small sample size limit with <math>p < n + m</math> then we are guaranteed that the top <math>m + n - p</math> correlations will be identically 1 and hence the test is meaningless.<ref>Yang Song, Peter J. Schreier, David Ram´ırez, and Tanuj Hasija ''Canonical correlation analysis of high-dimensional data with very small sample support'' {{ArXiv|1604.02047}}</ref>