Tracy–Widom test — tw • algatr

This function is adapted from the tw function in the archived CRAN package AssocTests. The implementation and original documentation were authored by Lin Wang, Wei Zhang, and Qizhai Li. The AssocTests package is archived on CRAN; see the CRAN archive for details.

Usage

tw(eigenvalues, eigenL, criticalpoint = 2.0234)

Arguments

eigenvalues: A numeric vector of eigenvalues (sorted decreasing).
eigenL: The number of eigenvalues.
criticalpoint: A numeric value corresponding to the significance level. If the significance level is 0.05, 0.01, 0.005, or 0.001, set criticalpoint to 0.9793, 2.0234, 2.4224, or 3.2724, respectively. Default: 2.0234.

Value

A list with class "htest":

`statistic`			a vector of Tracy–Widom statistics.
`alternative`			description of the alternative hypothesis.
`method`			test name.
`data.name`			name of the input data.
`SigntEigenL`			number of significant eigenvalues.

Details

Find the significant eigenvalues of a matrix (from AssocTests).

This function implements the Tracy–Widom test to determine how many leading eigenvalues of a matrix are statistically significant. This function and its documentation are from the tw function in the archived CRAN package AssocTests (version 1.0-1).

The input eigenvalues should be sorted in descending order. The test statistic is computed for each leading eigenvalue and compared to a supplied Tracy–Widom critical value.

References

Lin Wang, Wei Zhang, and Qizhai Li. AssocTests: An R Package for Genetic Association Studies. Journal of Statistical Software. 2020; 94(5): 1–26.

N. Patterson, A. L. Price, and D. Reich. Population Structure and Eigenanalysis. PLoS Genetics. 2006; 2(12): 2074–2093.

C. A. Tracy and H. Widom. Level-Spacing Distributions and the Airy Kernel. Communications in Mathematical Physics. 1994; 159(1): 151–174.

A. Bejan. Tracy–Widom and Painlevé II: Computational Aspects and Realisation in S-Plus. In First Workshop of the ERCIM Working Group on Computing and Statistics. 2008, Neuchâtel, Switzerland.

A. Bejan. Largest eigenvalues and sample covariance matrices. MSc Dissertation, University of Warwick. 2005. (This function was written by A. Bejan and made publicly available.)

Author

Original authors: Lin Wang, Wei Zhang, and Qizhai Li (AssocTests). Documentation adaptation: package maintainers of this project.

Examples

tw(eigenvalues = c(5, 3, 1, 0), eigenL = 4, criticalpoint = 2.0234)
#> 
#> 	Tracy-Widom test
#> 
#> data:  c(5, 3, 1, 0)
#> TW1 = -0.82427, TW2 = -0.60186, TW3 = -0.55525, TW4 = NaN
#> alternative hypothesis: the eigenvalue is significant
#>