Roy's Largest Root Test Under Rank-One Alternatives

Roy's largest root is a common test statistic in a variety of hypothesis testing problems. Despite its popularity, obtaining accurate tractable approximations to its distribution under the alternative has been a longstanding open problem in multivariate statistics. In this paper, assuming Gaussian observations and a rank one alternative, also known as {\em concentrated non-centrality}, we derive simple yet accurate approximations for the distribution of Roy's largest root test for five of the most common settings. These include signal detection in noise, multivariate analysis of variance and canonical correlation analysis. Our main result is that in all five cases Roy's test can be approximated using simple combinations of standard univariate distributions, such as central and non-central $\chi^2$ and $F$. Our results allow approximate power calculations for Roy's test, as well as estimates of sample size required to detect given (rank-one) effects by this test, both of which are important quantities in hypothesis-driven research.