  Roy's largest root is a common test statistic in multivariate analysis,
statistical signal processing and allied fields. Despite its ubiquity,
provision of accurate and tractable approximations to its distribution under
the alternative has been a longstanding open problem. Assuming Gaussian
observations and a rank one alternative, or concentrated non-centrality, we
derive simple yet accurate approximations for the most common low-dimensional
settings. These include signal detection in noise, multiple response
regression, multivariate analysis of variance and canonical correlation
analysis. A small noise perturbation approach, perhaps underused in statistics,
leads to simple combinations of standard univariate distributions, such as
central and non-central $\chi^2$ and $F$. Our results allow approximate power
and sample size calculations for Roy's test for rank one effects, which is
precisely where it is most powerful.
