# Roy’s Largest Root (Criterion): Definition

Roy’s Largest Root is a positive-valued, multivariate test statistic obtained in a hypothesis test. The test, along with similar statistics (e.g. Wilks’ Lambda or Pillai’s Trace) rely on eigenvalues. Where Roy’s Largest Root differs is that the focus is on extreme eigenvalues: It is the largest eigenvalue in a generated test matrix. Out of the common tests, Roy’s has the most statistical power when the noncentrality is heavily concentrated in a single root.

## What Does Roy’s Largest Root Test Tell You?

Increasing values for the statistic indicate increasing contributions by effects to the model in question. In general, large values indicate you should reject the null hypothesis. While tables are available in many texts, these have become somewhat obsolete due to the abundance of available statistical software.

## Comparison to Similar Statistics

Unlike several other common summary statistics (e.g. Hotelling’s and Pillai’s) that report explained variance across all discriminant functions, Roy’s Largest Root only explains the variance for one discriminant function:

Roy’s Largest Root = λ1 (1 + λ1)

Discriminant functions create weighted linear composites of quantitative variables.

Roy’s largest root is always smaller, or equal to, Hotelling’s trace. If they are equal, it may mean one of the following:

• The effect is mostly associated with one dependent variable,
• There is a strong correlation between dependent variables,
• The effect has a negligible contribution to the model.

## Other Names for Roy’s Largest Root

Roy’s Largest Root goes by many other names, including:

• Roy’s Largest Characteristic Root,
• Roy’s Greatest Root,
• Roy’s maximum root,
• Roy’s criterion.

## References

I. M. Johnstone, B. Nadler. Roy’s largest root test under rank-one alternatives. Biometrika. 2017 Mar; 104(1): 181–193. Published online 2017 Jan 13. doi: 10.1093/biomet/asw060
Warner, R. (2013). Applied Statistics: From Bivariate Through Multivariate Techniques: From Bivariate Through Multivariate Techniques. SAGE.

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Statistical concepts explained visually - Includes many concepts such as sample size, hypothesis tests, or logistic regression, explained by Stephanie Glen, founder of StatisticsHowTo.