It is very important to be able to answer the question: Does the data show evidence of additional population structure over and above what has already been detected? The test we propose is extremely simple. If our matrix X has eigenvalues and we already have declared the top k eigenvalues to be significant, then we simply test λk +1,…λm as though X was a (m′ − k) × (m′ − k) Wishart matrix. Johnstone shows [22, Proposition 1.2] that this procedure is conservative, at least for a true Wishart matrix. We tested this by generating data in which there is one eigenvalue that is overwhelmingly significant, and examined the distribution of the second eigenvalue. As shown by the P–P plot of Figure 3, the fit is again very good, especially for small p-values. If an eigenvalue is not significant, then further testing of smaller eigenvalues should not be done.
