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Summary  Factor analysis and principal component analysis are used in many application areas. The first step, choosing the number of components, remains a serious challenge. Our work proposes improved methods for this important problem. One of the most popular state of the art methods is  parallel analysis  (PA), which compares the observed factor strengths with simulated strengths under a noise‐only model. The paper proposes improvements to PA. We first derandomize it, proposing  deterministic PA , which is faster and more reproducible than PA. Both PA and deterministic PA are prone to a  shadowing  phenomenon in which a strong factor makes it difficult to detect smaller but more interesting factors. We propose  deflation  to counter shadowing. We also propose to raise the decision threshold to improve estimation accuracy. We prove several consistency results for our methods, and test them in simulations. We also illustrate our methods on data from the human genome diversity project, where they significantly improve the accuracy.

Deterministic parallel analysis: an improved method for selecting factors and principal components