A Monte Carlo examination of the broken-stick distribution to identify components to retain in principal component analysis

ABSTRACT

The broken-stick (BS) is a popular stopping rule in ecology to determine the number of meaningful components of principal component analysis. However, its properties have not been systematically investigated. The purpose of the current study is to evaluate its ability to detect the correct dimensionality in a data set and whether it tends to over- or underestimate it. A Monte Carlo protocol was carried out. Two main correlation matrices deemed usual in practice were used with three levels of correlation (0, 0.10 and 0.30) between components (generating oblique structure) and with different sample sizes. Analyses of the population correlation matrices indicated that, for extremely large sample sizes, the BS method could be correct for only one of the six simulated structure. It actually failed to identify the correct dimensionality half the time with orthogonal structures and did even worse with some oblique ones. In harder conditions, results show that the power of the BS decreases as sample size increases: weakening its usefulness in practice. Since the BS method seems unlikely to identify the underlying dimensionality of the data, and given that better stopping rules exist it appears as a poor choice when carrying principal component analysis.