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Bootstrapping Principal Components Analysis: Reply to Mehlman Et Al.
Author(s) -
Jackson Donald A.
Publication year - 1995
Publication title -
ecology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.144
H-Index - 294
eISSN - 1939-9170
pISSN - 0012-9658
DOI - 10.2307/1941220
Subject(s) - bootstrapping (finance) , citation , computer science , principal (computer security) , library science , information retrieval , mathematics , econometrics , operating system
Mehlman et al. (1995) identify a condition that may arise in various multivariate procedures, i.e., the reflection or reversal of the axis direction. They suggest that this condition may have led to an underestimation of the correct multivariate dimensionality in Jackson's (1993a) study of principal components analysis (PCA). The reversal of a multivariate axis or component presents no problem in standard analyses and does not change the interpretation of the data summary. In Jackson (1993a) I was using bootstrapped PCA to evaluate the number of nontrivial or "significant" components from the frequency distribution and the confidence limits of bootstrapped eigenvalues and eigenvector coefficients. Mehlman et al. (1995) state that it is important to recognize that an axis can reverse in the bootstrapped PCAs. If left in this orientation, there would be a very real possibility of underestimating the number of components based on the criterion of whether the 95% confidence limits encompassed zero or not. It is possible that bimodal distributions could arise with one mode on either side of zero due to reversals, even with a strongly structured data set. One may falsely conclude the component to be uninformative unless some measure is taken to recognize and correct this condition. During the course of the study by Jackson (1993a), axis reversals occurred frequently. This is a well-documented feature (e.g., Gauch et al. 1981, Knox 1989, Knox and Peet 1989, Jackson 1993b) of various multivariate methods, a point identified by Mehlman et al. (1995). Because of this documentation, I assumed it to be implicit in the approach and failed to state explicitly that one must examine for reversals in the coefficients. I am grateful to Mehlman et al. for making this clear to readers. Mehlman et al. (1995) suggest an approach based on "fixing" the sign of the largest eigenvector coefficient on each axis when histograms suggest that reversals may have occurred. In their approach, a single variable having the largest eigenvector coefficient in one PCA is assumed to be representative of the results from each subsequent bootstrapped PCA. In some cases this may