When the factor analysis model holds, component loadings are linear combinations of factor loadings, and vice versa. This interrelation permits us to define new optimization criteria and estimation methods for exploratory factor analysis. Although this article is primarily conceptual in nature, an illustrative example and a small simulation show the methodology to be promising.

Concise formulas for the asymptotic standard errors of component loading estimates were derived. The formulas cover the cases of principal component analysis for unstandardized and standardized variables with orthogonal and oblique rotations. The formulas can be used under any distributions for observed variables as long as the asymptotic covariance matrix for sample covariances/correlations is available. The estimated standard errors in numerical examples were shown to be equivalent to those by the methods using information matrices.

In this paper, the statistical significance of the contribution of variables to the principal components in principal components analysis (PCA) is assessed nonparametrically by the use of permutation tests. We compare a new strategy to a strategy used in previous research consisting of permuting the columns (variables) of a data matrix independently and concurrently, thus destroying the entire correlational structure of the data. This strategy is considered appropriate for assessing the significance of the PCA solution as a whole, but is not suitable for assessing the significance of the contribution of single variables. Alternatively, we propose a strategy involving permutation of one variable at a time, while keeping the other variables fixed. We compare the two approaches in a simulation study, considering proportions of Type I and Type II error. We use two corrections for multiple testing: the Bonferroni correction and controlling the False Discovery Rate (FDR). To assess the significance of the variance accounted for by the variables, permuting one variable at a time, combined with FDR correction, yields the most favorable results. This optimal strategy is applied to an empirical data set, and results are compared with bootstrap confidence intervals.

The asymptotic correlations between the estimates of factor and component loadings are obtained for the exploratory factor analysis model with the assumption of a multivariate normal distribution for manifest variables. The asymptotic correlations are derived for the cases of unstandardized and standardized manifest variables with orthogonal and oblique rotations. Based on the above results, the asymptotic standard errors for estimated correlations between factors and components are derived. Further, the asymptotic standard error of the mean squared canonical correlation for factors and components, which is an overall index for the closeness of factors and components, is derived. The results of a Monte Carlo simulation are presented to show the usefulness of the asymptotic results in the data with a finite sample size.

Relationships between the results of factor analysis and component analysis are derived when oblique factors have independent clusters with equal variances of unique factors. The factor loadings are analytically shown to be smaller than the corresponding component loadings while the factor correlations are shown to be greater than the corresponding component correlations. The condition for the inequality of the factor/component contributions is derived in the case with different variances for unique factors. Further, the asymptotic standard errors of parameter estimates are obtained for a simplified model with the assumption of multivariate normality, which shows that the component loading estimate is more stable than the corresponding factor loading estimate.

The objective of the present study was to assess the reproducibility of data-driven dietary patterns in different samples extracted from similar populations. Dietary patterns were extracted by applying principal component analyses to the dietary information collected from a sample of 3550 women recruited from seven screening centres belonging to the Spanish breast cancer (BC) screening network (Determinants of Mammographic Density in Spain (DDM-Spain) study). The resulting patterns were compared with three dietary patterns obtained from a previous Spanish case–control study on female BC (Epidemiological study of the Spanish group for breast cancer research (GEICAM: grupo Español de investigación en cáncer de mama)) using the dietary intake data of 973 healthy participants. The level of agreement between patterns was determined using both the congruence coefficient (CC) between the pattern loadings (considering patterns with a CC≥0·85 as fairly similar) and the linear correlation between patterns scores (considering as fairly similar those patterns with a statistically significant correlation). The conclusions reached with both methods were compared. This is the first study exploring the reproducibility of data-driven patterns from two studies and the first using the CC to determine pattern similarity. We were able to reproduce the EpiGEICAM Western pattern in the DDM-Spain sample (CC=0·90). However, the reproducibility of the Prudent (CC=0·76) and Mediterranean (CC=0·77) patterns was not as good. The linear correlation between pattern scores was statistically significant in all cases, highlighting its arbitrariness for determining pattern similarity. We conclude that the reproducibility of widely prevalent dietary patterns is better than the reproducibility of more population-specific patterns. More methodological studies are needed to establish an objective measurement and threshold to determine pattern similarity.