Constructing second-order accurate confidence intervals for communalities in factor analysis.

In an effort to find accurate alternatives to the usual confidence intervals based on normal approximations, this paper compares four methods of generating second-order accurate confidence intervals for non-standardized and standardized communalities in exploratory factor analysis under the normality assumption. The methods to generate the intervals employ, respectively, the Cornish-Fisher expansion and the approximate bootstrap confidence (ABC), and the bootstrap-t and the bias-corrected and accelerated bootstrap (BC(a)). The former two are analytical and the latter two are numerical. Explicit expressions of the asymptotic bias and skewness of the communality estimators, used in the analytical methods, are derived. A Monte Carlo experiment reveals that the performance of central intervals based on normal approximations is a consequence of imbalance of miscoverage on the left- and right-hand sides. The second-order accurate intervals do not require symmetry around the point estimates of the usual intervals and achieve better balance, even when the sample size is not large. The behaviours of the second-order accurate intervals were similar to each other, particularly for large sample sizes, and no method performed consistently better than the others.