Incorporating Uncertainty Into Parallel Analysis for Choosing the Number of Factors via Bayesian Methods

A number of psychometricians have suggested that parallel analysis (PA) tends to yield more accurate results in determining the number of factors in comparison with other statistical methods. Nevertheless, all too often PA can suggest an incorrect number of factors, particularly in statistically unfavorable conditions (e.g., small sample sizes and low factor loadings). Because of this, researchers have recommended using multiple methods to make judgments about the number of factors to extract. Implicit in this recommendation is that, when the number of factors is chosen based on PA, uncertainty nevertheless exists. We propose a Bayesian parallel analysis (B-PA) method to incorporate the uncertainty with decisions about the number of factors. B-PA yields a probability distribution for the various possible numbers of factors. We implement and compare B-PA with a frequentist approach, revised parallel analysis (R-PA), in the contexts of real and simulated data. Results show that B-PA provides relevant information regarding the uncertainty in determining the number of factors, particularly under conditions with small sample sizes, low factor loadings, and less distinguishable factors. Even if the indicated number of factors with the highest probability is incorrect, B-PA can show a sizable probability of retaining the correct number of factors. Interestingly, when the mode of the distribution of the probabilities associated with different numbers of factors was treated as the number of factors to retain, B-PA was somewhat more accurate than R-PA in a majority of the conditions.