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Improving the Use of parallel analysis by Accounting for Sampling Variability of the Observed Correlation Matrix

EDUCATIONAL AND PSYCHOLOGICAL MEASUREMENT 2025年第1期85卷 114-133页

作者： Xia, Yan Zhou, Xinchang Univ Illinois Champaign IL USA

parallel analysis has been considered one of the most accurate methods for determining the number of factors in factor analysis. One major advantage of parallel analysis over traditional factor retention methods (e.g., Kaiser's rule) is that it addresses the sampling variability of eigenvalues obtained from the identity matrix, representing the correlation matrix for a zero-factor model. This study argues that we should also address the sampling variability of eigenvalues obtained from the observed data, such that the results would inform practitioners of the variability of the number of factors across random samples. Thus, this study proposes to revise the parallel analysis to provide the proportion of random samples that suggest k factors (k = 0, 1, 2, . . .) rather than a single suggested number. Simulation results support the use of the proposed strategy, especially for research scenarios with limited sample sizes where sampling fluctuation is concerning.

关键词： parallel analysis dimensionality assessment factor analysis sample size

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Comparison of the K1 Rule,parallel analysis,and the Bass-Ackward Method on Identifying the Number of Factors in Factor analysis

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Fudan Journal of the Humanities and Social Sciences 2025年第1期18卷 17-44页

作者： Lingbo Tong Wen Qu Zhiyong Zhang Department of Psychology University of Notre DameNotre DameIN46556USA Fudan Institute for Advanced Study in Social Sciences Fudan UniversityShanghai200433China

Factor analysis is widely utilized to identify latent factors underlying the observed *** paper presents a comprehensive comparative study of two widely used methods for determining the optimal number of factors in factor analysis,the K1 rule,and parallel analysis,along with a more recently developed method,the bass-ackward *** provide an in-depth exploration of these techniques,discussing their historical development,advantages,and *** a series of Monte Carlo simulations,we assess the efficacy of these methods in accurately determining the appropriate number of ***,we examine two cessation criteria within the bass-ackward framework:BA-maxLoading and *** findings offer nuanced insights into the performance of these methods under various conditions,illuminating their respective advantages and potential *** enhance accessibility,we create an online visualization tool tailored to the factor structures generated by the bass-ackward *** research enriches the understanding of factor analysis methodology,assists researchers in method selection,and facilitates comprehensive interpretation of latent factor structures.

关键词： Exploratory factor analysis Number of factors Bass-ackward method parallel analysis K1 rule

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parallel analysis of electromagnetic scattering from random rough surfaces

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ELECTRONICS LETTERS 2003年第9期39卷 710-712页

作者： Xia, MY Chan, CH Peking Univ Dept Elect Beijing 100871 Peoples R China City Univ Hong Kong Dept Elect Engn Hong Kong Hong Kong Peoples R China

An efficient approach for simulation of random rough surface scattering is developed based on using a single integral equation formulation and a multilevel sparse-matrix canonical-grid method. Merits of the scheme are... 详细信息

关键词： parallel analysis Electromagnetic waves: theory Numerical approximation and analysis electromagnetic scattering rough surfaces wind-driven ocean surfaces Electromagnetic wave propagation integral equations sparse matrices electromagnetic wave scattering random rough surfaces multilevel sparse-matrix canonical-grid method integral equation formulation Integral equations (numerical analysis) Linear algebra (numerical analysis)

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parallel analysis with unidimensional binary data

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EDUCATIONAL AND PSYCHOLOGICAL MEASUREMENT 2005年第5期65卷 791-810页

作者： Weng, LJ Cheng, CP Natl Taiwan Univ Dept Psychol Taipei 106 Taiwan

The present simulation investigated the performance of parallel analysis for unidimensional binary data. Single-factor models with 8 and 20 indicators were examined, and sample size (50, 100, 200, 500, and 1,000), factor loading (.45,.70, and.90), response ratio on two categories (50/50, 60/40, 70/30, 80/20, and 90/10), and types of correlation coefficients (phi and tetrachoric correlations) were manipulated. The results indicated that parallel analysis performed well in identifying the number of factors. The performance improved as factor loading and sample size increased and as the percentages of responses on two categories became close. Using the 95th and 99th percentiles of the random data eigenvalues as the criteria for comparison in parallel analysis yielded higher correct rate than using mean eigenvalues.

关键词： factor analysis parallel analysis dichotomous variables number of factors unidimensionality

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Microfluidic formation of single cell array for parallel analysis of Ca²⁺ release-activated Ca²⁺ (CRAC) channel activation and inhibition

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ANALYTICAL BIOCHEMISTRY 2010年第2期396卷 173-179页

作者： Xu, Tao Li, Cheuk-Wing Yao, Xinsheng Cai, Guoping Yang, Mengsu Tsinghua Univ Grad Sch Shenzhen Div Life Sci Shenzhen 518057 Peoples R China Tsinghua Univ Dept Biol Sci & Biotechnol Shenzhen 518057 Peoples R China City Univ Hong Kong Dept Biol & Chem Kowloon Hong Kong Peoples R China City Univ Hong Kong Shenzhen Biotech & Hlth Ctr Key Lab Biochip Technol Shenzhen Peoples R China

High-throughput single cell analysis is required for understanding and predicting the complex stochastic responses of individual cells in changing environments. We have designed a microfluidic device consisting of parallel, independent channels with cell-docking structures for the formation of an array of individual cells. The microfluidic cell array was used to quantify single cell responses and the distribution of response patterns of calcium channels among a population of individual cells. In this device, 15 cell-docking units in each channel were fabricated with each unit containing 5 sandbag structures, such that an array of individual cells was formed in 8 independent channels. Single cell responses to different treatments in different channels were monitored in parallel to study the effects of the specific activator and inhibitor of the Ca2+ release-activated Ca2+ (CRAC) channels. Multichannel detection was performed to obtain the response patterns of the population of cells within this single cell array. The results demonstrate that it is possible to acquire single cell features in multichannels simultaneously with passive structural control, which provides an opportunity for high-throughput single cell response analysis in a microfluidic chip. (C) 2009 Elsevier Inc. All rights reserved.

关键词： Microfluidics Single cell array parallel analysis CRAC channel

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Implementing Horn's parallel analysis for principal component analysis and factor analysis

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STATA JOURNAL 2009年第2期9卷 291-298页

作者： Dinno, Alexis Calif State Univ Hayward E Bay Dept Biol Sci Hayward CA USA

I present paran, an implementation of Horn's parallel analysis criteria for factor or component retention in common factor analysis or principal component analysis in Stata. The command permits classical parallel analysis and more recent extensions to it for the pca and factor commands. paran provides a needed extension to Stata's built-in factor- and component-retention criteria.

关键词： st0166 paran parallel analysis factor analysis principal component analysis factor retention component retention Horn's criterion

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A Proposed Solution to the Problem With Using Completely Random Data to Assess the Number of Factors With parallel analysis

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EDUCATIONAL AND PSYCHOLOGICAL MEASUREMENT 2012年第3期72卷 357-374页

作者： Green, Samuel B. Levy, Roy Thompson, Marilyn S. Lu, Min Lo, Wen-Juo Arizona State Univ Tempe AZ 85287 USA Univ Arkansas Fayetteville AR 72701 USA

A number of psychometricians have argued for the use of parallel analysis to determine the number of factors. However, parallel analysis must be viewed at best as a heuristic approach rather than a mathematically rigorous one. The authors suggest a revision to parallel analysis that could improve its accuracy. A Monte Carlo study is conducted to compare revised and traditional parallel analysis approaches. Five dimensions are manipulated in the study: number of observations, number of factors, number of measured variables, size of the factor loadings, and degree of correlation between factors. Based on the results, the revised parallel analysis method, using principal axis factoring and the 95th percentile eigenvalue rule, offers promise.

关键词： factor analysis number of factors parallel analysis

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Relative Accuracy of Two Modified parallel analysis Methods that Use the Proper Reference Distribution

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EDUCATIONAL AND PSYCHOLOGICAL MEASUREMENT 2018年第4期78卷 589-604页

作者： Green, Samuel Xu, Yuning Thompson, Marilyn S. Arizona State Univ Tempe AZ USA

parallel analysis (PA) assesses the number of factors in exploratory factor analysis. Traditionally PA compares the eigenvalues for a sample correlation matrix with the eigenvalues for correlation matrices for 100 comparison datasets generated such that the variables are independent, but this approach uses the wrong reference distribution. The proper reference distribution of eigenvalues assesses the kth factor based on comparison datasets with k-1 underlying factors. Two methods that use the proper reference distribution are revised PA (R-PA) and the comparison data method (CDM). We compare the accuracies of these methods using Monte Carlo methods by manipulating the factor structure, factor loadings, factor correlations, and number of observations. In the 17 conditions in which CDM was more accurate than R-PA, both methods evidenced high accuracies (i.e.,>94.5%). In these conditions, CDM had slightly higher accuracies (mean difference of 1.6%). In contrast, in the remaining 25 conditions, R-PA evidenced higher accuracies (mean difference of 12.1%, and considerably higher for some conditions). We consider these findings in conjunction with previous research investigating PA methods and concluded that R-PA tends to offer somewhat stronger results. Nevertheless, further research is required. Given that both CDM and R-PA involve hypothesis testing, we argue that future research should explore effect size statistics to augment these methods.

关键词： exploratory factor analysis parallel analysis psychometrics

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Accuracy of the parallel analysis Procedure With Polychoric Correlations

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EDUCATIONAL AND PSYCHOLOGICAL MEASUREMENT 2009年第5期69卷 748-759页

作者： Cho, Sun-Joo Li, Feiming Bandalos, Deborah Univ Calif Berkeley Berkeley CA 94720 USA Univ Georgia Athens GA 30602 USA

The purpose of this study was to investigate the application of the parallel analysis (PA) method for choosing the number of factors in component analysis for situations in which data are dichotomous or ordinal. Although polychoric correlations are sometimes used as input for component analyses, the random data matrices generated for use in PA typically consist of Pearson correlations. In this study, the authors matched the type of random data matrix to the type of input matrix. Analyses were conducted on both polychoric and Pearson correlation matrices, and random matrices of the same type ( polychoric or Pearson) were generated for the PA procedure. PA based on random Pearson correlations was found to perform at least as well as PA based on random polychoric correlations, for nearly all of the conditions studied.

关键词： parallel analysis Pearson correlation polychoric correlation

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On the Detection of the Correct Number of Factors in Two-Facet Models by Means of parallel analysis

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EDUCATIONAL AND PSYCHOLOGICAL MEASUREMENT 2021年第5期81卷 872-903页

作者： Beauducel, Andre Hilger, Norbert Univ Bonn Bonn Germany

Methods for optimal factor rotation of two-facet loading matrices have recently been proposed. However, the problem of the correct number of factors to retain for rotation of two-facet loading matrices has rarely been addressed in the context of exploratory factor analysis. Most previous studies were based on the observation that two-facet loading matrices may be rank deficient when the salient loadings of each factor have the same sign. It was shown here that full-rank two-facet loading matrices are, in principle, possible, when some factors have positive and negative salient loadings. Accordingly, the current simulation study on the number of factors to extract for two-facet models was based on rank-deficient and full-rank two-facet population models. The number of factors to extract was estimated from traditional parallel analysis based on the mean of the unreduced eigenvalues as well as from nine other rather traditional versions of parallel analysis (based on the 95th percentile of eigenvalues, based on reduced eigenvalues, based on eigenvalue differences). parallel analysis based on the mean eigenvalues of the correlation matrix with the squared multiple correlations of each variable with the remaining variables inserted in the main diagonal had the highest detection rates for most of the two-facet factor models. Recommendations for the identification of the correct number of factors are based on the simulation results, on the results of an empirical example data set, and on the conditions for approximately rank-deficient and full-rank two-facet models.

关键词： facet model exploratory factor analysis parallel analysis

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