Parallel analysis approach for determining dimensionality in canonical correlation analysis. (English) Zbl 07184806
Summary: Canonical correlations are maximized correlation coefficients indicating the relationships between pairs of canonical variates that are linear combinations of the two sets of original variables. The number of non-zero canonical correlations in a population is called its dimensionality. Parallel analysis (PA) is an empirical method for determining the number of principal components or factors that should be retained in factor analysis. An example is given to illustrate for adapting proposed procedures based on PA and bootstrap modified PA to the context of canonical correlation analysis (CCA). The performances of the proposed procedures are evaluated in a simulation study by their comparison with traditional sequential test procedures with respect to the under-, correct- and over-determination of dimensionality in CCA.
MSC:
| 62H20 | Measures of association (correlation, canonical correlation, etc.) |
| 65C05 | Monte Carlo methods |
References:
| [1] | Timm NH. Applied multivariate analysis. New York: Springer-Verlag; 2002. [Google Scholar] · Zbl 1002.62036 |
| [2] | Fujikoshi Y, Ulyanov VV, Shimizu R. Multivariate statistics: high-dimensional and large-sample approximations. Hoboken: John Wiley Sons, Inc.; 2010. [Crossref], [Google Scholar] · Zbl 1304.62016 |
| [3] | Caliński T, Krzyśko M, W Wołyński. A comparison of some tests for determining the number of nonzero canonical correlations. Comm Statist Simulation Comput. 2006;35:727-749. doi: 10.1080/03610910600716290[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1104.62070 |
| [4] | Fujikoshi Y, Veitch LG. Estimation of dimensionality in canonical correlation analysis. Biometrika. 1979;66:345-351. doi: 10.1093/biomet/66.2.345[Crossref], [Web of Science ®], [Google Scholar] · Zbl 0407.62033 |
| [5] | Aoki K, Sato Y. A test statistic in canonical correlation analysis for categorical variables. Behaviormetrika. 2007;34(1):59-74. doi: 10.2333/bhmk.34.59[Crossref], [Google Scholar] · Zbl 1354.62003 |
| [6] | Caliński T, Krzyśko M. A closed testing procedure for canonical correlations. Comm Statist Theory Methods. 2005;34:1105-1116. doi: 10.1081/STA-200056830[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1073.62053 |
| [7] | Fan X. Canonical correlation analysis and structural equation modeling: what do they have in common? Struct Equ Model. 1997;4(1):65-79. doi: 10.1080/10705519709540060[Taylor & Francis Online], [Web of Science ®], [Google Scholar] |
| [8] | Gunderson BK, Muirhead RJ. On estimating the dimensionality in canonical correlation analysis. J Multivariate Anal. 1997;62:121-136. doi: 10.1006/jmva.1997.1677[Crossref], [Web of Science ®], [Google Scholar] · Zbl 0874.62066 |
| [9] | Harris RJ. The invalidity of partitioned - U tests in canonical correlation and multivariate analysis of variance. Multivariate Behav Res. 1976;11:353-365. doi: 10.1207/s15327906mbr1103_6[Taylor & Francis Online], [Web of Science ®], [Google Scholar] |
| [10] | Lawley DN. Tests of significance in canonical analysis. Biometrika. 1959;28:321-377. [Google Scholar] · Zbl 0094.14305 |
| [11] | Lee HS. Canonical correlation analysis using small number of samples. Comm Statist Simulation Comput. 2007;36(5):973-985. doi: 10.1080/03610910701539443[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1126.62046 |
| [12] | Mendoza JL, Markos VH, Gonter R. A new perspective on sequential testing procedures in canonical analysis: a Monte Carlo evaluation. Multivariate Behav Res. 1978;13:371-382. doi: 10.1207/s15327906mbr1303_8[Taylor & Francis Online], [Web of Science ®], [Google Scholar] |
| [13] | Yamada T, Sugiyama T. On the permutation test in canonical correlation analysis. Comput Statist Data Anal. 2006;50:2111-2123. doi: 10.1016/j.csda.2005.03.006[Crossref], [Web of Science ®], [Google Scholar] · Zbl 1445.62128 |
| [14] | Marcus R, P Eric, Gabriel KR. On closed testing procedures with special reference to ordered analysis of variance. Biometrika. 1976;63:655-660. doi: 10.1093/biomet/63.3.655[Crossref], [Web of Science ®], [Google Scholar] · Zbl 0353.62037 |
| [15] | Fujikoshi Y. Asymptotic expansion of the distributions of the latent roots in MANOVA and the canonical correlations. J Multivariate Anal. 1977;7:386-396. doi: 10.1016/0047-259X(77)90080-X[Crossref], [Web of Science ®], [Google Scholar] · Zbl 0366.62062 |
| [16] | Horn JL. A rationale and test for the number of factors in factor analysis. Psychometrika. 1965;30:179-185. doi: 10.1007/BF02289447[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1367.62186 |
| [17] | Guttman L. Some necessary conditions for common factor analysis. Psychometrika. 1954;19:149-161. doi: 10.1007/BF02289162[Crossref], [Google Scholar] · Zbl 0058.13004 |
| [18] | Hayton JC, Allen DG, Scarpello V. Factor retention decisions in exploratory factor analysis: a tutorial on parallel analysis. Organ Res Methods. 2004;7:191-205. doi: 10.1177/1094428104263675[Crossref], [Web of Science ®], [Google Scholar] |
| [19] | Glorfeld LW. An improvement on Horn’s parallel analysis methodology for selecting the correct number of factors to retain. Educ Psychol Meas. 1995;55:377-393. doi: 10.1177/0013164495055003002[Crossref], [Web of Science ®], [Google Scholar] |
| [20] | SJ Cho, Li F, Bandalos D. Accuracy of the parallel analysis procedure with polychoric correlations. Educ Psychol Meas. 2009;69:748-759. doi: 10.1177/0013164409332229[Crossref], [Web of Science ®], [Google Scholar] |
| [21] | Green SB, Redell N, Thompson MS, Levy R. Accuracy of revised and traditional parallel analyses for assessing dimensionality with binary data. Educ Psychol Meas. 2016;76:5-21. doi: 10.1177/0013164415581898[Crossref], [Web of Science ®], [Google Scholar] |
| [22] | Lorenzo-Seva U, Timmerman ME, HAL Kiers. The hull method for selecting the number of common factors. Multivariate Behav Res. 2011;46:340-364. doi: 10.1080/00273171.2011.564527[Taylor & Francis Online], [Web of Science ®], [Google Scholar] |
| [23] | Zwick WR, Velicer WF. Comparison of five rules for determining the number of components to retain. Psychol Bull. 1986;99:432-442. doi: 10.1037/0033-2909.99.3.432[Crossref], [Web of Science ®], [Google Scholar] |
| [24] | Buja A, Eyuboglu N. Remarks on parallel analysis. Multivariate Behav Res. 1992;27:509-540. doi: 10.1207/s15327906mbr2704_2[Taylor & Francis Online], [Web of Science ®], [Google Scholar] |
| [25] | Turner NE. The effect of common variance and structure pattern on random data eigenvalues: implications for the accuracy of parallel analysis. Educ Psychol Meas. 1998;58:541-568. doi: 10.1177/0013164498058004001[Crossref], [Web of Science ®], [Google Scholar] |
| [26] | LJ Weng, CP Cheng. Parallel analysis with unidimensional binary data. Educ Psychol Meas. 2005;65:697-716. doi: 10.1177/0013164404273941[Crossref], [Web of Science ®], [Google Scholar] |
| [27] | Crawford AV, Green SB, Levy R, WJ Lo, Scott L, Svetina D, Thompson MS. Evaluation of parallel analysis methods for determining the number of factors. Educ Psychol Meas. 2010;70:885-901. doi: 10.1177/0013164410379332[Crossref], [Web of Science ®], [Google Scholar] |
| [28] | Humphreys LG, Ilgen DR. Note on a criterion for the number of common factors. Educ Psychol Meas. 1969;29:571-578. doi: 10.1177/001316446902900303[Crossref], [Web of Science ®], [Google Scholar] |
| [29] | Timmerman ME, Lorenzo-Seva U. Dimensionality assessment of ordered polytomous items with parallel analysis. Psychol Methods. 2011;16:209-220. doi: 10.1037/a0023353[Crossref], [PubMed], [Web of Science ®], [Google Scholar] |
| [30] | Garrido LE, Abad FJ, Ponsoda V. A new look at Horn’s parallel analysis with ordinal variables. Psychol Methods. 2013;18(4):454-474. doi: 10.1037/a0030005[Crossref], [PubMed], [Web of Science ®], [Google Scholar] |
| [31] | Tran US, Formann AK. Performance of parallel analysis in retrieving unidimensionality in the presence of binary data. Educ Psychol Meas. 2009;69:50-61. doi: 10.1177/0013164408318761[Crossref], [Web of Science ®], [Google Scholar] |
| [32] | Jackson DA. Stopping rules in principal components analysis: a comparison of heuristical and statistical approaches. Ecology. 1993;74:2204-2214. doi: 10.2307/1939574[Crossref], [Web of Science ®], [Google Scholar] |
| [33] | Larsen R, Warne RT. Estimating confidence intervals for eigenvalues in exploratory factor analysis. Behav Res Methods. 2010;42(3):871-876. doi: 10.3758/BRM.42.3.871[Crossref], [PubMed], [Web of Science ®], [Google Scholar] |
| [34] | Parmet Y, Schechtman E, Sherman M. Factor analysis revisited – how many factors are there? Comm Statist Simulation Comput. 2010;39:1893-1908. doi: 10.1080/03610918.2010.524332[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1205.62074 |
| [35] | Warne RT, Larsen R. Evaluating a proposed modification of the Guttman rule for determining the number of factors in an exploratory factor analysis. Psychol Test Assess Model. 2014;56:104-123. [Google Scholar] |
| [36] | Zientek LR, Thompson B. Applying the bootstrap to the multivariate case: bootstrap component/factor analysis. Behav Res Methods. 2007;39(2):318-325. doi: 10.3758/BF03193163[Crossref], [PubMed], [Web of Science ®], [Google Scholar] |
| [37] | Dinno A. Exploring the sensitivity of Horn’s parallel analysis to the distributional form of random data. Multivariate Behav Res. 2009;44(3):362-388. doi: 10.1080/00273170902938969[Taylor & Francis Online], [Web of Science ®], [Google Scholar] |
| [38] | Fabrigar LR, Wegener DT, MacCallum RC, Strahan EJ. Evaluating the use of exploratory factor analysis in psychological research. Psychol Methods. 1999;4:272-299. doi: 10.1037/1082-989X.4.3.272[Crossref], [Web of Science ®], [Google Scholar] |
| [39] | Henson RK, Roberts JK. Use of exploratory factor analysis in published research: common errors and some comment on improved practice. Educ Psychol Meas. 2006;66:393-416. doi: 10.1177/0013164405282485[Crossref], [Web of Science ®], [Google Scholar] |
| [40] | Humphreys LG, RG Montanelli Jr. An investigation of the parallel analysis criterion for determining the number of common factors. Multivariate Behav Res. 1975;10:193-205. doi: 10.1207/s15327906mbr1002_5[Taylor & Francis Online], [Web of Science ®], [Google Scholar] |
| [41] | PR Peres-Neto, DA Jackson, KM Somers. How many principal components? Stopping rules for determining the number of non-trivial axes revisited. Comput Statist Data Anal. 2005;49:974-997. doi: 10.1016/j.csda.2004.06.015[Crossref], [Web of Science ®], [Google Scholar] · Zbl 1429.62223 |
| [42] | Velicer WF, Eaton CA, Fava JL. Construct explication through factor or component analysis: a review and evaluation of alternative procedures for determining the number of factors or components. In Goffin RD, Helmes E, editors. Problems and solutions in human assessment: honoring Douglas N. Jackson at seventy. New York, NY: Kluwer Academic/Plenum Publishers; 2000. p. 41-71. [Google Scholar] |
| [43] | Dinno A. Implementing Horn’s parallel analysis for principal component analysis and factor analysis. Stata J. 2009b;9:291-298. [Web of Science ®], [Google Scholar] |
| [44] | Ledesma RD, Valero-Mora P. Determining the number of factors to retain in EFA: an easy-to-use computer program for carrying out parallel analysis. Pract Assess Res Eval. 2007;12(2):1-10. [Google Scholar] |
| [45] | Liu OL, Rijmen F. A modified procedure for parallel analysis of ordered categorical data. Behav Res Methods. 2008;40(2):556-562. doi: 10.3758/BRM.40.2.556[Crossref], [PubMed], [Web of Science ®], [Google Scholar] |
| [46] | O’Connor BP. SPSS and SAS programs for determining the number of components using parallel analysis and Velicer’s MAP test. Behav Res Methods Instrum Comput. 2000;32:396-402. doi: 10.3758/BF03200807[Crossref], [PubMed], [Google Scholar] |
| [47] | Efron B, Tibshirani RJ. An introduction to the bootstrap. Boca Raton: Chapman Hall/CRC; 1994. [Google Scholar] · Zbl 0835.62038 |
| [48] | Rencher A, Christensen WF. Methods of multivariate analysis. 3rd ed. Hoboken, New Jersey: Wiley; 2012. [Crossref], [Google Scholar] · Zbl 1275.62011 |
| [49] | Reddon JR, Marceau R, Jackson DN. An application of singular value decomposition to the factor analysis of MMPI items. Appl Psychol Meas. 1982;6:275-283. doi: 10.1177/014662168200600303[Crossref], [Web of Science ®], [Google Scholar] |
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