F A C T O R Unrestricted Factor Analysis Release Version 10.8.04 x64bits July, 2018 Rovira i Virgili University Tarragona, SPAIN Programming: Urbano Lorenzo-Seva Mathematical Specification: Urbano Lorenzo-Seva Pere J. Ferrando Date: Saturday, November 10, 2018 Time: 10:42:26 -------------------------------------------------------------------------------- DETAILS OF ANALYSIS Method to handle missing values : Hot-Deck Multiple Imputation in Exploratory Factor Analysis (Lorenzo-Seva & Van Ginkel, 2016) Missing code value : 999 Number of participants : 165 Number of variables : 15 Variables included in the analysis : ALL Variables excluded in the analysis : NONE Number of factors : 2 Number of second order factors : 0 Dispersion matrix : Polychoric Correlations Robust analyses : Bias-corrected and accelerated (BCa; Lambert, Wildt & Durand, 1991) Number of bootstrap samples : 500 Asymptotic Covariance/Variance matrix : estimated using bootstrap sampling Bootstrap confidence intervals : 95% Method for factor extraction : Robust Diagonally Weighted Least Squares (RDWLS) Correction for robust Chi square : Robust Mean and Variance-scaled (Asparouhov & Muthen, 2010) Rotation to achieve factor simplicity : Promax Value of parameter k : 4.0000 Clever rotation start : Raw Varimax Number of random starts : 10 Maximum mumber of iterations : 100 Convergence value : 0.00001000 Factor scores estimates : Estimates based on linear model -------------------------------------------------------------------------------- UNIVARIATE DESCRIPTIVES Variable Mean Confidence Interval Variance Skewness Kurtosis (95%) (Zero centered) V 1 1.667 ( 1.45 1.89) 1.204 -0.029 -1.073 V 2 1.255 ( 0.99 1.52) 1.778 0.683 -0.817 V 3 1.430 ( 1.18 1.68) 1.591 0.386 -1.050 V 4 0.576 ( 0.38 0.78) 0.996 1.741 2.243 V 5 1.400 ( 1.14 1.66) 1.707 0.442 -1.059 V 6 1.230 ( 0.95 1.51) 1.971 0.747 -0.832 V 7 2.315 ( 2.06 2.57) 1.634 -0.362 -0.880 V 8 1.648 ( 1.41 1.89) 1.440 0.218 -0.924 V 9 0.909 ( 0.67 1.15) 1.428 1.013 -0.332 V 10 0.885 ( 0.64 1.13) 1.447 1.106 -0.044 V 11 1.691 ( 1.43 1.95) 1.668 -0.037 -1.405 V 12 1.418 ( 1.17 1.67) 1.540 0.409 -1.023 V 13 1.200 ( 0.96 1.44) 1.445 0.725 -0.447 V 14 0.770 ( 0.54 1.00) 1.377 1.339 0.543 V 15 1.636 ( 1.40 1.87) 1.395 0.070 -1.051 Polychoric correlation is advised when the univariate distributions of ordinal items are asymmetric or with excess of kurtosis. If both indices are lower than one in absolute value, then Pearson correlation is advised. You can read more about this subject in: Muthén, B., & Kaplan D. (1985). A comparison of some methodologies for the factor analysis of non-normal Likert variables. British Journal of Mathematical and Statistical Psychology, 38, 171-189. Muthén, B., & Kaplan D. (1992). A comparison of some methodologies for the factor analysis of non-normal Likert variables: A note on the size of the model. British Journal of Mathematical and Statistical Psychology, 45, 19-30. BAR CHARTS FOR ORDINAL VARIABLES Variable 1 Value Freq | 0 29 | ************************* 1 46 | **************************************** 2 44 | ************************************** 3 43 | ************************************* 4 3 | ** +-----------+---------+---------+-----------+ 0 11.5 23.0 34.5 46.0 Variable 2 Value Freq | 0 69 | **************************************** 1 34 | ******************* 2 26 | *************** 3 23 | ************* 4 13 | ******* +-----------+---------+---------+-----------+ 0 17.3 34.5 51.8 69.0 Variable 3 Value Freq | 0 52 | **************************************** 1 40 | ****************************** 2 32 | ************************ 3 32 | ************************ 4 9 | ****** +-----------+---------+---------+-----------+ 0 13.0 26.0 39.0 52.0 Variable 4 Value Freq | 0 114 | **************************************** 1 21 | ******* 2 20 | ******* 3 6 | ** 4 4 | * +-----------+---------+---------+-----------+ 0 28.5 57.0 85.5 114.0 Variable 5 Value Freq | 0 58 | **************************************** 1 35 | ************************ 2 31 | ********************* 3 30 | ******************** 4 11 | ******* +-----------+---------+---------+-----------+ 0 14.5 29.0 43.5 58.0 Variable 6 Value Freq | 0 77 | *************************************** 1 27 | ************** 2 24 | ************ 3 20 | ********** 4 17 | ******** +-----------+---------+---------+-----------+ 0 19.3 38.5 57.8 77.0 Variable 7 Value Freq | 0 20 | ***************** 1 23 | ******************* 2 41 | ********************************** 3 47 | **************************************** 4 34 | **************************** +-----------+---------+---------+-----------+ 0 11.8 23.5 35.3 47.0 Variable 8 Value Freq | 0 34 | ****************************** 1 45 | **************************************** 2 42 | ************************************* 3 33 | ***************************** 4 11 | ********* +-----------+---------+---------+-----------+ 0 11.3 22.5 33.8 45.0 Variable 9 Value Freq | 0 91 | **************************************** 1 29 | ************ 2 18 | ******* 3 23 | ********** 4 4 | * +-----------+---------+---------+-----------+ 0 22.8 45.5 68.3 91.0 Variable 10 Value Freq | 0 94 | **************************************** 1 26 | *********** 2 21 | ******** 3 18 | ******* 4 6 | ** +-----------+---------+---------+-----------+ 0 23.5 47.0 70.5 94.0 Variable 11 Value Freq | 0 44 | ******************************** 1 32 | *********************** 2 27 | ******************* 3 55 | **************************************** 4 7 | ***** +-----------+---------+---------+-----------+ 0 13.8 27.5 41.3 55.0 Variable 12 Value Freq | 0 50 | **************************************** 1 45 | ************************************ 2 29 | *********************** 3 33 | ************************** 4 8 | ****** +-----------+---------+---------+-----------+ 0 12.5 25.0 37.5 50.0 Variable 13 Value Freq | 0 62 | **************************************** 1 43 | *************************** 2 34 | ********************* 3 17 | ********** 4 9 | ***** +-----------+---------+---------+-----------+ 0 15.5 31.0 46.5 62.0 Variable 14 Value Freq | 0 104 | **************************************** 1 22 | ******** 2 18 | ****** 3 15 | ***** 4 6 | ** +-----------+---------+---------+-----------+ 0 26.0 52.0 78.0 104.0 Variable 15 Value Freq | 0 37 | ******************************** 1 38 | ********************************* 2 45 | **************************************** 3 38 | ********************************* 4 7 | ****** +-----------+---------+---------+-----------+ 0 11.3 22.5 33.8 45.0 -------------------------------------------------------------------------------- MULTIVARIATE DESCRIPTIVES Analysis of the Mardia's (1970) multivariate asymmetry skewness and kurtosis. Coefficient Statistic df P Skewness 41.826 1150.206 680 1.0000 SKewness corrected for small sample 41.826 1173.771 680 1.0000 Kurtosis 282.284 7.759 0.0000** ** Significant at 0.05 -------------------------------------------------------------------------------- STANDARIZED VARIANCE / COVARIANCE MATRIX (POLYCHORIC CORRELATION) (Polychoric algorithm: Bayes modal estimation; Choi, Kim, Chen, & Dannels, 2011) Variable 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 V 1 1.000 V 2 0.314 1.000 V 3 0.248 0.545 1.000 V 4 0.487 0.321 0.331 1.000 V 5 0.345 0.297 0.299 0.358 1.000 V 6 0.420 0.311 0.344 0.559 0.439 1.000 V 7 0.291 0.272 0.452 0.085 0.332 0.450 1.000 V 8 0.293 0.269 0.265 0.267 0.167 0.367 0.137 1.000 V 9 0.163 0.438 0.342 0.377 0.556 0.381 0.423 0.111 1.000 V 10 0.358 0.310 0.339 0.704 0.399 0.582 0.237 0.406 0.366 1.000 V 11 0.254 0.468 0.450 0.262 0.324 0.324 0.413 0.330 0.466 0.320 1.000 V 12 0.305 0.295 0.408 0.415 0.470 0.508 0.533 0.348 0.421 0.432 0.421 1.000 V 13 0.424 0.304 0.263 0.585 0.299 0.420 0.197 0.444 0.300 0.545 0.354 0.540 1.000 V 14 0.147 0.359 0.295 0.197 0.450 0.295 0.333 0.078 0.574 0.274 0.346 0.374 0.263 1.000 V 15 0.325 0.298 0.253 0.353 0.470 0.554 0.249 0.423 0.421 0.547 0.303 0.424 0.482 0.356 1.000 -------------------------------------------------------------------------------- ADEQUACY OF THE CORRELATION MATRIX Determinant of the matrix = 0.007378663636436 Bartlett's statistic = 776.5 (df = 105; P = 0.000010) Kaiser-Meyer-Olkin (KMO) test = 0.87325 (good) BC Bootstrap 95% confidence interval of KMO = ( 0.873 0.904) -------------------------------------------------------------------------------- EXPLAINED VARIANCE BASED ON EIGENVALUES Variable Eigenvalue Proportion of Cumulative Proportion Variance of Variance 1 6.13931 0.40929 0.40929 2 1.62293 0.10820 0.51748 3 1.12376 0.07492 4 0.92918 0.06195 5 0.86967 0.05798 6 0.70178 0.04679 7 0.66516 0.04434 8 0.53891 0.03593 9 0.51271 0.03418 10 0.42695 0.02846 11 0.40171 0.02678 12 0.35877 0.02392 13 0.31073 0.02072 14 0.26718 0.01781 15 0.13125 0.00875 -------------------------------------------------------------------------------- ROBUST GOODNESS OF FIT STATISTICS Root Mean Square Error of Approximation (RMSEA) = 0.049; BC Bootstrap 95% confidence interval = ( 0.0329 0.0473) (between 0.010 and 0.050 : close) Estimated Non-Centrality Parameter (NCP) = 31.160 Degrees of Freedom = 76 Test of Approximate Fit H0 : RMSEA < 0.05; P = 0.977 Minimum Fit Function Chi Square with 76 degrees of freedom = 59.609 (P = 0.916931) Robust Mean and Variance-Adjusted Chi Square with 76 degrees of freedom = 105.515 (P = 0.014214) Chi-Square for independence model with 105 degrees of freedom = 2082.466 Non-Normed Fit Index (NNFI; Tucker & Lewis) = 0.979; BC Bootstrap 95% confidence interval = ( 0.974 0.994) Comparative Fit Index (CFI) = 0.985; BC Bootstrap 95% confidence interval = ( 0.981 0.996) (between 0.950 and 0.990 : close) Schwarz’s Bayesian Information Criterion (BIC) = 335.282; BC Bootstrap 95% confidence interval = (319.257 333.705) Goodness of Fit Index (GFI) = 1.000; BC Bootstrap 95% confidence interval = ( 1.000 1.000) Adjusted Goodness of Fit Index (AGFI) = 1.000; BC Bootstrap 95% confidence interval = ( 1.000 1.000) Goodness of Fit Index without diagonal values (GFI) = 1.000; BC Bootstrap 95% confidence interval = ( 1.000 1.000) Adjusted Goodness of Fit Index without diagonal values(AGFI) = 1.000; BC Bootstrap 95% confidence interval = ( 1.000 1.000) EIGENVALUES OF THE REDUCED CORRELATION MATRIX Variable Eigenvalue 1 5.619433559 2 1.110369426 3 0.536862736 4 0.395340947 5 0.281723161 6 0.169149651 7 0.114251736 8 -0.030236893 9 -0.058578816 10 -0.077563218 11 -0.139664549 12 -0.208490902 13 -0.225232525 14 -0.249016774 15 -0.364326551 -------------------------------------------------------------------------------- UNROTATED LOADING MATRIX Variable F 1 F 2 Communality V 1 0.513 -0.198 0.302 V 2 0.573 0.220 0.376 V 3 0.574 0.240 0.387 V 4 0.679 -0.449 0.663 V 5 0.624 0.142 0.409 V 6 0.719 -0.146 0.538 V 7 0.536 0.342 0.404 V 8 0.469 -0.237 0.276 V 9 0.660 0.367 0.570 V 10 0.726 -0.378 0.670 V 11 0.586 0.235 0.398 V 12 0.701 0.063 0.496 V 13 0.652 -0.318 0.525 V 14 0.538 0.355 0.415 V 15 0.650 -0.142 0.443 -------------------------------------------------------------------------------- FULL TARGET LOADING MATRIX Obtained from prerotation of the loading matrix Variable C 1 C 2 V 1 0.021 0.788 V 2 0.727 0.041 V 3 0.769 0.032 V 4 0.001 1.000 V 5 0.520 0.108 V 6 0.081 0.563 V 7 0.984 0.004 V 8 0.006 0.901 V 9 0.916 0.010 V 10 0.005 0.913 V 11 0.750 0.036 V 12 0.341 0.213 V 13 0.008 0.886 V 14 1.000 0.003 V 15 0.074 0.583 -------------------------------------------------------------------------------- ROTATED LOADING MATRIX Variable F 1 F 2 V 1 0.040 0.523 V 2 0.564 0.073 V 3 0.589 0.051 V 4 -0.166 0.909 V 5 0.500 0.193 V 6 0.212 0.580 V 7 0.688 -0.089 V 8 -0.028 0.543 V 9 0.784 -0.047 V 10 -0.056 0.853 V 11 0.590 0.063 V 12 0.449 0.328 V 13 -0.026 0.741 V 14 0.705 -0.103 V 15 0.180 0.537 ROTATED LOADING MATRIX (loadings lower than absolute 0.300 omitted) Variable F 1 F 2 V 1 0.523 V 2 0.564 V 3 0.589 V 4 0.909 V 5 0.500 V 6 0.580 V 7 0.688 V 8 0.543 V 9 0.784 V 10 0.853 V 11 0.590 V 12 0.449 0.328 V 13 0.741 V 14 0.705 V 15 0.537 EXPLAINED VARIANCE OF ROTATED FACTORS AND RELIABILITY OF PHI-INFORMATION OBLIQUE EAP SCORES Ferrando & Lorenzo-Seva (2016) Factor Variance ORION Factor Determinacy Index 1 3.280 0.865 0.930 2 3.594 0.897 0.947 The appropriate implementation of EAP score estimation in factor model involves to obtain point estimates that make use of the full prior information (in particular the inter-factor correlation matrix), and to complement the point estimates with measures of the reliability of these estimates. In order to achieve it, FACTOR computes: (1) the EAP score estimation named 'Fully-Informative Prior Oblique EAP scores'; and (2) the reliability estimates named ORION (acronim for 'Overall Reliability of fully-Informative prior Oblique N-EAP scores'). See Ferrando & Lorenzo-Seva (2016) for further details. -------------------------------------------------------------------------------- INTER-FACTORS CORRELATION MATRIX Factor F 1 F 2 1 1.000 2 0.633 1.000 -------------------------------------------------------------------------------- STRUCTURE MATRIX Variable F 1 F 2 V 1 0.372 0.549 V 2 0.611 0.431 V 3 0.621 0.424 V 4 0.410 0.804 V 5 0.622 0.509 V 6 0.580 0.715 V 7 0.632 0.347 V 8 0.316 0.525 V 9 0.754 0.450 V 10 0.484 0.817 V 11 0.629 0.436 V 12 0.657 0.612 V 13 0.443 0.725 V 14 0.639 0.343 V 15 0.520 0.651 -------------------------------------------------------------------------------- BIAS-CORRECTED AND ACCELERATED (BCa) BOOTSTRAP 95% CONFIDENCE INTERVALS FOR LOADING VALUES Variable F 1 BCa Confidence Interval V 1 0.040 ( -0.299 0.299) V 2 0.564 ( 0.292 0.833) V 3 0.589 ( 0.243 0.843) V 4 -0.166 ( -0.408 0.018) V 5 0.500 ( 0.244 0.756) V 6 0.212 ( -0.027 0.382) V 7 0.688 ( 0.386 0.939) V 8 -0.028 ( -0.344 0.237) V 9 0.784 ( 0.521 1.024) V 10 -0.056 ( -0.244 0.101) V 11 0.590 ( 0.315 0.825) V 12 0.449 ( 0.190 0.646) V 13 -0.026 ( -0.269 0.142) V 14 0.705 ( 0.355 0.942) V 15 0.180 ( -0.072 0.425) Variable F 2 BCa Confidence Interval V 1 0.523 ( 0.257 0.749) V 2 0.073 ( -0.277 0.359) V 3 0.051 ( -0.244 0.358) V 4 0.909 ( 0.705 1.120) V 5 0.193 ( -0.088 0.462) V 6 0.580 ( 0.386 0.782) V 7 -0.089 ( -0.421 0.192) V 8 0.543 ( 0.124 0.762) V 9 -0.047 ( -0.311 0.213) V 10 0.853 ( 0.630 1.042) V 11 0.063 ( -0.273 0.316) V 12 0.328 ( 0.068 0.554) V 13 0.741 ( 0.551 0.929) V 14 -0.103 ( -0.420 0.221) V 15 0.537 ( 0.199 0.737) -------------------------------------------------------------------------------- INDICES OF FACTOR SIMPLICITY Bentler (1977) & Lorenzo-Seva (2003) Bentler's simplicity index (S) = 0.99624 (Percentile 100) BC Bootstrap 95% confidence interval = ( 0.995 0.999) Loading simplicity index (LS) = 0.60402 (Percentile 100) BC Bootstrap 95% confidence interval = ( 0.573 0.703) -------------------------------------------------------------------------------- BIAS-CORRECTED BOOTSTRAP 95% CONFIDENCE INTERVALS FOR INTER-FACTORS CORRELATION VALUES 1 -- 2 0.633* ( 0.595 0.705) * Significantly different from zero at population -------------------------------------------------------------------------------- DISTRIBUTION OF RESIDUALS Number of Residuals = 105 Summary Statistics for Fitted Residuals Smallest Fitted Residual = -0.1577 Median Fitted Residual = -0.0079 Largest Fitted Residual = 0.1639 Mean Fitted Residual = -0.0080 Variance Fitted Residual = 0.0043 Root Mean Square of Residuals (RMSR) = 0.0658 BC Bootstrap 95% confidence interval of RMSR = ( 0.061 0.062) Expected mean value of RMSR for an acceptable model = 0.0781 (Kelley's criterion) (Kelley, 1935,page 146; see also Harman, 1962, page 21 of the 2nd edition) Weighted Root Mean Square Residual (WRMR) = 0.0588 (values under 1.0 have been recommended to represent good fit; Yu & Muthen, 2002) BC Bootstrap 95% confidence interval of WRMR = ( 0.053 0.058) Histogram for fitted residuals Value Freq | -0.1577 2 | *** -0.1256 5 | ********* -0.0934 9 | ***************** -0.0613 13 | ************************ -0.0291 21 | **************************************** 0.0031 20 | ************************************** 0.0352 13 | ************************ 0.0674 11 | ******************** 0.0995 9 | ***************** 0.1317 1 | * 0.1639 1 | * +-----------+---------+---------+-----------+ 0 5.3 10.5 15.8 21.0 Summary Statistics for Standardized Residuals Smallest Standardized Residual = -2.02 Median Standardized Residual = -0.10 Largest Standardized Residual = 2.10 Mean Standardized Residual = -0.10 Stemleaf Plot for Standardized Residuals -2 | 0 -1 | 96654433222221100 -0 | 9888777776666655555444433333332211111 0 | 00000011112222233333344566667777788888 1 | 01111223457 2 | 1 -------------------------------------------------------------------------------- DESCRIPTIVES RELATED TO MISSING DATA Missing value code : 999 No missing data was observed in your data -------------------------------------------------------------------------------- References Asparouhov, T., & Muthen, B. (2010). Simple second order chi-square correction. Unpublished manuscript. Available at https://www.statmodel.com/download/WLSMV_new_chi21.pdf. Bentler, P.M. (1977). Factor simplicity index and transformations. Psychometrika, 59, 567-579. Ferrando, P. J., & Lorenzo-Seva U. (2016). A note on improving EAP trait estimation in oblique factor-analytic and item response theory models. Psicologica, 37, 235-247. Ferrando, P. J., & Lorenzo-Seva U. (2017). Assessing score determinacy, measurement quality, and closeness to unidimensionality in exploratory item factor analysis. Educational and Psychological Measurement, 0013164417719308 Harman, H. H. (1962). Modern Factor Analysis, 2nd Edition. University of Chicago Press, Chicago. Kelley, T. L. (1935). Essential Traits of Mental Life, Harvard Studies in Education, vol. 26. Harvard University Press, Cambridge. Lambert, Z.V., Wildt, A.R., & Durand, R.M. (1991). Approximating confidence intervals for factor loadings. Multivariate behavioral research, 26(3), 421 - 434. Lorenzo-Seva, U. (2003). A factor simplicity index. Psychometrika, 68, 49-60. Lorenzo-Seva, U., & Van Ginkel, J. R. (2016). Multiple Imputation of missing values in exploratory factor analysis of multidimensional scales: estimating latent trait scores. Anales de Psicología/Annals of Psychology, 32(2), 596-608. McDonald, R.P. (1999). Test theory: A unified treatment. Mahwah, NJ: Lawrence Erlbaum. Mardia, K. V. (1970), Measures of multivariate skewnees and kurtosis with applications. Biometrika, 57, 519-530. Olsson, U. (1979a). Maximum likelihood estimation of the polychoric correlation coefficient. Psychometrika, 44, 443-460. Olsson, U. (1979b). On the robustness of factor analysis against crude classification of the observations. Multivariate Behavioral Research, 14, 485-500. Mislevy, R.J., & Bock, R.D. (1990). BILOG 3 Item analysis and test scoring with binary logistic models. Mooresville: Scientific Software. Ten Berge, J.M.F., Snijders, T.A.B. & Zegers, F.E. (1981). Computational aspects of the greatest lower bound to reliability and constrained minimum trace factor analysis. Psychometrika, 46, 201-213. Ten Berge, J.M.F., & Socan, G. (2004). The greatest lower bound to the reliability of a test and the hypothesis of unidimensionality. Psychometrika, 69, 613-625. Woodhouse, B. & Jackson, P.H. (1977). Lower bounds to the reliability of the total score on a test composed of nonhomogeneous items: II. A search procedure to locate the greatest lower bound. Psychometrika, 42, 579-591. Yu, C., & Muthen, B. (2002, April).Evaluation of model fit indices for latent variable models with categorical and continuous outcomes.Paper presented at the annual meeting of the American Educational Research Association, New Orleans, L.A. FACTOR is based on CLAPACK. Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., & Sorensen, D. (1999). LAPACK Users' Guide. Society for Industrial and Applied Mathematics. Philadelphia, PA FACTOR can be refered as: Lorenzo-Seva, U., & Ferrando, P.J. (2013). FACTOR 9.2 A Comprehensive Program for Fitting Exploratory and Semiconfirmatory Factor Analysis and IRT Models. Applied Psychological Measurement, 37(6), 497-498. Lorenzo-Seva, U., & Ferrando, P.J. (2006). FACTOR: A computer program to fit the exploratory factor analysis model.Behavioral Research Methods, Instruments and Computers, 38(1), 88-91. For furhter information and new releases go to: psico.fcep.urv.cat/utilitats/factor -------------------------------------------------------------------------------- FACTOR completed Computing time : 3.17 minutes. Matrices generated : 35266465 Our last advice: Distrust 5% of statistics, and 95% of statisticians. (Cal desconfiar un 5% de l'estadística, i un 95% de l'estadístic.)