xF A C T O R
Unrestricted Factor Analysis
Release Version 12.04.05 x64bits
October, 2023
Rovira i Virgili University
Tarragona, SPAIN
Programming:
Urbano Lorenzo-Seva
Mathematical Specification:
Urbano Lorenzo-Seva
Pere J. Ferrando
Date: Saturday, May 25, 2024
Time: 17:7:11
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DETAILS OF ANALYSIS
Participants' scores data file : C:\Users\mvaim\Desktop\
Analisis Finales\Factor Comunic Mayo.dat
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 : 892
Number of variables : 7
Variables included in the analysis : ALL
Variables excluded in the analysis : NONE
Number of factors : 1
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 Unweighted Least
Squares (RULS)
Correction for robust Chi square : LOSEFER empirical correction
(Lorenzo-Seva & Ferrando, 2023)
Rotation to achieve factor simplicity : Robust Promin (Lorenzo-Seva
& Ferrando, 2019b)
Clever rotation start : Weighted Varimax
Number of random starts : 100
Maximum mumber of iterations : 1000
Convergence value : 0.00001000
Factor scores estimates : Estimates based on linear
model
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SOLOMON: A METHOD FOR SPLITTING A SAMPLE INTO TWO EQUIVALENT SUBSAMPLES
Lorenzo-Seva, U. (2022)
When the sample is at least of 400 observations (i.e., the number of rows is larger
than 399), two subsamples are
computed using Solomon method. This method optimally splits the sample in two
equivalent halves, and guarantees the
representativeness of the subsamples (i.e., all possible sources of variance are
enclosed in the subsamples).
To assess how equivalents are the two subsamples, the Ratio Communatily Index is
reported: the closer its value to 1,
the most equivalent subsamples are.
Ratio Communatily Index = 0.985262
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GROUPS INCLUDED IN THE ANALYSIS
GROUP N N without missing values
First Solomon subsample 446 446
Second Solomon subsample 446 446
UNIVARIATE DESCRIPTIVES
Variable Mean Confidence Interval Variance Skewness Kurtosis
(95%) (Zero
centered)
V 1 2.127 ( 2.05 2.20) 0.826 0.618 -0.271
V 2 2.211 ( 2.12 2.30) 1.052 0.456 -0.886
V 3 2.974 ( 2.89 3.06) 1.039 -0.387 -1.192
V 4 2.491 ( 2.41 2.58) 0.999 0.207 -1.025
V 5 2.406 ( 2.32 2.49) 1.019 0.314 -0.974
V 6 2.522 ( 2.44 2.61) 0.983 0.190 -1.027
V 7 2.288 ( 2.20 2.38) 1.044 0.374 -0.941
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. doi:10.1111/j.2044-8317.1985.tb00832.x
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. doi:10.1111/j.2044-
8317.1992.tb00975.x
BAR CHARTS FOR ORDINAL VARIABLES
Variable 1
Value Freq
|
0 1 |
1 218 | *******************
2 438 | ****************************************
3 137 | ************
4 98 | ********
+-----------+---------+---------+-----------+
0 109.5 219.0 328.5 438.0
Variable 2
Value Freq
|
0 1 |
1 249 | ****************************
2 346 | ****************************************
3 153 | *****************
4 143 | ****************
+-----------+---------+---------+-----------+
0 86.5 173.0 259.5 346.0
Variable 3
Value Freq
|
0 1 |
1 65 | ******
2 274 | ****************************
3 168 | *****************
4 384 | ****************************************
+-----------+---------+---------+-----------+
0 96.0 192.0 288.0 384.0
Variable 4
Value Freq
|
0 1 |
1 136 | **************
2 374 | ****************************************
3 186 | *******************
4 195 | ********************
+-----------+---------+---------+-----------+
0 93.5 187.0 280.5 374.0
Variable 5
Value Freq
|
0 1 |
1 161 | ****************
2 388 | ****************************************
3 159 | ****************
4 183 | ******************
+-----------+---------+---------+-----------+
0 97.0 194.0 291.0 388.0
Variable 6
Value Freq
|
0 1 |
1 123 | *************
2 378 | ****************************************
3 189 | ********************
4 201 | *********************
+-----------+---------+---------+-----------+
0 94.5 189.0 283.5 378.0
Variable 7
Value Freq
|
0 1 |
1 216 | ************************
2 355 | ****************************************
3 165 | ******************
4 155 | *****************
+-----------+---------+---------+-----------+
0 88.8 177.5 266.2 355.0
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MULTIVARIATE DESCRIPTIVES
Analysis of the Mardia's (1970) multivariate asymmetry skewness and kurtosis.
Coefficient Statistic df
P
Skewness 3.176 472.191 84
1.0000
SKewness corrected for small sample 3.176 474.177 84
1.0000
Kurtosis 70.578 10.081
0.0000**
** Significant at 0.05
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STANDARIZED VARIANCE / COVARIANCE MATRIX (POLYCHORIC CORRELATION)
(Polychoric algorithm: Bayes modal estimation; Choi, Kim, Chen, & Dannels, 2011)
Variable 1 2 3 4 5 6 7
V 1 1.000
V 2 0.645 1.000
V 3 0.625 0.705 1.000
V 4 0.564 0.637 0.627 1.000
V 5 0.628 0.631 0.649 0.751 1.000
V 6 0.609 0.572 0.594 0.683 0.675 1.000
V 7 0.627 0.731 0.661 0.641 0.637 0.613 1.000
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ADEQUACY OF THE POLYCHORIC CORRELATION MATRIX
Determinant of the matrix = 0.007680653511747
Bartlett's statistic = 4322.9 (df = 21; P = 0.000000)
Kaiser-Meyer-Olkin (KMO) test = 0.92155 (very good)
Bootstrap 95% confidence interval of KMO = ( 0.904 0.928)
ITEM LOCATION AND ITEM ADEQUACY INDICES
Items QIM RDI Normed MSA Bootstrap 95%
Confidence interval
1 2 0.53167 0.93874 ( 0.909 0.951)
2 2 0.55269 0.90426 ( 0.871 0.922)
7 3 0.57203 0.92770 ( 0.903 0.940)
5 3 0.60146 0.91173 ( 0.884 0.928)
4 3 0.62276 0.90216 ( 0.871 0.922)
6 3 0.63061 0.93201 ( 0.904 0.945)
3 3 0.74355 0.93945 ( 0.919 0.947)
Number of items proposed to be removed based on MSA: NONE
Quartile of Ipsative Means (QIM): The means of the variables are placed in the
distribution
of the average of the values registered for each participant, and the quartile in
which
the means are situated is reported. In a normal-range test, few items should be
placed
in the extreme quartiles, whereas most of items should be placed in the central
quartiles.
Relative Difficulty Index (RDI): it assesses the position of the items. For a
normal-range
test, an optimal pool of items should have about 75% RDI values between .40 and .60
and the
remaining values evenly distributed in both tails.
In test intended for clinical screening or selection purposes, a larger amount of
more extreme
items in the appropriate direction is generally recommended.
Measure of Sampling Adequacy (MSA): Values of MSA below .50 suggest that the item
does not
measure the same domain as the remaining items in the pool, and so that it should
be removed.
When removing items from the pool, all these aspects should be taken into
account. Sometimes, the conclusion is that new items should be added to the pool
of items.
Lorenzo-Seva, U. & Ferrando, P.J. (2021) MSA: the forgotten index for identifying
inappropriate items
before computing exploratory item factor analysis. Methodology, in
press.
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EXPLAINED VARIANCE BASED ON EIGENVALUES
Variable Eigenvalue Proportion of Cumulative Proportion
Variance of Variance
1 4.86101 0.69443 0.69443
2 0.55132 0.07876
3 0.42489 0.06070
4 0.34968 0.04995
5 0.32537 0.04648
6 0.25748 0.03678
7 0.23025 0.03289
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CLOSENESS TO UNIDIMENSIONALITY ASSESSMENT
Ferrando & Lorenzo-Seva (2018)
ITEM-LEVEL ASSESSMENT
Variable I-UniCo BC Bootstrap 95% Confidence intervals I-ECV BC
Bootstrap 95% Confidence intervals IREAL BC Bootstrap 95% Confidence
intervals
V 1 1.000 ( 0.995 1.000) 0.981 (
0.910 1.000) 0.107 ( 0.007 0.239)
V 2 0.989 ( 0.947 0.998) 0.869 (
0.747 0.941) 0.326 ( 0.207 0.492)
V 3 1.000 ( 0.998 1.000) 0.977 (
0.941 0.999) 0.121 ( 0.022 0.197)
V 4 0.990 ( 0.930 0.999) 0.875 (
0.717 0.954) 0.318 ( 0.180 0.528)
V 5 0.999 ( 0.993 1.000) 0.950 (
0.892 0.992) 0.191 ( 0.068 0.281)
V 6 0.998 ( 0.986 1.000) 0.948 (
0.854 0.995) 0.183 ( 0.054 0.314)
V 7 1.000 ( 0.997 1.000) 0.969 (
0.926 0.995) 0.144 ( 0.060 0.228)
OVERALL ASSESSMENT
UniCo = 0.996 BC BOOTSTRAP 95% CONFIDENCE INTERVALS = ( 0.994 0.998)
ECV = 0.934 BC BOOTSTRAP 95% CONFIDENCE INTERVALS = ( 0.920 0.955)
MIREAL = 0.199 BC BOOTSTRAP 95% CONFIDENCE INTERVALS = ( 0.167 0.223)
A value of UniCo (Unidimensional Congruence) and I-Unico (Item Unidimensional
Congruence) larger than 0.95 suggests that data can be treated as essentially
unidimensional.
A value of ECV (Explained Common Variance) and I-ECV (Item Explained Common
Variance) larger than 0.85 suggests that data can be treated as essentially
unidimensional.
A value of MIREAL (Mean of Item REsidual Absolute Loadings) and I-REAL (Item
REsidual Absolute Loadings) lower than 0.300 suggests that data can be treated as
essentially unidimensional.
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ROBUST GOODNESS OF FIT STATISTICS AFTER LOSEFER CORRECTION
Lorenzo-Seva & Ferrando (2023)
Root Mean Square Error of Approximation (RMSEA) = 0.049;
Bootstrap 95% confidence interval = ( 0.043 0.058)
(between
0.010 and 0.050 : close)
Estimated Non-Centrality Parameter (NCP) = 31.185
Degrees of Freedom = 14
Test of Approximate Fit
H0 : RMSEA < 0.05; P = 0.999
Minimum Fit Function Chi Square with 14 degrees of freedom = 27.905
(P = 0.015610)
LOSEFER empirically corrected Chi-square with 14 degrees of freedom = 44.493
(P = 0.000056)
Chi-Square for independence model with 21 degrees of freedom = 7779.033
Non-Normed Fit Index (NNFI; Tucker & Lewis) = 0.994;
Bootstrap 95% confidence interval = ( 0.992 0.996)
(larger than 0.990 : excellent)
Comparative Fit Index (CFI) = 0.996;
Bootstrap 95% confidence interval = ( 0.994 0.997)
(larger
than 0.990 : excellent)
Schwarz s Bayesian Information Criterion (BIC) = 139.601;
Bootstrap 95% confidence interval = (132.634 150.750)
Goodness of Fit Index (GFI) = 0.997;
Bootstrap 95% confidence interval = ( 0.995 0.998)
Adjusted Goodness of Fit Index (AGFI) = 0.996;
Bootstrap 95% confidence interval = ( 0.993 0.998)
Goodness of Fit Index without diagonal values (GFI) = 0.996;
Bootstrap 95% confidence interval = ( 0.994 0.998)
Adjusted Goodness of Fit Index without diagonal values(AGFI) = 0.995;
Bootstrap 95% confidence interval = ( 0.990 0.997)
EIGENVALUES OF THE REDUCED CORRELATION MATRIX
Variable Eigenvalue
1 4.506593964
2 0.202334177
3 0.030765418
4 0.000227181
5 -0.042332763
6 -0.076538467
7 -0.114460707
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CLOSE-FIT TEST AND POWER ANALYSIS RESULTS AFTER LOSEFER CORRECTION
Lorenzo-Seva & Ferrando (2022)
Sample size = 892
alfa = 0.050
CLOSE-FIT TEST
Root Mean Square Error of Approximation (RMSEA) = 0.049
P = 0.49 (degrees of freedom = 14) FOR RMSEA < 0.05
POWER ANALYSIS RESULTS
H0: RMSEA = 0.05
H1: RMSEA = 0.08
Beta = 0.93 (degrees of freedom = 14)
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UNROTATED LOADING MATRIX
Variable F 1 Communality
V 1 0.762 0.581
V 2 0.818 0.668
V 3 0.803 0.644
V 4 0.814 0.662
V 5 0.830 0.688
V 6 0.774 0.599
V 7 0.815 0.664
EXPLAINED VARIANCE AND RELIABILITY OF EAP SCORES
Ferrando & Lorenzo-Seva (2016)
Factor Variance EAP Reliability Factor Determinacy Index
1 4.507 0.928 0.963
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BIAS-CORRECTED AND ACCELERATED (BCa) BOOTSTRAP 95% CONFIDENCE INTERVALS FOR
LOADING VALUES
Variable F 1 BCa Confidence Interval
V 1 0.762 ( 0.718 0.800)
V 2 0.818 ( 0.783 0.849)
V 3 0.803 ( 0.762 0.837)
V 4 0.814 ( 0.772 0.842)
V 5 0.830 ( 0.800 0.859)
V 6 0.774 ( 0.727 0.812)
V 7 0.815 ( 0.776 0.844)
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GREATEST LOWER BOUND (GLB) TO RELIABILITY
Woodhouse & Jackson (1977)
WARNING: The GLB and Omega can only be trusted in large samples, preferably 1,000
cases or more,
due to a positive sampling bias (ten Berge & Socan, 2004).
Greatest Lower Bound to Reliability = 0.959486
McDonald's ordinal Omega = 0.926694
Standardized Cronbach's alpha = 0.926568
Total observed variance = 7.000
Total Common Variance = 5.108
ASSOCIATED COMMUNALITIES
Variable Communality
V 1 0.652511
V 2 0.835212
V 3 0.652133
V 4 0.824190
V 5 0.763463
V 6 0.695160
V 7 0.685631
The greatest lower bound (glb) to reliability represents the smallest reliability
possible
given observed covariance matrix under the restriction that the sum of error
variances
is maximized for errors that correlate 0 with other variables (Ten Berge, Snijders,
& Zegers, 1981).
Omega can be interpreted as the square of the correlation between the scale score
and the latent
variable common to all the indicators in the infinite universe of indicators of
which the scale
indicators are a subset (McDonald, 1999, page 89).
McDonald, R.P. (1999).Test theory : A unified treatment.Mahwah.L.E.A.
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DISTRIBUTION OF RESIDUALS
Number of Residuals = 21
Summary Statistics for Fitted Residuals
Smallest Fitted Residual = -0.0606
Median Fitted Residual = -0.0105
Largest Fitted Residual = 0.0758
Mean Fitted Residual = 0.0000
Variance Fitted Residual = 0.0015
Root Mean Square of Residuals (RMSR) = 0.0386
BC Bootstrap 95% confidence interval of RMSR = ( 0.029 0.046)
Expected mean value of RMSR for an acceptable model = 0.0335 (Kelley's criterion)
(Kelley, 1935,page 13; see also Harman, 1962, page 21 of the 2nd edition)
Note: if the value of RMSR is much larger than Kelley's criterion value the model
cannot be considered as good
Weighted Root Mean Square Residual (WRMR) = 0.0513 (values under 1.0
have been recommended to represent good fit; Yu & Muthen, 2002)
BC Bootstrap 95% confidence interval of WRMR = ( 0.039 0.062)
Histogram for fitted residuals
Value Freq
|
-0.0606 2 | ****************
-0.0378 4 | ********************************
-0.0151 5 | ****************************************
0.0076 3 | ************************
0.0303 3 | ************************
0.0531 2 | ****************
0.0758 2 | ****************
+-----------+---------+---------+-----------+
0 1.2 2.5 3.8 5.0
Summary Statistics for Standardized Residuals
Smallest Standardized Residual = -1.81
Median Standardized Residual = -0.31
Largest Standardized Residual = 2.26
Mean Standardized Residual = 0.00
Stemleaf Plot for Standardized Residuals
-1 | 8742
-0 | 8887551
0 | 22466
1 | 0569
2 | 3
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DESCRIPTIVES RELATED TO MISSING DATA
Missing value code : 999
No missing data was observed in your data
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References
Ferrando, P. J., & Lorenzo-Seva U. (2018). Assessing the quality and
appropriateness of factor solutions and factor score estimates in exploratory item
factor analysis. Educational and Psychological Measurement, 78, 762-780.
doi:10.1177/0013164417719308
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.
doi:10.1207/s15327906mbr2603_3
Lorenzo-Seva, U. (2022). SOLOMON: a method for splitting a sample into equivalent
subsamples in factor analysis. Behavior Research Method, 54, pages2665–2677.
doi:10.3758/s13428-021-01750-y
Lorenzo-Seva, U., & Ferrando, P.J. (2019b). Robust Promin: a method for diagonally
weighted factor rotation. LIBERABIT, Revista Peruana de Psicología, 25, 99-106.
doi:10.24265/liberabit.2019.v25n1.08
Lorenzo-Seva, U., & Ferrando, P.J. (2023). A simulation-based scaled test statistic
for assessing model-data fit in least-squares unrestricted factor-analysis
solutions. Methodology, 19, 96-115. doi:10.5964/meth.9839
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.
doi:10.6018/analesps.32.2.215161
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. doi:10.2307/2334770
Olsson, U. (1979a). Maximum likelihood estimation of the polychoric correlation
coefficient. Psychometrika, 44, 443-460. doi:10.1007/bf02296207
Olsson, U. (1979b). On the robustness of factor analysis against crude
classification of the observations. Multivariate Behavioral Research, 14, 485-500.
doi:10.1207/s15327906mbr1404_7
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. doi:10.1007/bf02293900
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.
doi:10.1007/bf02289858
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. doi:10.1007/bf02295980
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:
Ferrando, P.J., & Lorenzo-Seva, U. (2017). Program FACTOR at 10: origins,
development and future directions. Psicothema, 29(2), 236-241. doi:
10.7334/psicothema2016.304
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. doi:10.1177/0146621613487794
Lorenzo-Seva, U., & Ferrando, P.J. (2006). FACTOR: A computer program to fit the
exploratory factor analysis model. Behavioral Research Methods, 38(1), 88-91.
10.3758/bf03192753
For further information and new releases go to:
psico.fcep.urv.cat/utilitats/factor
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FACTOR completed
Computing time : 19.72 minutes.
Matrices generated : 289980958
Our last advice: Distrust 5% of statistics, and 95% of statisticians. (Cal
desconfiar un 5% de l'estadistica, i un 95% de l'estadistic.)
