 FACTOR

Factor is a freeware program developed at the Rovira i Virgili University. Users are invited to download a DEMO and the program:

If you work with Excel, the following file can be used to preprocess the data file. Please note that that you must allow macros when opening the preprocessing.xlsm file:

We would greatly appreciate any suggestions for future improvements. Detailed reports of failures are also welcome.

Version of the program: 10.10.01 (15th October 2019)
This version implements:

• Least-squares exploratory factor analysis based on tetrachoric/polychoric correlations is a robust, defensible and widely used approach for performing item analysis. A relatively common problem in this scenario, however, is that the inter-item correlation matrix might fail to be positive definite. In order to correct not positive definite correlation matrices, FACTOR implements smoothing methods. The basic principle in the smoothing corrections is to change the relative weight of the diagonal elements of the correlation matrix with respect to the non-diagonal elements. The challenge of the smoothing methods is to change the relative weight of the diagonal elements of the correlation matrix while destroying as little variance as possible in the process. In the present release of FACTOR, Ridge and Sweet smoothing methods have been implemented. It must be noted that Ridge Smoothing is a linear smoothing method that impacts all the variables in the correlation matrix. To prevent this from happening, we propose Sweet Smoothing: the aim of this non-linear smoothing method is to focus the smoothing procedure only on the problematic variables while destroying as little variance as possible in the process. The researcher is allowed to choose between these two smoothing methods when analyzing a dataset.

• In previous releases of FACTOR, KMO index was only reported based on Pearson correlations (even when analyzing ordinal data). Now that most careful smoothing methods are available in FACTOR, KMO index is reported also based on tetrachoric/polychoric correlation matrices.

• McDonald's linear and ordinal omega reliabilities coefficients are implemented.

• When computing principal component analysis, participants’ scores on the components are carefully handled. For example, they can now be stored in a separate file, and are reported to be “component scores” and not “factor scores”.

• This version corrects some internal bugs. These bugs were reported by some users when analysing they own data. We are grateful to these users that help us to improve Factor.

Version of the program: 10.9.02 (2nd May 2019)
This version implements:

• The implementation of Bifactor model (PEBI) has been improved. Now the model can be computed without the rotation of the group factors. In addition, it has been observed that when a large number of group factors are defined by a low number of variables (that probably show low saturations in the group factors), PEBI outcome can frequently converge on local minima. In order to avoid it, a large number of random starts (i.e., 100) is used. In order to decide the final solution, the reported solution is the solution in which the worse defined group factor explains the largest amount of variance: this is done independently of the rotation criterion used, so that bifactor solution based on PEBI is more consistent among different analyses of the same data.

• This version corrects some internal bugs. These bugs were reported by some users when analysing they own data. We are grateful to these users that help us to improve Factor.

Version of the program: 10.9.01 (11th April 2019)
This version implements:

• Diagonally-weighted factor rotation. As with weighted robust schemas in the extraction stage of factor analysis, robust rotation is expected to be particularly advantageous when the sampling errors of the bivariate correlations are considerably different and these errors can be estimated with reasonable accuracy. Different sampling errors are more likely to occur if the input correlations are tetrachoric and polychoric, because in this case the correlation matrix is estimated not jointly but pairwise. In order to compute a diagonally weighted factor rotation with FACTOR, the user has to select: (1) the robust factor analysis option, and (2) one of these three rotation methods: Promin, Weighted Varimax, or Weighted Oblimin. The output of the program informs the researcher that a robust rotation has been computed.

• Conditional reliability function based on polychoric correlations. It consists on a graphic display of conditional reliabilities (7) against the estimated factor levels, and a minimally acceptable cut-off value for one of them (for example, a conditional reliability of 0.8). Graphically this cut-off is a horizontal line parallel to the factor scores axis. In order to compute the reliability function with an optimal precision, the number of nodes to compute EAP scores must be large. In FACTOR at least 20 nodes are recommended.

• A frequent source of difficulties appears when the tetrachoric/polychoric correlation matrix turns out to be non-positive definite. It means that one or more eigenvalues is negative. To solve it, a smoothing procedure can be applied. However, if the negative values are large, most of the information in the correlation matrix is destroyed. When this is the case, the new release of Factor prints a detailed out so that the user has the maximum of information in order to try to solve this difficulty. In addition, Factor refuses to analyze smothered correlation matrices, if more that 60% of information has been destroyed during the smothering procedure. In addition, the smoothing algorithm deletes lowest amounts of variance in each iteration (0.001/sqrt(N)) in order to save a bit more of variance.

• Some bibliographical references have been up dated, and doi numbers are included for all the references.

• Confidence interval is computed for the added-value indices.

• This release of Factor corrects some internal bugs. These bugs were reported by some users when analysing their own data. We are grateful to these users that help us to improve Factor. For example, in this release the user has a most efficient control of the variables to be included (or excluded) from the analysis.

• A literature corner has been added at the Documentation section.

Version of the program: 10.8.04 (22th July 2018)
This version implements:

• This version corrects some internal bugs that appeared with the last update of Windows operative system. These bugs were reported by some users when analysing they own data. We are grateful to these users that help us to improve Factor.

Version of the program: 10.8.03 (7th May 2018)
This version implements:

• When polychoric correlations are computed, the user is allowed how to decide whether to estimate EAP factor scores based on the linear model (faster, but less accurate) or the graded model. In the case of the graded model, the user is allowed to decide the number of nodes to be used: the larger the number of nodes, the more precise (and time consuming) are the factor score estimates. The default number of nodes is 20, and a maximum of 100 is allowed. Please note that different estimates of ORION reliability can be obtained when using the linear and the graded model in order to estimate EAP factor scores. All these feactures can be configured in the “Other specifications of factor model”.

• FACTOR expects the data file encoded as ANSI. In the present release, the user can indicate that UTF-8 encoding (the default encoding when exporting data using SPSS program) has been used. Please, note the UNICODE encoding is not (yet) allowed; maybe someday.

• When computing person reliabilities, individuals with reliability estimates under the threshold value of .10 are marked as Inconsistent Responder.

• Robust chi-square computed in order to assess the goodness-of-it of models based on covariance dispersion matrices has been corrected.

• This version corrects some internal bugs. These bugs were reported by some users when analysing they own data. We are grateful to these users that help us to improve Factor.

Version of the program: 10.8.01 (10th January 2018)
This version implements:

• Measures initially designed to be single-trait often yield data that is compatible with both an essentially unidimensional factor-analysis solution, and a correlated-factors solution. For these cases, new indices are implemented that aim at providing information for deciding which of both solutions is the most appropriate and useful. The procedures implemented are a factor analysis extension of the added-value procedures initially proposed for subscale scores in educational testing. They can be selected in FACTOR as "Added value of multiple factor score estimates" in the "Other specifications of factor model" menu.

• Pratt's importance measures. Wu & Zumbo (2017) propose to compute importance measures to indicate the proportions of the variation in each observed indicator that are attributable to the factors (an interpretation analogous to the effect size measure of eta-squared). The importance measures can further be transformed to eta correlations: a measure of unique directional correlation of each factor with an observed indicator. These indices can be selected in FACTOR as "Display eta-squared and Pratt's importance measures" in the "Other specifications of factor model" menu.

• Factor score estimates are allowed to be saved in a separate text file for further analyses.

Version of the program: 10.7.01 (22th November 2017)
This version implements:

• Objectively Refined Target Matrix (RETAM). When a target matrix is proposed by the user, RETAM helps to refine the target matrix allowing to free and to fixe elements of the target matrix. RETAM risks to capitalize a factor solution on chance (i.e., the factor model is fitted to the sample at hand, not to the population). In the RETAM crossvalidation study, the sample has been half-splitted in two random subsamples: RETAM procedure is applied in the first subsample in order to obtain a refined target matrix; and the refined target matrix is then taken as a fixed target matrix (without further refinements) for the second subsample. If the rotated loading matrix in the second subsample is congruent with the rotated loading matrix in the first subsample, then the researcher must be confident that the final solution has not just been fitted to the sample data, but also to the population data.

• Conditional reliabilities function reports the statistical information corresponding to each score level of the latent trait. It is interpreted as the test information function in IRT context. The graphic shows (1) the conditional reliabilities against the factor score estimates as '*' marks, and (2) the cut-off value of 0.80 as a vertical dotted line.

• This version corrects some internal bugs. These bugs were reported by some users when analysing they own data. We are grateful to these users that help us to improve Factor.

Version of the program: 10.6.01 (13th November 2017)
This version implements:

• Our new procedure for fitting a pure exploratory bifactor solution has been refined to be able to manage bifactor models with a single group factor. In our bifactor propousal the general factor is orthogonal to the group factors, but the loadings on the group factors can satisfy any orthogonal or oblique rotation criterion. The proposal combines Procrustes rotations with analytical rotations. The basis input is a semi-specified target matrix that can be (a) defined by the user, (b) obtained by using Schmid-Leiman orthogonalization, or (c) automatically built from a conventional unrestricted solution based on a prescribed number of factors. In order to compute an exploratory bifactor model, the user has to: (a) specify the number of group factors, (b) check “Exploratory Bifactor Model” in the “Other specifications of factor model” menu, and (c) select the rotation criterion for the group factors. In the outcome, the general factor is labeled as GF.

Version of the program: 10.5.03 (22nd June 2017)
This version implements:

• Two new indices to assess the quality and effectiveness of factor scores estimates: sensitivity ratio, and expected percentage of true differences. The sensitivity ratio (SR) can be interpreted as the number of different factor levels than can be differentiated on the basis of the factor score estimates. The expected percentage of true differences (EPTD) is the estimated percentage of differences between the observed factor score estimates that are in the same direction as the corresponding true differences.

Version of the program: 10.5.02 (29th May 2017)
This version implements:

• A new procedure for fitting a pure exploratory bifactor solution in which the general factor is orthogonal to the group factors, but the loadings on the group factors can satisfy any orthogonal or oblique rotation criterion. The proposal combines Procrustes rotations with analytical rotations. The basis input is a semi-specified target matrix that can be (a) defined by the user, (b) obtained by using Schmid-Leiman orthogonalization, or (c) automatically built from a conventional unrestricted solution based on a prescribed number of factors. In order to compute an exploratory bifactor model, the user has to: (a) specify the number of group factors, (b) check “Exploratory Bifactor Model” in the “Other specifications of factor model” menu, and (c) select the rotation criterion for the group factors. In the outcome, the general factor is labeled as GF.

• Confidence intervals for ORION reliabilities based on bootstrap sampling techniques.

• Equivalence testing for linear ML factor analysis (Yuan, Chan, Marcoulides, & Bentler, 2016).

• This version corrects some internal bugs. These bugs were reported by some users when analysing they own data. We are grateful to these users that help us to improve Factor.

Version of the program: 10.5.01 (20th Abril 2017)
This version implements:

• Robust goodness-of fit indices are computed based on (1) mean-corrected chi-square statistic, (2) mean and variance-corrected chi-square statistic (Satterthwaite, 1941), and (3) mean and variance-corrected chi-square statistic estimated as proposed by Asparouhov and Muthén (2010).
• Weighted Root Mean Square Residual (WRMR) index is computed in order to assess the model residuals.
• New person fit indices are implemented: Personal Correlation (rp) and Weighted Mean-Squared Index (WMSI) indices are computed using optimal threshold values to detect aberrant responses (Ferrando, Vigil-Colet, & Lorenzo-Seva, 2017).
• A new set of indices of factor determinacy, construct replicability and closeness to unidimensionality, aimed at assessing the strength and quality of the solution beyond pure model-data fit.
• A new menu to configurate advanced indices and computings.
• This version corrects some internal bugs. These bugs were reported by some users when analysing they own data. We are grateful to these users that help us to improve Factor.

Version of the program: 10.4.01 (21st October 2016)
This version implements:

• Bootstrap sampling in order to computed robust factor analysis. Bootstrap Confidence intervals are computed for a large number of indices.
• Implementation of Tetrachoric/Polychoric correlation based on unified Bayes modal estimation (MAP) approach.
• Robust exploratory factor analysis based on asymptotic variance covariance matrix for correlation coefficients is computed based on (a) analytical estimates, or (b) bootstrap sampling.
• Implementation of Robust Unweighted Least Squares factor analysis, Robust exploratory Maximum Likelihood factor analysis, and Diagonally Weighted Least Squares factor analysis.
• The number of factor to be retained is increased up to at least two variables per factor.
• BIC dimensionality test: Schwarz’s Bayesian Information Criterion is computed for a number of factors models, so that the model with the optimal number of factors (i.e., the model that corresponds to a lower BIC value) is detected.
• The user is allowed to disable all the procedures to assess the number of factors/components to be retained.
• New person fit indices are implemented: Personal Correlation (rp) and Weighted Mean-Squared Index (WMSI) indices are computed using optimal threshold values to detect aberrant responses (Ferrando, Vigil-Colet, & Lorenzo-Seva, 2017).
• This version corrects some internal bugs. These bugs were reported by some users when analysing they own data. We are grateful to these users that help us to improve Factor. In addition, the internal computing has been redesigned in order to increase computing speed: for example, polychoric correlation matrix is only computed one time in each analysis session (even if different analysis are carried out).
• Please note that Windows XP is not supported anymore.

Version of the program: 10.3.01 (7th July 2015)
This version implements:

• FACTOR is now compiled to run with Windows 64-bits. This feacture allows to analyse large datasets. We successfully tested FACTOR with a dataset of 10,000 cases, 500 variables, and 3 extracted factors. The user can decide which realease (32-bits or 64-bits) wants to download.
• Missing values in the dataset are allowed. Multiple Imputation in exploratory factor analysis is implemented based on Lorenzo-Seva & Van Ginkel (2015) proposal. Missing values must be identified using a numerical code.
• The implementation of Polychoric correlation has been polished to allow convergence even when some cathegories in a particular variable is never used.
• This version corrects some internal bugs. These bugs were reported by some users when analysing they own data. We are grateful to these users that help us to improve Factor.

Version of the program: 9.30.1 (January, 2015)
This version corrects an internal error in the management of the computer memory. This error was observed by some users that were analyzing large datasets. We are grateful to these users that help us to improve Factor.

Version of the program: 9.20 (February, 2013)
This version implements:

• Item Response Theory parameterization of factor solutions based on discrete variables.
• Expected a-posteriori (EAP) estimation of latent trait scores in IRT models.
• Semi-confirmatory factor analysis based on orthogonal and oblique rotation to a (partially) specified target.
• Assessment of the congruence between the target and the rotated loading matrix.

Version of the program: 8.10 (April, 2012)
This version implements:

• Greatest lower bound (glb) to reliability, and McDonald's Omega reliability index.
• GFI and AGFI are computed excluding the diagonal values of the variance/covariance matrix.
• Algorithm 462: Bivariate Normal Distribution by Donnelly (1973) is used to compute polychoric correlation matrix. In addition, polychoric correlation matrix is computed with more demanding convergence values.
• Tetrachoric correlation matrix is computed based on AS116 algorithm. This algorithm is more accurate accurate than the algorithm provided in previous versions of the program.
• Technical revisions to solve different errors that halted the analysis and that were reported by users.

Version of the program: 8.02 (March, 2011)
This version implements:

• A more friendly user reading data implementation. ASCII format data files can be separated using different characters, and missing values are eliminated from the data.
• Variable labels are allowed.
• The output data file can be specified.
• New analyses are implemented: Optimal Parallel Analysis, Hull method, and Person fit indices.
• Some analyses have been improved. For example, the polychoric correlations matrix is checked to be positive definite and smoothed (if necessary), and the non-convergent coefficients are changed by the corresponding Pearson coefficient.
• Technical revisions to solve different errors that halted the analysis and that were reported by users.

Version of the program: 7.00 (January, 2007)
This version implements:

• Univariate mean, variance, skewness, and kurtosis
• Multivariate skewness and kurtosis (Mardia, 1970)
• Var charts for ordinal variables
• Polychoric correlation matrix with optional Ridge estimates
• Structure matrix in oblique factor solutions
• Schmid-Leiman second-order solution (1957)
• Mean, variance and histogram of fitted and standardized residuals. Automatic detection of large standardized residuals.

In addition, a bug that halted the program during the execution has been detected and corrected.

Version of the program: 6.02 (June, 2006)

This version implements PA - MBS. It is an extension of Parallel Analysis that generates random correlation matrices using marginally bootstrapped samples (Lattin, Carroll, & Green, 2003).

In addition, indices of asymmetry and kurtosis related to the variables are computed. The inspection of these indices helps to decide if polychoric correlation is to be computed when ordinal variables are analyzed.

Version of the program: 6.01 (March, 2005)

This version implements the selection of variables to be included and excluded in the analysis. 