24th September, 2005
This is virtually a new program - called Orthosim 2 - which implements an orthogonal procrustes routine as well as a non-procrustes orthogonal rotation. I recompiled the Fortran code using a new compiler (Intel Fortran 9.1) which is fully XP compatible, along with a recompile of the interface in Delphi 2005. I've done my best with the Fortran but this is an old program (1989/1992) with a modern interface bolted on to make it "usable"!

From the introduction to the online help file  ..."The program calculates four different kinds of similarity coefficient between a comparison and target matrix - both "as they are" on input to the program, and after the comparison matrix has been orthogonally rotated against the target matrix so as to minimize the sum of squared deviations between the comparison matrix and target matrix values. For exploratory factor analysis (EFA) vectors (factors), the program allows an investigator to choose between two kinds of approaches to the problem - that using a "procrustes" approach whereby the matrices are expressed in the same unit-metric space (coordinates of -1 and +1 , irrespective of the initial sizes of the loadings/coordinate values - they are stretched or shrunk accordingly so as to occupy a joint unified metric space), and a non-procrustes approach whereby the comparison and target matrices remain unadjusted, and the comparison matrix is rotated against the target "as is". Both orthogonal or obliquely rotated matrices may be presented to the program. However, oblique matrices will be "transformed" to orthogonal ones, prior to the factor comparison methodology being applied to these orthogonal versions.

When submitting Multidimensional Scaling vectors (coordinate dimensions) for comparison, both matrices are initially centered (their coordinate-space origins are equated), are row-normalized (the "procrustes" approach which expresses each matrix in a normalized unit metric space which preserves the distance relations), then any coordinate "reflections" undone as part of the orthogonal rotation to maximum congruity. This is known as "configural similarity" (Borg and Groenen, 1997). The reason for these specific transformations is that MDS solutions are arbitrary in terms of their location, scale, and orientation of variables in geometric space.It is the distance relations between variables which are critical in MDS; such relations can be preserved while allowing the origin, scale, and reflection of solutions to vary. Hence, the extra transformations required prior to congruential rotation.

The EFA-specific procedures of the program are of use for exploratory factor comparison analyses, or where the number of variables is so large as to preclude a structural equation modeling/confirmatory factor analysis approach. or where an investigator wishes to simply rotate a set of data to a schematic target (1s and 0s as loadings/coordinate values). The Validimax routine published in McCrae and Costa (1989.,1994) and used in McCrae, Zonderman, Bond, Costa, & Paunonen (1996) is the same as the Orthosim non-procrustes routine, which is the Kaiser, Hunka, Bianchini (1971) algorithm without the initial row-normalisation of the matrices. All are based upon Schonemann's (1966) and Cliff's (1966) expositions, which in turn are based upon even earlier work such as Ahmavarra (1954)."

Orthosim 2 now dynamically "remembers" and uses the last subdirectory used to open files on a subsequent file selection occasion as well as catering for filepaths/filenames up to 180 characters in length (allowing for deep and long-named subdirectory structures).


SPSS to Orthosim Input File Converter:(19th September, 2006- version 1.2 - fixed a minor output format problem, added new error-handling, and added more online help). This program converts SPSS 11/12/13/and 14 factor loading and factor correlation matrices into the fixed-format .vf (simple ASCII text) files required for input into Orthosim. The SPSS tables are required to be exported (or cut and paste) from the SPSS output viewer (or .spo file) to an Excel format file. This may be achieved either by using the export facility in SPSS or simply cutting and pasting the tables between SPSS and Excel. Comprehensive step-by-step online HTML help is provided to assist the user. The zipped installation program is available for download here (7.2Mb unfortunately due to a large "graphical image" helpfile) - and puts a suitably labelled icon on the desktop.

Note: The STATISTICA routines for extracting and converting factor-loading/coordinate files from STATISTICA are available for download from the software page (so that you can immediately compare target/comparison files using Orthosim without bothering with any job-control/special file setup).

Factor Similarity Analysis - ORTHOSIM-2 -Factor and Multidimensional Scaling orthogonal vector Matrix comparison.12th October, 2005. A small version update to 2.01 - corrected the "clipped" output of the factor correlation matrices (above 10 factors) - no effect on calculations etc....

Orthosim Features
1. Orthogonal Congruential rotation for EFA factor vectors - with or without row normalisation (i.e. no Procrustes effect). The Orthosim Procrustes solution is the original Kaiser, Hunka, Bianichini algorithm which uses row normalization prior to configural matching (albeit modified by Barrett et al (1998) in the light of criticisms by Ten Berge (1996)).
2. Configural Similarity assessment for MDS coordinate dimension vectors.
3. Input matrices printed as standard.
4. Two levels of output - brief and long.
5. Computes the Root-Mean-Square Difference between input comparison and target matrices
6. Computes the Root-Mean-Square Difference between optimally rotated comparison and target matrices
7. Computes the overall solution congruence
8. Computes the overall solution double-scaled euclidean distance similarity (DSED) coefficient
9. Computes the overall solution kernel smoothed distance (KSD) similarity coefficient
10. Computes variable congruence similarity indices
11. Computes Factor/Coordinate DSED and KSD comparisons between the optimally rotated comparison and target matrices.
12. Computes Factor/Coordinate comparison matrices using Pearson and Congruence coefficients between the optimally rotated comparison and target matrices.
13. Computes Factor/Coordinate comparison matrices using Pearson and Congruence coefficients between the initial comparison and target matrices.

Important
- The limits of the program are 350 variables and 35 factors/MDS coordinate dimensions per matrix
- The program can cope with unequal numbers of factors – but, the target matrix must contain the higher number of factors.
- Both matrices must have the same number of variables.
- Filepaths/filenames up to 180 characters in length are now catered for

Especially Important!
Unless the target matrix was a Varimax/orthogonal rotated matrix – an obliquely rotated input comparison and/or target matrix gets "orthogonalised" before factor/MDS coordinate matching – which means mostly it is uninterpretable (as it is just an arbitrary orthogonalised version of the obliquely rotated matrix. This is because the obliquely rotated matrices are "orthogonalised" to an arbitrary orthogonal representation (not simple structure at all) – then the orthogonalised comparison matrix is compared to the orthogonalised target matrix. Remember, the point of this routine is to enable you to say how similar two orthogonal sets of vectors are to one another, prior to any rotational solution.

In reality, the maximally congruent matrix output is only of value when using an orthogonal target matrix as input. This is because the already orthogonal matrix remains unchanged (untransformed) as the target matrix– and the comparison matrix is then rigidly rotated toward it. The comparison matrix could of course be obliquely rotated – indeed, it could be the same orthogonal target matrix, obliquely rotated, so that you can see how similar the obliquely rotated factor loadings are to the orthogonal version factor loadings, as well as confirming the congruential congruence via the KHB coefficients. See Example #5, online help for more details.

Also, you can download the paper published in 1998 that modified the KHB procedure to properly reflect similarity of loadings after transformation rather than the transformation rotation shifts (The Eysenck Personality Questionnaire: an examination of the factorial similarity of P, E, N, and L across 34 countries). Click here to download.

Download and Installation Information
The installation file is 8.84Mb in size. It's a single installation setup file which contains a large online help file and an (24th January, 2006: now includes a chapter on what size congruence/DSD/KSD is recommended by investigators). 84-page pdf manual of this online help (with a fair amount of graphics and equations hence the rather large installation file). You will also might have to install 4 fixed-format font files which are created in the installation directory c:\facsim. The "Orthosim Output Fonts and Fontsizer info.doc" which is installed in c:\facsim explains how to do this using a step-by-step visual guide. An icon for the program will be created on your desktop. The zipped archive containing the setup file is available here.

The (24th January, 2006) help file only is available here for download (1.4Mb) in a zipped archive - unzip it and just save the file to the subdirectory c:\facsim on your machine, overwriting the old Orthosim.chm file. The new help file has an augmented title bar which now provides a version number as v.1.