3WAYPACK HELP FILE
Pieter M. Kroonenberg
Department of Education
Rijksuniversiteit Leiden
==============================================================================
This document provides an overview of the 3WAYPACK Help File.
The programs INTERFACE3, PREPROC3, and POSTPROC are available for IBM
PCs and compatibles only. The programs TUCKALS3, TUCKALS2, and TRILIN
are available for IBM PCs and compatibles. Under special conditions they
can be made available for other systems with an ISO fortran compiler.
Copies of the 3WAYPACK User's Manual, the printed version of this
document and the programs can be ordered from
Pieter M. Kroonenberg
The Three-Mode Company
Mail: Department of Education, Leiden University
Wassenaarseweg 52, 2333 AK Leiden, The Netherlands
Email: kroonenb@rulfsw.leidenuniv.nl
Tel: *31-71-5273446/5273434
Fax: *31-71-5273945
Text version 1.1
Version date: 1 September 1996
PREFACE
This document is primarily an annotated overview of all menu entries
in INTERFACE3, and as such it may serve as a manual for the program
package. In essence the document will eventually be transformed into
Help File of 3WAYPACK. Unfortunately, the software does not yet contain
the help structure. The entries in this manual are those that would pop
up on the screen after pressing [F1] if the help system had been in place.
The interface works properly under DOS, provided a standard version
is used. Problems have occurred when attempting to use the Japanese
version of DOS. Therefore, Japanese users should first switch to the
english-language version of DOS. If you have trouble doing this please
check with your local systems manager. So far no problems have been
observed running the program under OS/2, or in a DOS window under either
Windows 3.1 or Windows95.
The present version of the software is not compatible with the
previous version. If you want to test the new software first before
discarding the older version be sure to install the programs in a new
subdirectory. For more details see the 3WAYPACK User's Manual.
If a file name or directory name is enclosed in brackets, an actual
file name or directory name must be substituted. For instance, in
.job, for jobname the actual name must be substituted; the
fixed extension is job. Following DOS conventions, a full path name
consists of [\...]\..
Please do not type the brackets. Names of screens will have Capitalized
First Letters in headings and small upper-case letters in the text. Text
on the screen will be in Upper and Lower case as the need may be. Keys
to be pressed will be in square brackets, for instance, [Enter]. Text
to be typed by the user will also be in computer type. In some menus,
sometimes options will be grey rather than black. This indicates that
such options are not available given the state of the analysis, or have
not yet been implemented. In the latter case, one should see this as an
advertisement of what we have in mind in terms of extensions of the
package.
All menu items are numbered by [number] and correspond to those that
will be incorporated in the software. For clarity the delimiter - is
used after the third digit. The headings of the menus often have the
form: menu text: message, where menu text indicates the text appearing
on the menu line in the program and message the explanation appearing on
the bottom or message line of the screen. In places this may look silly,
because the message is hardly longer than the menu text. In some cases
the message text is deleted. Not all screens are shown in this document;
especially in TUCKALS2 and TRILIN the reader is frequently referred to
the identical menu item for TUCKALS3.
Please note that this manual is based on the use of INTERFACE3 for
accessing the programs. However, the six fortran programs (and the
stand-alone version of PREPROC3, NDIMIS3) can all be used in
stand-alone mode as well. In a separate document the command summaries
are supplied for stand-alone usage of the programs. However, that
particular document is not stand-alone and should be used in conjunction
with this manual.
3WAYPACK MENU STRUCTURE
TABLE OF CONTENTS - OVERVIEW
0. General utilities Utilities to select, view files, clean-up jobs
1. Job definition screen Definition of job name, data set, analysis type
2. Data definition screen Description of data, format, labels, etc.
3. Main menu Central menu for each program
31. Edit Job definition Define new jobs, amend present ones
32. Edit Data definition Change labels, (re)define missing data
33. Program options Define the particulars of an analysis
331. PREPROC3 options
332. TUCKALS3 options
333. TUCKALS2 options
334. TRILIN options
34. Execute Execute analysis
35. View/print output View and/or Print output
36. Edit output Edit output
37. POSTPROC Postprocessing of output
371. ROTATE options
372. RESIDUAL options
373. JOINTPLT options
38. End of Program Return to DOS
4. PREPROC3 options Preprocessing three-way data
5. TUCKALS3 options Three-mode PCA with full core matrix
6. TUCKALS2 options Three-mode PCA with extended core matrix
7. TRILIN options Parallel factor analysis
8. ROTATE options Rotations of components and core matrix
9. RESIDUAL options Analysis of residuals
10. JOINTPLT options Joint plots DETAILED TABLE OF CONTENTS
DETAILED TABLE OF CONTENTS
0. General Utilities
01. File select utility
02. File view utility
03. Housekeeping screen
1. Job Definition screen
11. Job name
111. Job list
12. Data file
13. Analysis
131. Type of analysis
131-1.PREPROC3 data preprocessing
131-2.TUCKALS3 three-mode PCA with full core matrix
131-3.TUCKALS2 three-mode PCA with extended core matrix
131-4.TRILIN trilinear analysis: parallel factors
131-5.TRIADIC trilinear analysis: triadic algorithm
131-6.MIXCLUS cluster analysis
14. Job title
2. Data Definition screen
21. Data title
22. Mode 1
221. Levels
222. Label
23. Mode 2
231. Levels
232. Label
24. Mode 3
241. Levels
242. Label
25. Missing values
26. Data structure
27. Data format
28. Labels for levels
281. Input labels
3. Main Menu
31. Edit Job definition
32. Edit Data definition
33. Program options
331. PREPROC3 options Preprocessing
332. TUCKALS3 options Three-mode PCA with full core
333. TUCKALS2 options Three-mode PCA with extended core
334. TRILIN options Trilinear analysis/Parallel factors
34. Execute Execute analysis
35. View/print output View and/or print output
36. Edit output Edit output
37. POSTPROC output Postprocessing output
371. ROTATE Rotations of components and core
372. RESIDUAL Analysis of residuals
373. JOINTPLT Joint plots
38. End of program Return to DOS
4. PREPROC3 options: Preprocessing three-way data
331. PREPROC3 options Program options
331-1.Swap data matrix Three-dimensional "transpose"
331-2.Estimate missing data How to estimate missing data
331-3.Centre data Centre data per row, column, etc.
331-4.Normalise data Normalize data per slice
331-5.Print options Print means and/or ANOVA
331-6.Return to Main menu Return when parameters are set
5. TUCKALS3 options: Three-mode PCA with full core matrix
332. TUCKALS3 options Program options
332-1.Number of components Number of components of each mode
332-2.Missing data Handling missing data
332-3.Centring options Centre data
332-4.Normalization options Normalize frontal or lateral slices
332-5.Analysis options Iterations, convergence, etc. etc.
332-6.Initial configurations Types of initial configurations
332-7.Print options Print notes, initial configuration etc.
332-8.Component plots Plot of components
332-9.Return to Main menu Return when parameters are set
6. TUCKALS2 options: Three-mode PCA with extended core matrix
333. TUCKALS2 options Program options
333-1.Number of components Number of components per mode
333-2.Missing data Handling missing data
333-3.Centring options Centre data
333-4.Normalization options Normalize frontal and lateral slices
333-5.Analysis options Iterations, convergence, etc. etc.
333-6.Initial configurations Types of initial configurations
333-7.Print options Print notes, initial configuration, etc.
333-8.Component plots Plot of components
333-9.Return to Main menu Return when parameters are set
7. TRILIN options: Parallel factor analysis
334. TRILIN options Program options
334-1.Number of components Number of components
334-2.Missing data Handling missing data
334-3.Centring options Centre data
334-4.Normalization options Normalize frontal or lateral slices
334-5.Analysis options Iterations, convergence, etc. etc.
334-6.Initial configurations Types of initial configurations
334-7.Print options Print notes, initial configuration, etc.
334-8.Component plots Plot of components
334-9.Return to Main menu Return when parameters are set
8. ROTATE options: Rotations of components and core matrix
371. ROTATE Rotations of components and core
371-1.Job title
371-2.Main menu
371-21. ROTATE options Program options
371-22. Execute Execute analysis
371-23. View/print output View and/or print output
371-24. End of POSTPROC Return to the Main program
9. RESIDUAL options Analysis of residuals
372. RESIDUAL Analysis of Residuals
372-1.Job title
372-2.Main menu
372-21. RESIDUAL options Program options
372-22. Execute Execute analysis
372-23. View/print output View and/or print output
372-24. End of POSTPROC Return to the Main program
10. JOINTPLT options: Joint plots
373. JOINTPLT Joint plots
373-1.Job title
373-2.Main menu
373-21. JOINTPLT options Program options
373-22. Execute Execute analysis
373-23. View/print output View and/or print output
373-24. End of POSTPROC Return to the Main program
============================================================================
Chapter 0
General Utilities
Most general utilities have their own (independent) help systems, which
are already operational.
[01] File select utility
This utility takes you to the current directory, in which you can
move around with the cursor keys. You can move up in the directory
tree by moving to the box with two dots (parent directory) and press
[Enter], and you can move down the tree by moving to a directory
indicated by . and pressing [Enter]. Note that
due to space limitations the last letter of the directory name is
not shown. For an example see chapter 1, section [12].
[02] File view utility
This utility lets you inspect a file, and is only useful for ascii
files. By pressing [F1] an explanation is provided how to move
around in the viewer. For an example see chapter 3; section [35]
[03] Housekeeping screen
When there are many .job files in the program directory,
you can choose to start INTERFACE3 with the option /c (thus type IF3
/c at the DOS prompt), and the interface will start up with the
housekeeping screen in which you can erase obsolete jobs by marking
them. Interface3 will erase the marked jobs, all related set-up
files and output files without, of course, touching your data file.
Mishaps only occur when you create files with same name as the
jobname and extensions known to INTERFACE3.
===========================================================================
Chapter 1
Job Definition screen
This screen is used to start new analyses or call up old ones. The basic
identifier of an analysis or job is its . All files created by
IF3 will have the file name .. A specific analysis can
only have one job name. Thus, once you have chosen a job name and a type
of analysis they will always be linked. The job title will be used as
header for all output created during the analysis. When all required
information has been supplied, pressing [F2] will save the information.
IF3 will proceed to the next screen, the data definition screen, if no
data definition is available yet, and to the main menu if the data have
been used in IF3 before.
[11] Job Name
The job name is the basic identifier of an analysis and is always
linked to a specific type of analysis, which once chosen can not be
changed. All files created by IF3 will have the file name
., where identifies the content of the file.
Please choose your name with care to avoid conflicts with already
existing names. If you have removed data sets from their original
location, renamed directories in which previously analyzed data
resided, or removed IF3 created copy of your data (e.g.
.if3), the jobs referring to those files will be grey and
cannot be selected. To remove these grey jobs start IF3 next time
with IF3 /c, so that the housekeeping screen will appear and you can
remove the jobs in question. A new job name can be entered by
typing. If you type an already existing job name, the information
connected with this job name will be retrieved and filled in on the
other lines. If previous jobs are available, they may be selected
from the job list screen after pressing [F3] from the Job name line.
[111] Job List
The job list screen will list the last 32 jobs. Select the
appropriate one, and press [Enter]. If you do not want any of the
old ones, press [Esc] and type in a new name. If you have removed
data from their original location, renamed directories in which
previously analyzed data reside, or removed IF3 created copies of
your data (i.e., .if3), the jobs referring to those files
will be grey and cannot be selected. To remove these grey jobs start
IF3 the next time with IF3 /c. The housekeeping screen will appear
and you can remove the jobs in question.
If you see any job names that may be deleted, start IF3 the next
time with IF3 /c), so that the housekeeping screen will appear in
which you can throw away old jobs and their related files created by
IF3. If all you want to keep is the output file, rename the file or
its extension. IF3 will not touch your renamed file. This is a
better procedure than performing the clean-up by hand. No user
created files will be touched, nor files with unknown extensions
(for details, see the Help with the housekeeping screen).
[12] Data File
On the Data file line the name of the data set to be analyzed is
entered, for instance .. By pressing [F3] you can
select the appropriate data set from a list, either from the current
directory or after a switch to another (sub)directory. The contents
of a file can be inspected by pressing [V] (=View). Other disk units
(for instance, ) may be called by first typing a: on the Data
file line.
[13] Analysis
On this line the type of analysis is to be specified. Either type
the name of a program or even better, press [F3] to obtain a list of
available programs.
[131] Type of Analysis
Screen with program names.
[131-0] Specify
A program name has to be selected before one can leave the type of
analysis screen.
[131-1] PREPROC3
PREPROC3 is a program for preprocessing three-way data. The program
can perform functions such as
1. Swapping the dimensions of a three-way array;
2. Estimating missing data according to Anova models;
3. Centring three-way arrays;
4. Normalising three-way arrays;
5. Printing all means from a three-way Anova model;
6. Computing and printing three-way Anova summary tables.
[131-2] TUCKALS3
TUCKALS3 is a program for three-mode principal components with
dimension reduc- tion along all three ways. Its core matrix will
have as its dimensions the number of components of each of the three
ways (i.e. PxQxR). The program can also be used for a simple
singular value decomposition, and a weighted (or replicated)
principal component analysis.
[131-3] TUCKALS2
TUCKALS2 is a program for three-mode principal components with
dimension reduc- tion along two of the three ways. Its extended core
matrix will have as its dimensions the number of components of way 1
and way 2, and the number of levels of way 3 (i.e. PxQxK). The
program can also be used for a simple singular value decomposition,
and the IDIOSCAL model, if the input is a set of K symmetric
matrices with (squared) distances.
[131-4] TRILIN
TRILIN is a program for three-mode component analysis without a core
matrix. It fits the parallel factor analysis model (PARAFAC), which
is also called the canonical decomposition model (candecomp). The
PARAFAC model is an extension of the principal component model in
which the components are equal for each level of the third way
except for proportionality constants. It can be seen as a Tucker2
model with diagonal core, or a Tucker3 model with superdiagonal core
given that all ways have equal numbers of components. The program
can also be used for the INDSCAL model, if the input is a set of K
symmetric matrices with (squared) distances.
[131-5] Triadic
Triadic is a program for three-mode component analysis without a
core matrix. The program fits the Parallel factor analysis model
(PARAFAC), but it uses a non-standard algorithm, so-called triadic
fitting. This procedure allows for easy inclusion of constraints on
the components. Unfortunately, it is not yet available for general
use, but it is hoped that it will be included in the next release of
3WAYPACK.
[131-6] MIXCLUS
MIXCLUS is a set of two cluster analysis programs which are based on
the mixture method of clustering model developed by Kaye E. Basford.
At present only the two-way version is more or less operational
under IF3, and only available on special request. Please check with
The Three-Mode Company.
[14] Job Title
The Job title is optional, but it is advised to supply a meaningful
title, as this will be used to label the output.
Chapter 2
Data Definition screen
In the data definition screen the particulars of the data set to be
analyzed are entered. Suppose its name is .. Information
about the data should be supplied such as a data title (optional),
number of levels in each mode (obligatory), the labels of the modes
(recommended), missing data codes (if necessary), data organization in
the data file - Frontal or Case mode and data format (obligatory), and
labels for the levels in the modes (recommended). When finished, press
[F2]; the information will be saved to a data dictionary file for later
usage, the data will be written to a new file (.if3 according
to an internally determined new format), a label file .lab
will be written (even if it already exists), and the coordinates of the
missing data will be written to the file .mis (if specified).
If all is well the program will proceed to the main menu screen. If the
input format is incompatible with the data or there are fewer data than
specified by the number of levels and the input format, the program will
crash because of a DOS error and you will have to start anew (see the
3WAYPACK User's Manual for more information). If there are more data
than specified by the levels of the modes and the input format the
program will issue a warning and proceed to the main menu screen.
[21] Data Title
The data title gives information on the data file itself, but the
title is not printed in the output, except if no job title has been
specified in the job definition screen.
[22] Mode 1
[221] Levels
Number of levels in the first mode, indicated in the output and
manual by I; see also the diagram on the screen. The minimum and
default number of levels in this mode is 2.
[222] Label
Please provide a label of up to 12 characters for the first mode.
This label will be used to identify the mode in the output.
[23] Mode 2
[231] Levels
Number of levels in the second mode, indicated in the output and
manual by J; see also the diagram on the screen. The minimum and
default number of levels in this mode is 2.
[232] Label
Please provide a label of up to 12 characters for the second mode.
This label will be used to identify the mode in the output.
[24] Mode 3
[241] Levels
Number of levels in the third mode, indicated in the output and
manual by K; see also the diagram on the screen. The minimum and
default number of levels in this mode is 1. With one level in the
third mode, the program effectively analyses a two-way data matrix.
As the programs are not specifically geared towards two-way
analyses, the output might look a bit strange in places.
[242] Label
Please provide a label of up to 12 characters for the third mode.
This label will be used to identify the mode in the output.
[25] Missing Values
If a data set contains missing data you have several options to
treat such omissions. However, you must use the option Missing
values to let IF3 know how to handle them. You may specify as many
missing values as fit on the Missing values line. IF3 will scan the
data for these values and save a set of pointers to the location of
the missing values in the three-way array. The list of pointers will
be saved in a file called . and the name of this file
is passed to the three-way programs. If such a file already exists,
just press [F3] to tell IF3 of its existence. Please read the
3WAYPACK User's Manual for further information, especially in
relation to previous PREPROC3 runs.
[26] Data Structure
IF3 must know how the data are organized in the data file
. before it can read them. 3WAYPACK caters for the
two most common forms. The appropriate one can be selected by
pressing either [F] (frontal slice form) or [C] (case form).
- Case form: In statistical packages such as SPSS, SAS, the basic
data form is the rectangular data matrix where rows (levels of the
first mode) generally consist of cases (for instance, subjects,
genotypes), and the columns (levels of the second mode) of variables
(or attributes, items). When the variables are measured under
several conditions (levels of the third mode), all data for each
case are on the same record or possibly two or more consecutive
records. This type of arrangement is called case form. The three-way
programs cannot handle such data directly; they have to be converted
to frontal slice form, which is done by the interface. Note that in
the three-way case the data are contained in an I by JxK data matrix
(J running faster than K), and that in general the data for several
conditions are on the same record.
- Frontal slice form: In this form, the input data consist of a
series of rectangular data matrices underneath each other, one for
each condition. The rows of the data matrix for a single condition
represent the cases and the columns contain the variables. Note that
in this form, the data for each condition are on separate records.
[27] Data Format
A valid fortran-style input format must be provided. IF3 will check
the syntactic validity of the format. If you are not sure about the
correct format of your data, press [F3] to view your data file
.. Data formats which are inconsistent with the data
in the data file may lead to incorrect analyses, or an exit to DOS
with a so-called run-time error. If there are more data in the file
than specified, a warning will be issued and IF3 will proceed. If
there are fewer data than specified the program will exit to DOS
with a run-time error. Once a valid format has been specified for a
data set it cannot be altered. You have to make a copy of the
original data file with a new name or directory to specify a
different format, for instance if you want to skip certain columns.
Alternatively, you may edit the data file so that it will have a new
time stamp. A last possibility is to delete the .if3 file
from your data directory. In the last two cases your original data
description file will be lost.
[28] Labels for Levels
In order to facilitate reading the output, the levels within the
three modes may be labelled. It is strongly recommended to do so.
Labels may be supplied for any of the modes. After entering, labels
will be stored in a label file .lab.
[281] Input Labels
There are two ways of entering labels, either via an existing file
.lab, which is called by pressing [F3] on any yes or no,
or by typing yes on a no on a Label line. The Label fields in the
label screen allow for 6 characters. Pressing [F2] after all labels
have been supplied will return you to the data definition screen.
============================================================================
Chapter 3
Main Menu
Each program has its own main menu, but they all have the same
structure. From the menu the options for a program can be specified, the
program can be executed, and its output viewed, printed, and edited. The
output may be further analyzed or processed. After the execution of a
task, the program always returns to the main menu to perform the next
task or modify a previous one.
[31] Edit Job Definition
The data definition of the current data may be adapted by returning
to the job definition screen; in particular, labels and missing data
may be (re)specified.
[32] Edit Data Definition
New jobs may be started, or old jobs recalled, by going back to the
data definition screen.
[33] Program Options
In order to execute a program, several options need to be specified,
at least the number of components. Most programs can perform an
analysis with only default settings. However, this will mean only 1
component per mode. In any case one has to enter the program options
menu at least once to allow IF3 to create a set-up file,
.set.
[331] PREPROC3 options
PREPROC3 is a program for preprocessing three-way data. The program
can perform functions such as
1. Swapping the dimensions of a three-way array;
2. Estimating missing data according to Anova models;
3. Centring three-way arrays;
4. Normalising three-way arrays;
5. Printing all means from a three-way Anova model;
6. Computing and printing three-way Anova summary tables;
7. Creating a new data set in accordance with the changes specified.
If PREPROC3 changes the data in any way it creates a new data file
in the same directory where the original data file resides. The name
of this data set is .pp3. Simultaneously, PREPROC3 also
creates a data definition file for this data set, so that the
modified data can directly be used in IF3. A new data file is
created whenever the options 1, 2, 3, 4 from the above list are
requested.
[332] TUCKALS3 options
TUCKALS3 is a program for three-mode principal components with
dimension reduc- tion along all three ways. Its core matrix will
have as its dimensions the number of components of each of the three
ways (i.e. PxQxR). The program can also be used for a simple
singular value decomposition if K=1, and a weighted (or replicated)
principal component analysis if R=1. For details see the TUCKALS3
section.
[333] TUCKALS2 options
TUCKALS2 is a program for three-mode principal components with
dimension reduc- tion along two of the three ways. Its extended core
matrix will have as its dimensions the numbers of components of way
1 and way 2, and the number of levels of way 3 (i.e. PxQxK). The
program can also be used for a simple singular value decomposition
if K=1, or can be used to fit the IDIOSCAL model. For details see
chapter 6.
[334] TRILIN options
TRILIN is a program for three-mode component analysis without a core
matrix. It fits the parallel factor analysis model (PARAFAC), which
is also called the canonical decomposition model (candecomp). The
PARAFAC model is an extension of the principal component model in
which the components are equal for each level of the third way
except for proportionality constants. It can be seen as a Tucker2
model with diagonal core, or a Tucker3 model with superdiagonal core
given that all ways have equal numbers of components. The program
can also be used for the INDSCAL model, if the input is a set of K
symmetric matrices with (squared) distances. For details see chapter
7.
[34] Execute
After specifying the options for one of the three-mode programs,
pressing [Enter] will start execution of the program. If an output
file for the present job already exists you will be warned and asked
to confirm whether the old file may be overwritten. If not, IF3 will
make a new output file with the same name but a different extension
(.ou1, .ou2, etc.).
[35] View/Print Output
After a successful execution, you can view the output with the file
viewer. Once you are in the File Viewer you can print the output by
pressing [P], provided a printer is available. For details about the
file viewer, press Help - [F1] (see also section [02] on the file
view utility).
[36] Edit Output
This option is only available if an ascii editor has been specified
in the file IF3.CIG. IF3 calls this editor with the name of the
output file (.out) as the sole parameter. The edit output
option also provides the exit to DOS via your exit in the editor.
You can, for instance, use the editor to rename output files.
[37] POSTPROC output: Postprocessing output
With the programs in POSTPROC, the output of the three-way analysis
programs can be analysed further. The components and core matrix
(TUCKALS) or component weights (TRILIN), and the overall sum of
squares are written to the file .cpc by the three-way
analysis programs. This information is the basis for the three
postprocessing programs ROTATE (rotation/transformation of component
matrices), RESIDUAL (computations of residuals and residual-fitted
data plots), and JOINTPLT (joint (bi)plots of components of two ways
given a component of the third), which can be selected from the type
of POSTPROC screen. If you want to use more than one postprocessing
program, you have to select the postprocessing option in the main
menu screen of the relevant three-way analysis program every time.
No rotations or joint plots can be made for output from TRILIN, as
such operations are incompatible with the Parafac model fitted by
TRILIN.
[372] ROTATE
This program will rotate the components of any of the three ways.
The rotations (or rather) transformations available are
1. Varimax rotation on the orthonormal components (equivalent to the
Harris-Kaiser cluster rotation);
2. Oblimin transformation of the orthonormal components;
3. Optimally constant first component. This option can sometimes be
used to facilitate interpretation, especially of reference modes
for joint plots.
[373] RESIDUAL
This program will calculate standardized residuals and plot these
against the fitted data. You may choose to do the plotting per
frontal slice or for the entire data set. This program also provides
an option to write either the residuals and/or the fitted data to a
file called .res and .fit respectively. Note that
the fitted data are such that they perfectly fit the model, and thus
can be used for simulation etc.
[374] JOINTPLT
This program constructs joint (bi)plots for two modes given the
third, so that the third is the reference mode. In addition, it can
compute the coordinates for interactive biplots, but not yet the
interactive biplots themselves. These coordinates can in some data
sets be considered component scores.
[38] End of Program
Selecting this option will end the program and return you to DOS or
Windows.
===========================================================================
Chapter 4
PREPROC3 options
PREPROC3 is a program for preprocessing three-way data. The program can
perform functions such as
1. Swapping the dimensions of a three-way array;
2. Estimating missing data according to Anova models;
3. Centring three-way arrays;
4. Normalising three-way arrays;
5. Printing all possible means;
6. Computing and printing three-way Anova summary tables;
7. Creating a new data set in accordance with the changes specified.
If PREPROC3 changes the data in any way (i.e. options 1, 2, 3, 4 from
the above list are requested) it creates a new data file in the same
directory where the original data file resides. The name of this data
set is .pp3. Simultaneously PREPROC3 creates a data definition
file for this data set, so that the modified data can be used directly
in IF3. The screen below shows the job definition screen, after the
option Edit JOB definition has been selected in the main menu.
[331-1] Swap data matrix: Three-dimensional "transpose"
In some applications it is desirable to have a specific mode (1) as
the first mode, for instance subjects or judges; or (2) as the third
mode, for instance, time points. In such cases the modes have to be
swapped or transposed. Such swapping is particularly relevant for
the Tucker2 model because it is not symmetric in all modes.
Moreover, by judicious swapping allows TUCKALS3 to compute the
correct core covariances. PREPROC3 can swap a three-way array in any
desired manner.
[331-11] None (ABC): no swapping
Default
[331-12] ACB: transpose horizontal slices
Swap 2nd and 3rd modes
[331-13] BAC: transpose frontal slices
Swap 1st and 2nd modes
[331-14] CBA: transpose lateral slices
Swap 1st and 3rd modes
[331-15] BCA: 2nd 1st; 3rd 2nd; 1st 3rd way
Two-step cyclic permutation
[331-16] CAB: 3rd 1st; 1st 2nd; 2nd 3rd way
One-step cyclic permutation
[331-2] Estimate missing data: When and how to estimate missing data
Some three-way programs have only limited capabilities of handling
missing data. Even if a particular estimation procedure is included,
PREPROC3 can provide good starting values. The missing data
estimation procedures make use of terms from three-way analysis of
variance decompositions or from multivariate two-way analysis of
variance decompositions. The algorithms used to estimate missing
data are convergent if the data matrix is not singular, that is, if
there are no submatrices completely filled with missing data.
[331-21] None: no estimation of missing data
Default. Note that if there are missing data, but you decide not to
estimate them in the program, no centring or normalization can be
performed.
[331-22] Use two-way means
Estimates are based on the three-way analysis of variance model with
one observation per cell minus the three-way interaction term.
Missing value: x^(ijk) = m+a(i)+b(j)+c(k)+ab(ij)+ac(ik)+bc(jk)
[331-23] Use one-way marginal means
Estimates are based on the three-way main effects model.
Missing value: x^(ijk) = m+a(i)+b(j)+c(k)
[331-24] Per level of <1st mode>
Estimates to be used if the levels of the first mode have
incompatible measurement levels.
Missing value: x^(ijk) = m(i)+ab(ij)+ac(ik)
[331-25] Per level of <2nd mode>
Estimates to be used if the levels of the second mode have
incompatible measurement levels.
Missing value: x^(ijk) = m(j)+ab(ij)+bc(jk)
[331-26] Per level of <3rd mode>
Estimates to be used if the levels of the third mode have
incompatible measurement levels.
Missing value: x^(ijk) = m(k)+ac(ik)+bc(jk)
[331-3] Centre data: Centre data per row, column, etc.
Centring is a linear transformation of the data; specifically, the
mean is subtracted from the data. In order to keep track of the
proper mode in case of swapping, it was decided to define the
centring with respect to the original arrangement of the data, i.e.
before swapping. However, the program will take into account the
swapping performed. Different combinations of centring are possible.
The program will correctly handle a situation where redundant
centrings have been specified, for instance if both overall centring
and centring per row has been specified.
[331-31] Across <1st mode>: centring per column of original matrix
Definition: x^(ijk) = x(ijk) - mean(.jk)
[331-32] Across <2nd mode>: centring per row of original matrix
Definition: x^(ijk) = x(ijk) - mean(i.k)
[331-33] Across <3rd mode>: centring per tube of original matrix
Definition: x^(ijk) = x(ijk) - mean(ij.)
[331-34] Per <1st mode> slice: centring per horizontal slice of original matrix
Definition: x^(ijk) = x(ijk) - mean(i..)
[331-35] Per <2nd mode> slice: centring per lateral slice of original matrix
Definition: x^(ijk) = x(ijk) - mean(.j.)
[331-36] Per <3rd mode> slice: centring per frontal slice of original matrix
Definition: x^(ijk) = x(ijk) - mean(..k)
[331-37] Overall: centre entire data array
Definition: x^(ijk) = x(ijk) - mean(...)
[331-4] Normalize data: Normalize data per slice
Normalization is the rescaling of the original or previously centred
data, such that the sum of the squared data elements in a submatrix
is 1. Note that normalization is independent of centring.
Normalization in itself will not produce z-scores (mean 0, standard
deviation 1) unless the data in the normalized submatrix have a zero
mean. Moreover, the normalization results in sums of squares equal
to 1 rather than mean squares equal to 1. This is primarily done
because sums of squares may be partitioned into independent sums of
squares, while mean squares cannot be partitioned in that way. If
normalization is attempted perpendicular to a previously performed
centring, the normalization would destroy the centring. Therefore
such combinations are not allowed in the program. Only one
normalization can be selected at a time. Multiple nor- malizations
require an iterative process.
[331-41] <1st mode> slice: normalising original horizontal slice
Definition: x^(ijk) = x(ijk)/s(i..)
[331-42] <2nd mode> slice: normalising original lateral slice
Definition: x^(ijk) = x(ijk)/s(.j.)
[331-43] <3rd mode> slice: normalising original frontal slice
Definition: x^(ijk) = x(ijk)/s(..k)
[331-5] Print options: Print means and/or anova
PREPROC3 will print means at various junctures of the program and
will also print the results of a fixed effects three-way analysis of
variance with one observation per cell.
[331-51] Print MEANS
If requested and applicable, all means will be printed (1) after
swapping, (2) before and after estimation of missing data, (3)
before and after centring, and (4) after normalization.
[331-52] Print ANOVA: 3-way ANOVA (before and after centring/normalization)
If requested and applicable, a three-way anova is printed (1) before
and after estimation of missing data, and (2) after normalization.
As there is no real error term, the three-way interaction is used as
the within sum of squares in the calculation of the F statistic.
However, in large data sets it is expected that a substantial part
of the three-way interaction can be modelled with only a few
multiplicative components and thus a limited number of
degrees-of-freedom. The `real' mean error sum of squares is
generally expected to be smaller than the full three-way interaction
mean square, so that the F values are only a lower bound.
[331-6] Return to Main Menu
When the options have been selected, you can return to the main menu
for execution of the program. After returning to the Main Menu
proceed as described in chapter 3, section [34].
=============================================================================
Chapter 5
TUCKALS3 options
TUCKALS3 is a program for three-mode principal components with dimension
reduction along all three ways. Its core matrix will have as its
dimensions the number of components of each of the three ways (i.e.
PxQxR). The program can also be used for a singular value decomposition,
and a weighted (or replicated) principal component analysis.
[332-1] Number of components
By default each mode has one component. Note that selecting this
option without selecting any other option is sufficient for a
(rather trivial) analysis. Specifying more components than levels
will cause the program to beep at you. During execution of the
program, it may turn out that you have requested more components
than the data can support. In that case the program will
automatically reduce the number of components. Appropriate messages
will appear in the output.
[332-11] Number of components 1st mode
Specify the number of components (P) for the first mode. P should be
less than or equal to the number of levels (I).
[332-12] Number of components 2nd mode
Specify the number of components (Q) for the second mode. Q should
be less than or equal to the number of levels (J).
[332-13] Number of components 3rd mode
Specify the number of components (R) for the third mode. R should be
less than or equal to K, the number of levels.
[332-2] Missing data
TUCKALS3 does not recognise missing data codes, but needs to have
the locations (coordinates) of the missing data in the three-way
array. These coordinates can be read in from a special file,
., which contains as many lines as there are missing
data points. IF3 automatically creates such a file, if you have
specified missing data codes on the data definition screen. If you
press [F3], it will read an existing file with missing data
coordinates. On the Distribution disk a special program is supplied
to generate a missing-data coordinates file (misindex.exe) in case
you do not have missing data codes, but know the locations of the
missing data. [332-21] None: no estimation, all data interpreted as
valid Default. Note that if there are missing data, but you decide
not to estimate them in the program, no centring or normalization
can be performed.
[332-22] Substitute slice mean: use (frontal) slice means as initial estimates
For all missing data the mean of the appropriate frontal slice will
be substituted. The implication of this substitution is that the
data within a slice are comparable. If they are not, you either have
to use PREPROC3 to process the missing data, or if applicable swap
the three-way data around (with PREPROC3) in such a way that they
are comparable. For instance, if variables have different
measurement characteristics, they should be levels of the third mode
in order to use this option.
[332-23] User provided values or initial estimates
This option assumes that you already have sensible values for the
missing data (for instance calculated with one of the methods in
PREPROC3), and that TUCKALS3 may use these values as initial
estimates for the missing data.
[332-3] Centring options
Centring is the process of removing one or more means from the data.
For three-way data there are several possibilities of which the more
useful are included in TUCKALS3. Some additional options are
available in PREPROC3. Centring across a mode will cause the
components of that mode to be centred. Note that centring
fundamentally changes the data, so that it is virtually impossible
to compare analyses of differently centred data.
[332-31] None: no centring
Default
[332-32] Across <1st mode>: fiber centring per column
Centring across the first mode, also called centring per column,
indicates that for all KxJ columns the means are calculated by
summing over the index i and dividing by I, and that these means are
subtracted from the data values, so that all columns are in
deviation scores. As variables are often in the second mode, this is
a common option for profile data.
[332-33] Across <2nd mode>: fiber centring per row
Centring across the second mode, also called centring per row,
indicates that for all IxK rows the means are calculated by summing
over the index j and dividing by J. These means are subtracted from
the data values, so that all rows are in deviation scores. As
variables are often in the second mode, this is a common option for
profile data.
[332-34] Across <3rd mode>: fiber centring per tube
Centring across the third mode, also called centring per tube, means
that for all IxJ tubes the means are calculated by summing over the
index k and dividing by K. These means are subtracted from the data
values, so that all tubes are in deviation scores. This option is
especially useful if subjects are in the third mode.
[332-35] Across <1st mode> and <2nd mode>: per column and per row
Centring across the first and second mode, also called double
centring, means that for both the KxI and the JxK means are
calculated by summing over the index j plus dividing by J, and
summing over i plus dividing by I, respectively. These means are
subtracted from the data values (and the overall mean is added in
again to maintain the overall size of the data), so that all rows
and all columns are in deviation scores. This option is particulary
useful to transform (squared) distances or dissimilarities into
scalar products, for instance if one wants to perform a three-mode
scaling analysis.
[332-36] Per <3rd mode> slice: slice centring: per frontal slice
For each frontal slice the mean is calculated by summing over all
indices (i,j) and dividing by IJ. Within each slice, this mean is
subtracted from the data values.
[332-37] Overall centring
The mean is calculated over all data values, and subsequently
subtracted. This option is rarely used.
[332-38] Print removed means
To check whether the centring was carried out properly the removed
means may be listed. The form of the output is, however, not very
sophisticated.
[332-4] Normalization options: Normalize frontal or lateral slices
Normalization means that all frontal (lateral) slices end up having
a total sum of squares of 1. Note that the normalization is entirely
independent of the centring, so that the removed values do not
generally represent variances. Moreover, no attempt is made to
divide the sum of squares by degrees of freedom, so that the removed
values are not mean squares either. This means, for instance, that
the total sum of squares of the normalized three-way data is equal
to I (frontal-slice normalization) or equal to J (lateral-slice
normalization). For horizontal-slice normalization the ways should
be swapped via PREPROC3, which program can also perform the required
normalizations. Note that when both centring and normalization are
selected, centring is always carried out first.
[332-41] None: no normalization
Default
[332-42] Per <3rd mode> slice: normalize frontal slice
Each frontal slice is normalized such that the sum of squares of its
elements is equal to 1.
[332-43] Per <2nd mode> slice: normalize lateral slice
Each lateral slice is normalized such that the sum of squares of its
elements is equal to 1.
[332-44] Across <1st mode>: normalize column fibers
All JxK columns are normalized such that their sums of squares are
equal to 1. THIS OPTION IS NOT YET OPERATIONAL
[332-45] Print removed SSQ: print removed sums of squares
To check whether the normalization was carried out properly the
removed sums of squares may be listed. The form of the output is,
however, not very sophisticated.
[332-5] Analysis options: Iterations, convergence, acceleration, etc.
The analysis options control various (technical) aspects of the
analysis. Increasing the maximum number of iterations will be the
most frequent activity in this screen.
[332-51] Maximum number of iterations
When the maximum number of iterations is reached, the program will
finish as if it had converged and will print all relevant
information. The default is set at 50, the maximum value is 99999.
[332-52] Convergence criterion for iterative process
The default for the convergence of the discrepancy or loss function
is 10**-8 (10 to the power -8). From this value the criterion for
the norm of component matrices is derived. To change the convergence
criterion specify the size of the exponent, e.g. 10 for 10**-10. On
most machines the criterion should not be less than roughly 10**-16.
[332-53] Use acceleration technique (Yes/No)
Even though the program is usually fast enough, some facilities have
been built in to speed up convergence. One of those is the use of
successive overrelaxations. The overrelaxation should converge, but
in case of suspicious behaviour, such as negative differences of the
discrepancy function, please rerun the analysis without the
acceleration. Another method used to speed up the process is limited
convergence checking and reduced output during iteration.
[332-54] Random seed (0=internally determined)
This option is only useful if a random initial configuration is
requested. If the random seed is internally determined, the same
random number will be used for each run of an analysis. This option
allows changing the initial seed to create different initial
configurations.
[332-6] Initial configurations
TUCKALS3 needs initial configurations for the component matrices A,
B, and C. If there is no special reason to do otherwise, it is
suggested to stay with the default option as it will provide
excellent starting values. In fact, if there is a perfect solution,
it will be found by the default procedure.
[332-61] First mode:
Types of initial configurations for first mode The options are
identical for the three modes, and are therefore only listed for the
first mode.
[332-611] Tucker, not fixed (default): initial configuration derived from data
By default, the program computes an initial configuration for each
mode by calculating ten steps towards a solution as described in
Tucker (1966). This involves stacking the slices underneath each
other into a `long' matrix, e.g. JxK by I (a process sometimes
called unfolding), and performing a pca on this long matrix.
Generally the TUCKALS solution will be very close to the Tucker
solution, especially with a limited number of components.
[332-612] External, fixed: external configuration not modified during iteration
If an (orthonormal) solution is available from another analysis or
from the literature, this configuration may be read in and kept
fixed during the analysis. In other words, the rest of the solution
is fitted around the fixed configuration. Comparing the fits without
and with the fixed solution will allow an assessment of how well the
external configuration fits the data. The external solution is
assumed to be orthonormal (orthogonal with unit-length components).
If this is not the case the program will issue a warning, but will
proceed as if the configuration is correct. In such a case the
output will not necessarily be interpretable; especially the
partitioning of the fit might no longer be valid. The external
configuration should be in a file accessible to the program, and
should be in free format, i.e. there should be at least one blank
between numbers.
[332-613] External, not fixed: external configuration: modified during iteration
If an (orthonormal) solution is available from another analysis or
from the literature, you may request the program to read this
configuration and use it as a starting configuration. The external
configuration should be in a file accessible to the program, and
should be in free format, i.e. there should be at least one blank
between numbers.
[332-614] Random, not fixed: random initial configuration (orthonormalized)
This option is primarily useful for research purposes. It could for
instance be used to assess the sensitivity of the program to
different starting solutions. (The answer is 'not sensitive'.) To
create different starting configurations for each run, change the
random seed (cf. section [332-54]).
[332-62] Second mode: Types of initial configurations for second mode
[332-63] Third mode: Types of initial configurations for third mode
(options as for the first mode)
[332-64] Configuration file: Name of external configuration file
External configurations should be together in ONE external
configuration file in the order of the modes. Thus a first-mode
configuration should always precede a second-mode one and a
second-mode configuration a third-mode one. The program will request
the name of the file if you have indicated that there are external
configurations. If you do not specify a correct or valid file name,
the external configuration options will be reset to their defaults.
The numbers in the file should be in free format, i.e. there should
be at least one blank between numbers.
[332-65] Done: Return to TUCKALS3 Options
Selecting this option will return you to the TUCKALS3 options menu.
[332-7] Print options: Print notes, initial configuration etc.
The program can produce a large amount of output which is regulated
via the options on this screen. Even with the defaults, the output
can be considerable if there are many levels in one of the modes.
The output increases even further when more than 4 components are
specified for any one mode, due to the way the component matrices
are printed.
[332-71] Print notes: write notes and commentary in output
Throughout the program interpretational comments are available if
this option is active, which is the default.
[332-72] Initial configuration: write initial configuration
The default initial configuration is an approximation to an ordinary
PCA on combination-mode matrices (e.g. KxI by J), and corresponds to
the original procedure by Tucker. The initial configuration may be
inspected to investigate this Tucker solution (note that probably
the solution has not converged yet), or to ascertain the correct
reading of an external configuration.
[332-73] Iteration table: write history of iterations
Depending on your theoretical interest, you may want to see how
quickly the program converges. As the progress of the iteration is
not only written to the output file, but also printed on screen, you
have something to look at while the program is running. When the
acceleration technique is used you may want to check everything is
converging correctly. When there are missing data you may notice
that the function is not converging monotonically (negative signs
appear), but this is no reason for concern. This is due to an
inaccurate, but efficient, calculation of the loss function during
iteration; eventually the negative signs will disappear. If they do
not you are in trouble, possibly due to too many missing data
points. If the option is not active, a counter appears on the
screen, and in the output file only the final results of the
iterative process will be displayed.
[332-74] Core covariances (T3 only): write core covariances
The core covariance matrix is an IJ by IJ matrix with mean squares
for each combination of level i and j, summed over the levels of the
third mode (see the 3WAYPACK User's Manual for a detailed
explanation). This option determines whether they are calculated and
printed. The core covariance has not seen many practical uses, so
far.
[332-76] Contributions to SS-FIT
Sometimes you might not be interested to see how well each level of
each mode is fitted by the model. In such cases the printing of
these fit measures may be suppressed. Note that when external
configurations are used the contributions will not be printed,
because they do not partition the fit in this case.
[332-77] ASCII only: no extended ASCII in output
If you have a printer which cannot handle the extended ascii
character set used in the program, you may choose to convert these
characters to basic ascii by activating this option. Some printers
in Japan use the higher bit ascii codes for hiragana.
[332-78] UPPER CASE only: no lower case in text output
If your printer uses the lower case positions for other characters,
you may convert the entire output to UPPER CASE by activating this
option. Some printers in Japan use the lower-case symbols for
hiragana.
[332-8] Component plots
The components printed in the output are scaled in three different
ways, but only one of them can be plotted in a single run. When P
(or Q, or R) is less than 4, all components will be plotted against
one another. Components 5 and higher will only be plotted against
the first component.
[332-81] None: no component plots
Default
[332-82] Unit lengths: lengths components = 1
The lengths of all components are equal to one. Thus, the scatter in
the plots is not adjusted to the explained variability. This differs
from the situation for the component loadings in ordinary two-mode
pca, but it is the standard way for the component scores of the
subjects.
[332-83] Unit mean square: lengths components = number of levels
The lengths of all components are equal to the number of levels in a
mode. The outward appearance of the plots is identical to that of
the unit-length scaling above, due to the automatic adjustment of
the scales of the plots. This scaling has the advantage that the
values across modes are not influenced by the number of levels in a
mode.
[332-84] PCA scaled: components scaled as in standard PCA
The lengths of the components are equal to the square root of
standardized component weights. If the total sum of squares of the
data is equal to the number of levels of a mode, then this scaling
corresponds to the situation in regular two-mode PCA with
standardized variables. However, the values are not necessarily
comparable with regular `loadings' (variable-component
correlations), because this will depend on the centring and
normalization of the original variables. The effect of this scaling
is that the scatter in the plot will depend on the differences in
variability accounted for by the components.
[332-85] Plots screen size: ON: Plot size for screen; OFF: Plot size for printer
The plot size may be adapted to fit on the screen for cursory
inspection. However, in order to properly evaluate inner products
(correlations) from the plots, the axes should have the same
physical scales horizontally and vertically, which requires this
option to be OFF.
[332-9] Return to Main Menu: Return when parameters are set
When all parameters are set as required, selecting this option will
return you to the main menu to execute the program.
=============================================================================
Chapter 6
TUCKALS2 options
TUCKALS2 is a program for three-mode principal components with dimension
reduction along two of the three ways. Its extended core matrix will
have as its dimensions the number of components of the first and second
ways, and the number of levels of the third way (i.e. PxQxK). The
program can also be used for a simple singular value decomposition,
and also for the IDIOSCAL model if the input is a set of K symmetric
matrices with (squared) distances.
[333-1] Number of components
By default each mode has one component. Note that selecting this
option without selecting any other option is sufficient for a
(rather trivial) analysis. Specifying more components than levels
will cause the program to beep at you. During execution of the
program, it may turn out that you have requested more components
than the data can support. In that case the program will
automatically reduce the number of components. Appropriate messages
will appear in the output.
[333-11] Number of components 1st mode
Specify the number of components (P) for the first mode. P should be
less than or equal to the number of levels (I).
[333-12] Number of components 2nd mode
Specify the number of components (Q) for the second mode. Q should
be less than or equal to the number of levels (J).
[333-2] Missing data
TUCKALS2 does not recognise missing data codes, but needs to have
the locations (coordinates) of the missing data in the three-way
array. These coordinates can be read in from a special file,
., which contains as many lines as there are missing
data points. IF3 automatically creates such a file if you have
specified missing data codes on the data definition screen. If you
press [F3], the program will read an existing file with missing data
coordinates. On the Distribution disk a special program is supplied
to generate a missing-data coordinate file (misindex.exe) in case
you do not have missing data codes, but know the locations of the
missing data.
[333-21] None: no estimation: all data interpreted as valid
Default. Note that if there are missing data, but you decide not to
estimate them in the program, no centring or normalization can be
performed.
[333-22] Substitute slice mean: use (frontal) slice means as initial estimates
For all missing data the mean of the appropriate frontal slice will
be substituted. The implication of this substitution is that the
data within a slice are comparable. If they are not, you either have
to use PREPROC3 to process the missing data, or if applicable swap
the three-way data around (with PREPROC3) in such a way that they
are comparable. For instance, if variables have different
measurement characteristics, they should be levels of the third mode
in order to use this option.
[333-23] User provided values or initial estimates
This option assumes that you already have sensible values for the
missing data (for instance calculated with one of the methods in
PREPROC3), and that TUCKALS2 may use these values as initial
estimates for the missing data.
[333-3] Centring options
Centring is the process of removing one or more means from the data.
For three-way data there are several possibilities of which the more
useful are included in TUCKALS2. Some additional options are
available in PREPROC3. Centring across a mode will cause the
components of that mode to be centred. Note that centring
fundamentally changes the data, so that it is virtually impossible
to compare analyses of differently centred data.
[333-31] None: no centring
Default
[333-32] Across <1st mode>: fiber centring per column
Centring across the first mode, also called centring per column,
indicates that for all KxJ rows the means are calculated by summing
over the index i and dividing by I, and that these means are
subtracted from the data values, so that all columns are in
deviation scores. As variables are often in the second mode, this is
a common option for profile data.
[333-33] Across <2nd mode>: fiber centring per row
Centring across the second mode, also called centring per row,
indicates that for all IxK rows the means are calculated by summing
over the index j and dividing by J. These means are subtracted from
the data values, so that all rows are in deviation scores.
[333-34] Across <3rd mode>: fiber centring per tube
Centring across the third mode, also called centring per tube, means
that for all IxJ tubes the means are calculated by summing over the
index k and dividing by K. These means are subtracted from the data
values, so that all tubes are in deviation scores. This option is
especially useful if subjects constitute the third mode.
[333-35] Across <1st mode> and <2nd mode>: per column and per row
Centring across the first and second mode, also called double
centring, means that for both the KxI and the JxK means are
calculated by summing over the index j plus dividing by J, and
summing over i plus dividing by I, respectively. These means are
subtracted from the data values (and the overall mean is added in
again to maintain the overall size of the data), so that all rows
and all columns are in deviation scores. This option is particulary
useful to transform (squared) distances or dissimilarities into
scalar products, for instance if one wants to perform an IDIOSCAL
analysis.
[333-36] Per <3rd mode> slice: centring per frontal slice
For each frontal slice the mean is calculated by summing over all
indices (i,j) and then dividing by IJ. Within each slice, this mean
is subtracted from the data values.
[333-37] Overall centring
The mean is calculated over all data values, and subsequently
subtracted. This option is rarely used.
[333-38] Print removed means
To check whether the centring was carried out properly the removed
means may be listed. The form of the output is, however, not very
sophisticated.
[333-4] Normalization options: normalize frontal or lateral slices
Normalization means that all frontal (lateral) slices end up having
a total sum of squares of 1. Note that the normalization is entirely
independent of the centring, so that the removed values do not
generally represent variances. Moreover, no attempt is made to
divide the sum of squares by degrees of freedom, so that the removed
values are not mean squares either. This means, for instance, that
the total sum of squares of the normalized three-way data is equal
to I (frontal-slice normalization) or equal to J (lateral-slice
normalization). For horizontal-slice normalization the ways should
be swapped via PREPROC3, which program can also perform the required
normalizations. Note that when both centring and normalization are
requested centring is always carried out first.
[333-41] None: no normalization
Default
[333-42] Per <3rd mode> slice: normalize frontal slice
Each frontal slice is normalized in such a way that the sum of
squares of its elements is equal to 1.
[333-43] Per <2nd mode> slice: normalize lateral slice
Each lateral slice is normalized in such a way that the sum of
squares of its elements is equal to 1.
[333-44] Across <1st mode>: normalize column fibers
All JxK columns are normalized such that their sums of squares are
equal to 1. THIS OPTION IS NOT YET OPERATIONAL
[333-45] Print removed SSQ: print removed sums of squares
To check whether the normalization was carried out properly the
removed sums of squares may be listed. The form of the output is,
however, not very sophisticated.
[333-5] Analysis options: Iterations, convergence, acceleration, etc.
The analysis options control various (technical) aspects of the
analysis. Increasing the maximum number of iterations will be the
most frequent activity in this screen.
[333-51] Maximum number of iterations
When the maximum number of iterations is reached, the program will
finish as if it had converged and will print all relevant
information. The default is set at 50, the maximum value is 99999.
[333-52] Convergence criterion for iterative process
The default for the convergence of the discrepancy or loss function
is 10**-8 (10 to the power -8). From this value the criterion for
the norm of component matrices is derived. To change the convergence
criterion specify the size of the exponent, e.g. 10 for 10**-10. On
most machines the criterion should not be less than roughly 10**-16.
[333-53] Use acceleration technique (Yes/No)
Even though the program is usually fast enough, some facilities have
been built in to speed up convergence. One of those is the use of
successive overrelaxations. The overrelaxation should converge, but
in case of suspicious behaviour, such as negative differences of the
discrepancy function, please rerun the analysis without the ac-
celeration. Another method used to speed up the process is limited
convergence checking and reduced output during iteration.
[333-54] Random seed (0=internally determined)
This option is only useful if a random initial configuration is
requested. If the random seed is internally determined, the same
random number will be used for each run of an analysis. This option
allows changing the initial seed to create different initial
configurations.
[333-6] Initial configurations: Types of initial configurations
TUCKALS2 needs initial configurations for the component matrices A
and B. If there is no special reason to do otherwise, it is
suggested to stay with the default option as it will provide
excellent starting values. In fact, if there is a perfect solution,
it will be found by the default procedure.
[333-61] First mode: Types of initial configurations for first mode
The options are identical for the second mode, and are therefore
only listed for the first mode.
[333-611] Tucker, not fixed (default): initial configuration derived from data
By default, the program computes an initial configuration for the
first two modes by calculating ten steps towards a solution as
described in Tucker (1966). This involves stacking the slices
underneath each other into a `long' matrix, e.g. JxK by I (a process
sometimes called unfolding), and performing a pca on this long
matrix. Generally the TUCKALS solution will be very close to the
Tucker solution, especially with a limited number of components.
[333-612] External, fixed: external configuration not modified during
iterations
If an (orthonormal) solution is available from another analysis or
from the literature, this configuration may be read in and kept
fixed during the analysis. In other words, the rest of the solution
is fitted around the fixed configuration. Comparing fits without and
with the fixed solution will allow an assessment of how well the
external configuration fits the data. The program will assume the
external solution is orthonormal (orthogonal with unit-length
components). If this is not the case the program will issue a
warning, but will proceed as if the configuration is correct. The
output will not necessarily be interpretable; especially the
partitioning of the fit might no longer be valid. The external
configuration should be in a file accessible to the program, and
should be in free format, i.e. there should be at least one blank
between numbers.
[333-613] External, not fixed: external configuration: modified during iteration
If an (orthonormal) solution is available from another analysis or
from the literature, you may request the program to read this
configuration and use it as a starting configuration. The external
configuration should be in a file accessible to the program, and
should be in free format, i.e. there should be at least one blank
between numbers.
[333-614] Random, not fixed: random initial configuration (orthonormalized)
This option is primarily useful for research purposes. It could for
instance be used to assess the sensitivity of the program to
different starting solutions. (The answer is 'not sensitive'.) To
create different starting configurations for each run, change the
random seed (cf. section [333-54]).
[333-62] Second mode: Types of initial configurations for second mode
(options as for the first mode)
[333-64] Configuration file: Name of external configuration file
External configurations should be together in ONE external
configuration file in the order of the modes. Thus a first-mode
configuration should always precede a second- mode one. The program
will request the name of the file, if you have indicated that there
are external configurations. If you do not specify a correct or
valid file name, the external configuration options will be reset to
their defaults. The numbers in the file should be in free format,
i.e. there should be at least one blank between numbers.
[333-65] Done: Return to TUCKALS2 Options
Selecting this option will return you to the TUCKALS2 options menu.
[333-7] Print options: Print notes, initial configuration etc.
The program can produce a large amount of output which is regulated
via the options on this screen. Even with the defaults, the output
can be considerable if there are many levels in one of the modes.
The output increases even further when more than 4 components are
specified for any one mode, due to the way the component matrices
are printed.
[333-71] Print notes: write notes and commentary in output
Throughout the program interpretational comments are available if
this option is active, which is the default.
[333-72] Initial configuration: write initial configuration
The default initial configuration is an approximation to an ordinary
pca on com- bination-mode matrices (e.g. KxI by J), and corresponds
to the original procedure by Tucker. The initial configuration may
be inspected to investigate this Tucker solution, (note that
probably the solution has not converged yet) or to ascertain the
correct reading of the external configuration.
[333-73] Iteration table: write history of iterations
Depending on your theoretical interest, you may want to see how
quickly the program converges. As the progress of the iteration is
not only written to the output file, but also printed on screen, you
have something to look at while the program is running. When the
acceleration technique is used, you may want to check everything is
converging correctly. When there are missing data you may notice
that the function is not converging monotonically (negative signs
appear), but this is no reason for concern. This is due to an
inaccurate, but efficient, calculation of the loss function during
iteration; eventually the negative signs will disappear. If they do
not you are in trouble, possibly due to too many missing data
points. If the option is not active, a counter appears on the
screen, and in the output file only the final results of the
iterative process will be displayed.
[333-74] Print all core slices: no = compact printing of core matrix
The default option is to print the extended core matrix by stringing
out each frontal slice on a line, and printing the lines of the K
third mode levels underneath each other. This greatly facilitates
inspection of the same elements, but hinders the view on the
structure within a frontal core slice. By activating this option the
slices are printed as small matrices rather than lines.
[333-76] Contributions to SS-FIT
Sometimes you might not be interested to see how well each level of
each mode is fitted by the model. In such cases the printing of
these fit measures may be suppressed. Note that when external
configurations are used the contributions will not be printed,
because they do not partition the fit in this case.
[333-77] ASCII only: no extended ASCII in output
If you have a printer which cannot handle the extended ascii
character set used in the program, you may choose to convert these
characters to basic ascii by activating this option. Some printers
in Japan use the higher bit ascii codes for hiragana.
[333-78] UPPER CASE only: no lower-case in text output
If your printer uses the lower case positions for other characters,
you may convert the entire output to UPPER CASE by activating this
option. Some printers in Japan use the lower-case symbols for
hiragana.
[333-8] Component plots
The components printed in the output are scaled in three different
ways, but only one of them can be plotted in a single run. When P
(or Q) is less than 4, all components will be plotted against one
another. The components 5 and higher will only be plotted against
the first component.
[333-81] None: no component plots
Default
[333-82] Unit lengths: lengths components = 1
The lengths of all components are equal to one. Thus, the scatter in
the plots is not adjusted to the explained variability. This differs
from the situation for component loadings in ordinary two-mode pca,
but it is the standard way for the component scores of the subjects.
[333-83] Unit mean square: lengths components = number of levels
The lengths of all components are equal to the number of levels in a
mode. The outward appearance of the plots is identical to that of
the unit-length scaling above, due to the automatic adjustment of
the scales of the plots. This scaling has the advantage that the
values across modes are not influenced by the number of levels in a
mode.
[333-84] PCA scaled: components scaled as in standard PCA
The lengths of the components are equal to the square root of
standardized component weights. If the total sum of squares of the
data is equal to the number of levels of a mode, then this scaling
corresponds to the situation in regular two-mode pca with
standardized variables. However, the values are not necessarily
comparable with regular `loadings' (variable-component
correlations), because this will depend on the centring and
normalization of the original variables. The effect of this scaling
is that the scatter in the plot will depend on the differences in
variability accounted for by the components.
[333-85] Plots screen size: ON: Plot size for screen; OFF: Plot size for printer
The plot size may be adapted to fit on the screen for cursory
inspection. However, in order to properly evaluate inner products
(correlations) from the plots the axes should have the same physical
scales horizontally and vertically, which requires this option to be
OFF.
[333-9] Return to Main Menu: Return when parameters are set
When all parameters are set as required, selecting this option will
return you to the main menu to execute the program.
============================================================================
Chapter 7
TRILIN options
TRILIN is a program for three-mode component analysis without a core
matrix. It fits the parallel factor analysis model (PARAFAC), also
called the canonical decomposition model (CANDECOMP). The PARAFAC model
is an extension of the principal component model in which the components
are equal for each level of the third way except for proportionality
constants. No rotations can be performed on the components without
deterioration of the fit of the model. It can be seen as a Tucker2 model
with diagonal core, or a Tucker3 model with superdiagonal core given
that all ways have equal numbers of components. The program can also be
used for the INDSCAL model, if the input is a set of K symmetric
matrices with (squared) distances. The program can also be used to fit
the INDSCAL model, if the input is a set of K symmetric matrices with
(squared) distances.
[334-1] Number of components
Because the number of components in TRILIN is the same for all
modes, only the number of components for the first mode need to be
specified and the other two are adjusted automatically. By default
the modes have only one component. Note that selecting this option
without selecting any other option is sufficient for a (rather
trivial) analysis. In theory, you should be able to specify more
components than the smallest number of levels but this option has
not been thoroughly tested.
[334-11] Number of components for all modes
Specify the number of components (S) for all modes. Preferably, S
should be less than or equal to the smallest number of levels:
min(I,J,K).
[334-2] Missing data
TRILIN does not recognise missing data codes, but needs to have the
locations (coordinates) of the missing data in the three-way array.
These coordinates can be read in from a special file,
., which contains as many lines as there are missing
data points. IF3 automatically creates such a file, if you have
specified missing data codes on the data definition screen. If you
press [F3], it will read an existing file with missing data
coordinates. On the Distribution disk a special program is supplied
to generate a missing-data coordinate file (misindex.exe) in case
you do not have missing data codes, but know the locations of the
missing data.
[334-21] None : no estimation: all data interpreted as valid
Default. Note that if there are missing data, but you decide not to
estimate them in the program, no centring or normalization can be
performed.
[334-22] Substitute slice mean: use (frontal) slice means as initial estimates
For all missing data the mean of the appropriate frontal slice will
be substituted. The implication of this substitution is that the
data within a slice are comparable. If they are not, you either have
to use PREPROC3 to process the missing data, or if applicable swap
the three-way data around (with PREPROC3) in such a way that they
are comparable. For instance, if variables have different
measurement characteristics they should be levels of the third mode
in order to use this option.
[334-23] User provided values: user provided initial estimates
This option assumes that you already have sensible values for the
missing data (for instance calculated with one of the methods in
PREPROC3), and that TRILIN may use these values as initial estimates
for the missing data.
[334-3] Centring options
Centring is the process of removing one or more means from the data.
For three-way data there are several possibilities of which the more
useful are included in TRILIN. Some additional options are available
in PREPROC3. Centring across a mode will cause the components of
that mode to be centred. Note that centring fundamentally changes
the data, so that it is virtually impossible to compare analyses of
differently centred data.
[334-31] None: no centring
Default
[334-32] Across <1st mode>: fiber centring per column
Centring across the first mode, also called centring per column,
indicates that for all JxK columns the means are calculated by
summing over the index i and dividing by I. These means are
subtracted from the data values, so that all columns are in
deviation scores. As variables are often in the second mode, this is
a common option for profile data.
[334-33] Across <2nd mode>: fiber centring per row
Centring across the second mode, also called centring per row,
indicates that for all KxI rows the means are calculated by summing
over the index j and dividing by J, and that these means are
subtracted from the data values, so that all rows are in deviation
scores.
[334-34] Across <3rd mode>: fiber centring per tube
Centring across the third mode, also called centring per tube, means
that for all IxJ tubes the means are calculated by summing over the
index k and dividing by K. These means are subtracted from the data
values, so that all tubes are in deviation scores. This option is
especially useful if subjects are in the third mode.
[334-35] Across <1st mode> and <2nd mode>: per column and per row
Centring across the first and second mode, also called double
centring, means that for both the KxI and the JxK means are
calculated by summing over the index j plus dividing by J, and
summing over i plus dividing by I, respectively. These means are
subtracted from the data values (and the overall mean is added in
again to maintain the overall size of the data), so that all rows
and all columns are in deviation scores. This option is particulary
useful to transform (squared) distances or dissimilarities into
scalar products, for instance for an INDSCAL analysis.
[334-36] Per <3rd mode> slice: centring per frontal slice
For each frontal slice the mean is calculated by summing over all
indices (i,j) and dividing by IJ. Within each slice, this mean is
subtracted from the data values.
[334-37] Overall centring
The mean is calculated over all data values, and subsequently
subtracted. This option is rarely used.
[334-38] Print removed means
To check whether the centring was carried out properly the removed
means may be listed. The form of the output is, however, not very
sophisticated.
[334-4] Normalization options: Normalize frontal or lateral slices
Normalization means that all frontal (lateral) slices end up having
a total sum of squares of 1. Note that the normalization is entirely
independent of the centring, so that the removed values do not
generally represent variances. Moreover, no attempt is made to
divide the sum of squares by degrees of freedom, so that the removed
values are not mean squares either. This means, for instance, that
the total sum of squares of the normalized three-way data is equal
to I (frontal-slice normalization) or equal to J (lateral-slice
normalization). For horizontal-slice normalization the ways should
be swapped via PREPROC3, which program can also perform the required
normalizations. Note that when both centring and normalization are
requested centring is always carried out first.
[334-41] None: no normalization
Default
[334-42] Per <3rd mode> slice: normalize frontal slice
Each frontal slice is normalized in such a way that the sum of
squares of its elements is equal to 1.
[334-43] Per <2nd mode> slice: normalize lateral slice
Each lateral slice is normalized in such a way that the sum of
squares of its elements is equal to 1.
[334-44] Across <1st mode>: normalize column fibers
All JxK columns are normalized in such a way that their sums of
squares are equal to 1. THIS OPTION IS NOT YET OPERATIONAL.
[334-45] Print removed SSQ
To check whether the normalization was carried out properly the
removed sums of squares may be listed. The form of the output is,
however, not very sophisticated.
[334-5] Analysis options: Iterations, convergence, acceleration, etc.
The analysis options control various (technical) aspects of the
analysis. Increasing the maximum number of iterations will be the
most frequent activity in this screen.
[334-51] Maximum number of iterations
When the maximum number of iterations is reached, the program will
finish as if it had converged and will print all relevant
information. The default is set at 100, the maximum value is 99999.
[334-52] Convergence criterion for iterative process
The default for the convergence of the discrepancy or loss function
is 10**-8 (10 to the power -8). From this value the criterion for
the norm of component matrices is derived. To change the convergence
criterion specify the size of the exponent, e.g. 10 for 10**-10. On
most machines the criterion should not be less than roughly 10**-16.
[334-53] Use acceleration technique (Yes/No)
Because TRILIN usually requires a fair number of iteration, some
facilities have been built in to speed up convergence. One of those
is the use of successive overrelaxations. It is advised to use the
acceleration. The overrelaxation should converge, but in case of
suspicious behaviour, such as negative differences of the
discrepancy function, please rerun the analysis without the
acceleration. Another method used to speed up the process is limited
convergence checking and reduced output during iteration.
[334-54] Random seed (0=internally determined)
This option is only useful if a random initial configuration is
requested. If the random seed is internally determined, the same
random number will be used for each run of an analysis. This option
allows changing the initial seed to create different initial
configurations.
[334-55] Number of repeated analyses (1...20)
The TRILIN algorithm is rather sensitive to local minima; moreover
degenerate solutions occur not infrequently. In order to be able to
spot these, it is generally necessary to run several analyses. To
facilitate multiple runs, the program can perform several analyses
changing the initial configurations by using different initial
random seeds. The program will produce a summary table to help you
choose the best analysis.
[334-56] Automatic choice of the best analysis
The search for the best analysis, i.e. the analysis with the highest
fitted sum of squares, can also be left to the program. If this
option is active, minimal output of the multiple analyses will be
printed. However, the best analysis is rerun so that all the
requested output can be computed. If you would like to inspect the
full output of all analyses this option should be inactive.
[334-57] Restrictions on first mode
TRILIN contains three different algorithms to estimate the
parameters which may be used independently for each mode. The
default option (0) has no restrictions and is the basic algorithm
for the PARAFAC model. The second option (1) restricts the
components of a mode to be orthogonal, and in the third option (2)
all elements in a component matrix of a mode are restricted to be
non-negative.
[334-58] Restrictions on second mode
[334-59] Restrictions on third mode
(as above)
[334-6] Initial configurations
TRILIN needs initial configurations for the component matrices A, B,
and C. The default option is a random initial configuration for each
mode. If you have selected orthogonal components the Tucker initial
configuration might speed up convergence.
[334-61] First mode: Types of initial configurations for first mode
The options are identical for the three modes, and are therefore
only listed for the first mode.
[334-611] Random, not fixed: random initial configuration
This option is the default. To create different starting
configurations for each run change the random seed. Due to the
sensitivity of the algorithm to local solutions, several runs with
different random starts generally need to be made.
[334-612] External, fixed: external configuration not modified during iterations
If a solution is available from another analysis or from the
literature, this configuration may be read in and kept fixed during
the analysis. In other words, the rest of the solution is fitted
around the fixed configuration. Comparing fits without and with the
fixed solution will allow an assessment of how well the external
configuration fits the data. The external solution is assumed to
have unit-length components. If this is not the case the program
will issue a warning, and set the lengths equal to one. Not much
experience has been obtained with this option, and no guidance is
available with respect to degeneracies and local minima, but the
assumption is that the external con- figuration stabilizes the
solution. The external configuration should be in a file accessible
to the program, and should be in free format, i.e. there should be
at least one blank between numbers.
[334-613] External, not fixed: external configuration: modified during iterations
If you have a solution available from another analysis or from the
literature, you may request the program to read this configuration
and use it as a starting configuration. The external configuration
should be in a file accessible to the program, and should be in free
format, i.e. there should be at least one blank between numbers.
[334-614] Tucker, not fixed: initial configuration derived from data
The program can compute an initial configuration for each mode by
calculating ten steps towards a solution as described in Tucker
(1966). This involves stacking the slices underneath each other into
a 'long' matrix, e.g. JxK by I (a process sometimes called
unfolding), and performing a pca on this long matrix.
[334-62] Second mode: Types of initial configurations for second mode
[334-63] Third mode: Types of initial configurations for third mode
(options as for the first mode)
[334-64] Configuration file: Name of external configuration file
External configurations should be together in ONE external
configuration file in the order of the modes. Thus a first-mode
configuration should always precede a second-mode one and a
second-mode configuration a third-mode one. The program will request
the name of the file if you have indicated that there are external
configurations. If you do not specify a correct or valid file name,
the external configuration options will be reset to their defaults.
The numbers in the file should be in free format, i.e. there should
be at least one blank between numbers.
[334-65] Done: Return to TRILIN Options
Selecting this option will return you to the TRILIN options menu.
[334-7] Print options: Print notes, initial configuration etc.
The program can produce a large amount of output which is regulated
via the options on this screen. Even with the defaults, the output
can be considerable if there are many levels in one of the modes.
This increases even further when more than 4 components are
specified for any one mode, due to the way the component matrices
are printed.
[334-71] Print notes: write notes and commentary in output
Throughout the program interpretational comments are available if
this option is active, which is the default.
[334-72] Initial configuration: write initial configuration
This option may be selected to inspect the initial configuration,
which is especially useful to ascertain the correct reading of an
external configuration, or to inspect a Tucker initial solution, but
note that generally the latter solution has not completely converged
yet.
[334-73] Iteration table: write history of iterations
Depending on your theoretical interest, you may want to see how
quickly the program converges. As the progress of the iteration is
not only written to the output file, but also printed on screen, you
have something to look at while the program is running. When the
acceleration technique is used you may want to check everything is
converging correctly. When there are missing data you may notice
that the function is not converging monotonically (negative signs
appear), but this is no reason for concern. This is due to an
inaccurate, but efficient, calculation of the loss function during
iteration; eventually the negative signs will disappear. If they do
not you are in trouble, possibly due to too many missing data
points. If the option is not active, a counter appears on the
screen, and in the output file only the final results of the
iterative process will be displayed.
[334-74] Contributions to SS-FIT
Sometimes you might not be interested to see how well each level of
each mode is fitted by the model. In such cases the printing of
these fit measures may be suppressed.
[334-75] ASCII only: no extended ASCII in output
If you have a printer which cannot handle the extended ascii
character set used in the program, you may choose to convert these
characters to basic ascii by activating this option. Some printers
in Japan use the higher bit ascii codes for hiragana.
[334-76] UPPER CASE only: no lower case in text output
If your printer uses the lower case positions for other characters,
you may convert the entire output to UPPER CASE by activating this
option. Some printers in Japan use the lower-case symbols for
hiragana.
[334-8] Component plots
The components printed in the output are scaled in three different
ways, but only one of them can be plotted in a single run. When S is
less than 4, all components will be plotted against one another. The
components 5 and higher will only be plotted against the first
component.
[334-81] None: no component plots
Default
[334-82] Unit lengths: lengths components = 1
The lengths of all components are equal to one. Thus, the scatter in
the plots is not adjusted to the explained variability. This differs
from the situation for the component loadings in ordinary two-mode
pca, but it is the standard way for the component scores of the
subjects.
[334-83] Unit mean square: lengths components = number of levels
The lengths of all components are equal to the number of levels in a
mode. The outward appearance of the plots is identical to that of
the unit-length scaling above, due to the automatic adjustment of
the scales of the plots. This scaling has the advantage that the
values across modes are not influenced by the number of levels in a
mode. Especially in TRILIN this is important, because components
have the same meaning for each mode.
[334-84] PCA scaled: components scaled as in standard PCA
The lengths of the components are equal to the square root of
standardized component weights. If the total sum of squares of the
data is equal to the number of levels of a mode, then this scaling
corresponds to the situation in regular two-mode PCA with
standardized variables. However, the values are not necessarily
comparable with regular `loadings' (variable-component
correlations), because this will depend on the centring and
normalization of the original variables. The effect of this scaling
is that the scatter in the plot will depend on the differences in
variability accounted for by the components.
[334-85] Plots screen size: ON: Plot size for screen; OFF: Plot size for printer
The plot size may be adapted to fit on the screen for cursory
inspection. However, in order to properly evaluate inner products
(correlations) from the plots, the axes should have the same
physical scales horizontally and vertically, which requires this
option to be OFF.
[334-9] Return to Main Menu: Return when parameters are set
When all parameters are set as required, selecting this option will
return you to the main menu to execute the program.
===========================================================================
Chapter 8
ROTATE: Rotations of components and core
This program will rotate any or all TUCKALS2 and TUCKALS3 components. No
rotations are available for the TRILIN output, because such rotations
are incompatible with the Parafac model. The rotations (or rather
transformations) available are: 1. Varimax rotation on the orthonormal
components (equivalent to the Harris-Kaiser independent cluster
rotation); 2. Oblimin transformation of the orthonormal components; 3.
Optimally constant first component. This option can sometimes be used to
facilitate interpretation, especially of reference modes for joint
plots.
[371-1] Job Title
A title may be specified to identify the job in the output. If no
title is specified the title of the main job, if available, is used
for this purpose.
[371-2] Main Menu
From the menu the options for ROTATE can be specified, the program
can be executed, its output viewed and printed. After the execution
of a task you are returned to the main menu to perform the next task
or modify a previous one.
[371-21] ROTATE options
The component matrices of the three modes can be independently
transformed, either by varimax, or oblimin, or in such a way that
the first component of a component matrix is as constant as possible
by a nonsingular transformation. Note that the basis for
transformation is always the orthonormal component matrix. This is
different from two-mode pca where the basis is the matrix with
component-variable correlations (loadings)
[371-211] 1st mode: type of rotation for the first mode
There are four choices: no rotation, varimax, oblimin, and constant
first component.
[371-2111] No rotation
Default
[371-2112] Varimax rotation
The Varimax routine used is based on the code by Velleman. It
operates on the orthonormal component matrix A. Because of the
orthonormality, the Varimax is equivalent to the Harris-Kaiser
independent cluster rotation (see 3WAYPACK User's Manual for
details).
[371-2113] Oblimin: oblique rotation
The Oblimin rotation or rather transformation is based on an
algorithm by Jennrich. It operates on the orthonormal component
matrix. The correlations between the com- ponents are printed as
well. Due to the nonsingularity of the transformation and hence the
nonsingularity of the inverse transformation on the core matrix, the
sums of squares partitioning of the core matrix is no longer valid.
[371-2113] Constant First Component:
optimize equality of first component coefficients
The Constant First Component transformation is based on code by
Arbuckle and Friendly. The output will give a measure of the
improvement of equality of the elements in the first component.
[371-212] 2nd mode: type of rotation for the second mode
[371-2121] None
[371-2122] Varimax
[371-2123] Oblimin (as above)
[371-2113] Constant First Component
[371-213] 3rd mode: type of rotation for the third mode
[371-2131] None
[371-2132] Varimax
[371-2133] Oblimin (as above)
[371-2113] Constant First Component
[371-214] Output options
The output options are of two different kinds: those that regulate
the plot sizes and those that regulate the appearance of the output,
i.e. whether it is in standard ascii only and/or whether the output
contains lower-case letters.
[371-2141] No plots
Default
[371-2142] Plots to printer
[371-2143] Plots to screen
The plot size may be adapted to fit on the screen for cursory
inspection. However, in order to properly evaluate inner products
(correlations) from the plots, the axes should have the same
physical scales horizontally and vertically, which requires this
option to be OFF.
[371-2144] ASCII only: no extended ASCII in output
If you have a printer which cannot handle the extended ascii
character set used in the program, you may choose to convert these
characters to basic ascii by activating this option. Some printers
in Japan use the higher bit ascii codes for hiragana.
[371-2145] UPPER CASE only: no lower case in text output
If your printer uses the lower case positions for other characters,
you may convert the entire output to UPPER CASE by activating this
option. Some printers in Japan use the lower-case symbols for
hiragana.
[371-215] Return to Main Menu
When all parameters are set as required, selecting this option will
return you to the main menu to execute the program.
[371-22] EXECUTE: Execute analysis
After specifying the options for ROTATE, pressing [Enter] will start
execution of the program.
[371-23] View/Print output: View and/or print output
After a successful execution, you can view the output with the file
viewer. Once you are in the File Viewer you can print the output by
pressing [P], provided a printer is available. For details about the
workings of the file viewer, see section [02] on the file view
utility.
[371-24] End of POSTPROC: Return to continue with Main program
After you are finished with ROTATE, select this option to leave
POSTPROC and to return to the main menu of the three-way analysis
program.
=============================================================================
Chapter 9
RESIDUAL: Analysis of Residuals
This program will calculate standardized residuals and plot these
against the fitted data. You may choose to do the plotting per frontal
slice or for the entire data set. This program also offers the option to
write either the residuals and/or the fitted data to files called
.res and .fit respectively. Note that the fitted data
are such that they fit the model perfectly, and thus can be used for
simulation etc.
[372-1] Job Title
A title may be specified to identify the job in the output. If no
title is specified the title of the main job, if available, is used
for this purpose.
[372-2] Main Menu
From the menu the options for RESIDUAL can be specified, the program
can be executed, its output viewed and printed. After the execution
of a task you are returned to the main menu to perform the next task
or modify a previous one.
[372-21] RESIDUAL options
You can specify a global plot or frontal slice plots, the plot size,
and whether residuals and/or fitted data will be written to a file.
The menu has two sections: in the top section you can select one of
the options, in the bottom section you can select any option.
[372-211] None: no plots, no residuals listed
Default. Note that the residuals cannot be written to a file,
either.
[372-212] Global output: global plots and listings of residuals
The residuals are calculated for the entire data set, and the
standardized residuals are calculated on the basis of all residuals.
One single plot will show the standardized residuals against the raw
fitted data. A list of extreme residuals will be printed.
[372-213] Output per frontal slice: plots of residuals per frontal slice
The residuals are calculated and plotted for each frontal slice
separately. If you want residuals per lateral or per horizontal
slices, the only way to do this within IF3 is to let PREPROC3 swap
the data array in an appropriate way. The standardized residuals are
calculated for the data of each slice. A list of extreme residuals
will be printed for each slice.
[372-214] Plots screen size:ON: Plot size for screen; OFF: Plot size for printer
The plot size may be adapted to fit on the screen for direct
inspection. In this case there is no great necessity to have the
same physical scales horizontally and vertically, because the units
of the horizontal and vertical axes are different.
[372-215] ASCII only: no extended ASCII in output
If you have a printer, which cannot handle the extended ascii
character set used in the program, you may choose to convert these
characters to basic ascii by activating this option. Some printers
in Japan use the higher bit ascii codes for hiragana.
[372-216] UPPER CASE only: no lower case in text output
If your printer uses the lower case positions for other characters,
you may convert the entire output to UPPER CASE by activating this
option. Some printers in Japan use the lower-case symbols for
hiragana.
[372-217] Output residuals: residuals written to file .res
The residuals may be written to a file .res, so that they
can be processed by other programs.
[372-218] Output fitted data: fitted data written to file .fit
The fitted data may be written to a file .fit, so that they
can be processed by other programs. The fitted or implied data fit
the three-way model perfectly. Therefore, they may be used for
research purposes. In particular, the fitted data can be used as a
basis for a simulation study.
[372-22] EXECUTE: Execute analysis
After specifying the options for RESIDUAL, pressing [Enter] will
start execution of the program.
[372-23] View/Print output: View and/or print output
After a successful execution, you can view the output with the file
viewer. Once you are in the File Viewer you can print the output by
pressing [P], provided a printer is available. For details about the
workings of the file viewer, see section [02] on the file view
utility.
[372-24] End of POSTPROC: Return to continue with Main program
After you are finished with RESIDUAL, you have to select this option
to leave POSTPROC and to return to the main menu of the three-way
analysis program.
==========================================================================
Chapter 10
JOINTPLT: Joint Plots
This program constructs joint (bi)plots for two modes, given the third
(the third is the reference mode). In addition, it can compute the
coordinates for interactive biplots, but it cannot yet produce the
interactive biplots themselves. These coordinates can in some data sets
be considered component scores. No joint plots are available for TRILIN
as this is incompatible with the model used.
[373-1] Job Title
A title may be specified to identify the job in the output. If no
title is specified the title of the main job, if available, is used
for this purpose.
[373-2] Main Menu
From the menu the options for JOINTPLT can be specified, the program
can be executed, its output viewed and printed. After the execution
of a task you are returned to the main menu to perform the next task
or modify a previous one.
[373-21] JOINTPLT options: Program options
The program allows the choice of one type of joint plot or all three
types. If you want two types the program should be run twice. Note
that often the latter axes of the joint plots have very small
amounts of explained variability and should be ignored. Some
subjective judgement may be required for this. For each component of
the reference mode the inner products may be requested. This menu
has three sections: only one option can be selected from the first
section, and any option may be selected from the next two sections.
If the three-way analysis program is TUCKALS2 or only one matrix is
being analysed (K=1), only one type of joint plot is possible.
[373-211] Modes 1 and 2: joint plot of modes 1 and 2
Mode 3 is reference mode, and R sets of joint plots will be made,
one set for each component of mode 3. The number of axes of a joint
plot is the minimum of P and Q.
[373-212] Modes 2 and 3: joint plot of modes 2 and 3
Mode 1 is reference mode, and P sets of joint plots will be made,
one set for each component of mode 1. Each joint plot has min (Q,R)
axes.
[373-213] Modes 3 and 1: joint plot of modes 3 and 1
Mode 2 is reference mode, and Q sets of joint plots will be made,
one set for each component of mode 2. Each joint plot has min (R,P)
axes.
[373-214] All Modes: joint plots of all modes
Joint plots will be made for all combinations of modes. Note that
this will produce a large amount of output especially if any or all
of P, Q, and R are not small. Moreover, it will seldom be useful to
look at all three combinations.
[373-215] Inner products: output of inner products
The inner product of a row marker and a column marker indicates the
strength of the association of the two markers. Its size is the
product of the length of the projection of a row marker on the
vector of the column marker and the length of the column marker. A
zero value indicates no relation (projection into the origin), a
positive (negative) value a positive (negative) association.
Especially in joint plots with more than two dimensions, they can be
useful to check the importance of the projection of a row marker on
the vector of the column marker and vice versa. The inner products
are also the coordinates for interactive biplots (or the component
scores on the components of the reference mode). Unfortunately, in
the present version of the program such interactive biplots cannot
(yet) be made.
[373-216] Type of Scaling: OFF: symmetric scaling; ON: PCA scaling
By default the joint (bi)plots are scaled such that lengths of the
axes for the rows and column markers are equal (symmetric scaling).
However, such a scaling does not allow for a correlation
interpretation of the angles between the column markers, as is
usually desired in two-mode pca. The pca-scaling option allows for
the correlation interpretation in the two-way case (i.e. K = 1),
provided the data have been normalized per column (are z-scores). In
the three-way case the correlation interpretation is much more
difficult to realize and can in general only be approximate, even
with pca scaling.
[373-217] Plots screen size: ON: Plot size for screen; OFF: Plot size for printer
The plot size may be adapted to fit on the screen for cursory
inspection. However, in order to properly evaluate inner products
(correlations) from the plots, the axes should have the same
physical scales horizontally and vertically, which requires this
option to be OFF.
[373-218] ASCII only: no extended ASCII in output
If you have a printer which cannot handle the extended ascii
character set used in the program, you may choose to convert these
characters to basic ascii by activating this option. Some printers
in Japan use the higher bit ascii codes for hiragana.
[373-219] UPPER CASE only: no lower case in text output
If your printer uses the lower case positions for other characters,
you may convert the entire output to UPPER CASE by activating this
option. Some printers in Japan use the lower-case symbols for
hiragana.
[373-22] EXECUTE: Execute analysis
After specifying the options for JOINTPLT pressing [Enter] will
start execution of the program.
[373-23] View/Print output: View and/or print output
After a successful execution, you can view the output with the File
Viewer. Once you are in the File Viewer you can print the output by
pressing [P], provided a printer is available. For details about the
workings of the File Viewer, see section [02] on the file view
utility.
[373-24] End of POSTPROC: Return to continue with the Main program
After you are finished with JOINTPLT, select this option to leave
POSTPROC and to return to the main menu of the three-way analysis
program.
===========================================================================
Related publications
===================
Pieter M. Kroonenberg
3WAYPACK User's Manual
Leiden: Department of Education, Leiden University, 1996.
[Available from The Three-Mode Company, Vakgroep Algemene Pedagogiek,
Wassenaarseweg 52, 2333 AK Leiden, The Netherlands]
Pieter M. Kroonenberg
Three-mode principal component analysis: Theory and Applications.
Leiden: DSWO Press, 1983, reprint 1989 (+ errata), 2nd reprint 1995.
Henk A.L. Kiers
Three-way methods for the analysis of qualitative and quantitative two-way data.
Leiden: DSWO Press, 1990.
Wim P. Krijnen
The analysis of three-way arrays by constrained PARAFAC methods.
Leiden: DSWO Press, 1993.
[Available from DSWO Press, Wassenaarseweg 52, 2333 AK Leiden,
The Netherlands]
Richard A. Harshman & Margaret E. Lundy
Parafac reference manual.
London, Ontario: Scientific Software Associates, 1986.
[Available from R.A. Harshman, Department of Psychology, University of
Western Ontario, London, Ontario N6A 5C2, Canada].
==============================================================================
Acknowledgments.
3WAYPACK has been developed by Pieter M. Kroonenberg, Rijksuniversiteit
Leiden. Piet Brouwer was invaluable to the development of the program
suite. He designed and programmed the interface and related programs,
but sadly did not live to see the release of this version. Bart Smit
assisted in finalising the interface for release. The program has been
developed with financial support from Leiden University, and all those
persons and institutions who purchased previous versions.