Software development in this project is aimed
to expand and adapt existing tools and to develop
new software. Those tasks range from providing tools for data
analyses via model validation to
visualizations of the results. In close teamwork of software
developer, mathematician, and
psychologist, existing software has to be evaluated, various
tools to support developing
mathematical models (i.e., simulations, data and structure
visualizations), tools for validating
models (i.e., parameter estimation and hypotheses testing),
and a professional databank have to be developed. Furthermore
a clear, integrative user interface for the relevant software
components has to be created. The development of a professional
website as a platform to
publish the software packages and for presenting the project
and its results is another aim.
Finally, an important issue is documentation and dissemination
of the newly developed software
packages.
Simulation Tools

» 
Resampling Methods 


The Jackknife and the Bootstrap are nonparametric methods
for assessing the errors in a statistical estimation
problem. They provide several advantages over the traditional
parametric approach: the methods are easy to describe
and they apply to arbitrarily complicated situations;
distribution assumptions, such as normality, are never
made. These methods, however, can be computationally
very intensive.
For the current project,
we implemented Jackknife methods (deleteone, deleted)
and Boostrap with different confidence interval methods
(i.e. normal approximation and percentile). For the
implementations we used the .NETbased SasquatchSoftware
as frontend. In the course of the project, revised
versions of these resampling methods will be implemented
in the ePsytplatform.
Further readings:
» Biswal B.B., Taylor P.A., & Ulmer J.L. (2001).
Journal of Computer Assisted Tomographie,
25(1),:11320. [Abstract]
» Cheng, R.C.H. (2000). Analysis of simulation
experiments by bootstrap resampling. In B. A.
Peters, J. S. Smith, D. J. Medeiros,
and M. W. Rohrer (Eds.), Proceedings of the 2001
Winter
Simulation Conference, WSC 2001,
December 912, 2001, Arlington, VA, USA. [PDF]
» Dixon, P.M. Bootstrap Resampling. [PDF]
» Efron, B. (1982). The Jackknife, the Bootstrap,
and Other Resampling Plans. Philadelphia:
SIAM.
» Friedl, H., & Stampfer, E. (2000). Resampling
Methods. [PS]
» Helmers, R., & Putter, H. Bootstrap Resampling:
A survey of recent research in the
Netherlands. [PDF]
» Shimodaira, H. (2004). Approximately unbiased
tests of region using multistepmultiscale
bootstrap resampling. The Annals
of Statistics, 32 (6), 2616–2641. [PDF]
» Wan, J. (1998). A parametric version of JackknifeafterBootstrap.
In D.J. Medeiros, E.F.
Watson, J.S. Carson, and M.S. Manivannan
(Eds.), Proceedings of the 1998 Winter Simulation
Conference, WSC 1998, December 1316,
1998, Washington DC, USA. [PDF]





» 
Binary and Modelbased Simulations 


A central issue for generating and validating models
in the framework of the current project is simulating
data sets. We developed a number of simulation methods
based on random binary simulations and on modelbased
simulations. For the implementations we used the .NETbased
SasquatchSoftware as frontend.
In the course of the project, revised versions of these
simulation methods will be implemented in the ePsytplatform.
» Sasquatch
Simulations Manual (KickmeierRust, M.D., 2005;
Draft, restricted access)





» 
Random Number Generators 


A central issue of computerbased simulations are (pseudo)
random number generators (RNG). In the context of the
current project, we evaluated a number of RNGs, e.g.
Rand, MotherofAll, or Mersenne Twister [Summary].
For random number generations in all simulation tools
we implemented the Mersenne Twister by Makoto Matsumoto
and Takuji Nishimura.
Links and further readings:
» Mersenne
Twister Homepage
» pLAB
(University of Salzburg)
» A
comparison of algorithms.
» Fog, A. Chaotic Random Number Generators
with Random Cycle Lengths. [PDF]
» L’Ecuyer, P., & Touzin, R. (2003).
On the DengLin Random Number Generators and Related
Methods. [PDF]

Optimization Tools

» 
Praxis / EPM (Extended Principal
Axis Minimization) 


Numerical optimization (i.e.function minimization resp.
maximization) is an impoprtant aim for software development
in the framework of PKSIRT. We evaluated common minimization
methods and algorithms, e.g. EMalgorithm, the NewtonRaphson
method, or Brent's Principal Axis Minimization (PRAXIS).
In this course, we developed
EPM, an extension of the C implementation of
PRAXIS by Karl Gegenfurtner. The EPM approach automatically
generates and tests multiple sets of start values, randomly
drawn from user specified weighted primary and secondary
intervals (Round 1), and it refines and stabilizes the
smallest Round 1 approximated minimum by means of certain
iterative minimization loops and a stopping criterion
on the change in approximated smallest minimum found
in previous minimization runs and the minimum of the
current run (Rounds 2 and 3). EPM is availaible in two
versions; one uses the double data type, one long double.
» Ünlü, A., & KickmeierRust, M.D.
(2005). EPM: a strategy for principal axis minimization.
Proceedings of the 14th Hellenic European
Conference on Computer Mathematics & its
Applications, accepted for publication.
Athens, Greece: Lea Publishers. [PDF,
restricted access]
» Download EPM (double version) [ZIP,
16kB]
» Download EPM (long double version) [ZIP,
17kB]
Further readings:
» Brent, R.P. (1973). Algorithms for Minimization
Without Derivatives. Englewood Cliffs, NJ:
PrenticeHall.
» Gegenfurtner, K.R. (1992). PRAXIS: Brent's algorithm
for function minimization. Behavior
Research Methods,Instruments, & Computers,
24, 560564.
» Powell, M.J.D. (1964). An efficient method of
finding the minimum of a function of several
variables without calculating derivatives.
Computer Journal, 7, 155162.

Software Evaluation


To support mathematical modelling in the framework
of PKSIRT existing software applications in the fields
of Latent Class Analysis, Item Response Theory, and
Knowledge Space Theory must be evaluated. The Listing
[DOC, 37kB] shows the relevant evaluated applications.

Latent Class Modeling with Random Effects

Response error (careless error
and lucky guess) rates are modeled and estimated using
the general class of latent class models with random effects.
In this class, we have the general latent class model
with random effects (LCMRE), the special LCMRE with itemindependent,
classspecific random effects parameters (LCMRE_bu), and,
as another important special case, the traditional unrestricted
twoclasses latent class model (TLCM). Related C sources
are available for download below. Please note that each
of the three model types requires the matrix operation
tools!


» 
General Latent Class Model
with Random Effects (LCMRE) 


Parameter estimation (incl. variances) and goodnessoffit
testing [ZIP,
32 kB]



Data simulation 1 (BoxMuller
implementation of N(0,1)distribution) [ZIP,
8 kB] 


Data simulation 2 (GaussHermite
discretization of N(0,1)distribution) [ZIP,
9 kB] 


Data simulation 3 (Multinomial
probabilities calculated from model parameters) [ZIP,
9 kB] 


Summary statistics (see Info.txt
included in the ZipFile) [ZIP,
9 kB] 




» 
LCMRE with itemindependent,
classspecific random effects parameters (LCMRE_bu) 


Parameter estimation (incl.
variances) and goodnessoffit testing [ZIP,
33 kB] 


Data simulation 1 (BoxMuller
implementation of N(0,1)distribution) [ZIP,
8 kB] 


Data simulation 2 (GaussHermite
discretization of N(0,1)distribution) [ZIP,
9 kB] 


Data simulation 3 (Multinomial
probabilities calculated from model parameters) [ZIP,
9 kB] 


Summary statistics (see Info.txt
included in the ZipFile) [ZIP,
9 kB] 




» 
Traditional unrestricted
twoclasses latent class model (TLCM) 


Parameter estimation (incl.
variances) and goodnessoffit testing [ZIP,
31 kB] 


Data simulations using a general
basic local independence model (Note: data simulations
with the TLCM are obtained as special cases.) [ZIP,
7 kB] 


Summary statistics (see Info.txt
included in the ZipFile) [ZIP,
9 kB] 




» 
Matrix operation tools 


Small Matrix Toolbox for C
programmers version 0.42 (Support Unix and DOS)
by Patrick Ko Shupui. Copyright (c) 1992, 1993, 1994.
All Rights Reserved.
Modified to long double data type [ZIP,
17 kB]

