DAKS: An R Package for Data Analysis Methods in Knowledge Space Theory

Knowledge space theory is part of psychometrics and provides a theoretical framework for the modeling, assessment, and training of knowledge. It utilizes the idea that some pieces of knowledge may imply others, and is based on order and set theory. We introduce the R package DAKS for performing basic and advanced operations in knowledge space theory. This package implements three inductive item tree analysis algorithms for deriving quasi orders from binary data, the original, corrected, and minimized corrected algorithms, in sample as well as population quantities. It provides functions for computing population and estimated asymptotic variances of and one and two sample Z tests for the diff fit measures, and for switching between test item and knowledge state representations. Other features are a function for computing response pattern and knowledge state frequencies, a data (based on a finite mixture latent variable model) and quasi order simulation tool, and a Hasse diagram drawing device. We describe the functions of the package and demonstrate their usage by real and simulated data examples.

Kochen-Specker Sets and Generalized Orthoarguesian Equations

Every set (finite or infinite) of quantum vectors (states) satisfies generalized orthoarguesian equations ($n$OA). We consider two 3-dim Kochen-Specker (KS) sets of vectors and show how each of them should be represented by means of a Hasse diagram---a lattice, an algebra of subspaces of a Hilbert space--that contains rays and planes determined by the vectors so as to satisfy $n$OA. That also shows why they cannot be represented by a special kind of Hasse diagram called a Greechie diagram, as has been erroneously done in the literature. One of the KS sets (Peres') is an example of a lattice in which 6OA pass and 7OA fails, and that closes an open question of whether the 7oa class of lattices properly contains the 6oa class. This result is important because it provides additional evidence that our previously given proof of noa =< (n+1)oa can be extended to proper inclusion noa < (n+1)oa and that nOA form an infinite sequence of successively stronger equations.Comment: 16 pages and 5 figure

Limit theorems for a random directed slab graph

We consider a stochastic directed graph on the integers whereby a directed edge between $i$ and a larger integer $j$ exists with probability $p_{j-i}$ depending solely on the distance between the two integers. Under broad conditions, we identify a regenerative structure that enables us to prove limit theorems for the maximal path length in a long chunk of the graph. The model is an extension of a special case of graphs studied by Foss and Konstantopoulos, Markov Process and Related Fields, 9, 413-468. We then consider a similar type of graph but on the `slab' $\Z \times I$, where $I$ is a finite partially ordered set. We extend the techniques introduced in the in the first part of the paper to obtain a central limit theorem for the longest path. When $I$ is linearly ordered, the limiting distribution can be seen to be that of the largest eigenvalue of a $|I| \times |I|$ random matrix in the Gaussian unitary ensemble (GUE).Comment: 26 pages, 3 figure

DAKS: An R Package for Data Analysis Methods in Knowledge Space Theory

Knowledge space theory is part of psychometrics and provides a theoretical framework for the modeling, assessment, and training of knowledge. It utilizes the idea that some pieces of knowledge may imply others, and is based on order and set theory. We introduce the R package DAKS for performing basic and advanced operations in knowledge space theory. This package implements three inductive item tree analysis algorithms for deriving quasi orders from binary data, the original, corrected, and minimized corrected algorithms, in sample as well as population quantities. It provides functions for computing population and estimated asymptotic variances of and one and two sample Z tests for the diff fit measures, and for switching between test item and knowledge state representations. Other features are a function for computing response pattern and knowledge state frequencies, a data (based on a finite mixture latent variable model) and quasi order simulation tool, and a Hasse diagram drawing device. We describe the functions of the package and demonstrate their usage by real and simulated data examples