Content

The students will learn some of the mathematical, probabilistic, and statistical methods used in mathematical psychology. They will also learn how to model psychological phenomena by means of mathematical methods. The seminar reviews some of the elementary concepts from discrete mathematics, probability theory, and statistics. It discusses probabilistic knowledge space theory,
with an emphasis on probabilistic substructure response data models. The issue of parameter estimation and assessment of model-data fit is also discussed. Other topics of the seminar are model selection
and measures of association.

This course is the first part of a series of seminars on Research Methods: Mathematical Psychology. It is continued with Part II, Nonparametric Item Response Theory. See also the web page on Introduction to Exploratory Latent Class Analysis.

Method

Presentations by me, and interspersed exercises for students. Active in-class participation.
Oral exam in the session before last (June 24, 2004) of the seminar.

Oral Exam

Thursday, June 24, 2004
11.15am - 12.45am
HS 02.21
Questions

Literature

[1]  Doignon, J.-P. & Falmagne, J.-C. (1999). Knowledge Spaces. Springer.
[2]  Goodman, L.A. & Kruskal W.H. (1954-1972). Measures of association for Cross-Classifications I-IV.       Journal of the American Statistical Association.
[3]  Bishop, Y.M.M., Fienberg, S.E., & Holland, P.W. (1975). Discrete Multivariate Analysis: Theory
      and Practice. MIT.
[4]  Liebetrau, A.M. (1983). Measures of Association. Sage.
[5]  Linhart, H. & Zucchini, W. (1986). Model selection. Wiley.
[6]  Journal of Mathematical Psychology (2000). Special Issue on Model Selection.

Script

Title Page: Probability, Statistics, and Knowledge Structures
Contents
1 Probabilistic Knowledge Space Theory (KST)
        1.1  Knowledge Structures and Spaces, and Surmise Relations
    1.2  Probabilistic Knowledge Structures
               1.2.1  Why Probabilities ? - The Standard Example
           1.2.2  Summary - The General Definitions
    1.3  State Probabilities and Substructures
    1.4  Response Data Substructure Models
    1.5  Additional Reading
    1.6  Exercises for Chapter 1
2 Parameter Estimation and Assessment of Model-Data Fit
        2.1  Two Classical Goodness-Of-Fit Tests - The Standard Example
                2.1.1  Model, Data, and the Statistics X^2 and G^2
            2.1.2  A Glimpse of Birch's Regularity Conditions
            2.1.3  The Testing Problem and Neave's Step Method
                2.1.4  Step 1: Data and Multinomial Distribution
                2.1.5  Step 2: Null and Alternative Hypotheses
                2.1.6  Step 3: Test Statistics X^2 and G^2
                2.1.7  Step 4: Critical Region and Standard Example
                2.1.8  Step 5: Decision Rule and Standard Example
    2.6  Additional Reading
        2.7  Exercises for Chapter 2
A  Binary Relations on Sets
B  Functions
Bibliography
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Complete script (485 KB)

Model Selection
(Journal of Mathematical Psychology, 44(1), 2000. Special Issue on Model Selection)

An introduction to model selection (Zucchini, 2000)
How to assess a model's testability and identifiability (Bamber & Santen, 2000)
Akaike's information criterion and recent developments in information complexity (Bozdogan, 2000)
Bayesian model selection and model averaging (Wasserman, 2000)
The importance of complexity in model selection (In Jae Myung, 2000)
Key concepts in model selection: performance and generalizability (Forster, 2000)

Additional Material

Glossary of Mathematical Concepts (Pp. 12-15 of Doignon & Falmagne, 1999)
Birkhoff-Theorem (Pp. 38-40 of Doignon & Falmagne, 1999)
Generalization of deterministic KST to polychotomous items (Schrepp, 1997)
Alternative representations for knowledge spaces (Koppen, 1998)
Tutorial on maximum likelihood estimation (In Jae Myung, 2003)