Applications
CTT: continuous \(Z\) and \(X\) Equivalence of Classical Test Theory With item randomization and definitions of \(\xi\) for the discrete cases, now the application of these definitions to CTT is investigated.
What is measuring?
Several observations on a unit are combined to estimate an attribute as number(s) with meaning in reality. What is measuring? Elements of a Measurement/Assessment Observations are assessed in order to measure some attribute (e.g. mass, intelligence) of a unit (a person or an object). Typically measurements are slightly off the “true” value of the attribute: measurement error. Attributes change in the course of time. Examples Hubble Deep Field (1995) 10 day sensor-exposure to an unknown and dark part of the universe1. 3000 new galaxies were discovered. Some attributes of these galaxies were estimated from the observed images. Intelligence Testing Item structure: Rasch/CTT model assumptions Items are thoroughly designed and refined Controlled assessment procedure Measurement and the Social Sciences “Hard Sciences” Direct correspondence of assessed data and attribute.
Classical Test/Item Response Theory
Measurement Theory: Latent Variables in CTT Assessment can be described mathematically in terms of probability theory as a random experiment (Steyer): A person \(u \in \Omega_U\) is drawn from a population (manifest unit random variable \(U : \Omega \rightarrow \Omega_U\)). An observation \(y \in \Omega_o\) is made of this person \(u\) (manifest random variables \(Y : \Omega \rightarrow \mathbb R\)) Measurement of a Latent Variable A well-defined numerical value for the attribute \(Y\) of object \(u\) is the object conditional expectation \(E(Y | U=u)\) (Steyer).
Measuring with Text
Text violates conditional independence For CTT and Rasch models good test design can result in items \(\Ym\) not violating assumptions of s.i. conditional on latent variables. For text there does not exist a latent variable explaining the stochastic dependence of words in a text (because text has syntactical structure, among other reasons).