Measurements in mental tests through person space
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Measurements in mental tests through person space Satyendra Nath Chakrabartty 1
# Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract If a test is administered to n-persons, the test score can be viewed as a point in n – dimensional person space where each person is considered in an axis and the variables are represents as points or vectors. The orthogonality of the axes is better indication since individuals are assumed to be independent. Such presentation helps to make useful inferences about the subjects and various parameters of the test, item statistics along with geometrical interpretation of such computations. For a test consisting of oneright-rest wrong type items, the paper gives theoretical formulation of mental testing through person space, primarily in terms of length of observed score vector and angle separating the vector with the vector representing the maximum possible score in the test. In addition, method is described to obtain (i) test reliability, as per the theoretical definition, via a single administration of the test, which is extended to find reliability of a battery of tests (ii) difficulty, discriminating values of items and test along with their relationships. Empirical verification of the proposed methods are undertaken. Keywords n –dimensional person space and variable space . Error variance . Reliability . Difficulty and discriminating values . Test battery
Introduction The most commonly used geometry is the geometry in variable space which considers the variables in orthogonal axes and values of individuals or performance of individuals as points in the Euclidean space defined by the axes. However, it is possible to treat each person in an axis and represent the variables as points or vectors in the Euclidean space. This type of presentation is called “subject space” or “person space” or “ndimensional person space” for n– individuals (Reyment and Joreskog 1993; Huberty 1994; Yu et al. 2002). The orthogonality of the axes in such n-dimensional person space is better indication since different individuals are assumed to be independent of one another. In the geometry of the variable space, on the other hand, the axes are shown as orthogonal despite the fact the variables may have correlations of different degrees. If n- individuals take a test, then the observed test scores can be viewed as a point or a vector in n-dimensional person space. This type of presentation helps to make useful inferences about the sample as well as various parameters of the
* Satyendra Nath Chakrabartty [email protected] 1
Indian Ports Association, Indian Maritime University, N 304, VivekVihar, Sector 82, Noida 201304, India
test including item statistics along with geometrical interpretation of such computations. For example, length of a vector in person space is equal to the variance of the variable if components of the vector are deviation scores; cosine of the angle between two vectors of deviation scores equals the Pearson correlation between the variables. The perpendicul
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