Angular similarity in test parameters

Abstract

Through N-dimensional person space, the article gives measures of test parameters and item statistics, including difficulty/discriminating value of test, correlations between a pair of items, and item-total correlations with binary items using angular similarity between two vectors. Relationships between difficulty value and discriminating value of items and test were derived, including relationship between test reliability and test discriminating value. Reliability of a test as per theoretical definition in terms of length of score vectors of two parallel subtests and angle between such vectors was derived. The method was extended to find reliability of a battery of tests. Reliability and discriminating value of a Likert-type item and scale was found in terms of angular similarity without involving assumptions of continuous nature or linearity or normality for the observed variables, or the underlying variable being measured. The proposed methods also avoid test of unidimensionality or assumption of normality or bivariate normality associated with the polychoric correlations. Thus, the proposed methods are in fact nonparametric and considered as improvement over the existing ones. Reliability as a measure of association of two vectors and discrimination as a measure of distance between the vectors are likely to show a negative relationship.