古典的テスト理論のベクトルによる解釈 I

信頼性と妥当性

抄録

1. Vector analysis has been widely used in such fields as factor analysis, regression analysis, and the analysis of variance. It is also applicable to the field of test theory.
2. Let the transpose of an N-column vector X^T={X₁, X₂, …X_i, …X_N}^T whose i-th element is the score of the i-th person on the test under consideration.
By assuming that the mean of test scores for N persons is zero and by adjusting the unit of the vector, we can obtain the following relations: 1) The length of the vector is equal to the standard deviation of the test scores. 2) The cosine of the angle between two test vectors is equal to the correlation between the two sets of test scores. (See Fig. 1 on Top Page 2.)
3. The classical test theory starts from the following assumption and definition in terms of vector algebra: 1) A test vector X may be divided into two orthogonal components, X=T+E and T^T·E=0. (See Fig. 4 on Top Page 3.) 2) Two test vectors, X₁ and X₂, having such characteristics among components that T₁=T₂, E₁^T·E₂=T₁^T·E₂=E₁^T·T₂=0, and √E₁^T·E₁=√E₂^T·E₂ are called parallel test vectors in a sense that their true components are parallel. (See Fig. 5 and 6 on Top Page 3.)
Almost all the basic theorems given in the first nine chapters of Gulliksen's book, “Theory of Mental Tests (1950, Wiley)”, can be derived from the above assumptions without any complicated calculation.
4. For example, the reliability coefficient of a test is the square of the cosine of the angle between the test vector X and its true component T. This is equal to the cosine of the angle between two vectors of parallel tests, X₁ and X₂. (See Fig. 4 on Top Page 3.)
5. Spearman-Brown's prophecy formula is considered as the square of the cosine of the angle between the vector sum of parallel tests and the vector sum of their true components. (See Fig. 8 on Top Page 5.)
6. The validity coefficient between a test vector X and its criterion vector Y is defined as the cosine of the angle between X and Y. The formula for “Correction for Attenuation” is obtained by calculating the cosine of the angle between the true component of the test vector and that of the criterion vector. (See Fig. 9 on Top Page 7.)
7. Errors of measurement, substitution and prediction, pointed out by Gulliksen (1950, Chapter 4), are also interpretable by the concept of vector geometry. (See Fig. 7 on Top Page 5.)
8. The use of vector method has at least two advantages; the first is the algebraic aspect which facilitates the solution of complex problems by the use of simple rules of vector operation, and the second is the geometric aspect which helps us in an intuitive understanding of the nature of problems by the use of geometric figures.