### Reliability of latent dimensions of social status of judo athletes

#### Abstract

Coefficients γp vary in the range (0,1) and can take the value of 1 if and only if Ρ = I, i.e. if all the variables are measured without error, and the value of 0 if and only if Ρ = 0 and R = I, i.e. if the total variance of all the variables consists only of measurement error variance and variables from V have a spherical normal distribution.

Proof:

If the total variance of each variable from a set of variables consists only of measurement error variance, then, necessarily E2 = I and R = I and all the coefficients γp are equal to zero. The first part of the proposition is evident from the definition of coefficients γp. This means that reliability of each latent dimension, regardless of how the latent dimension is determined, equals 1if the variables from which the dimension is derived are measured without error.

However, matrix of reliability coefficients Ρ = (ρj) is often unknown, so measurement error variance matrix E2 is also unknown. But if variables from V are selected to represent a universe of variables U with the same field of meaning, the upper limit of measurement error variances is defined by elements of matrix U2 (Guttman, 1945), that is, unique variances of these variables. Therefore, in this case, the lower limit of reliability of latent dimensions can be estimated by the coefficients βp = 1 - (qp tU2qp)(qp tRqp)-1 p = 1,...,k which are derived using a method identical to that by which coefficients γp are derived under the definition E2 = U2, that is, the same procedure through which Guttman derived his measure λ6.

Keywords: / matrix / dimension / value / variance / variable /

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