Normal-Ogive Multidimensional Model

Abstract

In an attempt to provide a unified foundation for common factor analysis, true score theory, and latent trait (item response) theory, McDonald (1962a, 1962b, 1967) defined a general strong principle of local independence and described a general latent trait model, as follows: Let U be a n × 1 random vector of manifest variables—test or possibly binary item scores and θ a k × 1 random vector of latent traits—not yet defined. The strong principle of local independence, which defines θ and the dimension k of the vector U, states that

$$g\{ U\} \theta \} = \prod\limits_{i = 1}^k {{g_i}} \{ \left. {{U_i}} \right|\theta \} $$

(1)

where g{ } is the conditional density of U and g _i { } is the conditional density of the ith component. (Note that θ is not necessarily continuous and may consist of a dummy variable defining a latent class model.)